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- Many-valued logic 2 From Wikipedia, the free encyclopedia
- Contents 1 Alfred Tarski 1 1.1 Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Mathematician . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Logician . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Truth in formalized languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.5 Logical consequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.6 What are logical notions? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.7 Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.10 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Aristotle 10 2.1 Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Thought . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2.1 Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2.2 Aristotle’s epistemology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2.3 Geology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.4 Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.5 Metaphysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.6 Biology and medicine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.7 Psychology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2.8 Practical philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2.9 Views on women . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.3 Loss and preservation of his works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4 Legacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.4.1 Later Greek philosophers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.4.2 Inﬂuence on Byzantine scholars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.4.3 Inﬂuence on Islamic theologians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.4.4 Inﬂuence on Western Christian theologians . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.4.5 Post-Enlightenment thinkers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.5 List of works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 i
- ii CONTENTS 2.6 Eponym . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.8 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3 Emil Leon Post 46 3.1 Early work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 Recursion theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.3 Polyadic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.4 Selected papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4 Four-valued logic 49 4.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.1.1 Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.1.2 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.2 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5 Fuzzy logic 52 5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.1.1 Applying truth values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.1.2 Linguistic variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.2 Early applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.3.1 Hard science with IF-THEN rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.3.2 Deﬁne with multiply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.3.3 Deﬁne with sigmoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.4 Logical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.4.1 Propositional fuzzy logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.4.2 Predicate fuzzy logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.4.3 Decidability issues for fuzzy logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.5 Fuzzy databases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.6 Comparison to probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.7 Relation to ecorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.8 Compensatory fuzzy logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
- CONTENTS iii 5.11 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6 Hans Reichenbach 61 6.1 Life and work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6.2 Selected publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 6.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 6.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 6.5 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 6.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 7 Jan Łukasiewicz 65 7.1 Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 7.2 Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 7.3 Recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 7.4 Chronology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 7.5 Selected works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 7.5.1 Books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 7.5.2 Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 7.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 7.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 7.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 7.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 7.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 8 Many-valued logic 70 8.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 8.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 8.2.1 Kleene (strong) K3 and Priest logic P3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 8.2.2 Bochvar’s internal three-valued logic (also known as Kleene’s weak three-valued logic) . . . 71 8.2.3 Belnap logic (B4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 8.2.4 Gödel logics Gk and G∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 8.2.5 Łukasiewicz logics Lv and L∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 8.2.6 Product logic Π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 8.2.7 Post logics Pm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 8.3 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 8.3.1 Matrix semantics (logical matrices) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 8.4 Proof theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 8.5 Relation to classical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 8.5.1 Suszko’s thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 8.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 8.7 Research venues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
- iv CONTENTS 8.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 8.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 8.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 8.11 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 8.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 9 Principle of bivalence 76 9.1 Relationship with the law of the excluded middle . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 9.2 Classical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 9.3 Suszko’s thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 9.4 Criticisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 9.4.1 Future contingents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 9.4.2 Vagueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 9.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 9.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 9.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 9.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 10 Probabilistic logic 81 10.1 Historical context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 10.2 Modern proposals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 10.3 Possible application areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 10.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 10.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 10.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 10.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 11 Problem of future contingents 85 11.1 Aristotle’s solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 11.2 Leibniz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 11.3 20th century . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 11.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 11.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 11.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 11.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 12 Stephen Cole Kleene 89 12.1 Biography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 12.2 Important publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 12.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 12.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 12.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
- CONTENTS v 13 Term logic 91 13.1 Aristotle’s system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 13.2 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 13.3 Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 13.4 Proposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 13.5 Singular terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 13.6 Inﬂuence on philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 13.7 Decline of term logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 13.8 Revival . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 13.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 13.10Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 13.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 13.12External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 14 Three-valued logic 97 14.1 Representation of values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 14.2 Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 14.2.1 Kleene and Priest logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 14.2.2 Łukasiewicz logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 14.2.3 Bochvar logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 14.2.4 ternary Post logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 14.2.5 Modular algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 14.3 Application in SQL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 14.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 14.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 14.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 14.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 14.8 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 101 14.8.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 14.8.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 14.8.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
- Chapter 1 Alfred Tarski Alfred Tarski (/ˈtɑrski/; January 14, 1901 – October 26, 1983) was a Polish logician, mathematician and philosopher. Educated at the University of Warsaw and a member of the Lwów–Warsaw school of logic and the Warsaw school of mathematics and philosophy, he emigrated to the USA in 1939 where he became a naturalized citizen in 1945, and taught and carried out research in mathematics at the University of California, Berkeley from 1942 until his death.[1] A proliﬁc author best known for his work on model theory, metamathematics, and algebraic logic, he also contributed to abstract algebra, topology, geometry, measure theory, mathematical logic, set theory, and analytic philosophy. His biographers Anita and Solomon Feferman state that, “Along with his contemporary, Kurt Gödel, he changed the face of logic in the twentieth century, especially through his work on the concept of truth and the theory of models.”[2] 1.1 Life Alfred Tarski was born Alfred Teitelbaum (Polish spelling: “Tajtelbaum”), to parents who were Polish Jews in comfortable circumstances. He ﬁrst manifested his mathematical abilities while in secondary school, at Warsaw’s Szkoła Mazowiecka.[3] Nevertheless, he entered the University of Warsaw in 1918 intending to study biology.[4] After Poland regained independence in 1918, Warsaw University came under the leadership of Jan Łukasiewicz, Stanisław Leśniewski and Wacław Sierpiński and quickly became a world-leading research institution in logic, foun- dational mathematics, and the philosophy of mathematics. Leśniewski recognized Tarski’s potential as a mathe- matician and encouraged him to abandon biology.[4] Henceforth Tarski attended courses taught by Łukasiewicz, Sierpiński, Stefan Mazurkiewicz and Tadeusz Kotarbiński, and became the only person ever to complete a doctorate under Leśniewski’s supervision. Tarski and Leśniewski soon grew cool to each other. However, in later life, Tarski reserved his warmest praise for Kotarbiński, as was mutual. In 1923, Alfred Teitelbaum and his brother Wacław changed their surname to “Tarski.” (Years later, Alfred met another Alfred Tarski in northern California.) The Tarski brothers also converted to Roman Catholicism, Poland’s dominant religion. Alfred did so even though he was an avowed atheist.[5] Tarski was a Polish nationalist who saw himself as a Pole and wished to be fully accepted as such — later, in America, he spoke Polish at home. After becoming the youngest person ever to complete a doctorate at Warsaw University, Tarski taught logic at the Polish Pedagogical Institute, mathematics and logic at the University, and served as Łukasiewicz’s assistant. Because these positions were poorly paid, Tarski also taught mathematics at a Warsaw secondary school;[6] before World War II, it was not uncommon for European intellectuals of research caliber to teach high school. Hence between 1923 and his departure for the United States in 1939, Tarski not only wrote several textbooks and many papers, a number of them ground-breaking, but also did so while supporting himself primarily by teaching high-school mathematics. In 1929 Tarski married fellow teacher Maria Witkowska, a Pole of Catholic background. She had worked as a courier for the army in the Polish-Soviet War. They had two children; a son Jan who became a physicist, and a daughter Ina who married the mathematician Andrzej Ehrenfeucht.[7] Tarski applied for a chair of philosophy at Lwów University, but on Bertrand Russell's recommendation it was awarded to Leon Chwistek.[8] In 1930, Tarski visited the University of Vienna, lectured to Karl Menger's colloquium, and met Kurt Gödel. Thanks to a fellowship, he was able to return to Vienna during the ﬁrst half of 1935 to work with Menger’s research group. From Vienna he traveled to Paris to present his ideas on truth at the ﬁrst meeting of the Unity of 1
- 2 CHAPTER 1. ALFRED TARSKI Science movement, an outgrowth of the Vienna Circle. In 1937, Tarski applied for a chair at Poznań University but the chair was abolished.[9] Tarski’s ties to the Unity of Science movement likely saved his life, because they resulted in his being invited to address the Unity of Science Congress held in September 1939 at Harvard University. Thus he left Poland in August 1939, on the last ship to sail from Poland for the United States before the German and Soviet invasion of Poland and the outbreak of World War II. Tarski left reluctantly, because Leśniewski had died a few months before, creating a vacancy which Tarski hoped to ﬁll. Oblivious to the Nazi threat, he left his wife and children in Warsaw. He did not see them again until 1946. During the war, nearly all his extended family died at the hands of the German occupying authorities. Once in the United States, Tarski held a number of temporary teaching and research positions: Harvard University (1939), City College of New York (1940), and thanks to a Guggenheim Fellowship, the Institute for Advanced Study in Princeton (1942), where he again met Gödel. In 1942, Tarski joined the Mathematics Department at the University of California, Berkeley, where he spent the rest of his career. Tarski became an American citizen in 1945.[10] Although emeritus from 1968, he taught until 1973 and supervised Ph.D. candidates until his death.[11] At Berkeley, Tarski acquired a reputation as an awesome and demanding teacher, a fact noted by many observers: His seminars at Berkeley quickly became famous in the world of mathematical logic. His students, many of whom became distinguished mathematicians, noted the awesome energy with which he would coax and cajole their best work out of them, always demanding the highest standards of clarity and precision.[12] Tarski was extroverted, quick-witted, strong-willed, energetic, and sharp-tongued. He preferred his research to be collaborative — sometimes working all night with a colleague — and was very fastidious about priority.[13] A charismatic leader and teacher, known for his brilliantly precise yet suspenseful expository style, Tarski had intimidatingly high standards for students, but at the same time he could be very encouraging, and particularly so to women — in contrast to the general trend. Some students were frightened away, but a circle of disciples remained, many of whom became world-renowned leaders in the ﬁeld.[14] Tarski supervised twenty-four Ph.D. dissertations including (in chronological order) those of Andrzej Mostowski, Bjarni Jónsson, Julia Robinson, Robert Vaught, Solomon Feferman, Richard Montague, James Donald Monk, Haim Gaifman, Donald Pigozzi and Roger Maddux, as well as Chen Chung Chang and Jerome Keisler, authors of Model Theory (1973),[15] a classic text in the ﬁeld.[16][17] He also strongly inﬂuenced the dissertations of Alfred Lindenbaum, Dana Scott, and Steven Givant. Five of Tarski’s students were women, a remarkable fact given that men represented an overwhelming majority of graduate students at the time.[17] Tarski lectured at University College, London (1950, 1966), the Institut Henri Poincaré in Paris (1955), the Miller Institute for Basic Research in Science in Berkeley (1958–60), the University of California at Los Angeles (1967), and the Pontiﬁcal Catholic University of Chile (1974–75). Among many distinctions garnered over the course of his career, Tarski was elected to the United States National Academy of Sciences, the British Academy and the Royal Netherlands Academy of Arts and Sciences, received honorary degrees from the Pontiﬁcal Catholic University of Chile in 1975, from Marseilles' Paul Cézanne University in 1977 and from the University of Calgary, as well as the Berkeley Citation in 1981. Tarski presided over the Association for Symbolic Logic, 1944–46, and the International Union for the History and Philosophy of Science, 1956–57. He was also an honorary editor of Algebra Universalis.[18] 1.2 Mathematician Tarski’s mathematical interests were exceptionally broad for a mathematical logician. His collected papers run to about 2,500 pages, most of them on mathematics, not logic. For a concise survey of Tarski’s mathematical and logical accomplishments by his former student Solomon Feferman, see “Interludes I–VI” in Feferman and Feferman.[19] Tarski’s ﬁrst paper, published when he was 19 years old, was on set theory, a subject to which he returned throughout his life. In 1924, he and Stefan Banach proved that, if one accepts the Axiom of Choice, a ball can be cut into a ﬁnite number of pieces, and then reassembled into a ball of larger size, or alternatively it can be reassembled into two balls whose sizes each equal that of the original one. This result is now called the Banach–Tarski paradox. In A decision method for elementary algebra and geometry, Tarski showed, by the method of quantiﬁer elimination, that the ﬁrst-order theory of the real numbers under addition and multiplication is decidable. (While this result
- 1.3. LOGICIAN 3 appeared only in 1948, it dates back to 1930 and was mentioned in Tarski (1931).) This is a very curious result, because Alonzo Church proved in 1936 that Peano arithmetic (the theory of natural numbers) is not decidable. Peano arithmetic is also incomplete by Gödel’s incompleteness theorem. In his 1953 Undecidable theories, Tarski et al. showed that many mathematical systems, including lattice theory, abstract projective geometry, and closure algebras, are all undecidable. The theory of Abelian groups is decidable, but that of non-Abelian groups is not. In the 1920s and 30s, Tarski often taught high school geometry. Using some ideas of Mario Pieri, in 1926 Tarski de- vised an original axiomatization for plane Euclidean geometry, one considerably more concise than Hilbert’s. Tarski’s axioms form a ﬁrst-order theory devoid of set theory, whose individuals are points, and having only two primitive relations. In 1930, he proved this theory decidable because it can be mapped into another theory he had already proved decidable, namely his ﬁrst-order theory of the real numbers. In 1929 he showed that much of Euclidean solid geometry could be recast as a ﬁrst-order theory whose individuals are spheres (a primitive notion), a single primitive binary relation “is contained in”, and two axioms that, among other things, imply that containment partially orders the spheres. Relaxing the requirement that all individuals be spheres yields a formalization of mereology far easier to exposit than Lesniewski's variant. Near the end of his life, Tarski wrote a very long letter, published as Tarski and Givant (1999), summarizing his work on geometry. Cardinal Algebras studied algebras whose models include the arithmetic of cardinal numbers. Ordinal Algebras sets out an algebra for the additive theory of order types. Cardinal, but not ordinal, addition commutes. In 1941, Tarski published an important paper on binary relations, which began the work on relation algebra and its metamathematics that occupied Tarski and his students for much of the balance of his life. While that exploration (and the closely related work of Roger Lyndon) uncovered some important limitations of relation algebra, Tarski also showed (Tarski and Givant 1987) that relation algebra can express most axiomatic set theory and Peano arithmetic. For an introduction to relation algebra, see Maddux (2006). In the late 1940s, Tarski and his students devised cylindric algebras, which are to ﬁrst-order logic what the two-element Boolean algebra is to classical sentential logic. This work culminated in the two monographs by Tarski, Henkin, and Monk (1971, 1985). 1.3 Logician Tarski’s student, Vaught, has ranked Tarski as one of the four greatest logicians of all time — along with Aristotle, Gottlob Frege, and Kurt Gödel.[2][20][21] However, Tarski often expressed great admiration for Charles Sanders Peirce, particularly for his pioneering work in the logic of relations. Tarski produced axioms for logical consequence, and worked on deductive systems, the algebra of logic, and the theory of deﬁnability. His semantic methods, which culminated in the model theory he and a number of his Berkeley students developed in the 1950s and 60s, radically transformed Hilbert’s proof-theoretic metamathematics. In [Tarski’s] view, metamathematics became similar to any mathematical discipline. Not only its concepts and results can be mathematized, but they actually can be integrated into mathematics. ... Tarski destroyed the borderline between metamathematics and mathematics. He objected to restricting the role of metamathematics to the foundations of mathematics.[22] Tarski’s 1936 article “On the concept of logical consequence” argued that the conclusion of an argument will follow logically from its premises if and only if every model of the premises is a model of the conclusion. In 1937, he published a paper presenting clearly his views on the nature and purpose of the deductive method, and the role of logic in scientiﬁc studies. His high school and undergraduate teaching on logic and axiomatics culminated in a classic short text, published ﬁrst in Polish, then in German translation, and ﬁnally in a 1941 English translation as Introduction to Logic and to the Methodology of Deductive Sciences. Tarski’s 1969 “Truth and proof” considered both Gödel’s incompleteness theorems and Tarski’s undeﬁnability theo- rem, and mulled over their consequences for the axiomatic method in mathematics. 1.4 Truth in formalized languages In 1933, Tarski published a very long paper in Polish, titled “Pojęcie prawdy w językach nauk dedukcyjnych”,[23] setting out a mathematical deﬁnition of truth for formal languages. The 1935 German translation was titled “Der
- 4 CHAPTER 1. ALFRED TARSKI Wahrheitsbegriﬀ in den formalisierten Sprachen”, “The concept of truth in formalized languages”, sometimes short- ened to “Wahrheitsbegriﬀ”. An English translation appeared in the 1956 ﬁrst edition of the volume Logic, Semantics, Metamathematics. This collection of papers from 1923 to 1938 is an event in 20th-century analytic philosophy, a contribution to symbolic logic, semantics, and the philosophy of language. For a brief discussion of its content, see Convention T (and also T-schema). Some recent philosophical debate examines the extent to which Tarski’s theory of truth for formalized languages can be seen as a correspondence theory of truth. The debate centers on how to read Tarski’s condition of material adequacy for a truth deﬁnition. That condition requires that the truth theory have the following as theorems for all sentences p of the language for which truth is being deﬁned: “p” is true if and only if p. (where p is the proposition expressed by “p”) The debate amounts to whether to read sentences of this form, such as “Snow is white” is true if and only if snow is white as expressing merely a deﬂationary theory of truth or as embodying truth as a more substantial property (see Kirkham 1992). It is important to realize that Tarski’s theory of truth is for formalized languages, so examples in natural language are not illustrations of the use of Tarski’s theory of truth. 1.5 Logical consequence In 1936, Tarski published Polish and German versions of a lecture he had given the preceding year at the International Congress of Scientiﬁc Philosophy in Paris. A new English translation of this paper, Tarski (2002), highlights the many diﬀerences between the German and Polish versions of the paper, and corrects a number of mistranslations in Tarski (1983). This publication set out the modern model-theoretic deﬁnition of (semantic) logical consequence, or at least the basis for it. Whether Tarski’s notion was entirely the modern one turns on whether he intended to admit models with varying domains (and in particular, models with domains of diﬀerent cardinalities). This question is a matter of some debate in the current philosophical literature. John Etchemendy stimulated much of the recent discussion about Tarski’s treatment of varying domains.[24] Tarski ends by pointing out that his deﬁnition of logical consequence depends upon a division of terms into the logical and the extra-logical and he expresses some skepticism that any such objective division will be forthcoming. “What are Logical Notions?" can thus be viewed as continuing “On the Concept of Logical Consequence”. 1.6 What are logical notions? Another theory of Tarski’s attracting attention in the recent philosophical literature is that outlined in his “What are Logical Notions?" (Tarski 1986). This is the published version of a talk that he gave originally in 1966 in London and later in 1973 in Buﬀalo; it was edited without his direct involvement by John Corcoran. It became the most cited paper in the journal History and Philosophy of Logic.[25] In the talk, Tarski proposed a demarcation of the logical operations (which he calls “notions”) from the non-logical. The suggested criteria were derived from the Erlangen programme of the German 19th century Mathematician, Felix Klein. Mautner, in 1946, and possibly an article by the Portuguese mathematician Sebastiao e Silva, anticipated Tarski in applying the Erlangen Program to logic. That program classiﬁed the various types of geometry (Euclidean geometry, aﬃne geometry, topology, etc.) by the type of one-one transformation of space onto itself that left the objects of that geometrical theory invariant. (A one-to-one transformation is a functional map of the space onto itself so that every point of the space is associated with or mapped to one other point of the space. So, “rotate 30 degrees” and “magnify by a factor of 2” are intuitive descriptions of simple uniform one-one transformations.) Continuous transformations give rise to the objects of topology, similarity transformations to those of Euclidean geometry, and so on.
- 1.6. WHAT ARE LOGICAL NOTIONS? 5 As the range of permissible transformations becomes broader, the range of objects one is able to distinguish as preserved by the application of the transformations becomes narrower. Similarity transformations are fairly narrow (they preserve the relative distance between points) and thus allow us to distinguish relatively many things (e.g., equilateral triangles from non-equilateral triangles). Continuous transformations (which can intuitively be thought of as transformations which allow non-uniform stretching, compression, bending, and twisting, but no ripping or glueing) allow us to distinguish a polygon from an annulus (ring with a hole in the centre), but do not allow us to distinguish two polygons from each other. Tarski’s proposal was to demarcate the logical notions by considering all possible one-to-one transformations (automorphisms) of a domain onto itself. By domain is meant the universe of discourse of a model for the semantic theory of a logic. If one identiﬁes the truth value True with the domain set and the truth-value False with the empty set, then the following operations are counted as logical under the proposal: 1. Truth-functions: All truth-functions are admitted by the proposal. This includes, but is not limited to, all n-ary truth-functions for ﬁnite n. (It also admits of truth-functions with any inﬁnite number of places.) 2. Individuals: No individuals, provided the domain has at least two members. 3. Predicates: � the one-place total and null predicates, the former having all members of the domain in its extension and the latter having no members of the domain in its extension � two-place total and null predicates, the former having the set of all ordered pairs of domain members as its extension and the latter with the empty set as extension � the two-place identity predicate, with the set of all order-pairs in its extension, where a is a member of the domain � the two-place diversity predicate, with the set of all order pairs where a and b are distinct members of the domain � n-ary predicates in general: all predicates deﬁnable from the identity predicate together with conjunction, disjunction and negation (up to any ordinality, ﬁnite or inﬁnite) 4. Quantiﬁers: Tarski explicitly discusses only monadic quantiﬁers and points out that all such numerical quanti- ﬁers are admitted under his proposal. These include the standard universal and existential quantiﬁers as well as numerical quantiﬁers such as “Exactly four”, “Finitely many”, “Uncountably many”, and “Between four and 9 million”, for example. While Tarski does not enter into the issue, it is also clear that polyadic quantiﬁers are admitted under the proposal. These are quantiﬁers like, given two predicates Fx and Gy, “More(x, y)", which says “More things have F than have G.” 5. Set-Theoretic relations: Relations such as inclusion, intersection and union applied to subsets of the domain are logical in the present sense. 6. Set membership: Tarski ended his lecture with a discussion of whether the set membership relation counted as logical in his sense. (Given the reduction of (most of) mathematics to set theory, this was, in eﬀect, the question of whether most or all of mathematics is a part of logic.) He pointed out that set membership is logical if set theory is developed along the lines of type theory, but is extralogical if set theory is set out axiomatically, as in the canonical Zermelo–Fraenkel set theory. 7. Logical notions of higher order: While Tarski conﬁned his discussion to operations of ﬁrst-order logic, there is nothing about his proposal that necessarily restricts it to ﬁrst-order logic. (Tarski likely restricted his attention to ﬁrst-order notions as the talk was given to a non-technical audience.) So, higher-order quantiﬁers and predicates are admitted as well. In some ways the present proposal is the obverse of that of Lindenbaum and Tarski (1936), who proved that all the logical operations of Russell and Whitehead's Principia Mathematica are invariant under one-to-one transformations of the domain onto itself. The present proposal is also employed in Tarski and Givant (1987). Solomon Feferman and Vann McGee further discussed Tarski’s proposal in work published after his death. Feferman (1999) raises problems for the proposal and suggests a cure: replacing Tarski’s preservation by automorphisms with preservation by arbitrary homomorphisms. In essence, this suggestion circumvents the diﬃculty Tarski’s proposal has in dealing with sameness of logical operation across distinct domains of a given cardinality and across domains
- 6 CHAPTER 1. ALFRED TARSKI of distinct cardinalities. Feferman’s proposal results in a radical restriction of logical terms as compared to Tarski’s original proposal. In particular, it ends up counting as logical only those operators of standard ﬁrst-order logic without identity. McGee (1996) provides a precise account of what operations are logical in the sense of Tarski’s proposal in terms of expressibility in a language that extends ﬁrst-order logic by allowing arbitrarily long conjunctions and disjunctions, and quantiﬁcation over arbitrarily many variables. “Arbitrarily” includes a countable inﬁnity. 1.7 Works Anthologies and collections � 1986. The Collected Papers of Alfred Tarski, 4 vols. Givant, S. R., and McKenzie, R. N., eds. Birkauser. � Givant, Steven, 1986. “Bibliography of Alfred Tarski”, Journal of Symbolic Logic 51: 913-41. � 1983 (1956). Logic, Semantics, Metamathematics: Papers from 1923 to 1938 by Alfred Tarski, Corcoran, J., ed. Hackett. 1st edition edited and translated by J. H. Woodger, Oxford Uni. Press.[26] This collection contains translations from Polish of some of Tarski’s most important papers of his early career, including The Concept of Truth in Formalized Languages and On the Concept of Logical Consequence discussed above. Original publications of Tarski � 1930 Une contribution a la theorie de la mesure. Fund Math 15 (1930), 42-50. � 1930. (with Jan Łukasiewicz). “Untersuchungen uber den Aussagenkalkul” ["Investigations into the Sentential Calculus"], Comptes Rendus des seances de la Societe des Sciences et des Lettres de Varsovie, Vol, 23 (1930) Cl. III, pp. 31–32 in Tarski (1983): 38-59. � 1931. “Sur les ensembles déﬁnissables de nombres réels I”, Fundamenta Mathematica 17: 210-239 in Tarski (1983): 110-142. � 1936. “Grundlegung der wissenschaftlichen Semantik”, Actes du Congrès international de philosophie scien- tiﬁque, Sorbonne, Paris 1935, vol. III, Language et pseudo-problèmes, Paris, Hermann, 1936, pp. 1–8 in Tarski (1983): 401-408. � 1936. "Über den Begriﬀ der logischen Folgerung”, Actes du Congrès international de philosophie scientiﬁque, Sorbonne, Paris 1935, vol. VII, Logique, Paris: Hermann, pp. 1–11 in Tarski (1983): 409-420. � 1936 (with Adolf Lindenbaum). “On the Limitations of Deductive Theories” in Tarski (1983): 384-92. � 1994 (1941).[27][28] Introduction to Logic and to the Methodology of Deductive Sciences. Dover. � 1941. “On the calculus of relations”, Journal of Symbolic Logic 6: 73-89. � 1944. "The Semantical Concept of Truth and the Foundations of Semantics," Philosophy and Phenomenolog- ical Research 4: 341-75. � 1948. A decision method for elementary algebra and geometry. Santa Monica CA: RAND Corp.[29] � 1949. Cardinal Algebras. Oxford Univ. Press.[30] � 1953 (with Mostowski and Raphael Robinson). Undecidable theories. North Holland.[31] � 1956. Ordinal algebras. North-Holland. � 1965. “A simpliﬁed formalization of predicate logic with identity”,Archiv fürMathematische Logik undGrund- lagenforschung 7: 61-79 � 1969. "Truth and Proof", Scientiﬁc American 220: 63-77. � 1971 (with Leon Henkin and Donald Monk). Cylindric Algebras: Part I. North-Holland.
- 1.8. SEE ALSO 7 � 1985 (with Leon Henkin and Donald Monk). Cylindric Algebras: Part II. North-Holland. � 1986. “What are Logical Notions?", Corcoran, J., ed., History and Philosophy of Logic 7: 143-54. � 1987 (with Steven Givant). A Formalization of Set Theory Without Variables. Vol.41 of American Mathemati- cal Society colloquium publications. Providence RI: American Mathematical Society. ISBN 978-0821810415. Review � 1999 (with Steven Givant). “Tarski’s system of geometry”, Bulletin of Symbolic Logic 5: 175-214. � 2002. “On the Concept of Following Logically” (Magda Stroińska and David Hitchcock, trans.) History and Philosophy of Logic 23: 155-96. 1.8 See also � List of things named after Alfred Tarski 1.9 References [1] Feferman A. [2] Feferman & Feferman, p.1 [3] Feferman & Feferman, pp.17-18 [4] Feferman & Feferman, p.26 [5] Feferman & Feferman, p.294 [6] “The Newsletter of the Janusz Korczak Association of Canada” (PDF). September 2007. Number 5. Retrieved 8 February 2012. [7] Feferman & Feferman (2004), pp. 239–242. [8] Feferman & Feferman, p. 67 [9] Feferman & Feferman, pp. 102-103 [10] Feferman & Feferman, Chap. 5, pp. 124-149 [11] Robert Vaught; John Addison; Benson Mates; Julia Robinson (1985). “Alfred Tarski, Mathematics: Berkeley”. University of California (System) Academic Senate. Retrieved 2008-12-26. [12] Obituary in Times, reproduced here [13] Gregory Moore, “Alfred Tarski” in Dictionary of Scientiﬁc Biography [14] Feferman [15] Chang, C.C., and Keisler, H.J., 1973. Model Theory. North-Holland, Amsterdam. American Elsevier, New York. [16] Alfred Tarski at the Mathematics Genealogy Project [17] Feferman & Feferman, pp. 385-386 [18] O'Connor, John J.; Robertson, Edmund F., “Alfred Tarski”, MacTutor History of Mathematics archive, University of St Andrews. [19] Feferman & Feferman, pp. 43-52, 69-75, 109-123, 189-195, 277-287, 334-342 [20] Vaught, Robert L. (Dec 1986). “Alfred Tarski’s Work in Model Theory”. Journal of Symbolic Logic (ASL) 51 (4): 869– 882. doi:10.2307/2273900. JSTOR 2273900. [21] Restall, Greg (2002–2006). “Great Moments in Logic”. Archived from the original on 6 December 2008. Retrieved 2009-01-03.
- 8 CHAPTER 1. ALFRED TARSKI [22] Sinaceur, Hourya (2001). “Alfred Tarski: Semantic Shift, Heuristic Shift in Metamathematics”. Synthese (Springer Verlag) 126 (1–2): 49–65. doi:10.1023/A:1005268531418. ISSN 0039-7857. [23] Alfred Tarski, “POJĘCIE PRAWDY W JĘZYKACH NAUK DEDUKCYJNYCH”, Towarszystwo Naukowe Warsza- wskie, Warszawa, 1933. (Text in Polish in the Digital Library WFISUW-IFISPAN-PTF). [24] Etchemendy, John (1999). The Concept of Logical Consequence. Stanford CA: CSLI Publications. ISBN 1-57586-194-1. [25] http://www.tandfonline.com/action/showMostCitedArticles?journalCode=thpl20#.UkH58D_-kQs [26] Halmos, Paul (1957). “Review: Logic, semantics, metamathematics. Papers from 1923 to 1938 by Alfred Tarski; translated by J. H. Woodger” (PDF). Bull. Amer. Math. Soc. 63 (2): 155–156. [27] Quine, W. V. (1938). “Review: Einführung in die mathematische Logik und in die Methodologie der Mathematik by Alfred Tarski. Vienna, Springer, 1937. x+166 pp.” (PDF). Bull. Amer. Math. Soc. 44 (5): 317–318. [28] Curry, Haskell B. (1942). “Review: Introduction to Logic and to the Methodology of Deductive Sciences by Alfred Tarski” (PDF). Bull. Amer. Math. Soc. 48 (7): 507–510. [29] McNaughton, Robert (1953). “Review: A decision method for elementary algebra and geometry by A. Tarski” (PDF). Bull. Amer. Math. Soc. 59 (1): 91–93. [30] Birkhoﬀ, Garrett (1950). “Review: Cardinal algebras by A. Tarski” (PDF). Bull. Amer. Math. Soc. 56 (2): 208–209. [31] Gál, Ilse Novak (1954). “Review: Undecidable theories by Alfred Tarski in collaboration with A. Mostowsku and R. M. Robinson” (PDF). Bull. Amer. Math. Soc. 60 (6): 570–572. 1.10 Further reading Biographical references � Feferman, Anita Burdman (1999). “Alfred Tarski”. American National Biography 21. Oxford University Press. pp. 330–332. ISBN 978-0-19-512800-0. � Feferman, Anita Burdman; Feferman, Solomon (2004). Alfred Tarski: Life and Logic. Cambridge University Press. ISBN 978-0-521-80240-6. OCLC 54691904. � Givant, Steven, 1991. “A portrait of Alfred Tarski”, Mathematical Intelligencer 13: 16-32. � Patterson, Douglas. Alfred Tarski: Philosophy of Language and Logic (Palgrave Macmillan; 2012) 262 pages; biography focused on his work from the late-1920s to the mid-1930s, with particular attention to inﬂuences from his teachers Stanislaw Lesniewski and Tadeusz Kotarbinski. Logic literature � The December 1986 issue of the Journal of Symbolic Logic surveys Tarski’s work on model theory (Robert Vaught), algebra (Jonsson), undecidable theories (McNulty), algebraic logic (Donald Monk), and geometry (Szczerba). The March 1988 issue of the same journal surveys his work on axiomatic set theory (Azriel Levy), real closed ﬁelds (Lou Van Den Dries), decidable theory (Doner and Wilfrid Hodges), metamathematics (Blok and Pigozzi), truth and logical consequence (John Etchemendy), and general philosophy (Patrick Suppes). � Blok, W. J.; Pigozzi, Don, “Alfred Tarski’s Work on General Metamathematics”, The Journal of Symbolic Logic, Vol. 53, No. 1 (Mar., 1988), pp. 36–50 � Chang, C.C., and Keisler, H.J., 1973. Model Theory. North-Holland, Amsterdam. American Elsevier, New York. � Corcoran, John, and Sagüillo, José Miguel, 2011. “The Absence of Multiple Universes of Discourse in the 1936 Tarski Consequence-Deﬁnition Paper”, History and Philosophy of Logic 32: 359–80. � Corcoran, John, and Weber, Leonardo, 2015. “Tarski’s convention T: condition beta”, South American Journal of Logic. 1, 3–32.
- 1.11. EXTERNAL LINKS 9 � Etchemendy, John, 1999. The Concept of Logical Consequence. Stanford CA: CSLI Publications. ISBN 1- 57586-194-1 � Feferman, Solomon, 1999. "Logic, Logics, and Logicism," Notre Dame Journal of Formal Logic 40: 31-54. � Grattan-Guinness, Ivor, 2000. The Search for Mathematical Roots 1870-1940. Princeton Uni. Press. � Kirkham, Richard, 1992. Theories of Truth. MIT Press. � Maddux, Roger D., 2006. Relation Algebras, vol. 150 in “Studies in Logic and the Foundations of Mathemat- ics”, Elsevier Science. � Mautner, F. I., 1946. “An Extension of Klein’s Erlanger Program: Logic as Invariant-Theory”, American Journal of Mathematics 68: 345-84. � McGee, Van, 1996. “Logical Operations”, Journal of Philosophical Logic 25: 567-80. � Popper, Karl R., 1972, Rev. Ed. 1979, “Philosophical Comments on Tarski’s Theory of Truth”, with Adden- dum, Objective Knowledge, Oxford: 319-340. � Sinaceur, H., 2001. “Alfred Tarski: Semantic shift, heuristic shift in metamathematics”, Synthese 126: 49-65. � Smith, James T., 2010. “Deﬁnitions and Nondeﬁnability in Geometry”, American Mathematical Monthly 117:475–89. � Wolenski, Jan, 1989. Logic and Philosophy in the Lvov–Warsaw School. Reidel/Kluwer. 1.11 External links Media related to Alfred Tarski at Wikimedia Commons � Stanford Encyclopedia of Philosophy: � Tarski’s Truth Deﬁnitions by Wilfred Hodges. � Alfred Tarski by Mario Gómez-Torrente. � Propositional Consequence Relations and Algebraic Logic by Ramon Jansana. Includes a fairly detailed discussion of Tarski’s work on these topics. � Tarski’s Semantic Theory on the Internet Encyclopedia of Philosophy.
- Chapter 2 Aristotle For other uses, see Aristotle (disambiguation). Aristotle (/ˈærɪˌstɒtəl/;[1] Greek: Ἀριστοτέλης [aristotélɛːs], Aristotélēs; 384 – 322 BC)[2] was a Greek philosopher and scientist born in the Macedonian city of Stagira, Chalkidice, on the northern periphery of Classical Greece. His father, Nicomachus, died when Aristotle was a child, whereafter Proxenus of Atarneus became his guardian.[3] At eighteen, he joined Plato’s Academy in Athens and remained there until the age of thirty-seven (c. 347 BC). His writings cover many subjects – including physics, biology, zoology, metaphysics, logic, ethics, aesthetics, poetry, the- ater, music, rhetoric, linguistics, politics and government – and constitute the ﬁrst comprehensive system of Western philosophy. Shortly after Plato died, Aristotle left Athens and, at the request of Philip of Macedon, tutored Alexander the Great starting from 343 BC.[4] According to the Encyclopædia Britannica, “Aristotle was the ﬁrst genuine scientist in history ... [and] every scientist is in his debt.”[5] Teaching Alexander the Great gave Aristotle many opportunities and an abundance of supplies. He established a library in the Lyceum which aided in the production of many of his hundreds of books. The fact that Aristotle was a pupil of Plato contributed to his former views of Platonism, but, following Plato’s death, Aristotle immersed himself in empirical studies and shifted from Platonism to empiricism.[6] He believed all peoples’ concepts and all of their knowledge was ultimately based on perception. Aristotle’s views on natural sciences represent the groundwork underlying many of his works. Aristotle’s views on physical science profoundly shaped medieval scholarship. Their inﬂuence extended into the Renaissance and were not replaced systematically until the Enlightenment and theories such as classical mechanics. Some of Aristotle’s zoological observations, such as on the hectocotyl (reproductive) arm of the octopus, were not conﬁrmed or refuted until the 19th century. His works contain the earliest known formal study of logic, which was incorporated in the late 19th century into modern formal logic. In metaphysics, Aristotelianism profoundly inﬂuenced Judeo-Islamic philosophical and theological thought during the Middle Ages and continues to inﬂuence Christian theology, especially the scholastic tradition of the Catholic Church. Aristotle was well known among medieval Muslim intellectuals and revered as “The First Teacher” (Arabic: لوألا ملعملا ). His ethics, though always inﬂuential, gained renewed interest with the modern advent of virtue ethics. All aspects of Aristotle’s philosophy continue to be the object of active academic study today. Though Aristotle wrote many elegant treatises and dialogues – Cicero described his literary style as “a river of gold”[7] – it is thought that only around a third of his original output has survived.[8] The sum of his work’s inﬂuence often ranks him among the world’s top personalities of all time with the greatest inﬂuence, along with his teacher Plato, and his pupil Alexander the Great.[9][10] 2.1 Life Aristotle, whose name means “the best purpose”,[11] was born in 384 BC in Stagira, Chalcidice, about 55 km (34 miles) east of modern-day Thessaloniki.[12] His father Nicomachus was the personal physician to King Amyntas of Macedon. Although there is little information on Aristotle’s childhood, he probably spent some time within the 10
- 2.1. LIFE 11 School of Aristotle in Mieza, Macedonia Macedonian palace, making his ﬁrst connections with the Macedonian monarchy.[13] At about the age of eighteen, Aristotle moved to Athens to continue his education at Plato’s Academy. He remained there for nearly twenty years before leaving Athens in 348/47 BC. The traditional story about his departure records that he was disappointed with the Academy’s direction after control passed to Plato’s nephew Speusippus, although it is possible that he feared anti-Macedonian sentiments and left before Plato had died.[14] Aristotle then accompanied Xenocrates to the court of his friend Hermias of Atarneus in Asia Minor. There, he traveled with Theophrastus to the island of Lesbos, where together they researched the botany and zoology of the island. Aristotle married Pythias, either Hermias’s adoptive daughter or niece. She bore him a daughter, whom they also named Pythias. Soon after Hermias’ death, Aristotle was invited by Philip II of Macedon to become the tutor to his son Alexander in 343 BC.[4] Aristotle was appointed as the head of the royal academy of Macedon. During that time he gave lessons not only to Alexander, but also to two other future kings: Ptolemy and Cassander.[15] Aristotle encouraged Alexander toward eastern conquest and his attitude towards Persia was unabashedly ethnocentric. In one famous example, he counsels Alexander to be “a leader to the Greeks and a despot to the barbarians, to look after the former as after friends and relatives, and to deal with the latter as with beasts or plants”.[15] By 335 BC, Artistotle had returned to Athens, establishing his own school there known as the Lyceum. Aristotle conducted courses at the school for the next twelve years. While in Athens, his wife Pythias died and Aristotle became involved with Herpyllis of Stagira, who bore him a son whom he named after his father, Nicomachus. According to the Suda, he also had an eromenos, Palaephatus of Abydus.[16] This period in Athens, between 335 and 323 BC, is when Aristotle is believed to have composed many of his works.[4] He wrote many dialogues of which only fragments have survived. Those works that have survived are in treatise form and were not, for the most part, intended for widespread publication; they are generally thought to be lecture aids for his students. His most important treatises include Physics, Metaphysics, Nicomachean Ethics, Politics, De Anima (On the Soul) and Poetics. Aristotle not only studied almost every subject possible at the time, but made signiﬁcant contributions to most of
- 12 CHAPTER 2. ARISTOTLE “Aristotle” by Francesco Hayez (1791–1882) them. In physical science, Aristotle studied anatomy, astronomy, embryology, geography, geology, meteorology, physics and zoology. In philosophy, he wrote on aesthetics, ethics, government, metaphysics, politics, economics, psychology, rhetoric and theology. He also studied education, foreign customs, literature and poetry. His combined works constitute a virtual encyclopedia of Greek knowledge. Near the end of his life, Alexander and Aristotle became estranged over Alexander’s relationship with Persia and Persians. A widespread tradition in antiquity suspected Aristotle of playing a role in Alexander’s death, but there is little evidence.[17]
- 2.2. THOUGHT 13 Following Alexander’s death, anti-Macedonian sentiment in Athens was rekindled. In 322 BC, Eurymedon the Hi- erophant denounced Aristotle for not holding the gods in honor, prompting him to ﬂee to his mother’s family estate in Chalcis, explaining: “I will not allow the Athenians to sin twice against philosophy”[18][19] – a reference to Athens’s prior trial and execution of Socrates. He died in Euboea of natural causes later that same year, having named his student Antipater as his chief executor and leaving a will in which he asked to be buried next to his wife.[20] Charles Walston argues that the tomb of Aristotle is located on the sacred way between Chalcis and Eretria and to have contained two styluses, a pen, a signet-ring and some terra-cottas as well as what is supposed to be the earthly remains of Aristotle in the form of some skull fragments.[21] In general, the details of the life of Aristotle are not well-established. The biographies of Aristotle written in ancient times are often speculative and historians only agree on a few salient points.[22] 2.2 Thought 2.2.1 Logic Main article: Term logic For more details on this topic, see Non-Aristotelian logic. With the Prior Analytics, Aristotle is credited with the earliest study of formal logic,[23] and his conception of it was the dominant form of Western logic until 19th century advances in mathematical logic.[24] Kant stated in the Critique of Pure Reason that Aristotle’s theory of logic completely accounted for the core of deductive inference. History Aristotle “says that 'on the subject of reasoning' he 'had nothing else on an earlier date to speak of'".[25] However, Plato reports that syntax was devised before him, by Prodicus of Ceos, who was concerned by the correct use of words. Logic seems to have emerged from dialectics; the earlier philosophers made frequent use of concepts like reductio ad absurdum in their discussions, but never truly understood the logical implications. Even Plato had diﬃculties with logic; although he had a reasonable conception of a deductive system, he could never actually construct one, thus he relied instead on his dialectic.[26] Plato believed that deduction would simply follow from premises, hence he focused on maintaining solid premises so that the conclusion would logically follow. Consequently, Plato realized that a method for obtaining conclusions would be most beneﬁcial. He never succeeded in devising such a method, but his best attempt was published in his book Sophist, where he introduced his division method.[27] Analytics and the Organon Main article: Organon What we today call Aristotelian logic, Aristotle himself would have labeled “analytics”. The term “logic” he reserved to mean dialectics. Most of Aristotle’s work is probably not in its original form, because it was most likely edited by students and later lecturers. The logical works of Aristotle were compiled into six books in about the early 1st century CE: 1. Categories 2. On Interpretation 3. Prior Analytics 4. Posterior Analytics 5. Topics 6. On Sophistical Refutations
- 14 CHAPTER 2. ARISTOTLE Aristotle portrayed in the 1493 Nuremberg Chronicle as a scholar of the 15th century AD. The order of the books (or the teachings from which they are composed) is not certain, but this list was derived from analysis of Aristotle’s writings. It goes from the basics, the analysis of simple terms in the Categories, the analysis of propositions and their elementary relations in On Interpretation, to the study of more complex forms, namely, syllogisms (in the Analytics) and dialectics (in the Topics and Sophistical Refutations). The ﬁrst three treatises form the core of the logical theory stricto sensu: the grammar of the language of logic and the correct rules of reasoning. There
- 2.2. THOUGHT 15 is one volume of Aristotle’s concerning logic not found in the Organon, namely the fourth book of Metaphysics.[26] 2.2.2 Aristotle’s epistemology Plato (left) and Aristotle (right), a detail of The School of Athens, a fresco by Raphael. Aristotle gestures to the earth, representing his belief in knowledge through empirical observation and experience, while holding a copy of his Nicomachean Ethics in his hand, whilst Plato gestures to the heavens, representing his belief in The Forms, while holding a copy of Timaeus Like his teacher Plato, Aristotle’s philosophy aims at the universal. Aristotle’s ontology, however, ﬁnds the universal
- 16 CHAPTER 2. ARISTOTLE in particular things, which he calls the essence of things, while in Plato’s ontology, the universal exists apart from particular things, and is related to them as their prototype or exemplar. For Aristotle, therefore, epistemology is based on the study of particular phenomena and rises to the knowledge of essences, while for Plato epistemology begins with knowledge of universal Forms (or ideas) and descends to knowledge of particular imitations of these. For Aristotle, “form” still refers to the unconditional basis of phenomena but is “instantiated” in a particular substance (see Universals and particulars, below). In a certain sense, Aristotle’s method is both inductive and deductive, while Plato’s is essentially deductive from a priori principles.[28] In Aristotle’s terminology, “natural philosophy” is a branch of philosophy examining the phenomena of the natural world, and includes ﬁelds that would be regarded today as physics, biology and other natural sciences. In modern times, the scope of philosophy has become limited to more generic or abstract inquiries, such as ethics and metaphysics, in which logic plays a major role. Today’s philosophy tends to exclude empirical study of the natural world by means of the scientiﬁc method. In contrast, Aristotle’s philosophical endeavors encompassed virtually all facets of intellectual inquiry. In the larger sense of the word, Aristotle makes philosophy coextensive with reasoning, which he also would describe as “science”. Note, however, that his use of the term science carries a diﬀerent meaning than that covered by the term “scientiﬁc method”. For Aristotle, “all science (dianoia) is either practical, poetical or theoretical” (Metaphysics 1025b25). By practical science, he means ethics and politics; by poetical science, he means the study of poetry and the other ﬁne arts; by theoretical science, he means physics, mathematics and metaphysics. If logic (or “analytics”) is regarded as a study preliminary to philosophy, the divisions of Aristotelian philosophy would consist of: (1) Logic; (2) Theoretical Philosophy, including Metaphysics, Physics and Mathematics; (3) Practical Philosophy and (4) Poetical Philosophy. In the period between his two stays in Athens, between his times at the Academy and the Lyceum, Aristotle conducted most of the scientiﬁc thinking and research for which he is renowned today. In fact, most of Aristotle’s life was devoted to the study of the objects of natural science. Aristotle’s metaphysics contains observations on the nature of numbers but he made no original contributions to mathematics. He did, however, perform original research in the natural sciences, e.g., botany, zoology, physics, astronomy, chemistry, meteorology, and several other sciences. Aristotle’s writings on science are largely qualitative, as opposed to quantitative. Beginning in the 16th century, scientists began applying mathematics to the physical sciences, and Aristotle’s work in this area was deemed hopelessly inadequate. His failings were largely due to the absence of concepts like mass, velocity, force and temperature. He had a conception of speed and temperature, but no quantitative understanding of them, which was partly due to the absence of basic experimental devices, like clocks and thermometers. His writings provide an account of many scientiﬁc observations, a mixture of precocious accuracy and curious errors. For example, in his History of Animals he claimed that human males have more teeth than females.[29] In a similar vein, John Philoponus, and later Galileo, showed by simple experiments that Aristotle’s theory that a heavier object falls faster than a lighter object is incorrect.[30] On the other hand, Aristotle refuted Democritus's claim that the Milky Way was made up of “those stars which are shaded by the earth from the sun’s rays,” pointing out (correctly, even if such reasoning was bound to be dismissed for a long time) that, given “current astronomical demonstrations” that “the size of the sun is greater than that of the earth and the distance of the stars from the earth many times greater than that of the sun, then ... the sun shines on all the stars and the earth screens none of them.”[31] In places, Aristotle goes too far in deriving 'laws of the universe' from simple observation and over-stretched reason. Today’s scientiﬁc method assumes that such thinking without suﬃcient facts is ineﬀective, and that discerning the validity of one’s hypothesis requires far more rigorous experimentation than that which Aristotle used to support his laws. Aristotle also had some scientiﬁc blind spots. He posited a geocentric cosmology that we may discern in selections of the Metaphysics, which was widely accepted up until the 16th century. From the 3rd century to the 16th century, the dominant view held that the Earth was the rotational center of the universe. Because he was perhaps the philosopher most respected by European thinkers during and after the Renaissance, these thinkers often took Aristotle’s erroneous positions as given, which held back science in this epoch.[32] However, Aristotle’s scientiﬁc shortcomings should not mislead one into forgetting his great advances in the many scientiﬁc ﬁelds. For instance, he founded logic as a formal science and created foundations to biology that were not superseded for two millennia. Moreover, he introduced the fundamental notion that nature is composed of things that change and that studying such changes can provide useful knowledge of underlying constants.
- 2.2. THOUGHT 17 2.2.3 Geology As quoted from Charles Lyell’s Principles of Geology: He [Aristotle] refers to many examples of changes now constantly going on, and insists emphatically on the great results which they must produce in the lapse of ages. He instances particular cases of lakes that had dried up, and deserts that had at length become watered by rivers and fertilized. He points to the growth of the Nilotic delta since the time of Homer, to the shallowing of the Palus Maeotis within sixty years from his own time ... He alludes ... to the upheaving of one of the Eolian islands, previous to a volcanic eruption. The changes of the earth, he says, are so slow in comparison to the duration of our lives, that they are overlooked; and the migrations of people after great catastrophes, and their removal to other regions, cause the event to be forgotten. He says [12th chapter of his Meteorics] 'the distribution of land and sea in particular regions does not endure throughout all time, but it becomes sea in those parts where it was land, and again it becomes land where it was sea, and there is reason for thinking that these changes take place according to a certain system, and within a certain period.' The concluding observation is as follows: 'As time never fails, and the universe is eternal, neither the Tanais, nor the Nile, can have ﬂowed for ever. The places where they rise were once dry, and there is a limit to their operations, but there is none to time. So also of all other rivers; they spring up and they perish; and the sea also continually deserts some lands and invades others The same tracts, therefore, of the earth are not some always sea, and others always continents, but every thing changes in the course of time.'[33] 2.2.4 Physics Main article: Physics (Aristotle) Five elements Main article: Classical element Aristotle proposed a ﬁfth element, aether, in addition to the four proposed earlier by Empedocles. � Earth, which is cold and dry; this corresponds to the modern idea of a solid. � Water, which is cold and wet; this corresponds to the modern idea of a liquid. � Air, which is hot and wet; this corresponds to the modern idea of a gas. � Fire, which is hot and dry; this corresponds to the modern ideas of plasma and heat. � Aether, which is the divine substance that makes up the heavenly spheres and heavenly bodies (stars and plan- ets). Each of the four earthly elements has its natural place. All that is earthly tends toward the center of the universe, i.e., the center of the Earth. Water tends toward a sphere surrounding the center. Air tends toward a sphere surrounding the water sphere. Fire tends toward the lunar sphere (in which the Moon orbits). When elements are moved out of their natural place, they naturally move back towards it. This is “natural motion”—motion requiring no extrinsic cause. So, for example, in water, earthy bodies sink while air bubbles rise up; in air, rain falls and ﬂame rises. Outside all the other spheres, the heavenly, ﬁfth element, manifested in the stars and planets, moves in the perfection of circles. Motion Main article: potentiality and actuality Aristotle deﬁned motion as the actuality of a potentiality as such.[34] Aquinas suggested that the passage be under- stood literally; that motion can indeed be understood as the active fulﬁllment of a potential, as a transition toward a
- 18 CHAPTER 2. ARISTOTLE potentially possible state. Because actuality and potentiality are normally opposites in Aristotle, other commentators either suggest that the wording which has come down to us is erroneous, or that the addition of the “as such” to the deﬁnition is critical to understanding it.[35] Causality, the four causes Main article: Four causes Aristotle suggested that the reason for anything coming about can be attributed to four diﬀerent types of simultane- ously active causal factors: � Material cause describes the material out of which something is composed. Thus the material cause of a table is wood, and the material cause of a car is rubber and steel. It is not about action. It does not mean one domino knocks over another domino. � The formal cause is its form, i.e., the arrangement of that matter. It tells us what a thing is, that any thing is determined by the deﬁnition, form, pattern, essence, whole, synthesis or archetype. It embraces the account of causes in terms of fundamental principles or general laws, as the whole (i.e., macrostructure) is the cause of its parts, a relationship known as the whole-part causation. Plainly put, the formal cause is the idea existing in the ﬁrst place as exemplar in the mind of the sculptor, and in the second place as intrinsic, determining cause, embodied in the matter. Formal cause could only refer to the essential quality of causation. A simple example of the formal cause is the mental image or idea that allows an artist, architect, or engineer to create his drawings. � The eﬃcient cause is “the primary source”, or that from which the change under consideration proceeds. It identiﬁes 'what makes of what is made and what causes change of what is changed' and so suggests all sorts of agents, nonliving or living, acting as the sources of change or movement or rest. Representing the current understanding of causality as the relation of cause and eﬀect, this covers the modern deﬁnitions of “cause” as either the agent or agency or particular events or states of aﬀairs. So, take the two dominoes, this time of equal weighting, the ﬁrst is knocked over causing the second also to fall over. � The ﬁnal cause is its purpose, or that for the sake of which a thing exists or is done, including both purposeful and instrumental actions and activities. The ﬁnal cause or teleos is the purpose or function that something is supposed to serve. This covers modern ideas of motivating causes, such as volition, need, desire, ethics, or spiritual beliefs. Additionally, things can be causes of one another, causing each other reciprocally, as hard work causes ﬁtness and vice versa, although not in the same way or function, the one is as the beginning of change, the other as the goal. (Thus Aristotle ﬁrst suggested a reciprocal or circular causality as a relation of mutual dependence or inﬂuence of cause upon eﬀect). Moreover, Aristotle indicated that the same thing can be the cause of contrary eﬀects; its presence and absence may result in diﬀerent outcomes. Simply it is the goal or purpose that brings about an event. Our two dominoes require someone or something to intentionally knock over the ﬁrst domino, because it cannot fall of its own accord. Aristotle marked two modes of causation: proper (prior) causation and accidental (chance) causation. All causes, proper and incidental, can be spoken as potential or as actual, particular or generic. The same language refers to the eﬀects of causes, so that generic eﬀects assigned to generic causes, particular eﬀects to particular causes, operating causes to actual eﬀects. Essentially, causality does not suggest a temporal relation between the cause and the eﬀect. Optics Aristotle held more accurate theories on some optical concepts than other philosophers of his day. The second oldest written evidence of a camera obscura (after Mozi c. 400 BC) can be found in Aristotle’s documentation of such a device in 350 BC in Problemata. Aristotle’s apparatus contained a dark chamber that had a single small hole, or aperture, to allow for sunlight to enter. Aristotle used the device to make observations of the sun and noted that no matter what shape the hole was, the sun would still be correctly displayed as a round object. In modern cameras, this is analogous to the diaphragm. Aristotle also made the observation that when the distance between the aperture and the surface with the image increased, the image was magniﬁed.[36]
- 2.2. THOUGHT 19 Chance and spontaneity According to Aristotle, spontaneity and chance are causes of some things, distinguishable from other types of cause. Chance as an incidental cause lies in the realm of accidental things. It is “from what is spontaneous” (but note that what is spontaneous does not come from chance). For a better understanding of Aristotle’s conception of “chance” it might be better to think of “coincidence": Something takes place by chance if a person sets out with the intent of having one thing take place, but with the result of another thing (not intended) taking place. For example: A person seeks donations. That person may ﬁnd another person willing to donate a substantial sum. However, if the person seeking the donations met the person donating, not for the purpose of collecting donations, but for some other purpose, Aristotle would call the collecting of the donation by that particular donator a result of chance. It must be unusual that something happens by chance. In other words, if something happens all or most of the time, we cannot say that it is by chance. There is also more speciﬁc kind of chance, which Aristotle names “luck”, that can only apply to human beings, because it is in the sphere of moral actions. According to Aristotle, luck must involve choice (and thus deliberation), and only humans are capable of deliberation and choice. “What is not capable of action cannot do anything by chance”.[37] 2.2.5 Metaphysics Main article: Metaphysics (Aristotle) Aristotle deﬁnes metaphysics as “the knowledge of immaterial being,” or of “being in the highest degree of abstraction.” He refers to metaphysics as “ﬁrst philosophy”, as well as “the theologic science.” Substance, potentiality and actuality See also: Potentiality and actuality (Aristotle) Aristotle examines the concepts of substance and essence (ousia) in his Metaphysics (Book VII), and he concludes that a particular substance is a combination of both matter and form. In book VIII, he distinguishes the matter of the substance as the substratum, or the stuﬀ of which it is composed. For example, the matter of a house is the bricks, stones, timbers etc., or whatever constitutes the potential house, while the form of the substance is the actual house, namely 'covering for bodies and chattels’ or any other diﬀerentia (see also predicables) that let us deﬁne something as a house. The formula that gives the components is the account of the matter, and the formula that gives the diﬀerentia is the account of the form.[38] With regard to the change (kinesis) and its causes now, as he deﬁnes in his Physics and On Generation and Corruption 319b–320a, he distinguishes the coming to be from: 1. growth and diminution, which is change in quantity; 2. locomotion, which is change in space; and 3. alteration, which is change in quality. The coming to be is a change where nothing persists of which the resultant is a property. In that particular change he introduces the concept of potentiality (dynamis) and actuality (entelecheia) in association with the matter and the form. Referring to potentiality, this is what a thing is capable of doing, or being acted upon, if the conditions are right and it is not prevented by something else. For example, the seed of a plant in the soil is potentially (dynamei) plant, and if is not prevented by something, it will become a plant. Potentially beings can either 'act' (poiein) or 'be acted upon' (paschein), which can be either innate or learned. For example, the eyes possess the potentiality of sight (innate – being acted upon), while the capability of playing the ﬂute can be possessed by learning (exercise – acting). Actuality is the fulﬁllment of the end of the potentiality. Because the end (telos) is the principle of every change, and for the sake of the end exists potentiality, therefore actuality is the end. Referring then to our previous example, we could say that an actuality is when a plant does one of the activities that plants do.
- 20 CHAPTER 2. ARISTOTLE “For that for the sake of which a thing is, is its principle, and the becoming is for the sake of the end; and the actuality is the end, and it is for the sake of this that the potentiality is acquired. For animals do not see in order that they may have sight, but they have sight that they may see.”[39] In summary, the matter used to make a house has potentiality to be a house and both the activity of building and the form of the ﬁnal house are actualities, which is also a ﬁnal cause or end. Then Aristotle proceeds and concludes that the actuality is prior to potentiality in formula, in time and in substantiality. With this deﬁnition of the particular substance (i.e., matter and form), Aristotle tries to solve the problem of the unity of the beings, for example, “what is it that makes a man one"? Since, according to Plato there are two Ideas: animal and biped, how then is man a unity? However, according to Aristotle, the potential being (matter) and the actual one (form) are one and the same thing.[40] Universals and particulars Main article: Aristotle’s theory of universals Aristotle’s predecessor, Plato, argued that all things have a universal form, which could be either a property, or a relation to other things. When we look at an apple, for example, we see an apple, and we can also analyze a form of an apple. In this distinction, there is a particular apple and a universal form of an apple. Moreover, we can place an apple next to a book, so that we can speak of both the book and apple as being next to each other. Plato argued that there are some universal forms that are not a part of particular things. For example, it is possible that there is no particular good in existence, but “good” is still a proper universal form. Bertrand Russell is a 20th-century philosopher who agreed with Plato on the existence of “uninstantiated universals”. Aristotle disagreed with Plato on this point, arguing that all universals are instantiated. Aristotle argued that there are no universals that are unattached to existing things. According to Aristotle, if a universal exists, either as a particular or a relation, then there must have been, must be currently, or must be in the future, something on which the universal can be predicated. Consequently, according to Aristotle, if it is not the case that some universal can be predicated to an object that exists at some period of time, then it does not exist. In addition, Aristotle disagreed with Plato about the location of universals. As Plato spoke of the world of the forms, a location where all universal forms subsist, Aristotle maintained that universals exist within each thing on which each universal is predicated. So, according to Aristotle, the form of apple exists within each apple, rather than in the world of the forms. 2.2.6 Biology and medicine In Aristotelian science, especially in biology, things he saw himself have stood the test of time better than his retelling of the reports of others, which contain error and superstition. He dissected animals but not humans; his ideas on how the human body works have been almost entirely superseded. Empirical research program Aristotle is the earliest natural historian whose work has survived in some detail. Aristotle certainly did research on the natural history of Lesbos, and the surrounding seas and neighbouring areas. The works that reﬂect this research, such as History of Animals, Generation of Animals, and Parts of Animals, contain some observations and interpretations, along with sundry myths and mistakes. The most striking passages are about the sea-life visible from observation on Lesbos and available from the catches of ﬁshermen. His observations on catﬁsh, electric ﬁsh (Torpedo) and angler-ﬁsh are detailed, as is his writing on cephalopods, namely, Octopus, Sepia (cuttleﬁsh) and the paper nautilus (Argonauta argo). His description of the hectocotyl arm, used in sexual reproduction, was widely disbelieved until its rediscovery in the 19th century. He separated the aquatic mammals from ﬁsh, and knew that sharks and rays were part of the group he called Selachē (selachians).[41] Another good example of his methods comes from the Generation of Animals in which Aristotle describes breaking open fertilized chicken eggs at intervals to observe when visible organs were generated. He gave accurate descriptions of ruminants' four-chambered fore-stomachs, and of the ovoviviparous embryological development of the hound shark Mustelus mustelus.[42]
- 2.2. THOUGHT 21 Octopus swimming Classiﬁcation of living things Aristotle distinguished about 500 species of birds, mammals and ﬁshes.[43] His classiﬁcation of living things contains some elements which still existed in the 19th century. What the modern zoologist would call vertebrates and inverte- brates, Aristotle called 'animals with blood' and 'animals without blood' (he did not know that complex invertebrates do make use of hemoglobin, but of a diﬀerent kind from vertebrates). Animals with blood were divided into live- bearing (mammals), and egg-bearing (birds and ﬁsh). Invertebrates ('animals without blood') are insects, crustacea (divided into non-shelled – cephalopods – and shelled) and testacea (molluscs). In some respects, this incomplete classiﬁcation is better than that of Linnaeus, who crowded the invertebrata together into two groups, Insecta and Vermes (worms). For Charles Singer, “Nothing is more remarkable than [Aristotle’s] eﬀorts to [exhibit] the relationships of living things as a scala naturae"[41] Aristotle’s History of Animals classiﬁed organisms in relation to a hierarchical "Ladder of Life" (scala naturae or Great Chain of Being), placing them according to complexity of structure and function so that higher organisms showed greater vitality and ability to move.[44] Aristotle believed that intellectual purposes, i.e., ﬁnal causes, guided all natural processes. Such a teleological view gave Aristotle cause to justify his observed data as an expression of formal design. Noting that “no animal has, at the same time, both tusks and horns,” and “a single-hooved animal with two horns I have never seen,” Aristotle suggested that Nature, giving no animal both horns and tusks, was staving oﬀ vanity, and giving creatures faculties only to such a degree as they are necessary. Noting that ruminants had multiple stomachs and weak teeth, he supposed the ﬁrst was to compensate for the latter, with Nature trying to preserve a type of balance.[45] In a similar fashion, Aristotle believed that creatures were arranged in a graded scale of perfection rising from plants on up to man, the scala naturae.[46] His system had eleven grades, arranged according “to the degree to which they are infected with potentiality”, expressed in their form at birth. The highest animals laid warm and wet creatures alive, the lowest bore theirs cold, dry, and in thick eggs.
- 22 CHAPTER 2. ARISTOTLE Torpedo fuscomaculata Leopard shark Aristotle also held that the level of a creature’s perfection was reﬂected in its form, but not preordained by that form. Ideas like this, and his ideas about souls, are not regarded as science at all in modern times. He placed emphasis on the type(s) of soul an organism possessed, asserting that plants possess a vegetative soul, responsible for reproduction and growth, animals a vegetative and a sensitive soul, responsible for mobility and sen- sation, and humans a vegetative, a sensitive, and a rational soul, capable of thought and reﬂection.[47] Aristotle, in contrast to earlier philosophers, but in accordance with the Egyptians, placed the rational soul in the heart, rather than the brain.[48] Notable is Aristotle’s division of sensation and thought, which generally went against previous philosophers, with the exception of Alcmaeon.[49]
- 2.2. THOUGHT 23 Successor: Theophrastus Main articles: Theophrastus and Historia Plantarum (Theophrastus) Aristotle’s successor at the Lyceum, Theophrastus, wrote a series of books on botany—the History of Plants—which survived as the most important contribution of antiquity to botany, even into the Middle Ages. Many of Theophrastus’ names survive into modern times, such as carpos for fruit, and pericarpion for seed vessel. Rather than focus on formal causes, as Aristotle did, Theophrastus suggested a mechanistic scheme, drawing analogies between natural and artiﬁcial processes, and relying on Aristotle’s concept of the eﬃcient cause. Theophrastus also recognized the role of sex in the reproduction of some higher plants, though this last discovery was lost in later ages.[50] Inﬂuence on Hellenistic medicine For more details on this topic, see Medicine in ancient Greece. After Theophrastus, the Lyceum failed to produce any original work. Though interest in Aristotle’s ideas survived, they were generally taken unquestioningly.[51] It is not until the age of Alexandria under the Ptolemies that advances in biology can be again found. The ﬁrst medical teacher at Alexandria, Herophilus of Chalcedon, corrected Aristotle, placing intelligence in the brain, and connected the nervous system to motion and sensation. Herophilus also distinguished between veins and arteries, noting that the latter pulse while the former do not.[52] Though a few ancient atomists such as Lucretius chal- lenged the teleological viewpoint of Aristotelian ideas about life, teleology (and after the rise of Christianity, natural theology) would remain central to biological thought essentially until the 18th and 19th centuries. Ernst Mayr claimed that there was “nothing of any real consequence in biology after Lucretius and Galen until the Renaissance.”[53] Aris- totle’s ideas of natural history and medicine survived, but they were generally taken unquestioningly.[54] 2.2.7 Psychology Aristotle’s psychology, given in his treatise On the Soul (peri psyche, often known by its Latin title De Anima), posits three kinds of soul (“psyches”): the vegetative soul, the sensitive soul, and the rational soul. Humans have a rational soul. This kind of soul is capable of the same powers as the other kinds: Like the vegetative soul it can grow and nourish itself; like the sensitive soul it can experience sensations and move locally. The unique part of the human, rational soul is its ability to receive forms of other things and compare them. For Aristotle, the soul (psyche) was a simpler concept than it is for us today. By soul he simply meant the form of a living being. Because all beings are composites of form and matter, the form of living beings is that which endows them with what is speciﬁc to living beings, e.g. the ability to initiate movement (or in the case of plants, growth and chemical transformations, which Aristotle considers types of movement).[55] Memory According to Aristotle, memory is the ability to hold a perceived experience in your mind and to have the ability to distinguish between the internal “appearance” and an occurrence in the past.[56] In other words, a memory is a mental picture (phantasm) in which Aristotle deﬁnes in De Anima, as an appearance which is imprinted on the part of the body that forms a memory. Aristotle believed an “imprint” becomes impressed on a semi-ﬂuid bodily organ that undergoes several changes in order to make a memory. A memory occurs when a stimuli is too complex that the nervous system (semi-ﬂuid bodily organ) cannot receive all the impressions at once. These changes are the same as those involved in the operations of sensation, common sense, and thinking .[57] The mental picture imprinted on the bodily organ is the ﬁnal product of the entire process of sense perception. It does not matter if the experience was seen or heard, every experience ends up as a mental image in memory [58] Aristotle uses the word “memory” for two basic abilities. First, the actual retaining of the experience in the mnemonic “imprint” that can develop from sensation. Second, the intellectual anxiety that comes with the “imprint” due to being impressed at a particular time and processing speciﬁc contents. These abilities can be explained as memory is neither sensation nor thinking because is arises only after a lapse of time. Therefore, memory is of the past, [59] prediction is of the future, and sensation is of the present. The retrieval of our “imprints” cannot be performed suddenly.
- 24 CHAPTER 2. ARISTOTLE A transitional channel is needed and located in our past experiences, both for our previous experience and present experience. Aristotle proposed that slow-witted people have good memory because the ﬂuids in their brain do not wash away their memory organ used to imprint experiences and so the “imprint” can easily continue. However, they cannot be too slow or the hardened surface of the organ will not receive new “imprints”. He believed the young and the old do not properly develop an “imprint”. Young people undergo rapid changes as they develop, while the elderly’s organs are beginning to decay, thus stunting new “imprints”. Likewise, people who are too quick-witted are similar to the young and the image cannot be ﬁxed because of the rapid changes of their organ. Because intellectual functions are not involved in memory, memories belong to some animals too, but only those in which have perception of time. Recollection Because Aristotle believes people receive all kinds of sense perceptions and people perceive them as images or “imprints”, people are continually weaving together new “imprints” of things they experience. In or- der to search for these “imprints”, people search the memory itself.[60] Within the memory, if one experience is oﬀered instead of a speciﬁc memory, that person will reject this experience until they ﬁnd what they are looking for. Recollection occurs when one experience naturally follows another. If the chain of “images” is needed, one memory will stimulate the other. If the chain of “images” is not needed, but expected, then it will only stimulate the other memory in most instances. When people recall experiences, they stimulate certain previous experiences until they have stimulated the one that was needed.[61] Recollection is the self-directed activity of retrieving the information stored in a memory “imprint” after some time has passed. Retrieval of stored information is dependent on the scope of mnemonic capabilities of a being (human or animal) and the abilities the human or animal possesses .[62] Only humans will remember “imprints” of intellectual activity, such as numbers and words. Animals that have perception of time will be able to retrieve memories of their past observations. Remembering involves only perception of the things remembered and of the time passed. Recol- lection of an “imprint” is when the present experiences a person remembers are similar with elements corresponding in character and arrangement of past sensory experiences. When an “imprint” is recalled, it may bring forth a large group of related “imprints”.[63] Aristotle believed the chain of thought, which ends in recollection of certain “imprints”, was connected systematically in three sorts of relationships: similarity, contrast, and contiguity. These three laws make up his Laws of Association. Aristotle believed that past experiences are hidden within our mind. A force operates to awaken the hidden material to bring up the actual experience. According to Aristotle, association is the power innate in a mental state, which operates upon the unexpressed remains of former experiences, allowing them to rise and be recalled.[64] Dreams Sleep Before understanding Aristotle’s take on dreams, ﬁrst his idea of sleep must be examined. Aristotle gives an account of his explanation of sleep in On Sleep and Wakefulness.[65] Sleep takes place as a result of overuse of the senses[66] or of digestion,[65] so it is vital to the body, including the senses, so it can be revitalized.[66] While a person is asleep, the critical activities, which include thinking, sensing, recalling and remembering, do not function as they do during wakefulness.[66] Since a person cannot sense during sleep they can also not have a desire, which is the result of a sensation.[66] However, the senses are able to work during sleep,[66] albeit diﬀerently than when a person is awake because during sleep a person can still have sensory experiences.[65] Also, all of the senses are not inactive during sleep, only the ones that are weary.[66] Theory of dreams Dreams do not involve actually sensing a stimulus because, as discussed, the senses do not work as they normally do during sleep.[66] In dreams, sensation is still involved, but in an altered manner than when awake.[66] Aristotle explains the phenomenon that occurs when a person stares at a moving stimulus such as the waves in a body of water.[65] When they look away from that stimulus, the next thing they look at appears to be moving in a wave like motion. When a person perceives a stimulus and the stimulus is no longer the focus of their attention, it leaves an impression.[65] When the body is awake and the senses are functioning properly, a person constantly encounters new stimuli to sense and so the impressions left from previously perceived stimuli become irrelevant.[66] However, during sleep the impressions stimuli made throughout the day become noticed because there are not new sensory experiences to distract from these impressions that were made.[65] So, dreams result from these lasting impressions. Since impressions are all that are left and not the exact stimuli, dreams will not resemble the actual experience that occurred when awake.[67] During sleep, a person is in an altered state of mind.[65] Aristotle compares a sleeping person to a person who is
- 2.2. THOUGHT 25 overtaken by strong feelings toward a stimulus.[65] For example, a person who has a strong infatuation with someone may begin to think they see that person everywhere because they are so overtaken by their feelings.[65] When a person is asleep, their senses are not acting as they do when they are awake and this results in them thinking like a person who is inﬂuenced by strong feelings.[65] Since a person sleeping is in this suggestible state, they become easily deceived by what appears in their dreams.[65] When asleep, a person is unable to make judgments as they do when they are awake[65] Due to the senses not func- tioning normally during sleep, they are unable to help a person judge what is happening in their dream.[65] This in turn leads the person to believe the dream is real.[65] Dreams may be absurd in nature but the senses are not able to discern whether they are real or not.[65] So, the dreamer is left to accept the dream because they lack the choice to judge it. One component of Aristotle’s theory of dreams introduces ideas that are contradictory to previously held beliefs.[68] He claimed that dreams are not foretelling and that they are not sent by a divine being.[68] Aristotle reasoned that instances in which dreams do resemble future events are happenstances not divinations.[68] These ideas were contra- dictory to what had been believed about dreams, but at the time in which he introduced these ideas more thinkers were beginning to give naturalistic as opposed to supernatural explanations to phenomena.[68] Aristotle also includes in his theory of dreams what constitutes a dream and what does not. He claimed that a dream is ﬁrst established by the fact that the person is asleep when they experience it.[67] If a person had an image appear for a moment after waking up or if they see something in the dark it is not considered a dream because they were awake when it occurred.[67] Secondly, any sensory experience that actually occurs while a person is asleep and is perceived by the person while asleep does not qualify as part of a dream.[67] For example, if, while a person is sleeping, a door shuts and in their dream they hear a door is shut, Aristotle argues that this sensory experience is not part of the dream.[67] The actual sensory experience is perceived by the senses, the fact that it occurred while the person was asleep does not make it part of the dream.[67] Lastly, the images of dreams must be a result of lasting impressions of sensory experiences had when awake.[67] 2.2.8 Practical philosophy Ethics Main article: Aristotelian ethics Aristotle considered ethics to be a practical rather than theoretical study, i.e., one aimed at becoming good and doing good rather than knowing for its own sake. He wrote several treatises on ethics, including most notably, the Nicomachean Ethics. Aristotle taught that virtue has to do with the proper function (ergon) of a thing. An eye is only a good eye in so much as it can see, because the proper function of an eye is sight. Aristotle reasoned that humans must have a function speciﬁc to humans, and that this function must be an activity of the psuchē (normally translated as soul) in accordance with reason (logos). Aristotle identiﬁed such an optimum activity of the soul as the aim of all human deliberate action, eudaimonia, generally translated as “happiness” or sometimes “well being”. To have the potential of ever being happy in this way necessarily requires a good character (ēthikē aretē), often translated as moral (or ethical) virtue (or excellence).[69] Aristotle taught that to achieve a virtuous and potentially happy character requires a ﬁrst stage of having the fortune to be habituated not deliberately, but by teachers, and experience, leading to a later stage in which one consciously chooses to do the best things. When the best people come to live life this way their practical wisdom (phronesis) and their intellect (nous) can develop with each other towards the highest possible human virtue, the wisdom of an accomplished theoretical or speculative thinker, or in other words, a philosopher.[70] Politics Main article: Politics (Aristotle) In addition to his works on ethics, which address the individual, Aristotle addressed the city in his work titled Politics. Aristotle considered the city to be a natural community. Moreover, he considered the city to be prior in importance to the family which in turn is prior to the individual, “for the whole must of necessity be prior to the part”.[71] He also famously stated that “man is by nature a political animal”. Aristotle conceived of politics as being like an
- 26 CHAPTER 2. ARISTOTLE organism rather than like a machine, and as a collection of parts none of which can exist without the others. Aristotle’s conception of the city is organic, and he is considered one of the ﬁrst to conceive of the city in this manner.[72] The common modern understanding of a political community as a modern state is quite diﬀerent from Aristotle’s understanding. Although he was aware of the existence and potential of larger empires, the natural community according to Aristotle was the city (polis) which functions as a political “community” or “partnership” (koinōnia). The aim of the city is not just to avoid injustice or for economic stability, but rather to allow at least some citizens the possibility to live a good life, and to perform beautiful acts: “The political partnership must be regarded, therefore, as being for the sake of noble actions, not for the sake of living together.” This is distinguished from modern approaches, beginning with social contract theory, according to which individuals leave the state of nature because of “fear of violent death” or its “inconveniences.”[73] Excerpt from a speech by the character ‘Aristotle’ in the book Protrepticus (Hutchinson and Johnson, 2015 p. 22)[74] For we all agree that the most excellent man should rule, i.e., the supreme by nature, and that the law rules and alone is authoritative; but the law is a kind of intelligence, i.e. a discourse based on intelligence. And again, what standard do we have, what criterion of good things, that is more precise than the intelligent man? For all that this man will choose, if the choice is based on his knowledge, are good things and their contraries are bad. And since everybody chooses most of all what conforms to their own proper dispositions (a just man choosing to live justly, a man with bravery to live bravely, likewise a self-controlled man to live with self-control), it is clear that the intelligent man will choose most of all to be intelligent; for this is the function of that capacity. Hence it’s evident that, according to the most authoritative judgment, intelligence is supreme among goods. Rhetoric and poetics Main articles: Rhetoric (Aristotle) and Poetics (Aristotle) Aristotle considered epic poetry, tragedy, comedy, dithyrambic poetry and music to be imitative, each varying in imitation by medium, object, and manner.[75] For example, music imitates with the media of rhythm and harmony, whereas dance imitates with rhythm alone, and poetry with language. The forms also diﬀer in their object of imitation. Comedy, for instance, is a dramatic imitation of men worse than average; whereas tragedy imitates men slightly better than average. Lastly, the forms diﬀer in their manner of imitation – through narrative or character, through change or no change, and through drama or no drama.[76] Aristotle believed that imitation is natural to mankind and constitutes one of mankind’s advantages over animals.[77] While it is believed that Aristotle’s Poetics comprised two books – one on comedy and one on tragedy – only the portion that focuses on tragedy has survived. Aristotle taught that tragedy is composed of six elements: plot-structure, character, style, thought, spectacle, and lyric poetry.[78] The characters in a tragedy are merely a means of driving the story; and the plot, not the characters, is the chief focus of tragedy. Tragedy is the imitation of action arousing pity and fear, and is meant to eﬀect the catharsis of those same emotions. Aristotle concludes Poetics with a discussion on which, if either, is superior: epic or tragic mimesis. He suggests that because tragedy possesses all the attributes of an epic, possibly possesses additional attributes such as spectacle and music, is more uniﬁed, and achieves the aim of its mimesis in shorter scope, it can be considered superior to epic.[79] Aristotle was a keen systematic collector of riddles, folklore, and proverbs; he and his school had a special interest in the riddles of the Delphic Oracle and studied the fables of Aesop.[80] 2.2.9 Views on women Main article: Aristotle’s views on women Aristotle’s analysis of procreation describes an active, ensouling masculine element bringing life to an inert, passive female element. On this ground, feminist metaphysics have accused Aristotle of misogyny[81] and sexism.[82] How- ever, Aristotle gave equal weight to women’s happiness as he did to men’s, and commented in his Rhetoric that the things that lead to happiness need to be in women as well as men.[83]
- 2.3. LOSS AND PRESERVATION OF HIS WORKS 27 2.3 Loss and preservation of his works See also: Corpus Aristotelicum See also: Recovery of Aristotle Modern scholarship reveals that Aristotle’s “lost” works stray considerably in characterization[84] from the surviving Aristotelian corpus. Whereas the lost works appear to have been originally written with an intent for subsequent publication, the surviving works do not appear to have been so.[84] Rather the surviving works mostly resemble lecture notes unintended for publication.[84] The authenticity of a portion of the surviving works as originally Aristotelian is also today held suspect, with some books duplicating or summarizing each other, the authorship of one book questioned and another book considered to be unlikely Aristotle’s at all.[84] Some of the individual works within the corpus, including the Constitution of Athens, are regarded by most scholars as products of Aristotle’s “school,” perhaps compiled under his direction or supervision. Others, such as On Colors, may have been produced by Aristotle’s successors at the Lyceum, e.g., Theophrastus and Straton. Still others acquired Aristotle’s name through similarities in doctrine or content, such as the De Plantis, possibly by Nicolaus of Damascus. Other works in the corpus include medieval palmistries and astrological and magical texts whose connections to Aristotle are purely fanciful and self-promotional.[85] According to a distinction that originates with Aristotle himself, his writings are divisible into two groups: the "exoteric" and the "esoteric".[86] Most scholars have understood this as a distinction between works Aristotle in- tended for the public (exoteric), and the more technical works intended for use within the Lyceum course / school (esoteric).[87] Modern scholars commonly assume these latter to be Aristotle’s own (unpolished) lecture notes (or in some cases possible notes by his students).[88] However, one classic scholar oﬀers an alternative interpretation. The 5th century neoplatonist Ammonius Hermiae writes that Aristotle’s writing style is deliberately obscurantist so that “good people may for that reason stretch their mind even more, whereas empty minds that are lost through carelessness will be put to ﬂight by the obscurity when they encounter sentences like these.”[89] Another common assumption is that none of the exoteric works is extant – that all of Aristotle’s extant writings are of the esoteric kind. Current knowledge of what exactly the exoteric writings were like is scant and dubious, though many of them may have been in dialogue form. (Fragments of some of Aristotle’s dialogues have survived.) Perhaps it is to these that Cicero refers when he characterized Aristotle’s writing style as “a river of gold";[90] it is hard for many modern readers to accept that one could seriously so admire the style of those works currently available to us.[88] However, some modern scholars have warned that we cannot know for certain that Cicero’s praise was reserved speciﬁcally for the exoteric works; a few modern scholars have actually admired the concise writing style found in Aristotle’s extant works.[91] One major question in the history of Aristotle’s works, then, is how were the exoteric writings all lost, and how did the ones we now possess come to us[92] The story of the original manuscripts of the esoteric treatises is described by Strabo in his Geography and Plutarch in his Parallel Lives.[93] The manuscripts were left from Aristotle to his successor Theophrastus, who in turn willed them to Neleus of Scepsis. Neleus supposedly took the writings from Athens to Scepsis, where his heirs let them languish in a cellar until the 1st century BC, when Apellicon of Teos discovered and purchased the manuscripts, bringing them back to Athens. According to the story, Apellicon tried to repair some of the damage that was done during the manuscripts’ stay in the basement, introducing a number of errors into the text. When Lucius Cornelius Sulla occupied Athens in 86 BC, he carried oﬀ the library of Apellicon to Rome, where they were ﬁrst published in 60 BC by the grammarian Tyrannion of Amisus and then by the philosopher Andronicus of Rhodes.[94][95] Carnes Lord attributes the popular belief in this story to the fact that it provides “the most plausible explanation for the rapid eclipse of the Peripatetic school after the middle of the third century, and for the absence of widespread knowl- edge of the specialized treatises of Aristotle throughout the Hellenistic period, as well as for the sudden reappearance of a ﬂourishing Aristotelianism during the ﬁrst century B.C.”[96] Lord voices a number of reservations concerning this story, however. First, the condition of the texts is far too good for them to have suﬀered considerable damage followed by Apellicon’s inexpert attempt at repair. Second, there is “incontrovertible evidence,” Lord says, that the treatises were in circulation during the time in which Strabo and Plutarch suggest they were conﬁned within the cellar in Scepsis. Third, the deﬁnitive edition of Aristotle’s texts seems to have been made in Athens some ﬁfty years before Andronicus supposedly compiled his. And fourth, ancient library catalogues predating Andronicus’ intervention list an Aristotelian corpus quite similar to the one we currently possess. Lord sees a number of post-Aristotelian interpolations in the Politics, for example, but is generally conﬁdent that the work has come down to us relatively intact. On the one hand, the surviving texts of Aristotle do not derive from ﬁnished literary texts, but rather from working
- 28 CHAPTER 2. ARISTOTLE drafts used within Aristotle’s school, as opposed, on the other hand, to the dialogues and other “exoteric” texts which Aristotle published more widely during his lifetime. The consensus is that Andronicus of Rhodes collected the esoteric works of Aristotle’s school which existed in the form of smaller, separate works, distinguished them from those of Theophrastus and other Peripatetics, edited them, and ﬁnally compiled them into the more cohesive, larger works as they are known today.[97] 2.4 Legacy More than 2300 years after his death, Aristotle remains one of the most inﬂuential people who ever lived. He con- tributed to almost every ﬁeld of human knowledge then in existence, and he was the founder of many new ﬁelds. According to the philosopher Bryan Magee, “it is doubtful whether any human being has ever known as much as he did”.[98] Among countless other achievements, Aristotle was the founder of formal logic,[99] pioneered the study of zoology, and left every future scientist and philosopher in his debt through his contributions to the scientiﬁc method.[100][101] Despite these achievements, the inﬂuence of Aristotle’s errors is considered by some to have held back science con- siderably. Bertrand Russell notes that “almost every serious intellectual advance has had to begin with an attack on some Aristotelian doctrine”. Russell also refers to Aristotle’s ethics as “repulsive”, and calls his logic “as deﬁ- nitely antiquated as Ptolemaic astronomy”. Russell notes that these errors make it diﬃcult to do historical justice to Aristotle, until one remembers how large of an advance he made upon all of his predecessors.[4] 2.4.1 Later Greek philosophers The immediate inﬂuence of Aristotle’s work was felt as the Lyceum grew into the Peripatetic school. Aristo- tle’s notable students included Aristoxenus, Dicaearchus, Demetrius of Phalerum, Eudemos of Rhodes, Harpalus, Hephaestion, Meno, Mnason of Phocis, Nicomachus, and Theophrastus. Aristotle’s inﬂuence over Alexander the Great is seen in the latter’s bringing with him on his expedition a host of zoologists, botanists, and researchers. He had also learned a great deal about Persian customs and traditions from his teacher. Although his respect for Aris- totle was diminished as his travels made it clear that much of Aristotle’s geography was clearly wrong, when the old philosopher released his works to the public, Alexander complained “Thou hast not done well to publish thy acroa- matic doctrines; for in what shall I surpass other men if those doctrines wherein I have been trained are to be all men’s common property?"[102] 2.4.2 Inﬂuence on Byzantine scholars Greek Christian scribes played a crucial role in the preservation of Aristotle by copying all the extant Greek language manuscripts of the corpus. The ﬁrst Greek Christians to comment extensively on Aristotle were John Philoponus, Elias, and David in the sixth century, and Stephen of Alexandria in the early seventh century.[103] John Philoponus stands out for having attempted a fundamental critique of Aristotle’s views on the eternity of the world, movement, and other elements of Aristotelian thought.[104] After a hiatus of several centuries, formal commentary by Eustratius and Michael of Ephesus reappears in the late eleventh and early twelfth centuries, apparently sponsored by Anna Comnena.[105] 2.4.3 Inﬂuence on Islamic theologians Aristotle was one of the most revered Western thinkers in early Islamic theology. Most of the still extant works of Aristotle,[106] as well as a number of the original Greek commentaries, were translated into Arabic and studied by Muslim philosophers, scientists and scholars. Averroes, Avicenna and Alpharabius, who wrote on Aristotle in great depth, also inﬂuenced Thomas Aquinas and other Western Christian scholastic philosophers. Alkindus con- sidered Aristotle as the outstanding and unique representative of philosophy[107] and Averroes spoke of Aristotle as the “exemplar” for all future philosophers.[108] Medieval Muslim scholars regularly described Aristotle as the “First Teacher”.[109] The title “teacher” was ﬁrst given to Aristotle by Muslim scholars, and was later used by Western philosophers (as in the famous poem of Dante) who were inﬂuenced by the tradition of Islamic philosophy.[110] In accordance with the Greek theorists, the Muslims considered Aristotle to be a dogmatic philosopher, the author of a closed system, and believed that Aristotle shared with Plato essential tenets of thought. Some went so far as to
- 2.5. LIST OF WORKS 29 credit Aristotle himself with neo-Platonic metaphysical ideas.[106] 2.4.4 Inﬂuence on Western Christian theologians With the loss of the study of ancient Greek in the early medieval Latin West, Aristotle was practically unknown there from c. AD 600 to c. 1100 except through the Latin translation of the Organon made by Boethius. In the twelfth and thirteenth centuries, interest in Aristotle revived and Latin Christians had translations made, both from Arabic translations, such as those by Gerard of Cremona,[111] and from the original Greek, such as those by James of Venice and William of Moerbeke. After Thomas Aquinas wrote his theology, working from Moerbeke’s translations, the demand for Aristotle’s writings grew and the Greek manuscripts returned to the West, stimulating a revival of Aristotelianism in Europe that continued into the Renaissance.[112] Aristotle is referred to as “The Philosopher” by Scholastic thinkers such as Thomas Aquinas. See Summa Theologica, Part I, Question 3, etc. These thinkers blended Aristotelian philosophy with Christianity, bringing the thought of Ancient Greece into the Middle Ages. It required a repudiation of some Aristotelian principles for the sciences and the arts to free themselves for the discovery of modern scientiﬁc laws and empirical methods. The medieval English poet Chaucer describes his student as being happy by having at his beddes heed Twenty bookes, clad in blak or reed, Of aristotle and his philosophie,[113] The Italian poet Dante says of Aristotle in the ﬁrst circles of hell, I saw the Master there of those who know, Amid the philosophic family, By all admired, and by all reverenced; There Plato too I saw, and Socrates, Who stood beside him closer than the rest.[114] 2.4.5 Post-Enlightenment thinkers The German philosopher Friedrich Nietzsche has been said to have taken nearly all of his political philosophy from Aristotle.[115] However implausible this is, it is certainly the case that Aristotle’s rigid separation of action from production, and his justiﬁcation of the subservience of slaves and others to the virtue – or arete – of a few justiﬁed the ideal of aristocracy. It is Martin Heidegger, not Nietzsche, who elaborated a new interpretation of Aristotle, intended to warrant his deconstruction of scholastic and philosophical tradition. Ayn Rand accredited Aristotle as “the greatest philosopher in history” and cited him as a major inﬂuence on her thinking. More recently, Alasdair MacIntyre has attempted to reform what he calls the Aristotelian tradition in a way that is anti-elitist and capable of disputing the claims of both liberals and Nietzscheans.[116] 2.5 List of works Main article: Corpus Aristotelicum The works of Aristotle that have survived from antiquity through medieval manuscript transmission are collected in the Corpus Aristotelicum. These texts, as opposed to Aristotle’s lost works, are technical philosophical treatises from within Aristotle’s school. Reference to them is made according to the organization of Immanuel Bekker's Royal Prussian Academy edition (Aristotelis Opera edidit Academia Regia Borussica, Berlin, 1831–1870), which in turn is based on ancient classiﬁcations of these works.
- 30 CHAPTER 2. ARISTOTLE 2.6 Eponym The Aristotle Mountains along the Oscar II Coast of Graham Land, Antarctica, are named after Aristotle. He was the ﬁrst person known to conjecture the existence of a landmass in the southern high-latitude region and call it “Antarctica”.[117] Aristoteles (crater) is a crater on the Moon bearing the classical form of Aristotle’s name. 2.7 See also � Aristotelian physics � Aristotelian society � Aristotelian theology � Conimbricenses � List of writers inﬂuenced by Aristotle � Otium � Philia � Pseudo-Aristotle 2.8 Notes and references [1] “Aristotle” entry in Collins English Dictionary, HarperCollins Publishers, 1998. [2] That these undisputed dates (the ﬁrst half of the Olympiad year 384/383 BC, and in 322 shortly before the death of Demosthenes) are correct was shown already by August Boeckh (Kleine Schriften VI 195); for further discussion, see Felix Jacoby on FGrHist 244 F 38. Ingemar Düring, Aristotle in the Ancient Biographical Tradition, Göteborg, 1957, p. 253. [3] “Biography of Aristotle”. Biography.com. Retrieved 12 March 2014. [4] Bertrand Russell, A History of Western Philosophy, Simon & Schuster, 1972. [5] Encyclopædia Britannica (2008). The Britannica Guide to the 100 Most Inﬂuential Scientists. Running Press. p. 12. ISBN 9780762434213. [6] Barnes 2007, p. 6. [7] Cicero, Marcus Tullius (106–43 BC). “Academica Priora”. Book II, chapter XXXVIII, §119. Retrieved 25 January 2007. veniet ﬂumen orationis aureum fundens Aristoteles Check date values in: |date= (help) [8] Jonathan Barnes, “Life and Work” in The Cambridge Companion to Aristotle (1995), p. 9. [9] “Guardian on Time Magazine’s 100 personalities of all time”. [10] “Ranker.com - The most inﬂuential people of all time”. [11] Campbell, Michael. “Behind the Name: Meaning, Origin and History of the Name “Aristotle"". Behind the Name: The Etymology and History of First Names. www.behindthename.com. Retrieved 6 April 2012. [12] McLeisch, Kenneth Cole (1999). Aristotle: The Great Philosophers. Routledge. p. 5. ISBN 0-415-92392-1. [13] Anagnostopoulos, G., “Aristotle’s Life” in A Companion to Aristotle (Blackwell Publishing, 2009), p. 4. [14] Carnes Lord, introduction to The Politics by Aristotle (Chicago: University of Chicago Press, 1984). [15] Peter Green, Alexander of Macedon, University of California Press Ltd. (Oxford, England) 1991, pp. 58–59 [16] William George Smith,Dictionary of Greek and Roman Biography and Mythology, vol. 3, p. 88 [17] Peter Green, Alexander of Macedon, University of California Press Ltd. (Oxford, England), 1991, p. 379 and 459.
- 2.8. NOTES AND REFERENCES 31 [18] Jones, W. T. (1980). The Classical Mind: A History of Western Philosophy. Harcourt Brace Jovanovich. p. 216. ISBN 0155383124. [19] Vita Marciana 41, cf. Aelian Varia historica 3.36, Ingemar Düring, Aristotle in the Ancient Biographical Tradition, Göte- borg, 1957, T44a-e. [20] Aristotle’s Will, Aufstieg und Niedergang der römischen Welt by Hildegard Temporini, Wolfgang Haase. [21] See The Politics of Aristotle translated by Ernest Barker, Oxford: Clarendom Press, 1946, p. xxiii, note 2, who refers to Corpus Inscriptionum Graecarum, vol. xii, fasc. ix, s.v. Eretria. [22] See Shields, C., “Aristotle’s Philosophical Life and Writings” in The Oxford Handbook of Aristotle (Oxford University Press, 2012), pp. 3–16. Düring, I., Aristotle in the Ancient Biographical Tradition (Göteborg, 1957) is a collection of [an overview of?] ancient biographies of Aristotle. [23] MICHAEL DEGNAN, 1994. Recent Work in Aristotle’s Logic. Philosophical Books 35.2 (April 1994): 81–89. [24] Corcoran, John (2009). “Aristotle’s Demonstrative Logic”. History and Philosophy of Logic, 30: 1–20. [25] Bocheński, I. M. (1951). Ancient Formal Logic. Amsterdam: North-Holland Publishing Company. [26] Bocheński, 1951. [27] Rose, Lynn E. (1968). Aristotle’s Syllogistic. Springﬁeld: Charles C Thomas Publisher. [28] Jori, Alberto (2003). Aristotele. Milano: Bruno Mondadori Editore. [29] Aristotle, History of Animals, 2.3. [30] “Stanford Encyclopedia of Philosophy”. Plato.stanford.edu. Retrieved 26 April 2009. [31] Aristotle, Meteorology 1.8, trans. E.W. Webster, rev. J. Barnes. [32] Burent, John. 1928. Platonism, Berkeley: University of California Press, pp. 61, 103–104. [33] Charles Lyell, Principles of Geology, 1832, p.17 [34] Physics 201a10–11, 201a27–29, 201b4–5 [35] Sachs, Joe (2005), “Aristotle: Motion and its Place in Nature”, Internet Encyclopedia of Philosophy [36] Michael Lahanas. “Optics and ancient Greeks”. Mlahanas.de. Archived from the original on 11 April 2009. Retrieved 26 April 2009. [37] Aristotle, Physics 2.6 [38] Aristotle, Metaphysics VIII 1043a 10–30 [39] Aristotle, Metaphysics IX 1050a 5–10 [40] Aristotle, Metaphysics VIII 1045a–b [41] Singer, Charles. A short history of biology. Oxford 1931. [42] Emily Kearns, “Animals, knowledge about,” in Oxford Classical Dictionary, 3rd ed., 1996, p. 92. [43] Carl T. Bergstrom; Lee Alan Dugatkin (2012). Evolution. Norton. p. 35. ISBN 978-0-393-92592-0. [44] Aristotle, of course, is not responsible for the later use made of this idea by clerics. [45] Mason, A History of the Sciences pp 43–44 [46] Mayr, The Growth of Biological Thought, pp 201–202; see also: Lovejoy, The Great Chain of Being [47] Aristotle, De Anima II 3 [48] Mason, A History of the Sciences pp 45 [49] Guthrie, A History of Greek Philosophy Vol. 1 pp. 348 [50] Mayr, The Growth of Biological Thought, pp 90–91; Mason, A History of the Sciences, p 46 [51] Annas, Classical Greek Philosophy pp 252
- 32 CHAPTER 2. ARISTOTLE [52] Mason, A History of the Sciences pp 56 [53] Mayr, The Growth of Biological Thought, pp 90–94; quotation from p 91 [54] Annas, Classical Greek Philosophy, p 252 [55] Stanford Encyclopedia of Philosophy, article “Psychology”. [56] Bloch, David (2007). Aristotle on Memory and Recollection. p. 12. ISBN 9004160469. [57] Bloch 2007, p. 61. [58] Carruthers, Mary (2007). The Book of Memory: A Study of Memory in Medieval Culture. p. 16. ISBN 9780521429733. [59] Bloch 2007, p. 25. [60] Warren, Howard (1921). A History of the Association Psychology. p. 30. [61] Warren 1921, p. 25. [62] Carruthers 2007, p. 19. [63] Warren 1921, p. 296. [64] Warren 1921, p. 259. [65] Holowchak, Mark (1996). “Aristotle on Dreaming: What Goes on in Sleep when the 'Big Fire' goes out”. Ancient Philos- ophy 16 (2): 405–423. Retrieved 7 November 2014. [66] Shute, Clarence (1941). The Psychology of Aristotle: An Analysis of the Living Being. Morningsdie Heights: New York: Columbia University Press. pp. 115–118. [67] Modrak, Deborah (2009). “Dreams and Method in Aristotle”. Skepsis: A Journal for Philosophy and Interdisciplinary Research 20: 169–181. [68] Webb, Wilse (1990). Dreamtime and dreamwork: Decoding the language of the night. New consciousness reader series. Los Angeles, CA, England: Jeremy P. Tarcher, Inc. pp. 174–184. ISBN 0-87477-594-9. [69] Nicomachean Ethics Book I. See for example chapter 7 1098a. [70] Nicomachean Ethics Book VI. [71] Politics 1253a19–24 [72] Ebenstein, Alan; William Ebenstein (2002). Introduction to Political Thinkers. Wadsworth Group. p. 59. [73] For a diﬀerent reading of social and economic processes in the Nicomachean Ethics and Politics see Polanyi, K. (1957) “Aristotle Discovers the Economy” in Primitive, Archaic and Modern Economies: Essays of Karl Polanyi ed. G. Dalton, Boston 1971, 78–115 [74] D. S. Hutchinson and Monte Ransome Johnson (25 January 2015). “New Reconstruction, includes Greek text”. [75] Aristotle, Poetics I 1447a [76] Aristotle, Poetics III [77] Aristotle, Poetics IV [78] Aristotle, Poetics VI [79] Aristotle, Poetics XXVI [80] Temple, Olivia, and Temple, Robert (translators), The Complete Fables By Aesop Penguin Classics, 1998. ISBN 0-14- 044649-4 Cf. Introduction, pp. xi–xii. [81] Freeland, Cynthia A. (1998). Feminist Interpretations of Aristotle. Penn State University Press. ISBN 0-271-01730-9. [82] Morsink, Johannes (Spring 1979). “Was Aristotle’s Biology Sexist?". Journal of the History of Biology 12 (1): 83–112. doi:10.1007/bf00128136. [83] Aristotle; Roberts, W. Rhys (translator). Honeycutt, Lee, ed. Rhetoric. pp. Book I, Chapter 5. Where, as among the Lacedaemonians, the state of women is bad, almost half of human life is spoilt.
- 2.8. NOTES AND REFERENCES 33 [84] Terence Irwin and Gail Fine, Cornell University,Aristotle: Introductory Readings. Indianapolis, Indiana: Hackett Publishing Company, Inc. (1996), Introduction, pp. xi–xii. [85] Lynn Thorndike, “Chiromancy in Medieval Latin Manuscripts,” Speculum 40 (1965), pp. 674–706; Roger A. Pack, “Pseudo-Arisoteles: Chiromantia,” Archives d'histoire doctrinale et littéraire du Moyen Âge 39 (1972), pp. 289–320; Pack, “A Pseudo-Aristotelian Chiromancy,” Archives d'histoire doctrinale et littéraire du Moyen Âge 36 (1969), pp. 189–241. [86] Jonathan Barnes, “Life and Work” in The Cambridge Companion to Aristotle (1995), p. 12; Aristotle himself: Nicomachean Ethics 1102a26–27. Aristotle himself never uses the term “esoteric” or “acroamatic”. For other passages where Aristotle speaks of exōterikoi logoi, see W. D. Ross, Aristotle’s Metaphysics (1953), vol. 2, pp. 408–410. Ross defends an interpre- tation according to which the phrase, at least in Aristotle’s own works, usually refers generally to “discussions not peculiar to the Peripatetic school", rather than to speciﬁc works of Aristotle’s own. [87] Humphry House (1956). Aristotles Poetics. p. 35. [88] Barnes, “Life and Work”, p. 12. [89] Ammonius (1991). On Aristotle’s Categories. Ithaca, NY: Cornell University Press. ISBN 0-8014-2688-X. p. 15 [90] Cicero, Marcus Tullius (106 BC – 43 BC). "ﬂumen orationis aureum fundens Aristoteles". Academica Priora. Retrieved 25 January 2007. Check date values in: |date= (help) [91] Barnes, “Roman Aristotle”, in Gregory Nagy, Greek Literature, Routledge 2001, vol. 8, p. 174 n. 240. [92] .The deﬁnitive, English study of these questions is Barnes, “Roman Aristotle”. [93] “Sulla.” [94] Ancient Rome: from the early Republic to the assassination of Julius Caesar – Page 513, Matthew Dillon, Lynda Garland [95] The Encyclopedia Americana, Volume 22 – Page 131, Grolier Incorporated – Juvenile Nonﬁction [96] Lord, Carnes (1984). Introduction to the Politics, by Aristotle. Chicago: Chicago University Press. p. 11. [97] Anagnostopoulos, G., “Aristotle’s Works and Thoughts”, A Companion to Aristotle (Blackwell Publishing, 2009), p. 16. See also, Barnes, J., “Life and Work”, The Cambridge Companion to Aristotle (Cambridge University Press, 1995), pp. 10–15. [98] Magee, Bryan (2010). The Story of Philosophy. Dorling Kindersley. p. 34. [99] W. K. C. Guthrie (1990). "A history of Greek philosophy: Aristotle : an encounter". Cambridge University Press. p.156. ISBN 0-521-38760-4 [100] “Aristotle (Greek philosopher) – Britannica Online Encyclopedia”. Britannica.com. Archived from the original on 22 April 2009. Retrieved 26 April 2009. [101] Durant, Will (2006) [1926]. The Story of Philosophy. United States: Simon & Schuster, Inc. p. 92. ISBN 978-0-671- 73916-4. [102] Plutarch, Life of Alexander [103] Richard Sorabji, ed. Aristotle Transformed London, 1990, 20, 28, 35–36. [104] Richard Sorabji, ed. Aristotle Transformed (London, 1990) 233–274. [105] Richard Sorabji, ed. Aristotle Transformed (London, 1990) 20–21; 28–29, 393–406; 407–408. [106] Encyclopedia of Islam, Aristutalis [107] Rasa'il I, 103, 17, Abu Rida [108] Comm. Magnum in Aristotle, De Anima, III, 2, 43 Crawford [109] al-mua'llim al-thani, Aristutalis [110] Nasr, Seyyed Hossein (1996). The Islamic Intellectual Tradition in Persia. Curzon Press. pp. 59–60. ISBN 0-7007-0314-4. [111] Inﬂuence of Arabic and Islamic Philosophy on the Latin West entry in the Stanford Encyclopedia of Philosophy [112] Aristotelianism in the Renaissance entry in the Stanford Encyclopedia of Philosophy [113] Geoﬀrey Chaucer, The Canterbury Tales, Prologue, lines 295–295
- 34 CHAPTER 2. ARISTOTLE [114] vidi 'l maestro di color che sanno seder tra ﬁlosoﬁca famiglia. Tutti lo miran, tutti onor li fanno: quivi vid'ïo Socrate e Platone che 'nnanzi a li altri più presso li stanno; Dante, L'Inferno (Hell), Canto IV. Lines 131–135 [115] Durant, p. 86 [116] Kelvin Knight, Aristotelian Philosophy, Polity Press, 2007, passim. [117] Aristotle Mountains. SCAR Composite Antarctic Gazetteer. 2.9 Further reading The secondary literature on Aristotle is vast. The following references are only a small selection. � Ackrill J. L. (1997). Essays on Plato and Aristotle, Oxford University Press, USA. � Ackrill, J. L. (1981). Aristotle the Philosopher. Oxford and New York: Oxford University Press. � Adler, Mortimer J. (1978). Aristotle for Everybody. New York: Macmillan. A popular exposition for the general reader. � Ammonius (1991). Cohen, S. Marc; Matthews, Gareth B, eds. On Aristotle’s Categories. Ithaca, NY: Cornell University Press. ISBN 0-8014-2688-X. � Aristotle (1908–1952). The Works of Aristotle Translated into English Under the Editorship of W. D. Ross, 12 vols. Oxford: Clarendon Press. These translations are available in several places online; see External links. � Bakalis Nikolaos. (2005). Handbook of Greek Philosophy: From Thales to the Stoics Analysis and Fragments, Traﬀord Publishing ISBN 1-4120-4843-5 � Barnes J. (1995). The Cambridge Companion to Aristotle, Cambridge University Press. � Bocheński, I. M. (1951). Ancient Formal Logic. Amsterdam: North-Holland Publishing Company. � Bolotin, David (1998). An Approach to Aristotle’s Physics: With Particular Attention to the Role of His Manner of Writing. Albany: SUNY Press. A contribution to our understanding of how to read Aristotle’s scientiﬁc works. � Burnyeat, M. F. et al. (1979). Notes on Book Zeta of Aristotle’s Metaphysics. Oxford: Sub-faculty of Philos- ophy. � Cantor, Norman F.; Klein, Peter L., eds. (1969). Ancient Thought: Plato and Aristotle. Monuments of Western Thought 1. Waltham, Mass: Blaisdell Publishing Co. � Chappell, V. (1973). Aristotle’s Conception of Matter, Journal of Philosophy 70: 679–696. � Code, Alan. (1995). Potentiality in Aristotle’s Science and Metaphysics, Paciﬁc Philosophical Quarterly 76. � Ferguson, John (1972). Aristotle. New York: Twayne Publishers. � De Groot, Jean (2014). Aristotle’s Empiricism: Experience and Mechanics in the 4th Century BC, Parmenides Publishing, ISBN 978-1-930972-83-4 � Frede, Michael. (1987). Essays in Ancient Philosophy. Minneapolis: University of Minnesota Press. � Fuller, B.A.G. (1923). Aristotle. History of Greek Philosophy 3. London: Cape. � Gendlin, Eugene T. (2012). Line by Line Commentary on Aristotle’s De Anima, Volume 1: Books I & II; Volume 2: Book III. Spring Valley, New York: The Focusing Institute. Available online in PDF. � Gill, Mary Louise. (1989). Aristotle on Substance: The Paradox of Unity. Princeton: Princeton University Press.
- 2.9. FURTHER READING 35 � Guthrie, W. K. C. (1981). A History of Greek Philosophy, Vol. 6. Cambridge University Press. � Halper, Edward C. (2009). One and Many in Aristotle’s Metaphysics, Volume 1: Books Alpha – Delta, Par- menides Publishing, ISBN 978-1-930972-21-6. � Halper, Edward C. (2005). One andMany in Aristotle’s Metaphysics, Volume 2: The Central Books, Parmenides Publishing, ISBN 978-1-930972-05-6. � Irwin, T. H. (1988). Aristotle’s First Principles. Oxford: Clarendon Press, ISBN 0-19-824290-5. � Jaeger, Werner (1948). Robinson, Richard, ed. Aristotle: Fundamentals of the History of His Development (2nd ed.). Oxford: Clarendon Press. � Jori, Alberto. (2003). Aristotele, Milano: Bruno Mondadori Editore (Prize 2003 of the "International Academy of the History of Science") ISBN 88-424-9737-1. � Kiernan, Thomas P., ed. (1962). Aristotle Dictionary. New York: Philosophical Library. � Knight, Kelvin. (2007). Aristotelian Philosophy: Ethics and Politics from Aristotle to MacIntyre, Polity Press. � Lewis, Frank A. (1991). Substance and Predication in Aristotle. Cambridge: Cambridge University Press. � Lloyd, G. E. R. (1968). Aristotle: The Growth and Structure of his Thought. Cambridge: Cambridge Univ. Pr., ISBN 0-521-09456-9. � Lord, Carnes. (1984). Introduction to The Politics, by Aristotle. Chicago: Chicago University Press. � Loux, Michael J. (1991). Primary Ousia: An Essay on Aristotle’s Metaphysics Ζ and Η. Ithaca, NY: Cornell University Press. � Maso, Stefano (Ed.), Natali, Carlo (Ed.), Seel, Gerhard (Ed.). (2012) Reading Aristotle: Physics VII.3: What is Alteration? Proceedings of the International ESAP-HYELE Conference, Parmenides Publishing. ISBN 978- 1-930972-73-5 � McKeon, Richard (1973). Introduction to Aristotle (2d ed.). Chicago: University of Chicago Press. � Owen, G. E. L. (1965c). “The Platonism of Aristotle”. Proceedings of the British Academy 50: 125–150. [Reprinted in J. Barnes, M. Schoﬁeld, and R. R. K. Sorabji, eds.(1975). Articles on Aristotle Vol 1. Science. London: Duckworth 14–34.] � Pangle, Lorraine Smith (2003). Aristotle and the Philosophy of Friendship. Cambridge: Cambridge University Press. Aristotle’s conception of the deepest human relationship viewed in the light of the history of philosophic thought on friendship. � Plato (1979). Allen, Harold Joseph; Wilbur, James B, eds. The Worlds of Plato and Aristotle. Buﬀalo: Prometheus Books. � Reeve, C. D. C. (2000). Substantial Knowledge: Aristotle’s Metaphysics. Indianapolis: Hackett. � Rose, Lynn E. (1968). Aristotle’s Syllogistic. Springﬁeld: Charles C Thomas Publisher. � Ross, Sir David (1995). Aristotle (6th ed.). London: Routledge. A classic overview by one of Aristotle’s most prominent English translators, in print since 1923. � Scaltsas, T. (1994). Substances and Universals in Aristotle’s Metaphysics. Ithaca: Cornell University Press. � Strauss, Leo (1964). “On Aristotle’s Politics", in The City and Man, Chicago; Rand McNally. � Swanson, Judith (1992). The Public and the Private in Aristotle’s Political Philosophy. Ithaca: Cornell University Press. � Taylor, Henry Osborn (1922). “Chapter 3: Aristotle’s Biology”. Greek Biology and Medicine. Archived from the original on 11 February 2006. � Veatch, Henry B. (1974). Aristotle: A Contemporary Appreciation. Bloomington: Indiana U. Press. For the general reader. � Woods, M. J. (1991b). “Universals and Particular Forms in Aristotle’s Metaphysics”. Aristotle and the Later Tradition. Oxford Studies in Ancient Philosophy. Suppl. pp. 41–56.
- 36 CHAPTER 2. ARISTOTLE 2.10 External links � Aristotle at PhilPapers. � Aristotle at the Indiana Philosophy Ontology Project. � At the Internet Encyclopedia of Philosophy: � Aristotle (general article) � Biology � Ethics � Logic � Metaphysics � Motion and its Place in Nature � Poetics � Politics � From the Stanford Encyclopedia of Philosophy: � Aristotle (general article) � Aristotle in the Renaissance � Biology � Causality � Commentators on Aristotle � Ethics � Logic � Mathematics � Metaphysics � Natural philosophy � Non-contradiction � Political theory � Psychology � Rhetoric � General article at The Catholic Encyclopedia � Diogenes Laërtius, Life of Aristotle, translated by Robert Drew Hicks (1925). � Works by Aristotle at Open Library. � Timeline of Aristotle’s life � Aristotle at PlanetMath.org.. Collections of works � At the Massachusetts Institute of Technology (primarily in English). � Works by Aristotle at Project Gutenberg � Works by or about Aristotle at Internet Archive � Works by Aristotle at LibriVox (public domain audiobooks) � (English) (Greek) Perseus Project at Tufts University. � At the University of Adelaide (primarily in English). � (Greek) (French) P. Remacle
- 2.10. EXTERNAL LINKS 37 � The 11-volume 1837 Bekker edition of Aristotle’s Works in Greek (PDF · DJVU) � Bekker’s Prussian Academy of Sciences edition of the complete works of Aristotle at Archive.org: � vol. 1 � vol. 2 � vol. 3 � vol. 4 � vol. 5 � (English) Aristotle Collection (translation).
- 38 CHAPTER 2. ARISTOTLE The frontispiece to a 1644 version of the expanded and illustrated edition of Historia Plantarum (ca. 1200), which was originally written around 300 BC.
- 2.10. EXTERNAL LINKS 39 Aristotle’s classiﬁcation of constitutions
- 40 CHAPTER 2. ARISTOTLE First page of a 1566 edition of the Nicomachean Ethics in Greek and Latin
- 2.10. EXTERNAL LINKS 41 “Aristotle” by Jusepe de Ribera
- 42 CHAPTER 2. ARISTOTLE “Aristotle with a bust of Homer" by Rembrandt.
- 2.10. EXTERNAL LINKS 43 An early Islamic portrayal of Aristotle (right) and Alexander the Great.
- 44 CHAPTER 2. ARISTOTLE Statue by Cipri Adolf Bermann (1915) at the University of Freiburg Freiburg im Breisgau
- 2.10. EXTERNAL LINKS 45 “ARISTOTLE” near the ceiling of the Great Hall in the Library of Congress.
- Chapter 3 Emil Leon Post This article is about the logician. For the writer on etiquette, see Emily Post. Emil Leon Post (February 11, 1897 – April 21, 1954) was a Polish-born American mathematician and logician. He is best known for his work in the ﬁeld that eventually became known as computability theory. 3.1 Early work Post was born in Augustów, Suwałki Governorate, Russian Empire (now Poland) into a Polish-Jewish family that immigrated to America when he was a child. His parents were Arnold and Pearl Post.[2] He attended the Townsend Harris High School and continued on to graduate from City College of New York in 1917 with a B.S. in Mathematics.[1] After completing his Ph.D. in mathematics at Columbia University, he did a post-doctorate at Princeton University. While at Princeton, he came very close to discovering the incompleteness of Principia Mathematica, which Kurt Gödel proved in 1931. Post then became a high school mathematics teacher in New York City. In his doctoral thesis, Post proved, among other things, that the propositional calculus of Principia Mathematica was complete: all tautologies are theorems, given the Principia axioms and the rules of substitution and modus ponens. Post also devised truth tables independently of Wittgenstein and C.S. Peirce and put them to good mathematical use. Jean Van Heijenoort's well-known source book on mathematical logic (1966) reprinted Post’s classic article setting out these results. In 1936, he was appointed to the mathematics department at the City College of New York. He died in 1954 of a heart attack following electroshock treatment for depression;[3][4] he was 57. 3.2 Recursion theory In 1936, Post developed, independently of Alan Turing, a mathematical model of computation that was essentially equivalent to the Turing machine model. Intending this as the ﬁrst of a series of models of equivalent power but increasing complexity, he titled his paper Formulation 1. This model is sometimes called “Post’s machine” or a Post- Turing machine, but is not to be confused with Post’s tag machines or other special kinds of Post canonical system, a computational model using string rewriting and developed by Post in the 1920s but ﬁrst published in 1943. Post’s rewrite technique is now ubiquitous in programming language speciﬁcation and design, and so with Church’s lambda- calculus is a salient inﬂuence of classical modern logic on practical computing. Post devised a method of 'auxiliary symbols’ by which he could canonically represent any Post-generative language, and indeed any computable function or set at all. The unsolvability of his Post correspondence problem turned out to be exactly what was needed to obtain unsolvability results in the theory of formal languages. In an inﬂuential address to the American Mathematical Society in 1944, he raised the question of the existence of an 46
- 3.3. POLYADIC GROUPS 47 uncomputable recursively enumerable set whose Turing degree is less than that of the halting problem. This question, which became known as Post’s problem, stimulated much research. It was solved in the aﬃrmative in the 1950s by the introduction of the powerful priority method in recursion theory. 3.3 Polyadic groups Post made a fundamental and still inﬂuential contribution to the theory of polyadic, or n-ary, groups in a long paper published in 1940. His major theorem showed that a polyadic group is the iterated multiplication of elements of a normal subgroup of a group, such that the quotient group is cyclic of order n − 1. He also demonstrated that a polyadic group operation on a set can be expressed in terms of a group operation on the same set. The paper contains many other important results. 3.4 Selected papers � Post, Emil Leon (1936). “Finite Combinatory Processes - Formulation 1”. Journal of Symbolic Logic 1: 103– 105. � Post, Emil Leon (1940). “Polyadic groups”. Transactions of the American Mathematical Society 48: 208–350. � Post, Emil Leon (1943). “Formal Reductions of the General Combinatorial Decision Problem”. American Journal of Mathematics 65: 197–215. � Post, Emil Leon (1944). “Recursively enumerable sets of positive integers and their decision problems”. Bul- letin of the American Mathematical Society 50: 284–316. Introduces the important concept of many-one re- duction. 3.5 See also � Arithmetical hierarchy � Functional completeness � List of multiple discoveries � Post’s inversion formula � Post’s lattice � Post’s theorem 3.6 Notes [1] Urquhart (2008) [2] O'Connor, John J.; Robertson, Edmund F., “Emil Leon Post”, MacTutor History of Mathematics archive, University of St Andrews. [3] Baaz, Matthias, ed. (2011). Kurt Gödel and the Foundations of Mathematics: Horizons of Truth (1st ed.). Cambridge University Press. ISBN 9781139498432. [4] Urquhart (2008), p. 430. 3.7 References � Urquhart, Alasdair (2008). “Emil Post” (PDF). In Gabbay, Dov M.; Woods, John Woods. Logic from Russell to Church. Handbook of the History of Logic 5. Elsevier BV.
- 48 CHAPTER 3. EMIL LEON POST 3.8 Further reading � Anshel, Iris Lee; Anshel, Michael (November 1993). “From the Post-Markov Theorem Through Decision Problems to Public-Key Cryptography”. The American Mathematical Monthly (Mathematical Association of America) 100 (9): 835–844. Dedicated to Emil Post and contains special material on Post. This includes “Post’s Relation to the Cryptology and Cryptographists of his Era: ... Steven Brams, the noted game theorist and political scientist, has remarked to us that the life and legacy of Emil Post represents one aspect of New York intellectual life during the ﬁrst half of the twentieth century that is very much in need of deeper exploration. The authors hope that this paper serves to further this pursuit”. (pp. 842–843) � Davis, Martin, ed. (1993). The Undecidable. Dover. pp. 288–406. ISBN 0-486-43228-9. Reprints several papers by Post. � Davis, Martin (1994). “Emil L. Post: His Life and Work”. Solvability, Provability, Deﬁnability: The Collected Works of Emil L. Post. Birkhäuser. pp. xi—xxviii. A biographical essay. � Jackson, Allyn (May 2008). “An interview with Martin Davis”. Notices of the AMS 55 (5): 560–571. Much material on Emil Post from his ﬁrst-hand recollections. 3.9 External links � Emil Leon Post at the Mathematics Genealogy Project � Emil Leon Post Papers 1927-1991, American Philosophical Society, Philadelphia, Pennsylvania.
- Chapter 4 Four-valued logic In logic, a four-valued logic is used to model signal values in digital circuits: the four values are Z, X and the boolean values 1 and 0. Z stands for high impedance or open circuit, while X stands for “unknown”. There is also a 9-valued logic standard by the IEEE called IEEE 1164. There are other types of four value logic, such as Belnap’s four-valued relevance logic: the possible values are 1) true, 2) false, 3) both true and false, and 4) neither true nor false. Belnap’s logic is designed to cope with multiple information sources such that if only true is found then true is assigned, if only false is found then false is assigned, if some sources say true and others say false then both is assigned, and if no information is given by any information source then neither is assigned. 4.1 Applications 4.1.1 Electronics Among the distinct logic values supported by digital electronics theory (as deﬁned in VHDL's std_logic) are such diverse elements as: � 1 or High, usually representing TRUE. � 0 or Low, usually representing FALSE. � X representing a “Conﬂict”. � U representing “Unassigned” or “Unknown”. � - representing "Don't Care". � Z representing "high impedance", undriven line. � H, L and W are other high-impedance values, the weak pull to “High”, “Low” and “Don't Know” correspond- ingly. The “U” value does not exist in real-world circuits, it is merely a placeholder used in simulators and for design purposes. Some simulators support representation of the “Z” value, others do not. The “Z” value does exist in real-world circuits but only as an output state. Use of “U” value in simulation Many hardware description language (HDL) simulation tools, such as Verilog and VHDL, support an unknown value like that shown above during simulation of digital electronics. The unknown value may be the result of a design error, which the designer can correct before synthesis into an actual circuit. The unknown also represents uninitialised memory values and circuit inputs before the simulation has asserted what the real input value should be. HDL synthesis tools usually produce circuits that operate only on binary logic. 49
- 50 CHAPTER 4. FOUR-VALUED LOGIC Use of “X” value in digital design When designing a digital circuit, some conditions may be outside the scope of the purpose that the circuit will perform. Thus, the designer does not care what happens under those conditions. In addition, the situation occurs that inputs to a circuit are masked by other signals so the value of that input has no eﬀect on circuit behaviour. In these situations, it is traditional to use “X” as a placeholder to indicate "Don't Care" when building truth tables. This is especially common in state machine design and Karnaugh map simpliﬁcation. The “X” values provide additional degrees of freedom to the ﬁnal circuit design, generally resulting in a simpliﬁed and smaller circuit.[1] Once the circuit design is complete and a real circuit is constructed, the “X” values will no longer exist. They will become some tangible “0” or “1” value but could be either depending on the ﬁnal design optimisation. Use of “Z” value for high impedance Main article: three-state Some digital devices support a form of three-state logic on their outputs only. The three states are “0”, “1”, and “Z”. Commonly referred to as tristate [2] logic (a trademark of National Semiconductor), it comprises the usual true and false states, with a third transparent high impedance state (or 'oﬀ-state') which eﬀectively disconnects the logic output. This provides an eﬀective way to connect several logic outputs to a single input, where all but one are put into the high impedance state, allowing the remaining output to operate in the normal binary sense. This is commonly used to connect banks of computer memory and other similar devices to a common data bus; a large number of devices can communicate over the same channel simply by ensuring only one is enabled at a time. It is important to note that while outputs can have one of three states, inputs can only recognise two. Hence the kind of relations shown in the table above do not occur. Although it could be argued that the high-impedance state is eﬀectively an “unknown”, there is absolutely no provision in the vast majority of normal electronics to interpret a high-impedance state as a state in itself. Inputs can only detect “0” and “1”. When a digital input is left disconnected (i.e., when it is given a high impedance signal), the digital value interpreted by the input depends on the type of technology used. TTL technology will reliably default to a “1” state. On the other hand CMOS technology will temporarily hold the previous state seen on that input (due to the capacitance of the gate input). Over time, leakage current causes the CMOS input to drift in a random direction, possibly causing the input state to ﬂip. Disconnected inputs on CMOS devices can pick up noise, they can cause oscillation, the supply current may dramatically increase (crowbar power) or the device may completely destroy itself. Exotic ternary-logic devices True three-valued logic can be implemented in electronics, although the complexity of design has thus far made it uneconomical to pursue commercially and interest has been primarily conﬁned to research (see Setun); 'Normal' binary logic is simply cheaper to implement and in most cases can easily be conﬁgured to emulate ternary systems. There are, however, useful applications in fuzzy logic and error correction, and several true ternary logic devices have been manufactured. 4.1.2 Software Vehicle technology In the SAE J1939 standard, used for CAN data transmission in heavy road vehicles, there are four logical (boolean) values, False, True, Error Condition, and Not installed (represented by values 0-3). Error Condition means there is a technical problem obstacling data acquisition. The logics for that is for example True and Error Condition=Error Condition. Not installed is used for a feature which does not exist in this vehicle, and should be disregarded for logical calculation. On CAN, usually ﬁxed data messages are sent containing many signal values each, so a signal representing a not-installed feature will be sent anyway.
- 4.2. NOTES 51 4.2 Notes [1] Wakerly, John F (2001). Digital Design Principles & Practices. Prentice Hall. ISBN 0-13-090772-3. [2] National Semiconductor (1993), LS TTL Data Book, National Semiconductor Corporation
- Chapter 5 Fuzzy logic Fuzzy logic is a form of many-valued logic in which the truth values of variables may be any real number between 0 and 1. By contrast, in Boolean logic, the truth values of variables may only be 0 or 1. Fuzzy logic has been extended to handle the concept of partial truth, where the truth value may range between completely true and completely false.[1] Furthermore, when linguistic variables are used, these degrees may be managed by speciﬁc functions.[2] The term “fuzzy logic” was introduced with the 1965 proposal of fuzzy set theory by Lotﬁ A. Zadeh.[3][4] Fuzzy logic has been applied to many ﬁelds, from control theory to artiﬁcial intelligence. Fuzzy logic had, however, been studied since the 1920s, as inﬁnite-valued logic—notably by Łukasiewicz and Tarski.[5] 5.1 Overview Classical logic only permits propositions having a value of truth or falsity. The notion of whether 1+1=2 is an absolute, immutable and mathematical truth. However, there exist certain propositions with variable answers, such as asking various people to identify a colour. The notion of truth doesn't fall by the wayside, but rather on a means of representing and reasoning over partial knowledge when aﬀorded, by aggregating all possible outcomes into a dimensional spectrum. Both degrees of truth and probabilities range between 0 and 1 and hence may seem similar at ﬁrst. For example, let a 100 ml glass contain 30 ml of water. Then we may consider two concepts: empty and full. The meaning of each of them can be represented by a certain fuzzy set. Then one might deﬁne the glass as being 0.7 empty and 0.3 full. Note that the concept of emptiness would be subjective and thus would depend on the observer or designer. Another designer might, equally well, design a set membership function where the glass would be considered full for all values down to 50 ml. It is essential to realize that fuzzy logic uses truth degrees as a mathematical model of the vagueness phenomenon while probability is a mathematical model of ignorance. 5.1.1 Applying truth values A basic application might characterize various sub-ranges of a continuous variable. For instance, a temperature measurement for anti-lock brakes might have several separate membership functions deﬁning particular temperature ranges needed to control the brakes properly. Each function maps the same temperature value to a truth value in the 0 to 1 range. These truth values can then be used to determine how the brakes should be controlled. In this image, the meanings of the expressions cold,warm, and hot are represented by functions mapping a temperature scale. A point on that scale has three “truth values” — one for each of the three functions. The vertical line in the image represents a particular temperature that the three arrows (truth values) gauge. Since the red arrow points to zero, this temperature may be interpreted as “not hot”. The orange arrow (pointing at 0.2) may describe it as “slightly warm” and the blue arrow (pointing at 0.8) “fairly cold”. 52
- 5.2. EARLY APPLICATIONS 53 1 0 cold warm hot temperature Fuzzy logic temperature 5.1.2 Linguistic variables While variables in mathematics usually take numerical values, in fuzzy logic applications, the non-numeric are often used to facilitate the expression of rules and facts.[6] A linguistic variable such as age may have a value such as young or its antonym old. However, the great utility of linguistic variables is that they can be modiﬁed via linguistic hedges applied to primary terms. These linguistic hedges can be associated with certain functions. 5.2 Early applications The Japanese were the ﬁrst to utilize fuzzy logic for practical applications. The ﬁrst notable application was on the high-speed train in Sendai, in which fuzzy logic was able to improve the economy, comfort, and precision of the ride.[7] It has also been used in recognition of hand written symbols in Sony pocket computers; ﬂight aid for helicopters; controlling of subway systems in order to improve driving comfort, precision of halting, and power economy; improved fuel consumption for auto mobiles; single-button control for washing machines, automatic motor control for vacuum cleaners with recognition of surface condition and degree of soiling; and prediction systems for early recognition of earthquakes through the Institute of Seismology Bureau of Metrology, Japan.[8] 5.3 Example 5.3.1 Hard science with IF-THEN rules Fuzzy set theory deﬁnes fuzzy operators on fuzzy sets. The problem in applying this is that the appropriate fuzzy operator may not be known. For example, a simple temperature regulator that uses a fan might look like this: IF temperature IS very cold THEN stop fan IF temperature IS cold THEN turn down fan IF temperature IS normal THEN maintain fan IF temperature IS hot THEN speed up fan There is no “ELSE” – all of the rules are evaluated, because the temperature might be “cold” and “normal” at the same time to diﬀerent degrees. The AND, OR, and NOT operators of boolean logic exist in fuzzy logic, usually deﬁned as the minimum, maximum, and complement; when they are deﬁned this way, they are called the Zadeh operators. So for the fuzzy variables x and y: NOT x = (1 - truth(x)) x AND y = minimum(truth(x), truth(y)) x OR y = maximum(truth(x), truth(y))
- 54 CHAPTER 5. FUZZY LOGIC There are also other operators, more linguistic in nature, called hedges that can be applied. These are generally adverbs such as “very”, or “somewhat”, which modify the meaning of a set using a mathematical formula. 5.3.2 Deﬁne with multiply x AND y = x*y x OR y = 1-(1-x)*(1-y) 1-(1-x)*(1-y) comes from this: x OR y = NOT( AND( NOT(x), NOT(y) ) ) x OR y = NOT( AND(1-x, 1-y) ) x OR y = NOT( (1-x)*(1-y) ) x OR y = 1-(1-x)*(1-y) 5.3.3 Deﬁne with sigmoid sigmoid(x)=1/(1+e^-x) sigmoid(x)+sigmoid(-x) = 1 (sigmoid(x)+sigmoid(-x))*(sigmoid(y)+sigmoid(-y))*(sigmoid(z)+sigmoid(- z)) = 1 5.4 Logical analysis In mathematical logic, there are several formal systems of “fuzzy logic"; most of them belong among so-called t-norm fuzzy logic. 5.4.1 Propositional fuzzy logics The most important propositional fuzzy logics are:- � Monoidal t-norm-based propositional fuzzy logic MTL is an axiomatization of logic where conjunction is deﬁned by a left continuous t-norm and implication is deﬁned as the residuum of the t-norm. Its models correspond to MTL-algebras that are pre-linear commutative bounded integral residuated lattices. � Basic propositional fuzzy logic BL is an extension of MTL logic where conjunction is deﬁned by a continuous t-norm, and implication is also deﬁned as the residuum of the t-norm. Its models correspond to BL-algebras. � Łukasiewicz fuzzy logic is the extension of basic fuzzy logic BL where standard conjunction is the Łukasiewicz t-norm. It has the axioms of basic fuzzy logic plus an axiom of double negation, and its models correspond to MV-algebras. � Gödel fuzzy logic is the extension of basic fuzzy logic BL where conjunction is Gödel t-norm. It has the axioms of BL plus an axiom of idempotence of conjunction, and its models are called G-algebras. � Product fuzzy logic is the extension of basic fuzzy logic BL where conjunction is product t-norm. It has the axioms of BL plus another axiom for cancellativity of conjunction, and its models are called product algebras. � Fuzzy logic with evaluated syntax (sometimes also called Pavelka’s logic), denoted by EVŁ, is a further gen- eralization of mathematical fuzzy logic. While the above kinds of fuzzy logic have traditional syntax and many-valued semantics, in EVŁ is evaluated also syntax. This means that each formula has an evaluation. Ax- iomatization of EVŁ stems from Łukasziewicz fuzzy logic. A generalization of classical Gödel completeness theorem is provable in EVŁ. 5.4.2 Predicate fuzzy logics These extend the above-mentioned fuzzy logics by adding universal and existential quantiﬁers in a manner similar to the way that predicate logic is created from propositional logic. The semantics of the universal (resp. existential) quantiﬁer in t-norm fuzzy logics is the inﬁmum (resp. supremum) of the truth degrees of the instances of the quantiﬁed subformula.
- 5.5. FUZZY DATABASES 55 5.4.3 Decidability issues for fuzzy logic The notions of a “decidable subset” and "recursively enumerable subset” are basic ones for classical mathematics and classical logic. Thus the question of a suitable extension of these concepts to fuzzy set theory arises. A ﬁrst proposal in such a direction was made by E.S. Santos by the notions of fuzzy Turing machine, Markov normal fuzzy algorithm and fuzzy program (see Santos 1970). Successively, L. Biacino and G. Gerla argued that the proposed deﬁnitions are rather questionable and therefore they proposed the following ones. Denote by Ü the set of rational numbers in [0,1]. Then a fuzzy subset s : S ! [0,1] of a set S is recursively enumerable if a recursive map h : S×N ! Ü exists such that, for every x in S, the function h(x,n) is increasing with respect to n and s(x) = lim h(x,n). We say that s is decidable if both s and its complement –s are recursively enumerable. An extension of such a theory to the general case of the L-subsets is possible (see Gerla 2006). The proposed deﬁnitions are well related with fuzzy logic. Indeed, the following theorem holds true (provided that the deduction apparatus of the considered fuzzy logic satisﬁes some obvious eﬀectiveness property). Theorem. Any axiomatizable fuzzy theory is recursively enumerable. In particular, the fuzzy set of logically true formulas is recursively enumerable in spite of the fact that the crisp set of valid formulas is not recursively enumerable, in general. Moreover, any axiomatizable and complete theory is decidable. It is an open question to give supports for a Church thesis for fuzzy mathematics the proposed notion of recursive enumerability for fuzzy subsets is the adequate one. To this aim, an extension of the notions of fuzzy grammar and fuzzy Turing machine should be necessary (see for example Wiedermann’s paper). Another open question is to start from this notion to ﬁnd an extension of Gödel's theorems to fuzzy logic. It is known that any boolean logic function could be represented using a truth table mapping each set of variable values into set of values f0; 1g . The task of synthesis of boolean logic function given in tabular form is one of basic tasks in traditional logic that is solved via disjunctive (conjunctive) perfect normal form. Each fuzzy (continuous) logic function could be represented by a choice table containing all possible variants of comparing arguments and their negations. A choice table maps each variant into value of an argument or a negation of an argument. For instance, for two arguments a row of choice table contains a variant of comparing values x1 , :x1 , x2 , :x2 and the corresponding function value f(x2 � :x1 � x1 � :x2) = :x1 . The task of synthesis of fuzzy logic function given in tabular form was solved in.[9] New concepts of constituents of minimum and maximum were introduced. The suﬃcient and necessary conditions that a choice table deﬁnes a fuzzy logic function were derived. 5.5 Fuzzy databases Once fuzzy relations are deﬁned, it is possible to develop fuzzy relational databases. The ﬁrst fuzzy relational database, FRDB, appeared in Maria Zemankova’s dissertation. Later, some other models arose like the Buckles-Petry model, the Prade-Testemale Model, the Umano-Fukami model or the GEFRED model by J.M. Medina, M.A. Vila et al. In the context of fuzzy databases, some fuzzy querying languages have been deﬁned, highlighting the SQLf by P. Bosc et al. and the FSQL by J. Galindo et al. These languages deﬁne some structures in order to include fuzzy aspects in the SQL statements, like fuzzy conditions, fuzzy comparators, fuzzy constants, fuzzy constraints, fuzzy thresholds, linguistic labels and so on. Much progress has been made to take fuzzy logic database applications to the web and let the world easily use them, for example: http://sullivansoftwaresystems.com/cgi-bin/fuzzy-logic-match-algorithm.cgi?SearchString=garia This en- ables fuzzy logic matching to be incorporated into a database system or application. 5.6 Comparison to probability Fuzzy logic and probability address diﬀerent forms of uncertainty. While both fuzzy logic and probability theory can represent degrees of certain kinds of subjective belief, fuzzy set theory uses the concept of fuzzy set membership, i.e., how much a variable is in a set (there is not necessarily any uncertainty about this degree), and probability theory uses the concept of subjective probability, i.e., how probable is it that a variable is in a set (it either entirely is or entirely is not in the set in reality, but there is uncertainty around whether it is or is not). The technical consequence of this distinction is that fuzzy set theory relaxes the axioms of classical probability, which are themselves derived from adding uncertainty, but not degree, to the crisp true/false distinctions of classical Aristotelian logic.
- 56 CHAPTER 5. FUZZY LOGIC Bruno de Finetti argues that only one kind of mathematical uncertainty, probability, is needed, and thus fuzzy logic is unnecessary. However, Bart Kosko shows in Fuzziness vs. Probability that probability theory is a subtheory of fuzzy logic, as questions of degrees of belief in mutually-exclusive set membership in probability theory can be represented as certain cases of non-mutually-exclusive graded membership in fuzzy theory. In that context, he also derives Bayes’ theorem from the concept of fuzzy subsethood. Lotﬁ A. Zadeh argues that fuzzy logic is diﬀerent in character from probability, and is not a replacement for it. He fuzziﬁed probability to fuzzy probability and also generalized it to possibility theory. (cf.[10]) More generally, fuzzy logic is one of many diﬀerent extensions to classical logic intended to deal with issues of uncertainty outside of the scope of classical logic, the inapplicability of probability theory in many domains, and the paradoxes of Dempster-Shafer theory. See also probabilistic logics. 5.7 Relation to ecorithms Leslie Valiant, a winner of the Turing Award, uses the term “ecorithms” to describe how many less exact systems and techniques like fuzzy logic (and “less robust” logic) can be applied to learning algorithms. Valiant essentially redeﬁnes machine learning as evolutionary. Ecorithms and fuzzy logic also have the common property of dealing with possibilities more than probabilities, although feedback and feed forward, basically stochastic “weights,” are a feature of both when dealing with, for example, dynamical systems. In general use, ecorithms are algorithms that learn from their more complex environments (Hence Eco) to generalize, approximate and simplify solution logic. Like fuzzy logic, they are methods used to overcome continuous variables or systems too complex to completely enumerate or understand discretely or exactly. See in particular p. 58 of the reference comparing induction/invariance, robust, mathematical and other logical limits in computing, where techniques including fuzzy logic and natural data selection (à la “computational Darwinism”) can be used to short-cut computational complexity and limits in a “practical” way (such as the brake temperature example in this article).[11] 5.8 Compensatory fuzzy logic Compensatory fuzzy logic (CFL) is a branch of fuzzy logic with modiﬁed rules for conjunction and disjunction. When the truth value of one component of a conjunction or disjunction is increased or decreased, the other component is decreased or increased to compensate. This increase or decrease in truth value may be oﬀset by the increase or decrease in another component. An oﬀset may be blocked when certain thresholds are met. Proponents claim that CFL allows better semantic behavior. Compensatory Fuzzy Logic consists of four continuous operators: conjunction (c); disjunction (d); fuzzy strict order (or); and negation (n). The conjunction is the geometric mean and its dual as conjunctive and disjunctive operators.[12] 5.9 See also � Adaptive neuro fuzzy inference system (ANFIS) � Artiﬁcial neural network � Defuzziﬁcation � Expert system � False dilemma � Fuzzy architectural spatial analysis � Fuzzy classiﬁcation � Fuzzy concept � Fuzzy Control Language � Fuzzy control system
- 5.10. REFERENCES 57 � Fuzzy electronics � Fuzzy subalgebra � FuzzyCLIPS � High Performance Fuzzy Computing � IEEE Transactions on Fuzzy Systems � Interval ﬁnite element � Machine learning � Neuro-fuzzy � Noise-based logic � Rough set � Sorites paradox � Type-2 fuzzy sets and systems � Vector logic 5.10 References [1] Novák, V., Perﬁlieva, I. and Močkoř, J. (1999) Mathematical principles of fuzzy logic Dodrecht: Kluwer Academic. ISBN 0-7923-8595-0 [2] Ahlawat, Nishant, Ashu Gautam, and Nidhi Sharma (International Research Publications House 2014) “Use of Logic Gates to Make Edge Avoider Robot.” International Journal of Information & Computation Technology (Volume 4, Issue 6; page 630) ISSN 0974-2239 (Retrieved 27 April 2014) [3] “Fuzzy Logic”. Stanford Encyclopedia of Philosophy. Stanford University. 2006-07-23. Retrieved 2008-09-30. [4] Zadeh, L.A. (1965). “Fuzzy sets”. Information and Control 8 (3): 338–353. doi:10.1016/s0019-9958(65)90241-x. [5] Pelletier, Francis Jeﬀry (2000). “Review of Metamathematics of fuzzy logics" (PDF). The Bulletin of Symbolic Logic 6 (3): 342–346. JSTOR 421060. [6] Zadeh, L. A. et al. 1996 Fuzzy Sets, Fuzzy Logic, Fuzzy Systems, World Scientiﬁc Press, ISBN 981-02-2421-4 [7] Kosko, B (June 1, 1994). “Fuzzy Thinking: The New Science of Fuzzy Logic”. Hyperion. [8] Bansod, Nitin A., Marshall Kulkarni, and S.H. Patil (Bharati Vidyapeeth College of Engineering) “Soft Computing- A Fuzzy Logic Approach”. Soft Computing (Allied Publishers 2005) (page 73) [9] Zaitsev D.A., Sarbei V.G., Sleptsov A.I., Synthesis of continuous-valued logic functions deﬁned in tabular form, Cyber- netics and Systems Analysis, Volume 34, Number 2 (1998), 190-195. [10] Novák, V (2005). “Are fuzzy sets a reasonable tool for modeling vague phenomena?". Fuzzy Sets and Systems 156: 341– 348. doi:10.1016/j.fss.2005.05.029. [11] Valiant, Leslie, (2013) Probably Approximately Correct: Nature’s Algorithms for Learning and Prospering in a Complex World New York: Basic Books. ISBN 978-0465032716 [12] Cejas, Jesús, (2011) Compensatory Fuzzy Logic. La Habana: Revista de Ingeniería Industrial. ISSN 1815-5936
- 58 CHAPTER 5. FUZZY LOGIC 5.11 Bibliography � Arabacioglu, B. C. (2010). “Using fuzzy inference system for architectural space analysis”. Applied Soft Com- puting 10 (3): 926–937. doi:10.1016/j.asoc.2009.10.011. � Biacino, L.; Gerla, G. (2002). “Fuzzy logic, continuity and eﬀectiveness”. Archive for Mathematical Logic 41 (7): 643–667. doi:10.1007/s001530100128. ISSN 0933-5846. � Cox, Earl (1994). The fuzzy systems handbook: a practitioner’s guide to building, using, maintaining fuzzy systems. Boston: AP Professional. ISBN 0-12-194270-8. � Gerla, Giangiacomo (2006). “Eﬀectiveness and Multivalued Logics”. Journal of Symbolic Logic 71 (1): 137– 162. doi:10.2178/jsl/1140641166. ISSN 0022-4812. � Hájek, Petr (1998). Metamathematics of fuzzy logic. Dordrecht: Kluwer. ISBN 0-7923-5238-6. � Hájek, Petr (1995). “Fuzzy logic and arithmetical hierarchy”. Fuzzy Sets and Systems 3 (8): 359–363. doi:10.1016/0165-0114(94)00299-M. ISSN 0165-0114. � Halpern, Joseph Y. (2003). Reasoning about uncertainty. Cambridge, Mass: MIT Press. ISBN 0-262-08320- 5. � Höppner, Frank; Klawonn, F.; Kruse, R.; Runkler, T. (1999). Fuzzy cluster analysis: methods for classiﬁcation, data analysis and image recognition. New York: John Wiley. ISBN 0-471-98864-2. � Ibrahim, Ahmad M. (1997). Introduction to Applied Fuzzy Electronics. Englewood Cliﬀs, N.J: Prentice Hall. ISBN 0-13-206400-6. � Klir, George J.; Folger, Tina A. (1988). Fuzzy sets, uncertainty, and information. Englewood Cliﬀs, N.J: Prentice Hall. ISBN 0-13-345984-5. � Klir, George J.; St Clair, Ute H.; Yuan, Bo (1997). Fuzzy set theory: foundations and applications. Englewood Cliﬀs, NJ: Prentice Hall. ISBN 0-13-341058-7. � Klir, George J.; Yuan, Bo (1995). Fuzzy sets and fuzzy logic: theory and applications. Upper Saddle River, NJ: Prentice Hall PTR. ISBN 0-13-101171-5. � Kosko, Bart (1993). Fuzzy thinking: the new science of fuzzy logic. New York: Hyperion. ISBN 0-7868-8021- X. � Kosko, Bart; Isaka, Satoru (July 1993). “Fuzzy Logic”. Scientiﬁc American 269 (1): 76–81. doi:10.1038/scientiﬁcamerican0793- 76. � Lohani, A.K.; Goel, N.K.; Bhatia K.K.S. (2006). “Takagi–Sugeno fuzzy inference system for modeling stage– discharge relationship”. Journal of Hydrology 331 (1): 146–160. doi:10.1016/j.jhydrol.2006.05.007. � Lohani, A.K.; Goel, N.K.; Bhatia K.K.S. (2007). “Deriving stage–discharge–sediment concentration relation- ships using fuzzy logic”. Hydrological Sciences Journal 52 (4): 793–807. doi:10.1623/hysj.52.4.793. � Lohani, A.K.; Goel, N.K.; Bhatia K.K.S. (2011). “Comparative study of neural network, fuzzy logic and linear transfer function techniques in daily rainfall‐runoﬀ modelling under diﬀerent input domains”. Hydrological Processes 25 (2): 175–193. doi:10.1002/hyp.7831. � Lohani, A.K.; Goel, N.K.; Bhatia K.K.S. (2012). “Hydrological time series modeling: A comparison between adaptive neuro-fuzzy, neural network and autoregressive techniques”. Journal of Hydrology. 442-443 (6): 23–35. doi:10.1016/j.jhydrol.2012.03.031. � Malek Masmoudi and Alain Haït, Project scheduling under uncertainty using fuzzy modeling and solving tech- niques, Engineering Applications of Artiﬁcial Intelligence - Elsevier, July 2012. � Malek Masmoudi and Alain Haït, Fuzzy uncertainty modelling for project planning; application to helicopter maintenance, International Journal of Production Research, Vol 50, issue 24, November2012. � Montagna, F. (2001). “Three complexity problems in quantiﬁed fuzzy logic”. Studia Logica 68 (1): 143–152. doi:10.1023/A:1011958407631. ISSN 0039-3215.
- 5.11. BIBLIOGRAPHY 59 � Mundici, Daniele; Cignoli, Roberto; D'Ottaviano, Itala M. L. (1999). Algebraic foundations of many-valued reasoning. Dodrecht: Kluwer Academic. ISBN 0-7923-6009-5. � Novák, Vilém (1989). Fuzzy Sets and Their Applications. Bristol: Adam Hilger. ISBN 0-85274-583-4. � Novák, Vilém (2005). “On fuzzy type theory”. Fuzzy Sets and Systems 149 (2): 235–273. doi:10.1016/j.fss.2004.03.027. � Novák, Vilém; Perﬁlieva, Irina; Močkoř, Jiří (1999). Mathematical principles of fuzzy logic. Dordrecht: Kluwer Academic. ISBN 0-7923-8595-0. � Onses, Richard (1996). Second Order Experton: A new Tool for Changing Paradigms in Country Risk Calcula- tion. ISBN 84-7719-558-7. � Onses, Richard (1994). Détermination de l´incertitude inhérente aux investissements en Amérique Latine sur la base de la théorie des sous ensembles ﬂous. Barcelona. ISBN 84-475-0881-1. � Passino, Kevin M.; Yurkovich, Stephen (1998). Fuzzy control. Boston: Addison-Wesley. ISBN 0-201-18074- X. � Pedrycz, Witold; Gomide, Fernando (2007). Fuzzy systems engineering: Toward Human-Centerd Computing. Hoboken: Wiley-Interscience. ISBN 978-0-471-78857-7. � Pu, Pao Ming; Liu, Ying Ming (1980). “Fuzzy topology. I. Neighborhood structure of a fuzzy point and Moore- Smith convergence”. Journal of Mathematical Analysis and Applications 76 (2): 571–599. doi:10.1016/0022- 247X(80)90048-7. ISSN 0022-247X. � Sahoo, Bhabagrahi; Lohani, A.K.; Sahu, Rohit K. (2006). “Fuzzy multiobjective and linear programming based management models for optimal land-water-crop system planning”. Water resources management,Springer Netherlands 20 (1): 931–948. doi:10.1007/s11269-005-9015-x. � Santos, Eugene S. (1970). “Fuzzy Algorithms”. Information and Control 17 (4): 326–339. doi:10.1016/S0019- 9958(70)80032-8. � Scarpellini, Bruno (1962). “Die Nichaxiomatisierbarkeit des unendlichwertigen Prädikatenkalküls von Łukasiewicz”. Journal of Symbolic Logic (Association for Symbolic Logic) 27 (2): 159–170. doi:10.2307/2964111. ISSN 0022-4812. JSTOR 2964111. � Seising, Rudolf (2007). The Fuzziﬁcation of Systems. The Genesis of Fuzzy Set Theory and Its Initial Applica- tions -- Developments up to the 1970s. Springer-Verlag. ISBN 978-3-540-71795-9. � Steeb, Willi-Hans (2008). The Nonlinear Workbook: Chaos, Fractals, Cellular Automata, Neural Networks, Genetic Algorithms, Gene Expression Programming, Support Vector Machine, Wavelets, Hidden Markov Models, Fuzzy Logic with C++, Java and SymbolicC++ Programs: 4edition. World Scientiﬁc. ISBN 981-281-852-9. � Tsitolovsky, Lev; Sandler, Uziel (2008). Neural Cell Behavior and Fuzzy Logic. Springer. ISBN 978-0-387- 09542-4. � Wiedermann, J. (2004). “Characterizing the super-Turing computing power and eﬃciency of classical fuzzy Turing machines”. Theor. Comput. Sci. 317 (1–3): 61–69. doi:10.1016/j.tcs.2003.12.004. � Yager, Ronald R.; Filev, Dimitar P. (1994). Essentials of fuzzy modeling and control. New York: Wiley. ISBN 0-471-01761-2. � Van Pelt, Miles (2008). Fuzzy Logic Applied to Daily Life. Seattle, WA: No No No No Press. ISBN 0-252- 16341-9. � Von Altrock, Constantin (1995). Fuzzy logic and NeuroFuzzy applications explained. Upper Saddle River, NJ: Prentice Hall PTR. ISBN 0-13-368465-2. � Wilkinson, R.H. (1963). “A method of generating functions of several variables using analog diode logic”. IEEE Transactions on Electronic Computers 12 (2): 112–129. doi:10.1109/PGEC.1963.263419. � Zadeh, L.A. (1968). “Fuzzy algorithms”. Information and Control 12 (2): 94–102. doi:10.1016/S0019- 9958(68)90211-8. ISSN 0019-9958.
- 60 CHAPTER 5. FUZZY LOGIC � Zadeh, L.A. (1965). “Fuzzy sets”. Information and Control 8 (3): 338–353. doi:10.1016/S0019-9958(65)90241- X. ISSN 0019-9958. � Zemankova-Leech, M. (1983). “Fuzzy Relational Data Bases”. Ph. D. Dissertation. Florida State University. � Zimmermann, H. (2001). Fuzzy set theory and its applications. Boston: Kluwer Academic Publishers. ISBN 0-7923-7435-5. � Moghaddam, M. J., M. R. Soleymani, and M. A. Farsi. “Sequence planning for stamping operations in pro- gressive dies.” Journal of Intelligent Manufacturing(2013): 1-11. 5.12 External links � Formal fuzzy logic - article at Citizendium � Fuzzy Logic - article at Scholarpedia � Modeling With Words - article at Scholarpedia � Fuzzy logic - article at Stanford Encyclopedia of Philosophy � Fuzzy Math - Beginner level introduction to Fuzzy Logic � Fuzzylite - A cross-platform, free open-source Fuzzy Logic Control Library written in C++. Also has a very useful graphic user interface in QT4. � Online Calculator based upon Fuzzy logic – Gives online calculation in educational example of fuzzy logic model.
- Chapter 6 Hans Reichenbach Hans Reichenbach (September 26, 1891 – April 9, 1953) was a leading philosopher of science, educator, and proponent of logical empiricism. Reichenbach is best known for founding the Berlin Circle, and as the author of The Rise of Scientiﬁc Philosophy. 6.1 Life and work After completing secondary school in Hamburg, Hans studied civil engineering at the Technische Hochschule in Stuttgart, and physics, mathematics and philosophy at various universities, including Berlin, Erlangen, Göttingen and Munich. Among his teachers were Ernst Cassirer, David Hilbert, Max Planck, Max Born and Arnold Sommerfeld. Reichenbach was active in youth movements and student organizations, and published articles about the university reform, the freedom of research, and against anti-Semitic inﬁltrations in student organizations. His older brother Bernhard Reichenbach shared in this activism and went on to become a member of the Communist Workers’ Party of Germany, representing this organisation on the Executive Committee of the Communist International. He also worked with Alexander Schwab and [[]] at this time. Reichenbach received a degree in philosophy from the University of Erlangen in 1915 and his dissertation on the theory of probability, supervised by Paul Hensel and Max Noether, was published in 1916. Reichenbach served during World War I on the Russian front, in the German army radio troops. In 1917 he was removed from active duty, due to an illness, and returned in Berlin. While working as a physicist and engineer, Reichenbach attended Albert Einstein's lectures on the theory of relativity in Berlin from 1917 to 1920. In 1920 Reichenbach began teaching at the Technische Hochschule at Stuttgart as Privatdozent. In the same year, he published his ﬁrst book on the philosophical implications of the theory of relativity, The Theory of Relativity and A Priori Knowledge, which criticized the Kantian notion of synthetic a priori. He subsequently published Axiomatization of the Theory of Relativity (1924), From Copernicus to Einstein (1927) and The Philosophy of Space and Time (1928), the last stating the logical positivist view on the theory of relativity. In 1926, with the help of Albert Einstein, Max Planck and Max von Laue, Reichenbach became assistant professor in the physics department of Humboldt University of Berlin. He gained notice for his methods of teaching, as he was easily approached and his courses were open to discussion and debate. This was highly unusual at the time, although the practice is nowadays a common one. In 1928, Reichenbach founded the so-called "Berlin Circle" (German: Die Gesellschaft für empirische Philosophie; English: Society for Empirical Philosophy). Among its members were Carl Gustav Hempel, Richard von Mises, David Hilbert and Kurt Grelling. The Vienna Circle manifesto lists 30 of Reichenbach’s publications in a bibliography of closely related authors. In 1930 he and Rudolf Carnap began editing the journal Erkenntnis (“Knowledge”). When Adolf Hitler became Chancellor of Germany in 1933, Reichenbach was immediately dismissed from his ap- pointment at the University of Berlin under the government’s so called “Race Laws” due to his Jewish ancestry. Reichenbach himself did not practise Judaism, and his mother was a German Protestant, but nevertheless suﬀered problems. He thereupon emigrated to Turkey, where he headed the Department of Philosophy at the University of Istanbul. He introduced interdisciplinary seminars and courses on scientiﬁc subjects, and in 1935 he published The Theory of Probability. 61
- 62 CHAPTER 6. HANS REICHENBACH In 1938, with the help of Charles W. Morris, Reichenbach moved to the United States to take up a professorship at the University of California, Los Angeles in its Philosophy Department. His work on the philosophical foundations of quantum mechanics was published in 1944, followed by Elements of Symbolic Logic in 1947, and The Rise of Scientiﬁc Philosophy (his most popular book[1]) in 1951. Reichenbach helped establish UCLA as a leading philosophy department in the United States in the post-war period. Carl Hempel, Hilary Putnam, and Wesley Salmon are perhaps his most prominent students. Reichenbach died in Los Angeles on April 9, 1953, while working on problems in the philosophy of time and on the nature of scientiﬁc laws. As part of this he proposed a three part model of time in language, involving speech time, event time and - critically - reference time, which has been used by linguists since for describing tenses.[2] This work resulted in two books published posthumously: The Direction of Time and Nomological Statements and Admissible Operations. 6.2 Selected publications � 1916. Der Begriﬀ der Wahrscheinlichkeit für die mathematische Darstellung der Wirklichkeit. Ph.D. disserta- tion, Erlangen. � 1920. Relativitätstheorie und Erkenntnis apriori. English translation: 1965. The theory of relativity and a priori knowledge. University of California Press. � 1922. “Der gegenwärtige Stand der Relativitätsdiskussion.” English translation: “The present state of the dis- cussion on relativity” in Reichenbach (1959). � 1924. Axiomatik der relativistischen Raum-Zeit-Lehre. English translation: 1969. Axiomatization of the theory of relativity. University of California Press. � 1924. “Die Bewegungslehre bei Newton, Leibniz und Huyghens.” English translation: “The theory of motion according to Newton, Leibniz, and Huyghens” in Reichenbach (1959). � 1927. Von Kopernikus bis Einstein. DerWandel unseresWeltbildes. English translation: 1942, From Copernicus to Einstein. Alliance Book Co. � 1928. Philosophie der Raum-Zeit-Lehre. English translation: Maria Reichenbach, 1957, The Philosophy of Space and Time. Dover. ISBN 0-486-60443-8 � 1930. Atom und Kosmos. Das physikalische Weltbild der Gegenwart. English translation: 1932, Atom and cosmos: the world of modern physics. G. Allen & Unwin, ltd. � 1931. “Ziele und Wege der heutigen Naturphilosophie.” English translation: “Aims and methods of modern philosophy of nature” in Reichenbach (1959). � 1935. Wahrscheinlichkeitslehre : eine Untersuchung über die logischen und mathematischen Grundlagen der Wahrscheinlichkeitsrechnung. English translation: 1949, The theory of probability, an inquiry into the logical and mathematical foundations of the calculus of probability. University of California Press. � 1938. Experience and prediction: an analysis of the foundations and the structure of knowledge. University of Chicago Press. � 1942. From Copernicus to Einstein Dover 1980: ISBN 0-486-23940-3 � 1944. Philosophic Foundations of Quantum Mechanics. University of California Press. Dover 1998: ISBN 0-486-40459-5 � 1947. Elements of Symbolic Logic. Macmillan Co. Dover 1980: ISBN 0-486-24004-5 � 1948. “Philosophy and physics” in Faculty research lectures, 1946. University of California Press. � 1949. “The philosophical signiﬁcance of the theory of relativity” in Schilpp, P. A., ed., Albert Einstein: philosopher-scientist. Evanston : The Library of Living Philosophers. � 1951. The Rise of Scientiﬁc Philosophy. University of California Press. ISBN 978-0-520-01055-0
- 6.3. SEE ALSO 63 � 1954. Nomological statements and admissible operations. North Holland. � 1956. The Direction of Time. University of California Press. Dover 1971: ISBN 0-486-40926-0 � 1959. Modern philosophy of science: Selected essays by Hans Reichenbach. Routledge & Kegan Paul. Green- wood Press 1981: ISBN 0-313-23274-1 � 1978. Selected writings, 1909-1953: with a selection of biographical and autobiographical sketches (Vienna circle collection). Dordrecht: Reidel. Springer paperback vol 1: ISBN 90-277-0292-6 � 1979. Hans Reichenbach, logical empiricist (Synthese library). Dordrecht : Reidel. � 1991. Erkenntnis Orientated: ACentennial volume for Rudolf Carnap andHans Reichenbach. Kluwer. Springer 2003: ISBN 0-7923-1408-5 � 1991. Logic, language, and the structure of scientiﬁc theories : proceedings of the Carnap-Reichenbach centen- nial, University of Konstanz, 21–24 May 1991. University of Pittsburgh Press. 6.3 See also � American philosophy � List of American philosophers 6.4 References [1] [2] Derczynski, L; Gaizauskas, R (2013). “Empirical Validation of Reichenbach’s Tense Framework”. Proceedings of the International Conference on Computational Semantics. � Günther Sandner, The Berlin Group in the Making: Politics and Philosophy in the Early Works of Hans Re- ichenbach and Kurt Grelling. To appear in the Proceedings of 10th International Congress of the International Society for the History of Philosophy of Science (HOPOS), Ghent, July 2014. Abstract 6.5 Sources � Grünbaum, A., 1963, Philosophical Problems of Space and Time. Chpt. 3. � Carl Hempel, 1991, Hans Reichenbach remembered, Erkenntnis 35: 5-10. � Wesley Salmon, 1977, “The philosophy of Hans Reichenbach,” Synthese 34: 5-88. � Wesley Salmon, 1991, “Hans Reichenbach’s vindication of induction,” Erkenntnis 35: 99-122. 6.6 External links � Quotations related to Hans Reichenbach at Wikiquote � The Rise of Scientiﬁc Philosophy Descriptive summary & full searchable text at Google Book Search � O'Connor, John J.; Robertson, Edmund F., “Hans Reichenbach”, MacTutor History of Mathematics archive, University of St Andrews. � The Internet Encyclopedia of Philosophy: Hans Reichenbach by Mauro Murzi. � The Stanford Encyclopedia of Philosophy: Hans Reichenbach by Clark Glymour and Frederick Eberhardt.
- 64 CHAPTER 6. HANS REICHENBACH � The Stanford Encyclopedia of Philosophy: "Reichenbach’s Common Cause Principle" by Frank Arntzenius. � Guide to the Hans Reichenbach Collection at the University of Pittsburgh’s Archive of Scientiﬁc Philosophy: � Reichenbach’s theory of tense and its application to English
- Chapter 7 Jan Łukasiewicz Jan Łukasiewicz (Polish: [ˈjan wukaˈɕɛvʲitʂ]; 21 December 1878 – 13 February 1956) was a Polish logician and philosopher born in Lwów, which, before the Polish partitions, was in Poland, Galicia, then Austria-Hungary. His work centred on analytical philosophy, mathematical logic, and history of logic. He thought innovatively about tra- ditional propositional logic, the principle of non-contradiction and the law of excluded middle. Modern work on Aristotle’s logic builds on the tradition started in 1951 with the establishment by Lukasiewicz of a revolutionary paradigm. The Lukasiewicz approach was reinvigorated in the early 1970s in a series of papers by John Corcoran and Timothy Smiley--which inform modern translations of Prior Analytics by Robin Smith in 1989 and Gisela Striker in 2009.[1] Lukasiewicz is still regarded as one of the most important historians of logic. 7.1 Life He grew up in Lwów and was the only child of Paweł Łukasiewicz, a captain in the Austrian army, and Leopoldina (née Holtzer), the daughter of a civil servant. His family was Roman Catholic. He ﬁnished his gymnasium studies in philology and in 1897 went on to Lwów University, which, before the Polish partitions was in Poland, where he studied philosophy and mathematics. In philosophy he was a pupil of Kazimierz Twardowski. In 1902, he received a Doctor of Philosophy degree under the patronage of emperor Franz Joseph I of Austria who gave him a special doctor ring with diamonds. He spent three years as a private teacher, and in 1905 he received a scholarship to complete his philosophical studies at the University of Berlin and the University of Louvain in Belgium. Łukasiewicz continued studying for his habilitation qualiﬁcation and in 1906 submitted his thesis to the University of Lwów. In 1906 he was appointed a lecturer at the University of Lwów where he was eventually appointed Extraor- dinary Professor by Emperor Franz Joseph I. He taught there until the First World War. In 1915 he was invited to lecture as a full professor at the University of Warsaw which had re-opened after being closed down by the Tsarist government in the 19th century. In 1919 Łukasiewicz left the university to serve as Polish Minister of Religious Denominations and Public Education in the Paderewski government until 1920. Łukasiewicz led the development of a Polish curriculum replacing the Russian, German and Austrian curricula previously used in partitioned Poland. The Łukasiewicz curriculum emphasized the early acquisition of logical and mathematical concepts. In 1928 he married Regina Barwińska. He remained a professor at the University of Warsaw from 1920 until 1939 when the family house was destroyed by German bombs and the university was closed under German occupation. He had been a rector of the university twice. In this period Lukasiewicz and Stanisław Leśniewski founded the Lwów–Warsaw school of logic which was later made internationally famous by Alfred Tarski who had been Leśniewski’s student. At the beginning of World War II he worked at the Warsaw Underground University as part of the secret system of education in Poland during World War II. 65
- 66 CHAPTER 7. JAN ŁUKASIEWICZ He and his wife wanted to move to Switzerland but couldn't get permission from the German authorities. Instead, in the summer of 1944, they left Poland with the help of Heinrich Scholz and spent the last few months of the war in Münster, Germany hoping to somehow go on further, perhaps to Switzerland. Following the war, he emigrated to Ireland and worked at University College Dublin (UCD) until his death. 7.2 Work A number of axiomatizations of classical propositional logic are due to Łukasiewicz. A particularly elegant axioma- tization features a mere three axioms and is still invoked down to the present day. He was a pioneer investigator of multi-valued logics; his three-valued propositional calculus, introduced in 1917, was the ﬁrst explicitly axiomatized non-classical logical calculus. He wrote on the philosophy of science, and his approach to the making of scientiﬁc theories was similar to the thinking of Karl Popper. Łukasiewicz invented the Polish notation (named after his nationality) for the logical connectives around 1920. There is a quotation from his paper, Remarks on Nicod’s Axiom and on “Generalizing Deduction”, page 180; “I came upon the idea of a parenthesis-free notation in 1924. I used that notation for the ﬁrst time in my article [2] The reference cited by Łukasiewicz above is apparently a lithographed report in Polish. The referring paper by Łukasiewicz Remarks on Nicod’s Axiom and on “Generalizing Deduction”, originally published in Polish in 1931,[3] was later reviewed by H. A. Pogorzelski in the Journal of Symbolic Logic in 1965.[4] In Łukasiewicz 1951 book, Aristotle’s Syllogistic from the Standpoint of Modern Formal Logic, he mentions that the principle of his notation was to write the functors before the arguments to avoid brackets and that he had employed his notation in his logical papers since 1929.[5] He then goes on to cite, as an example, a 1930 paper he wrote with Alfred Tarski on the sentential calculus.[6] This notation is the root of the idea of the recursive stack, a last-in, ﬁrst-out computer memory store proposed by sev- eral researchers including Turing, Bauer and Hamblin, and ﬁrst implemented in 1957. In 1960, Łukasiewicz notation concepts and stacks were used as the basis of the Burroughs B5000 computer designed by Robert S. Barton and his team at Burroughs Corporation in Pasadena, California. The concepts also led to the design of the English Electric multi-programmed KDF9 computer system of 1963, which had two such hardware register stacks. A similar concept underlies the reverse Polish notation (RPN, a postﬁx notation) of the Friden EC-130 calculator and its successors, many Hewlett Packard calculators, the Forth programming language, or the PostScript page description language. 7.3 Recognition In 2008 the Polish Information Processing Society established the Jan Łukasiewicz Award, to be presented to the most innovative Polish IT companies.[7] From 1999-2004, the Department of Computer Science building at UCD was called the Łukasiewicz Building, until all campus buildings were renamed after the disciplines they housed. 7.4 Chronology � 1878 born � 1890–1902 studies with Kazimierz Twardowski in Lemberg (Lwów, L'viv) � 1902 doctorate (mathematics and philosophy), University of Lemberg with the highest distinction possible � 1906 habilitation thesis completed, University of Lemberg � 1906 becomes a lecturer � 1910 essays on the principle of non-contradiction and the excluded middle
- 7.5. SELECTED WORKS 67 � 1911 extraordinary professor at Lemberg � 1915 invited to the newly reopened University of Warsaw � 1916 new Kingdom of Poland declared � 1917 develops three-valued propositional calculus � 1919 Polish Minister of Education � 1920–1939 professor at Warsaw University founds with Stanisław Leśniewski the Lwów–Warsaw school of logic (see also Alfred Tarski, Stefan Banach, Hugo Steinhaus, Zygmunt Janiszewski, Stefan Mazurkiewicz) � 1928 marries Regina Barwińska � 1944 ﬂees to Germany and settles in Hembsen, where he was brought for his own safety. � 1946 exile in Belgium � 1946 oﬀered a chair by the Royal Irish Academy, held at University College Dublin � 1953 writes autobiography � 1956 dies in Dublin 7.5 Selected works 7.5.1 Books � Łukasiewicz, Jan (1951). Aristotle’s Syllogistic from the Standpoint of Modern Formal Logic. Oxford University Press. 2nd Edition, enlarged, 1957. Reprinted by Garland Publishing in 1987. ISBN 0-8240-6924-2 � Łukasiewicz, Jan (1958). Elementy logiki matematycznej (in Polish). Warsaw, Państwowe Wydawnictwo Naukowe. OCLC 11322101. � Łukasiewicz, Jan (1964) [1963]. Elements of Mathematical Logic. Translated from Polish by Olgierd Woj- tasiewicz. New York, Macmillan. OCLC 671498. � Łukasiewicz, Jan (1970). Ludwik Borkowski, ed. Selected Works. North-Holland Pub. Co. ISBN 0-7204- 2252-3. OCLC 115237. � Łukasiewicz, Jan (1998). Jacek Jadacki, ed. Logika i metaﬁzyka. Miscellanea (in Polish). Warsaw, WFiS UW. ISBN 83-910113-3-X. 7.5.2 Papers � 1903 “On Induction as Inversion of Deduction” � 1906 “Analysis and Construction of the Concept of Cause” � 1910 “On Aristotle’s Principle of Contradiction” � 1913 “On the Reversibility of the Relation of Ground and Consequence” � 1920 “On Three-valued Logic” � 1921 “Two-valued Logic” � 1922 “A Numerical Interpretation of the Theory of Propositions” � 1928 “Concerning the Method in Philosophy” � 1929 “Elements of Mathematical Logic” � 1929 “On Importance and Requirements of Mathematical Logic”
- 68 CHAPTER 7. JAN ŁUKASIEWICZ � 1930 “Philosophical Remarks on Many-Valued Systems of Propositional Logic” � 1930 “Investigations into the Sentential Calculus” ["Untersuchungen über den Aussagenkalkül"], with Alfred Tarski � 1931 “Comments on Nicod’s Axiom and the 'Generalizing Deduction'" � 1934 “On Science” � 1934 “Importance of Logical Analysis for Knowledge” � 1934 “Outlines of the History of the Propositional Logic” � 1936 “Logistic and Philosophy” � 1937 “In Defense of the Logistic” � 1938 “On Descartes’s Philosophy” � 1943 “The Shortest Axiom of the Implicational Calculus of Propositions” � 1951 “On Variable Functors of Propositional Arguments” � 1952 “On the Intuitionistic Theory of Deduction” � 1953 “A System of Modal Logic” � 1954 “On a Controversial Problem of Aristotle’s Modal Syllogistic” 7.6 See also � Łukasiewicz logic � History of philosophy in Poland � Stanisław Leśniewski � List of Poles � 27114 Lukasiewicz 7.7 Notes [1] � Review of “Aristotle, Prior Analytics: Book I, Gisela Striker (translation and commentary), Oxford UP, 2009, 268pp., $39.95 (pbk), ISBN 978-0-19-925041-7.” in the Notre Dame Philosophical Reviews, 2010.02.02. [2] Łukasiewicz(1), p. 610, footnote.” [3] Łukasiewicz, Jan, “Uwagi o aksjomacie Nicoda i 'dedukcji uogólniającej'", (“Remarks on Nicod’s Axiom and the “Gener- alizing Deduction”), Księga pamiątkowa Polskiego Towarzystwa Filozoﬁcznego, Lwów 1931. [4] Pogorzelski, H. A., “Reviewed work(s): Remarks on Nicod’s Axiom and on “Generalizing Deduction” by Jan Łukasiewicz; Jerzy Słupecki; Państwowe Wydawnictwo Naukowe”, The Journal of Symbolic Logic, Vol. 30, No. 3 (Sep. 1965), pp. 376–377. This paper by Jan Łukasiewicz was re-published in Warsaw in 1961 in a volume edited by Jerzy Słupecki. It had been published originally in 1931 in Polish. [5] Cf. Łukasiewicz, (1951) Aristotle’s Syllogistic from the Standpoint of Modern Formal Logic, Chapter IV “Aristotle’s System in Symbolic Form” (section on “Explanation of the Symbolism”), p.78 and on. [6] Łukasiewicz, Jan; Tarski, Alfred, “Untersuchungen über den Aussagenkalkül” ["Investigations into the sentential calculus"], Comptes Rendus des séances de la Société des Sciences et des Lettres de Varsovie, Vol. 23 (1930) Cl. III, pp. 31–32. This paper can be found translated into English in Chapter IV “Investigations into the Sentential Calculus”, pp.39-59, in Logic, Semantics, Metamathematics: Papers from 1923 to 1938 by Alfred Tarski, translated into English by J.H. Woodger, Oxford University Press, 1956; 2nd edition, Hackett Publishing Company, 1983 [7] “2009 International Multiconference on Computer Science and Information Technology (IMCSIT)", conference report
- 7.8. REFERENCES 69 7.8 References � “Curriculum Vitae of Jan Łukasiewicz”, Rome, Italy: Metalogicon journal, (1994) VII, 2 (July–December issue). � Craig, Edward (general editor), “Article: Jan Łukasiewicz”, Routledge Encyclopedia of Philosophy, 1998, Volume 5, pp. 860–863. 7.9 Further reading � Borkowski, L.; Słupecki, J., “The logical works of J. Łukasiewicz”, Studia Logica 8 (1958), 7–56. � Kotarbiński, T., “Jan Łukasiewicz’s works on the history of logic”, Studia Logica 8 (1958), 57–62. � Kwiatkowski, T., “Jan Łukasiewicz – A historian of logic”, Organon 16–17 (1980–1981), 169–188. � Marshall, D., "Łukasiewicz, Leibniz and the arithmetization of the syllogism”, Notre Dame Journal of Formal Logic 18 (2) (1977), 235–242. � Seddon, Frederick (1996). Aristotle & Łukasiewicz on the Principle of Contradiction. Ames, Iowa: Modern Logic Pub. ISBN 1-884905-04-8. OCLC 37533856. � Woleński, Jan (1994). Philosophical Logic in Poland. Kluwer Academic Publishers. ISBN 0-7923-2293-2. OCLC 27938071. � Woleński, Jan, “Jan Łukasiewicz on the Liar Paradox, Logical Consequence, Truth and Induction”, Modern Logic 4 (1994), 394–400. 7.10 External links � Jan Lukasiewicz entry by Peter Simons in the Stanford Encyclopedia of Philosophy � O'Connor, John J.; Robertson, Edmund F., “Jan Łukasiewicz”, MacTutor History of Mathematics archive, University of St Andrews. � Łukasiewicz entry at Polish Philosophy Page, ed. by F. Coniglione (University of Catania) � Jan Łukasiewicz at the Mathematics Genealogy Project
- Chapter 8 Many-valued logic In logic, a many-valued logic (also multi- or multiple-valued logic) is a propositional calculus in which there are more than two truth values. Traditionally, in Aristotle's logical calculus, there were only two possible values (i.e., “true” and “false”) for any proposition. Classical two-valued logic may be extended to n-valued logic for n greater than 2. Those most popular in the literature are three-valued (e.g., Łukasiewicz’s and Kleene’s, which accept the values “true”, “false”, and “unknown”), the ﬁnite-valued (ﬁnitely-many valued) with more than three values, and the inﬁnite-valued (inﬁnitely-many valued), such as fuzzy logic and probability logic. 8.1 History The ﬁrst known classical logician who didn't fully accept the law of excluded middle was Aristotle (who, ironically, is also generally considered to be the ﬁrst classical logician and the “father of logic”[1]). Aristotle admitted that his laws did not all apply to future events (De Interpretatione, ch. IX), but he didn't create a system of multi-valued logic to explain this isolated remark. Until the coming of the 20th century, later logicians followed Aristotelian logic, which includes or assumes the law of the excluded middle. The 20th century brought back the idea of multi-valued logic. The Polish logician and philosopher Jan Łukasiewicz began to create systems of many-valued logic in 1920, using a third value, “possible”, to deal with Aristotle’s paradox of the sea battle. Meanwhile, the American mathematician, Emil L. Post (1921), also introduced the formulation of additional truth degrees with n ≥ 2, where n are the truth values. Later, Jan Łukasiewicz and Alfred Tarski together formulated a logic on n truth values where n ≥ 2. In 1932 Hans Reichenbach formulated a logic of many truth values where n→inﬁnity. Kurt Gödel in 1932 showed that intuitionistic logic is not a ﬁnitely-many valued logic, and deﬁned a system of Gödel logics intermediate between classical and intuitionistic logic; such logics are known as intermediate logics. 8.2 Examples Main articles: Three-valued logic and Four-valued logic 8.2.1 Kleene (strong) K3 and Priest logic P3 Kleene's "(strong) logic of indeterminacy”K3 (sometimesKS3 ) and Priest’s “logic of paradox” add a third “undeﬁned” or “indeterminate” truth value I. The truth functions for negation (¬), conjunction (∧), disjunction (∨), implication (→K), and biconditional (↔K) are given by:[2] The diﬀerence between the two logics lies in how tautologies are deﬁned. In K3 only T is a designated truth value, while in P3 both T and I are (a logical formula is considered a tautology if it evaluates to a designated truth value). In Kleene’s logic I can be interpreted as being “underdetermined”, being neither true nor false, while in Priest’s logic I can be interpreted as being “overdetermined”, being both true and false. K3 does not have any tautologies, while P3 has the same tautologies as classical two-valued logic. 70
- 8.2. EXAMPLES 71 8.2.2 Bochvar’s internal three-valued logic (also known as Kleene’s weak three-valued logic) Another logic is Bochvar’s “internal” three-valued logic ( BI3 ) also called Kleene’s weak three-valued logic. Except for negation and biconditional, its truth tables are all diﬀerent from the above.[3] The intermediate truth value in Bochvar’s “internal” logic can be described as “contagious” because it propagates in a formula regardless of the value of any other variable.[4] 8.2.3 Belnap logic (B4) Belnap’s logic B4 combines K3 and P3. The overdetermined truth value is here denoted as B and the underdetermined truth value as N. 8.2.4 Gödel logics Gk and G∞ In 1932 Gödel deﬁned[5] a familyGk of many-valued logics, with ﬁnitely many truth values 0; 1k�1 ; 2k�1 ; : : : k�2k�1 ; 1 , for exampleG3 has the truth values 0; 12 ; 1 andG4 has 0; 13 ; 23 ; 1 . In a similar manner he deﬁned a logic with inﬁnitely many truth values, G1 , in which the truth values are all the real numbers in the interval [0; 1] . The designated truth value in these logics is 1. The conjunction ^ and the disjunction _ are deﬁned respectively as the maximum and minimum of the operands: � u ^ v := minfu; vg � u _ v := maxfu; vg Negation :G and implication !G are deﬁned as follows: � :Gu = ( 1; ifu = 0 0; ifu > 0 u!G v = ( 1; ifu � v 0; ifu > v Gödel logics are completely axiomatisable, that is to say it is possible to deﬁne a logical calculus in which all tautologies are provable. 8.2.5 Łukasiewicz logics Lv and L∞ Implication !L and negation :L were deﬁned by Jan Łukasiewicz through the following functions: � :Lu := 1� u � u!L v := minf1; 1� u+ vg At ﬁrst Łukasiewicz used these deﬁnition in 1920 for his three-valued logic L3 , with truth values 0; 12 ; 1 . In 1922 he developed a logic with inﬁnitely many values L1 , in which the truth values spanned the real numbers in the interval [0; 1] . In both cases the designated truth walue was 1.[6] By adopting truth values deﬁned in the same way as for Gödel logics 0; 1v�1 ; 2v�1 ; : : : ; v�2v�1 ; 1 , it is possible to create a ﬁnitely-valued family of logics Lv , the abovementioned L1 and the logic L@0 , in which the truth values are given by the rational numbers in the interval [0; 1] . The set of tautologies in L1 and L@0 is identical.
- 72 CHAPTER 8. MANY-VALUED LOGIC 8.2.6 Product logic Π In product logic we have truth values in the interval [0; 1] , a conjunction � and an implication !� , deﬁned as follows[7] � u� v := uv � u!� v := ( 1; ifu � v v u ; ifu > v Additionally there is a negative designated value 0 that denotes the concept of false. Through this value it is possible to deﬁne a negation :� and an additional conjunction ^� as follows: � :�u := u!� 0 � u ^� v := u� (u!� v) 8.2.7 Post logics Pm In 1921 Post deﬁned a family of logics Pm with (as in Lv and Gk ) the truth values 0; 1m�1 ; 2m�1 ; : : : ; m�2m�1 ; 1 . Negation :P and disjunction _P are deﬁned as follows: � :Pu := ( 1; ifu = 0 u� 1m�1 ; ifu 6= 0 � u _P v := maxfu; vg 8.3 Semantics 8.3.1 Matrix semantics (logical matrices) 8.4 Proof theory 8.5 Relation to classical logic Logics are usually systems intended to codify rules for preserving some semantic property of propositions across transformations. In classical logic, this property is “truth.” In a valid argument, the truth of the derived proposition is guaranteed if the premises are jointly true, because the application of valid steps preserves the property. However, that property doesn't have to be that of “truth"; instead, it can be some other concept. Multi-valued logics are intended to preserve the property of designationhood (or being designated). Since there are more than two truth values, rules of inference may be intended to preserve more than just whichever corresponds (in the relevant sense) to truth. For example, in a three-valued logic, sometimes the two greatest truth-values (when they are represented as e.g. positive integers) are designated and the rules of inference preserve these values. Precisely, a valid argument will be such that the value of the premises taken jointly will always be less than or equal to the conclusion. For example, the preserved property could be justiﬁcation, the foundational concept of intuitionistic logic. Thus, a proposition is not true or false; instead, it is justiﬁed or ﬂawed. A key diﬀerence between justiﬁcation and truth, in this case, is that the law of excluded middle doesn't hold: a proposition that is not ﬂawed is not necessarily justiﬁed; instead, it’s only not proven that it’s ﬂawed. The key diﬀerence is the determinacy of the preserved property: One may prove that P is justiﬁed, that P is ﬂawed, or be unable to prove either. A valid argument preserves justiﬁcation across transformations, so a proposition derived from justiﬁed propositions is still justiﬁed. However, there are proofs in classical logic that depend upon the law of excluded middle; since that law is not usable under this scheme, there are propositions that cannot be proven that way.
- 8.6. APPLICATIONS 73 8.5.1 Suszko’s thesis See also: Principle of bivalence § Suszko’s thesis 8.6 Applications Known applications of many-valued logic can be roughly classiﬁed into two groups.[8] The ﬁrst group uses many- valued logic domain to solve binary problems more eﬃciently. For example, a well-known approach to represent a multiple-output Boolean function is to treat its output part as a single many-valued variable and convert it to a single- output characteristic function. Other applications of many-valued logic include design of Programmable Logic Arrays (PLAs) with input decoders, optimization of ﬁnite state machines, testing, and veriﬁcation. The second group targets the design of electronic circuits which employ more than two discrete levels of signals, such as many-valued memories, arithmetic circuits, Field Programmable Gate Arrays (FPGA) etc. Many-valued circuits have a number of theoretical advantages over standard binary circuits. For example, the interconnect on and oﬀ chip can be reduced if signals in the circuit assume four or more levels rather than only two. In memory design, storing two instead of one bit of information per memory cell doubles the density of the memory in the same die size. Applications using arithmetic circuits often beneﬁt from using alternatives to binary number systems. For example, residue and redundant number systems can reduce or eliminate the ripple-through carries which are involved in normal binary addition or subtraction, resulting in high-speed arithmetic operations. These number systems have a natural implementation using many-valued circuits. However, the practicality of these potential advantages heavily depends on the availability of circuit realizations, which must be compatible or competitive with present-day standard technologies. 8.7 Research venues An IEEE International Symposium on Multiple-Valued Logic (ISMVL) has been held annually since 1970. It mostly caters to applications in digital design and veriﬁcation.[9] There is also a Journal of Multiple-Valued Logic and Soft Computing.[10] 8.8 See also Mathematical logic � Degrees of truth � Fuzzy logic � Gödel logic � Kleene logic � Kleene algebra (with involution) � Łukasiewicz logic � MV-algebra � Post logic � Principle of bivalence � A. N. Prior � Relevance logic Philosophical logic
- 74 CHAPTER 8. MANY-VALUED LOGIC � False dilemma � Mu Digital logic � MVCML, multiple-valued current-mode logic � IEEE 1164 a nine-valued standard for VHDL � IEEE 1364 a four-valued standard for Verilog � Noise-based logic 8.9 Notes 8.10 References [1] Hurley, Patrick. A Concise Introduction to Logic, 9th edition. (2006). [2] (Gottwald 2005, p. 19) [3] (Bergmann 2008, p. 80) [4] (Bergmann 2008, p. 80) [5] Gödel, Kurt (1932). “Zum intuitionistischen Aussagenkalkül”. Anzeiger Akademie der Wissenschaften Wien (69): 65f. [6] Kreiser, Lothar; Gottwald, Siegfried; Stelzner, Werner (1990). Nichtklassische Logik. Eine Einführung. Berlin: Akademie- Verlag. pp. 41ﬀ –– 45ﬀ. ISBN 978-3-05-000274-3. [7] Hajek, Petr: Fuzzy Logic. In: Edward N. Zalta: The Stanford Encyclopedia of Philosophy, Spring 2009. () [8] Dubrova, Elena (2002). Multiple-Valued Logic Synthesis and Optimization, in Hassoun S. and Sasao T., editors, Logic Synthesis and Veriﬁcation, Kluwer Academic Publishers, pp. 89-114 [9] http://www.informatik.uni-trier.de/~{}ley/db/conf/ismvl/index.html [10] http://www.oldcitypublishing.com/MVLSC/MVLSC.html 8.11 Further reading General � Béziau J.-Y. (1997), What is many-valued logic ? Proceedings of the 27th International Symposium onMultiple- Valued Logic, IEEE Computer Society, Los Alamitos, pp. 117–121. � Malinowski, Gregorz, (2001), Many-Valued Logics, in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic. Blackwell. � Bergmann, Merrie (2008), An introduction to many-valued and fuzzy logic: semantics, algebras, and derivation systems, Cambridge University Press, ISBN 978-0-521-88128-9 � Cignoli, R. L. O., D'Ottaviano, I, M. L., Mundici, D., (2000). Algebraic Foundations of Many-valued Reason- ing. Kluwer. � Malinowski, Grzegorz (1993). Many-valued logics. Clarendon Press. ISBN 978-0-19-853787-8. � S. Gottwald, A Treatise on Many-Valued Logics. Studies in Logic and Computation, vol. 9, Research Studies Press: Baldock, Hertfordshire, England, 2001. � Gottwald, Siegfried (2005). “Many-Valued Logics” (PDF).
- 8.12. EXTERNAL LINKS 75 � Miller, D. Michael; Thornton, Mitchell A. (2008). Multiple valued logic: concepts and representations. Syn- thesis lectures on digital circuits and systems 12. Morgan & Claypool Publishers. ISBN 978-1-59829-190-2. � Hájek P., (1998), Metamathematics of fuzzy logic. Kluwer. (Fuzzy logic understood as many-valued logic sui generis.) Speciﬁc � Alexandre Zinoviev, Philosophical Problems of Many-Valued Logic, D. Reidel Publishing Company, 169p., 1963. � Prior A. 1957, Time and Modality. Oxford University Press, based on his 1956 John Locke lectures � Goguen J.A. 1968/69, The logic of inexact concepts, Synthese, 19, 325–373. � Chang C.C. and Keisler H. J. 1966. Continuous Model Theory, Princeton, Princeton University Press. � Gerla G. 2001, Fuzzy logic: Mathematical Tools for Approximate Reasoning, Kluwer Academic Publishers, Dordrecht. � Pavelka J. 1979, On fuzzy logic I: Many-valued rules of inference, Zeitschr. f. math. Logik und Grundlagen d. Math., 25, 45–52. � Metcalfe, George; Olivetti, Nicola; Dov M. Gabbay (2008). Proof Theory for Fuzzy Logics. Springer. ISBN 978-1-4020-9408-8. Covers proof theory of many-valued logics as well, in the tradition of Hájek. � Hähnle, Reiner (1993). Automated deduction in multiple-valued logics. Clarendon Press. ISBN 978-0-19- 853989-6. � Azevedo, Francisco (2003). Constraint solving over multi-valued logics: application to digital circuits. IOS Press. ISBN 978-1-58603-304-0. � Bolc, Leonard; Borowik, Piotr (2003). Many-valued Logics 2: Automated reasoning and practical applications. Springer. ISBN 978-3-540-64507-8. � Stanković, Radomir S.; Astola, Jaakko T.; Moraga, Claudio (2012). Representation of Multiple-Valued Logic Functions. Morgan & Claypool Publishers. doi:10.2200/S00420ED1V01Y201205DCS037. ISBN 978-1- 60845-942-1. 8.12 External links � Gottwald, Siegfried (2009). “Many-Valued Logic”. Stanford Encyclopedia of Philosophy. � Stanford Encyclopedia of Philosophy: "Truth Values"—by Yaroslav Shramko and Heinrich Wansing. � IEEE Computer Society's Technical Committee on Multiple-Valued Logic � Resources for Many-Valued Logic by Reiner Hähnle, Chalmers University � Many-valued Logics W3 Server (archived) � Yaroslav Shramko and Heinrich Wansing (2014). “Suszko’s Thesis”. Stanford Encyclopedia of Philosophy. � Carlos Caleiro, Walter Carnielli, Marcelo E. Coniglio and João Marcos, Two’s company: “The humbug of many logical values” in Jean-Yves Beziau, ed. (2007). Logica Universalis: Towards a General Theory of Logic (2nd ed.). Springer Science & Business Media. pp. 174–194. ISBN 978-3-7643-8354-1.
- Chapter 9 Principle of bivalence This article is about logical principle. For chemical meaning (an atom with 2 bonds), see Bivalent (chemistry). In logic, the semantic principle (or law) of bivalence states that every declarative sentence expressing a proposition (of a theory under inspection) has exactly one truth value, either true or false.[1][2] A logic satisfying this principle is called a two-valued logic[3] or bivalent logic.[2][4] In formal logic, the principle of bivalence becomes a property that a semantics may or may not possess. It is not the same as the law of excluded middle, however, and a semantics may satisfy that law without being bivalent.[2] It may be written in the second-order sentence as: 8P 8x(x 2 P _ x /2 P ) , demonstrating similarity yet diﬀering mainly by quantiﬁed set elements. The principle of bivalence is studied in philosophical logic to address the question of which natural-language state- ments have a well-deﬁned truth value. Sentences which predict events in the future, and sentences which seem open to interpretation, are particularly diﬃcult for philosophers who hold that the principle of bivalence applies to all declarative natural-language statements.[2] Many-valued logics formalize ideas that a realistic characterization of the notion of consequence requires the admissibility of premises which, owing to vagueness, temporal or quantum inde- terminacy, or reference-failure, cannot be considered classically bivalent. Reference failures can also be addressed by free logics.[5] 9.1 Relationship with the law of the excluded middle The principle of bivalence is related to the law of excluded middle though the latter is a syntactic expression of the language of a logic of the form “P ∨ ¬P”. The diﬀerence between the principle and the law is important because there are logics which validate the law but which do not validate the principle.[2] For example, the three-valued Logic of Paradox (LP) validates the law of excluded middle, but not the law of non-contradiction, ¬(P ∧ ¬P), and its intended semantics is not bivalent.[6] In classical two-valued logic both the law of excluded middle and the law of non-contradiction hold.[1] Many modern logic programming systems replace the law of the excluded middle with the concept of negation as failure. The programmer may wish to add the law of the excluded middle by explicitly asserting it as true; however, it is not assumed a priori. 9.2 Classical logic The intended semantics of classical logic is bivalent, but this is not true of every semantics for classical logic. In Boolean-valued semantics (for classical propositional logic), the truth values are the elements of an arbitrary Boolean algebra, “true” corresponds to the maximal element of the algebra, and “false” corresponds to the minimal element. Intermediate elements of the algebra correspond to truth values other than “true” and “false”. The principle of bivalence holds only when the Boolean algebra is taken to be the two-element algebra, which has no intermediate elements. 76
- 9.3. SUSZKO’S THESIS 77 Assigning Boolean semantics to classical predicate calculus requires that the model be a complete Boolean algebra because the universal quantiﬁer maps to the inﬁmum operation, and the existential quantiﬁer maps to the supremum;[7] this is called a Boolean-valued model. All ﬁnite Boolean algebras are complete. 9.3 Suszko’s thesis In order to justify his claim that true and false are the only logical values, Suszko (1977) observes that every structural Tarskian many-valued propositional logic can be provided with a bivalent semantics.[8] 9.4 Criticisms 9.4.1 Future contingents Main article: Problem of future contingents A famous example[2] is the contingent sea battle case found in Aristotle's work, De Interpretatione, chapter 9: Imagine P refers to the statement “There will be a sea battle tomorrow.” The principle of bivalence here asserts: Either it is true that there will be a sea battle tomorrow, or it is false that there will be a sea battle tomorrow. Aristotle hesitated to embrace bivalence for such future contingents; Chrysippus, the Stoic logician, did embrace bivalence for this and all other propositions. The controversy continues to be of central importance in both the philosophy of time and the philosophy of logic. One of the early motivations for the study of many-valued logics has been precisely this issue. In the early 20th century, the Polish formal logician Jan Łukasiewicz proposed three truth-values: the true, the false and the as-yet- undetermined. This approach was later developed by Arend Heyting and L. E. J. Brouwer;[2] see Łukasiewicz logic. Issues such as this have also been addressed in various temporal logics, where one can assert that "Eventually, either there will be a sea battle tomorrow, or there won't be.” (Which is true if “tomorrow” eventually occurs.) 9.4.2 Vagueness Such puzzles as the Sorites paradox and the related continuum fallacy have raised doubt as to the applicability of classical logic and the principle of bivalence to concepts that may be vague in their application. Fuzzy logic and some other multi-valued logics have been proposed as alternatives that handle vague concepts better. Truth (and falsity) in fuzzy logic, for example, comes in varying degrees. Consider the following statement in the circumstance of sorting apples on a moving belt: This apple is red.[9] Upon observation, the apple is an undetermined color between yellow and red, or it is motled both colors. Thus the color falls into neither category " red " nor " yellow ", but these are the only categories available to us as we sort the apples. We might say it is “50% red”. This could be rephrased: it is 50% true that the apple is red. Therefore, P is 50% true, and 50% false. Now consider: This apple is red and it is not-red.
- 78 CHAPTER 9. PRINCIPLE OF BIVALENCE In other words, P and not-P. This violates the law of noncontradiction and, by extension, bivalence. However, this is only a partial rejection of these laws because P is only partially true. If P were 100% true, not-P would be 100% false, and there is no contradiction because P and not-P no longer holds. However, the law of the excluded middle is retained, because P and not-P implies P or not-P, since “or” is inclusive. The only two cases where P and not-P is false (when P is 100% true or false) are the same cases considered by two-valued logic, and the same rules apply. Example of a 3-valued logic applied to vague (undetermined) cases: Kleene 1952[10] (§64, pp. 332–340) oﬀers a 3-valued logic for the cases when algorithms involving partial recursive functions may not return values, but rather end up with circumstances “u” = undecided. He lets “t” = “true”, “f” = “false”, “u” = “undecided” and redesigns all the propositional connectives. He observes that: “We were justiﬁed intuitionistically in using the classical 2-valued logic, when we were using the con- nectives in building primitive and general recursive predicates, since there is a decision procedure for each general recursive predicate; i.e. the law of the excluded middle is proved intuitionistically to apply to general recursive predicates. “Now if Q(x) is a partial recursive predicate, there is a decision procedure for Q(x) on its range of deﬁnition, so the law of the excluded middle or excluded “third” (saying that, Q(x) is either t or f) applies intuitionistically on the range of deﬁnition. But there may be no algorithm for deciding, given x, whether Q(x) is deﬁned or not . . .. Hence it is only classically and not intuitionistically that we have a law of the excluded fourth (saying that, for each x, Q(x) is either t, f, or u). “The third “truth value” u is thus not on par with the other two t and f in our theory. Consideration of its status will show that we are limited to a special kind of truth table”. The following are his “strong tables":[11] For example, if a determination cannot be made as to whether an apple is red or not-red, then the truth value of the assertion Q: " This apple is red " is " u ". Likewise, the truth value of the assertion R " This apple is not-red " is " u ". Thus the AND of these into the assertion Q AND R, i.e. " This apple is red AND this apple is not-red " will, per the tables, yield " u ". And, the assertion Q OR R, i.e. " This apple is red OR this apple is not-red " will likewise yield " u ". 9.5 See also � Dualism � Exclusive disjunction � Degrees of truth � Anekantavada � Extensionality � False dilemma � Fuzzy logic � Logical disjunction � Logical equality � Logical value � Multi-valued logic � Propositional logic � Relativism
- 9.6. REFERENCES 79 � Supervaluationism � Truthbearer � Truthmaker � Truth-value link � Quantum logic � Perspectivism � Rhizome (philosophy) � True and false 9.6 References [1] Lou Goble (2001). The Blackwell guide to philosophical logic. Wiley-Blackwell. p. 309. ISBN 978-0-631-20693-4. [2] Paul Tomassi (1999). Logic. Routledge. p. 124. ISBN 978-0-415-16696-6. [3] Lou Goble (2001). The Blackwell guide to philosophical logic. Wiley-Blackwell. p. 4. ISBN 978-0-631-20693-4. [4] Mark Hürlimann (2009). Dealing with Real-World Complexity: Limits, Enhancements and New Approaches for Policy Makers. Gabler Verlag. p. 42. ISBN 978-3-8349-1493-4. [5] Dov M. Gabbay; John Woods (2007). The Many Valued and Nonmonotonic Turn in Logic. The handbook of the history of logic 8. Elsevier. p. vii. ISBN 978-0-444-51623-7. Retrieved 4 April 2011. [6] Graham Priest (2008). An introduction to non-classical logic: from if to is. Cambridge University Press. pp. 124–125. ISBN 978-0-521-85433-7. [7] Morten Heine Sørensen; Paweł Urzyczyn (2006). Lectures on the Curry-Howard isomorphism. Elsevier. pp. 206–207. ISBN 978-0-444-52077-7. [8] “Stanford Encyclopedia of Philosophy”. [9] Note the use of the (extremely) deﬁnite article: " This " as opposed to a more-vague " The ". " The " would have to be accompanied with a pointing-gesture to make it deﬁnitive. Ff Principia Mathematica (2nd edition), p. 91. Russell & Whitehead observe that this " this " indicates “something given in sensation” and as such it shall be considered “elementary”. [10] Stephen C. Kleene 1952 Introduction to Metamathematics, 6th Reprint 1971, North-Holland Publishing Company, Ams- terdam NY, ISBN 0-7294-2130-9. [11] “Strong tables” is Kleene’s choice of words. Note that even though " u " may appear for the value of Q or R, " t " or " f " may, in those occasions, appear as a value in " Q V R ", " Q & R " and " Q → R ". “Weak tables” on the other hand, are “regular”, meaning they have " u " appear in all cases when the value " u " is applied to either Q or R or both. Kleene notes that these tables are not the same as the original values of the tables of Łukasiewicz 1920. (Kleene gives these diﬀerences on page 335). He also concludes that " u " can mean any or all of the following: “undeﬁned”, “unknown (or value immaterial)", “value disregarded for the moment”, i.e. it is a third category that does not (ultimately) exclude " t " and " f " (page 335). 9.7 Further reading � Devidi, D.; Solomon, G. (1999). “On Confusions About Bivalence and Excluded Middle”. Dialogue (in French) 38 (4): 785–799. doi:10.1017/S0012217300006715.. � Betti Arianna (2002) The Incomplete Story of Łukasiewicz and Bivalence in T. Childers (ed.) The Logica 2002 Yearbook, Prague: The Czech Academy of Sciences—Filosoﬁa, pp. 21–26 � Jean-Yves Béziau (2003) "Bivalence, excluded middle and non contradiction", in The Logica Yearbook 2003, L.Behounek (ed), Academy of Sciences, Prague, pp. 73–84. � Font, J. M. (2009). “Taking Degrees of Truth Seriously”. Studia Logica 91 (3): 383–406. doi:10.1007/s11225- 009-9180-7.
- 80 CHAPTER 9. PRINCIPLE OF BIVALENCE 9.8 External links � Truth Values entry by Yaroslav Shramko, Heinrich Wansing in the Stanford Encyclopedia of Philosophy
- Chapter 10 Probabilistic logic The aim of a probabilistic logic (also probability logic and probabilistic reasoning) is to combine the capacity of probability theory to handle uncertainty with the capacity of deductive logic to exploit structure. The result is a richer and more expressive formalism with a broad range of possible application areas. Probabilistic logics attempt to ﬁnd a natural extension of traditional logic truth tables: the results they deﬁne are derived through probabilistic expressions instead. A diﬃculty with probabilistic logics is that they tend to multiply the computational complexities of their probabilistic and logical components. Other diﬃculties include the possibility of counter-intuitive results, such as those of Dempster-Shafer theory. The need to deal with a broad variety of contexts and issues has led to many diﬀerent proposals. 10.1 Historical context There are numerous proposals for probabilistic logics. Very roughly, they can be categorized into two diﬀerent classes: those logics that attempt to make a probabilistic extension to logical entailment, such as Markov logic networks, and those that attempt to address the problems of uncertainty and lack of evidence (evidentiary logics). That probability and uncertainty are not quite the same thing may be understood by noting that, despite the math- ematization of probability in the Enlightenment, mathematical probability theory remains, to this very day, entirely unused in criminal courtrooms, when evaluating the “probability” of the guilt of a suspected criminal.[1] More precisely, in evidentiary logic, there is a need to distinguish the truth of a statement from the conﬁdence in its truth: thus, being uncertain of a suspect’s guilt is not the same as assigning a numerical probability to the commission of the crime. A single suspect may be guilty or not guilty, just as a coin may be ﬂipped heads or tails. Given a large collection of suspects, a certain percentage may be guilty, just as the probability of ﬂipping “heads” is one- half. However, it is incorrect to take this law of averages with regard to a single criminal (or single coin-ﬂip): the criminal is no more “a little bit guilty”, just as a single coin ﬂip is “a little bit heads and a little bit tails": we are merely uncertain as to which it is. Conﬂating probability and uncertainty may be acceptable when making scientiﬁc measurements of physical quantities, but it is an error, in the context of “common sense” reasoning and logic. Just as in courtroom reasoning, the goal of employing uncertain inference is to gather evidence to strengthen the conﬁdence of a proposition, as opposed to performing some sort of probabilistic entailment. Historically, attempts to quantify probabilistic reasoning date back to antiquity. There was a particularly strong interest starting in the 12th century, with the work of the Scholastics, with the invention of the half-proof (so that two half-proofs are suﬃcient to prove guilt), the elucidation of moral certainty (suﬃcient certainty to act upon, but short of absolute certainty), the development of Catholic probabilism (the idea that it is always safe to follow the established rules of doctrine or the opinion of experts, even when they are less probable), the case-based reasoning of casuistry, and the scandal of Laxism (whereby probabilism was used to give support to almost any statement at all, it being possible to ﬁnd an expert opinion in support of almost any proposition.).[1] 10.2 Modern proposals Below is a list of proposals for probabilistic and evidentiary extensions to classical and predicate logic. 81
- 82 CHAPTER 10. PROBABILISTIC LOGIC � The term "probabilistic logic" was ﬁrst used in a paper by Nils Nilsson published in 1986, where the truth values of sentences are probabilities.[2] The proposed semantical generalization induces a probabilistic logical entailment, which reduces to ordinary logical entailment when the probabilities of all sentences are either 0 or 1. This generalization applies to any logical system for which the consistency of a ﬁnite set of sentences can be established. � The central concept in the theory of subjective logic[3] are opinions about some of the propositional variables involved in the given logical sentences. A binomial opinion applies to a single proposition and is represented as a 3-dimensional extension of a single probability value to express various degrees of ignorance about the truth of the proposition. For the computation of derived opinions based on a structure of argument opinions, the theory proposes respective operators for various logical connectives, such as e.g. multiplication (AND), comultiplication (OR), division (UN-AND) and co-division (UN-OR) of opinions [4] as well as conditional deduction (MP) and abduction (MT).[5] � Approximate reasoning formalism proposed by fuzzy logic can be used to obtain a logic in which the models are the probability distributions and the theories are the lower envelopes.[6] In such a logic the question of the consistency of the available information is strictly related with the one of the coherence of partial probabilistic assignment and therefore with Dutch book phenomenon. � Markov logic networks implement a form of uncertain inference based on the maximum entropy principle— the idea that probabilities should be assigned in such a way as to maximize entropy, in analogy with the way that Markov chains assign probabilities to ﬁnite state machine transitions. � Systems such as Pei Wang's Non-Axiomatic Reasoning System (NARS) or Ben Goertzel's Probabilistic Logic Networks (PLN) add an explicit conﬁdence ranking, as well as a probability to atoms and sentences. The rules of deduction and induction incorporate this uncertainty, thus side-stepping diﬃculties in purely Bayesian approaches to logic (including Markov logic), while also avoiding the paradoxes of Dempster-Shafer theory. The implementation of PLN attempts to use and generalize algorithms from logic programming, subject to these extensions. � In the theory of probabilistic argumentation,[7][8] probabilities are not directly attached to logical sentences. Instead it is assumed that a particular subsetW of the variables V involved in the sentences deﬁnes a probability space over the corresponding sub-σ-algebra. This induces two distinct probability measures with respect to V , which are called degree of support and degree of possibility, respectively. Degrees of support can be regarded as non-additive probabilities of provability, which generalizes the concepts of ordinary logical entailment (for V = fg ) and classical posterior probabilities (for V = W ). Mathematically, this view is compatible with the Dempster-Shafer theory. � The theory of evidential reasoning[9] also deﬁnes non-additive probabilities of probability (or epistemic prob- abilities) as a general notion for both logical entailment (provability) and probability. The idea is to augment standard propositional logic by considering an epistemic operator K that represents the state of knowledge that a rational agent has about the world. Probabilities are then deﬁned over the resulting epistemic universe Kp of all propositional sentences p, and it is argued that this is the best information available to an analyst. From this view, Dempster-Shafer theory appears to be a generalized form of probabilistic reasoning. 10.3 Possible application areas � Argumentation theory � Artiﬁcial intelligence � Artiﬁcial general intelligence � Bioinformatics � Formal epistemology � Game theory
- 10.4. SEE ALSO 83 � Philosophy of science � Psychology � Statistics 10.4 See also � Statistical relational learning � Bayesian inference, Bayesian networks, Bayesian probability � Cox’s theorem � Dempster-Shafer theory � Fréchet inequalities � Fuzzy logic � Imprecise probability � Logic, Deductive logic, Non-monotonic logic � Possibility theory � Probabilism, Half-proof, Scholasticism � Probabilistic database � Probability, Probability theory � Probabilistic argumentation � Reasoning � Subjective logic � Uncertainty � Uncertain inference � Upper and lower probabilities 10.5 References [1] James Franklin, The Science of Conjecture: Evidence and Probability before Pascal, 2001 The Johns Hopkins Press, ISBN 0-8018-7109-3 [2] Nilsson, N. J., 1986, “Probabilistic logic,” Artiﬁcial Intelligence 28(1): 71-87. [3] Jøsang, A., 2001, “A logic for uncertain probabilities,” International Journal of Uncertainty, Fuzziness and Knowledge- Based Systems 9(3):279-311. [4] Jøsang, A. and McAnally, D., 2004, “Multiplication and Comultiplication of Beliefs,” International Journal of Approximate Reasoning, 38(1), pp.19-51, 2004 [5] Jøsang, A., 2008, “Conditional Reasoning with Subjective Logic,” Journal of Multiple-Valued Logic and Soft Computing, 15(1), pp.5-38, 2008. [6] Gerla, G., 1994, “Inferences in Probability Logic,” Artiﬁcial Intelligence 70(1–2):33–52. [7] Kohlas, J., and Monney, P.A., 1995. A Mathematical Theory of Hints. An Approach to the Dempster-Shafer Theory of Evidence. Vol. 425 in Lecture Notes in Economics and Mathematical Systems. Springer Verlag. [8] Haenni, R, 2005, “Towards a Unifying Theory of Logical and Probabilistic Reasoning,” ISIPTA'05, 4th International Symposium on Imprecise Probabilities and Their Applications: 193-202. [9] Ruspini, E.H., Lowrance, J., and Strat, T., 1992, “Understanding evidential reasoning,” International Journal of Approxi- mate Reasoning, 6(3): 401-424.
- 84 CHAPTER 10. PROBABILISTIC LOGIC 10.6 Further reading � Adams, E. W., 1998. A Primer of Probability Logic. CSLI Publications (Univ. of Chicago Press). � Bacchus, F., 1990. “Representing and reasoning with Probabilistic Knowledge. A Logical Approach to Prob- abilities”. The MIT Press. � Carnap, R., 1950. Logical Foundations of Probability. University of Chicago Press. � Chuaqui, R., 1991. Truth, Possibility and Probability: New Logical Foundations of Probability and Statistical Inference. Number 166 in Mathematics Studies. North-Holland. � Haenni, H., Romeyn, JW, Wheeler, G., and Williamson, J. 2011. Probabilistic Logics and Probabilistic Net- works, Springer. � Hájek, A., 2001, “Probability, Logic, and Probability Logic,” in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic, Blackwell. � Jaynes, E., ~1998, “Probability Theory: The Logic of Science”, pdf and Cambridge University Press 2003. � Kyburg, H. E., 1970. Probability and Inductive Logic Macmillan. � Kyburg, H. E., 1974. The Logical Foundations of Statistical Inference, Dordrecht: Reidel. � Kyburg, H. E. & C. M. Teng, 2001. Uncertain Inference, Cambridge: Cambridge University Press. � Romeiyn, J. W., 2005. Bayesian Inductive Logic. PhD thesis, Faculty of Philosophy, University of Groningen, Netherlands. � Williamson, J., 2002, “Probability Logic,” in D. Gabbay, R. Johnson, H. J. Ohlbach, and J. Woods, eds., Handbook of the Logic of Argument and Inference: the Turn Toward the Practical. Elsevier: 397-424. 10.7 External links � Progicnet: Probabilistic Logic And Probabilistic Networks � Subjective logic demonstrations � The Society for Imprecise Probability
- Chapter 11 Problem of future contingents Aristotle: if a sea-battle will not be fought tomorrow, then it was also true yesterday that it will not be fought. But all past truths are necessary truths. Therefore it is not possible that the battle will be fought Future contingent propositions (or simply, future contingents) are statements about states of aﬀairs in the future that are neither necessarily true nor necessarily false. The problem of future contingents seems to have been ﬁrst discussed by Aristotle in chapter 9 of his On Inter- pretation (De Interpretatione), using the famous sea-battle example.[1] Roughly a generation later, Diodorus Cronus from the Megarian school of philosophy stated a version of the problem in his notorious Master Argument.[2] The problem was later discussed by Leibniz. The problem can be expressed as follows. Suppose that a sea-battle will not be fought tomorrow. Then it was also true yesterday (and the week before, and last year) that it will not be fought, since any true statement about what will be the case was also true in the past. But all past truths are now necessary truths; therefore it is now necessarily true that the battle will not be fought, and thus the statement that it will be fought is necessarily false. Therefore it is not possible that the battle will be fought. In general, if something will not be the case, it is not possible for it to be the case. “For a man may predict an event ten thousand years beforehand, and another may predict the reverse; that which was truly predicted at the moment in the past will of necessity take place in the fullness of time” (18 b35). This conﬂicts with the idea that of our own free choice: that we have the power to determine or control the course of 85
- 86 CHAPTER 11. PROBLEM OF FUTURE CONTINGENTS events in the future, which seems impossible if what happens, or does not happen, is necessarily going to happen, or not happen. As Aristotle says, if so there would be no need “to deliberate or to take trouble, on the supposition that if we should adopt a certain course, a certain result would follow, while, if we did not, the result would not follow”. 11.1 Aristotle’s solution Aristotle solved the problem by asserting that the principle of bivalence found its exception in this paradox of the sea battles: in this speciﬁc case, what is impossible is that both alternatives can be possible at the same time: either there will be a battle, or there won't. Both options can't be simultaneously taken. Today, they are neither true nor false; but if one is true, then the other becomes false. According to Aristotle, it is impossible to say today if the proposition is correct: we must wait for the contingent realization (or not) of the battle, logic realizes itself afterwards: One of the two propositions in such instances must be true and the other false, but we cannot say determi- nately that this or that is false, but must leave the alternative undecided. One may indeed be more likely to be true than the other, but it cannot be either actually true or actually false. It is therefore plain that it is not necessary that of an aﬃrmation and a denial, one should be true and the other false. For in the case of that which exists potentially, but not actually, the rule which applies to that which exists actually does not hold good. (§9) For Diodorus, the future battle was either impossible or necessary. Aristotle added a third term, contingency, which saves logic while in the same time leaving place for indetermination in reality. What is necessary is not that there will or that there won't be a battle tomorrow, but the dichotomy itself is necessary: A sea-ﬁght must either take place tomorrow or not, but it is not necessary that it should take place tomorrow, neither is it necessary that it should not take place, yet it is necessary that it either should or should not take place tomorrow. (De Interpretatione, 9, 19 a 30.) Thus, the event always comes in the form of the future, undetermined event; logic always comes afterwards. Hegel would say the same thing by claiming that wisdom came at dusk. For Aristotle, this is also a practical, ethical question: to pretend that the future is determined would have unacceptable consequences for man. 11.2 Leibniz Leibniz gave another response to the paradox in §6 of Discourse on Metaphysics: “That God does nothing which is not orderly, and that it is not even possible to conceive of events which are not regular.” Thus, even a miracle, the Event by excellence, does not break the regular order of things. What is seen as irregular is only a default of perspective, but does not appear so in relation to universal order. Possibility exceeds human logics. Leibniz encounters this paradox because according to him: Thus the quality of king, which belonged to Alexander the Great, an abstraction from the subject, is not suﬃciently determined to constitute an individual, and does not contain the other qualities of the same subject, nor everything which the idea of this prince includes. God, however, seeing the individual concept, or haecceity, of Alexander, sees there at the same time the basis and the reason of all the predicates which can be truly uttered regarding him; for instance that he will conquer Darius and Porus, even to the point of knowing a priori (and not by experience) whether he died a natural death or by poison,- facts which we can learn only through history. When we carefully consider the connection of things we see also the possibility of saying that there was always in the soul of Alexander marks of all that had happened to him and evidences of all that would happen to him and traces even of everything which occurs in the universe, although God alone could recognize them all. (§8) If everything which happens to Alexander derives from the haecceity of Alexander, then fatalism threatens Leibniz’s construction: We have said that the concept of an individual substance includes once for all everything which can ever happen to it and that in considering this concept one will be able to see everything which can truly be said
- 11.3. 20TH CENTURY 87 concerning the individual, just as we are able to see in the nature of a circle all the properties which can be derived from it. But does it not seem that in this way the diﬀerence between contingent and necessary truths will be destroyed, that there will be no place for human liberty, and that an absolute fatality will rule as well over all our actions as over all the rest of the events of the world? To this I reply that a distinction must be made between that which is certain and that which is necessary. (§13) Against Aristotle’s separation between the subject and the predicate, Leibniz states: “Thus the content of the subject must always include that of the predicate in such a way that if one understands perfectly the concept of the subject, he will know that the predicate appertains to it also.” (§8) The predicate (what happens to Alexander) must be completely included in the subject (Alexander) “if one un- derstands perfectly the concept of the subject”. Leibniz henceforth distinguishes two types of necessity: necessary necessity and contingent necessity, or universal necessity vs singular necessity. Universal necessity concerns universal truths, while singular necessity concerns something necessary which could not be (it is thus a “contingent necessity”). Leibniz hereby uses the concept of compossible worlds. According to Leibniz, contingent acts such as “Caesar cross- ing the Rubicon” or “Adam eating the apple” are necessary: that is, they are singular necessities, contingents and accidentals, but which concerns the principle of suﬃcient reason. Furthermore, this leads Leibniz to conceive of the subject not as a universal, but as a singular: it is true that “Caesar crosses the Rubicon”, but it is true only of this Caesar at this time, not of any dictator nor of Caesar at any time (§8, 9, 13). Thus Leibniz conceives of substance as plural: there is a plurality of singular substances, which he calls monads. Leibniz hence creates a concept of the individual as such, and attributes to it events. There is a universal necessity, which is universally applicable, and a singular necessity, which applies to each singular substance, or event. There is one proper noun for each singular event: Leibniz creates a logic of singularity, which Aristotle thought impossible (he considered that there could only be knowledge of generality). 11.3 20th century One of the early motivations for the study of many-valued logics has been precisely this issue. In the early 20th century, the Polish formal logician Jan Łukasiewicz proposed three truth-values: the true, the false and the as-yet- undetermined. This approach was later developed by Arend Heyting and L. E. J. Brouwer;[3] see Łukasiewicz logic. Issues such as this have also been addressed in various temporal logics, where one can assert that "Eventually, either there will be a sea battle tomorrow, or there won't be.” (Which is true if “tomorrow” eventually occurs.) The Modal Fallacy The error in the argument underlying the alleged “Problem of Future Contingents” lies in the assumption that “X is the case” implies that “necessarily, X is the case”. In logic, this is known as the Modal Fallacy.[4] By asserting “A sea-ﬁght must either take place tomorrow or not, but it is not necessary that it should take place tomorrow, neither is it necessary that it should not take place, yet it is necessary that it either should or should not take place tomorrow.” Aristotle is simply claiming “necessarily (a or not-a)”, which is correct. However, the next step in Aristotle’s reasoning seems to be: “If a is the case, then necessarily, a is the case”, which is a logical fallacy. Expressed in another way: (i) If a proposition is true, then it cannot be false. (ii) If a proposition cannot be false, then it is necessarily true. (iii) Therefore if a proposition is true, it is necessarily true. That is, there are no contingent propositions. Every proposition is either necessarily true or necessarily false. The fallacy arises in the ambiguity of the ﬁrst premise. If we interpret it close to the English, we get: (iv) P entails it is not possible that not-P (v) It is not possible that not-P entails it is necessary that P (vi) Therefore, P entails it is necessary that P However, if we recognize that the original English expression (i) is potentially misleading, that it assigns a necessity to what is simply nothing more than a necessary condition, then we get instead as our premises: (vii) It is not possible that (P and not P) (viii) (It is not possible that not P) entails (it is necessary that P) From these latter two premises, one cannot validly infer the conclusion:
- 88 CHAPTER 11. PROBLEM OF FUTURE CONTINGENTS (ix) P entails it is necessary that P 11.4 See also � Logical determinism � Free will � Principle of distributivity � Principle of plenitude � Truth-value link � In Borges' The Garden of Forking Paths, both alternatives happen, thus leading to what Deleuze calls “incom- possible worlds” 11.5 Notes [1] Dorothea Frede, The sea-battle reconsidered, Oxford Studies in Ancient Philosophy 1985, pp. 31-87. [2] Dialectical School entry by Susanne Bobzien in the Stanford Encyclopedia of Philosophy [3] Paul Tomassi (1999). Logic. Routledge. p. 124. ISBN 978-0-415-16696-6. [4] Norman Swartz, The Modal Fallacy 11.6 Further reading � Dorothea Frede (1985), The Sea-Battle Reconsidered, Oxford Studies in Ancient Philosophy 3, 31-87. � Peter Øhrstrøm; Per F. V. Hasle (1995). Temporal logic: from ancient ideas to artiﬁcial intelligence. Springer. ISBN 978-0-7923-3586-3. � Richard Gaskin (1995). The sea battle and the master argument: Aristotle and Diodorus Cronus on the meta- physics of the future. Walter de Gruyter. ISBN 978-3-11-014430-7. � Melvin Fitting; Richard L. Mendelsohn (1998). First-order modal logic. Springer. pp. 35–40. ISBN 978-0- 7923-5335-5. attempts to reconstruct both Aristotle’s and Diodorus’ arguments in propositional modal logic � John MacFarlane (2003), Sea Battles, Futures Contingents, and Relative Truth and Future Contingent and Relative Truth, The Philosophical Quarterly 53, 321-36 � Jules Vuillemin, Le chapitre IX du De Interpretatione d'Aristote - Vers une réhabilitation de l'opinion comme connaissance probable des choses contingentes, in Philosophiques, vol. X, n°1, April 1983 (French) 11.7 External links � Future Contingents entry by Peter Øhrstrøm and Per Hasle in the Stanford Encyclopedia of Philosophy � Medieval Theories of Future Contingents entry by Simo Knuuttila in the Stanford Encyclopedia of Philosophy � The Master Argument: The Sea Battle in De Intepretatione 9, Diodorus Cronus, Philo the Dialectician with a bibliography on Diodorus and the problem of future contingents
- Chapter 12 Stephen Cole Kleene Stephen Cole Kleene /ˈkliːniː/ KLEE-nee (January 5, 1909 – January 25, 1994) was an American mathematician. One of the students of Alonzo Church, Kleene, along with Alan Turing, Emil Post, and others, is best known as a founder of the branch of mathematical logic known as recursion theory, which subsequently helped to provide the foundations of theoretical computer science. Kleene’s work grounds the study of which functions are computable. A number of mathematical concepts are named after him: Kleene hierarchy, Kleene algebra, the Kleene star (Kleene closure), Kleene’s recursion theorem and the Kleene ﬁxpoint theorem. He also invented regular expressions, and made signiﬁcant contributions to the foundations of mathematical intuitionism. Although his last name is commonly pronounced /ˈkliːniː/ KLEE-nee or /ˈkliːn/ kleen, Kleene himself pronounced it /ˈkleɪniː/ KLAY-nee.[1] His son, Ken Kleene, wrote: “As far as I am aware this pronunciation is incorrect in all known languages. I believe that this novel pronunciation was invented by my father.”[2] 12.1 Biography Kleene was awarded the BA degree from Amherst College in 1930. He was awarded the Ph.D. in mathematics from Princeton University in 1934. His thesis, entitled A Theory of Positive Integers in Formal Logic, was supervised by Alonzo Church. In the 1930s, he did important work on Church’s lambda calculus. In 1935, he joined the mathematics department at the University of Wisconsin–Madison, where he spent nearly all of his career. After two years as an instructor, he was appointed assistant professor in 1937. While a visiting scholar at the Institute for Advanced Study in Princeton, 1939–40, he laid the foundation for recursion theory, an area that would be his lifelong research interest. In 1941, he returned to Amherst College, where he spent one year as an associate professor of mathematics. During World War II, Kleene was a lieutenant commander in the United States Navy. He was an instructor of navigation at the U.S. Naval Reserve’s Midshipmen’s School in New York, and then a project director at the Naval Research Laboratory in Washington, D.C. In 1946, Kleene returned to Wisconsin, becoming a full professor in 1948 and the Cyrus C. MacDuﬀee professor of mathematics in 1964. He was chair of the Department of Mathematics and Computer Science, 1962–63, and Dean of the College of Letters and Science from 1969 to 1974. The latter appointment he took on despite the considerable student unrest of the day, stemming from the Vietnam War. He retired from the University of Wisconsin in 1979. The mathematics library at the University of Wisconsin was renamed in his honour.[3] Kleene’s teaching at Wisconsin resulted in three texts in mathematical logic, Kleene (1952, 1967) and Kleene and Vesley (1965), often cited and still in print. Kleene (1952) wrote alternative proofs to the Gödel’s incompleteness theorems that enhanced their canonical status and made them easier to teach and understand. Kleene and Vesley (1965) is the classic American introduction to intuitionist logic and mathematics. Kleene served as president of the Association for Symbolic Logic, 1956–58, and of the International Union of History and Philosophy of Science,[4] 1961. In 1990, he was awarded the National Medal of Science. Kleene and his wife Nancy Elliott had four children. He had a lifelong devotion to the family farm in Maine. An avid mountain climber, he had a strong interest in nature and the environment, and was active in many conservation causes. 89
- 90 CHAPTER 12. STEPHEN COLE KLEENE 12.2 Important publications � 1952. Introduction to Metamathematics. New York: Van Nostrand. (Ishi Press: 2009 reprint).[5] � 1956. “Representation of Events in Nerve Nets and Finite Automata” in Automata Studies. Claude Shannon and John McCarthy, eds. � 1965 (with Richard Eugene Vesley). The Foundations of Intuitionistic Mathematics. North-Holland.[6] � 1967. Mathematical Logic. John Wiley. Dover reprint, 2001. ISBN 0-486-42533-9. � 1981. “Origins of Recursive Function Theory” in Annals of the History of Computing 3, No. 1. 12.3 See also � Kleene star � Kleene hierarchy � Kleene’s smn theorem � Realizability � Intuitionism � Kleene–Rosser paradox � Kleene’s algorithm � Kleene’s theorem 12.4 References � This article is based on material taken from the Free On-line Dictionary of Computing prior to 1 November 2008 and incorporated under the “relicensing” terms of the GFDL, version 1.3 or later. [1] Pace, Eric (January 27, 1994). “Stephen C. Kleene Is Dead at 85; Was Leader in Computer Science”. The New York Times. [2] In Entry “Stephen Kleene” at Free Online Dictionary of Computing. [3] “About the Kleene Mathematics Library”. UW - Madison Libraries. Retrieved 17 July 2012. [4] IUHPS website; also known as “International Union of the History and the Philosophy of Science”. A member of ICSU, the International Council for Science (formerly named International Council of Scientiﬁc Unions). [5] “WorldCat: editions for 'Introduction to metamathematics.'". Retrieved April 2, 2013. [6] Bishop, Errett (1965). “Review: The foundations of intuitionistic mathematics, by Stephen Cole Kleene and Richard Eugene Vesley”. Bull. Amer. Math. Soc. 71 (6): 850–852. doi:10.1090/s0002-9904-1965-11412-4. 12.5 External links � O'Connor, John J.; Robertson, Edmund F., “Stephen Cole Kleene”, MacTutor History of Mathematics archive, University of St Andrews. � Stephen Cole Kleene at the Mathematics Genealogy Project � Biographical memoir – by Saunders Mac Lane � Kleene bibliography. � Interview with Kleene and John Barkley Rosser about their experiences at Princeton
- Chapter 13 Term logic In philosophy, term logic, also known as traditional logic orAristotelian logic, is a loose name for the way of doing logic that began with Aristotle and that was dominant until the advent of modern predicate logic in the late nineteenth century. This entry is an introduction to the term logic needed to understand philosophy texts written before predicate logic came to be seen as the only formal logic of interest. Readers lacking a grasp of the basic terminology and ideas of term logic can have diﬃculty understanding such texts, because their authors typically assumed an acquaintance with term logic. 13.1 Aristotle’s system Aristotle’s logical work is collected in the six texts that are collectively known as the Organon. Two of these texts in particular, namely the Prior Analytics and De Interpretatione, contain the heart of Aristotle’s treatment of judgements and formal inference, and it is principally this part of Aristotle’s works that is about term logic. Modern work on Aristotle’s logic builds on the tradition started in 1951 with the establishment by Jan Lukasiewicz of a revolutionary paradigm.[1] The Jan Lukasiewicz approach was reinvigorated in the early 1970s by John Corcoran and Timothy Smiley — which informs modern translations of Prior Analytics by Robin Smith in 1989 and Gisela Striker in 2009.[2] 13.2 Basics The fundamental assumption behind the theory is that propositions are composed of two terms – hence the name “two-term theory” or “term logic” – and that the reasoning process is in turn built from propositions: � The term is a part of speech representing something, but which is not true or false in its own right, such as “man” or “mortal”. � The proposition consists of two terms, in which one term (the “predicate”) is “aﬃrmed” or “denied” of the other (the “subject”), and which is capable of truth or falsity. � The syllogism is an inference in which one proposition (the “conclusion”) follows of necessity from two others (the “premises”). A proposition may be universal or particular, and it may be aﬃrmative or negative. Traditionally, the four kinds of propositions are: � A-type: Universal and aﬃrmative (“Every philosopher is mortal”) � I-type: Particular and aﬃrmative (“Some philosopher is mortal”) � E-type: Universal and negative (“Every philosopher is immortal”) � O-type: Particular and negative (“Some philosopher is immortal”) 91
- 92 CHAPTER 13. TERM LOGIC This was called the fourfold scheme of propositions (see types of syllogism for an explanation of the letters A, I, E, and O in the traditional square). Aristotle’s original square of opposition, however, does not lack existential import: � A-type: Universal and aﬃrmative (“Every philosopher is mortal”) � I-type: Particular and aﬃrmative (“Some philosopher is mortal”) � E-type: Universal and negative (“No philosopher is mortal”) � O-type: Particular and negative (“Not every philosopher is mortal”) In the Stanford Encyclopedia of Philosophy article, “The Traditional Square of Opposition”, Terence Parsons explains: One central concern of the Aristotelian tradition in logic is the theory of the categorical syllogism. This is the theory of two-premised arguments in which the premises and conclusion share three terms among them, with each proposition containing two of them. It is distinctive of this enterprise that everybody agrees on which syllogisms are valid. The theory of the syllogism partly constrains the inter- pretation of the forms. For example, it determines that the A form has existential import, at least if the I form does. For one of the valid patterns (Darapti) is: Every C is B Every C is A So, some A is B This is invalid if the A form lacks existential import, and valid if it has existential import. It is held to be valid, and so we know how the A form is to be interpreted. One then naturally asks about the O form; what do the syllogisms tell us about it? The answer is that they tell us nothing. This is because Aristotle did not discuss weakened forms of syllogisms, in which one concludes a particular proposition when one could already conclude the corresponding universal. For example, he does not mention the form: No C is B Every A is C So, some A is not B If people had thoughtfully taken sides for or against the validity of this form, that would clearly be relevant to the understanding of the O form. But the weakened forms were typically ignored... One other piece of subject-matter bears on the interpretation of the O form. People were interested in Aristotle’s discussion of “inﬁnite” negation, which is the use of negation to form a term from a term instead of a proposition from a proposition. In modern English we use “non” for this; we make “non- horse,” which is true of exactly those things that are not horses. In medieval Latin “non” and “not” are the same word, and so the distinction required special discussion. It became common to use inﬁnite negation, and logicians pondered its logic. Some writers in the twelfth and thirteenth centuries adopted a principle called “conversion by contraposition.” It states that � 'Every S is P ' is equivalent to 'Every non-P is non-S ' � 'Some S is not P ' is equivalent to 'Some non-P is not non-S ' Unfortunately, this principle (which is not endorsed by Aristotle) conﬂicts with the idea that there may be empty or universal terms. For in the universal case it leads directly from the truth: Every man is a being to the falsehood: Every non-being is a non-man (which is false because the universal aﬃrmative has existential import, and there are no non-beings). And in the particular case it leads from the truth (remember that the O form has no existential import): A chimera is not a man to the falsehood: A non-man is not a non-chimera
- 13.3. TERM 93 These are [Jean] Buridan’s examples, used in the fourteenth century to show the invalidity of contraposi- tion. Unfortunately, by Buridan’s time the principle of contraposition had been advocated by a number of authors. The doctrine is already present in several twelfth century tracts, and it is endorsed in the thirteenth century by Peter of Spain, whose work was republished for centuries, by William Sherwood, and by Roger Bacon. By the fourteenth century, problems associated with contraposition seem to be well-known, and authors generally cite the principle and note that it is not valid, but that it becomes valid with an additional assumption of existence of things falling under the subject term. For example, Paul of Venice in his eclectic and widely published Logica Parva from the end of the fourteenth century gives the traditional square with simple conversion but rejects conversion by contraposition, essentially for Buridan’s reason.[3] —Terence Parsons, The Stanford Encyclopedia of Philosophy 13.3 Term A term (Greek horos) is the basic component of the proposition. The original meaning of the horos (and also of the Latin terminus) is “extreme” or “boundary”. The two terms lie on the outside of the proposition, joined by the act of aﬃrmation or denial. For early modern logicians like Arnauld (whose Port-Royal Logic was the best-known text of his day), it is a psychological entity like an “idea” or "concept". Mill considers it a word. To assert “all Greeks are men” is not to say that the concept of Greeks is the concept of men, or that word “Greeks” is the word “men”. A proposition cannot be built from real things or ideas, but it is not just meaningless words either. 13.4 Proposition In term logic, a “proposition” is simply a form of language: a particular kind of sentence, in which the subject and predicate are combined, so as to assert something true or false. It is not a thought, or an abstract entity. The word “propositio” is from the Latin, meaning the ﬁrst premise of a syllogism. Aristotle uses the word premise (protasis) as a sentence aﬃrming or denying one thing of another (Posterior Analytics 1. 1 24a 16), so a premise is also a form of words. However, as in modern philosophical logic, it means that which is asserted by the sentence. Writers before Frege and Russell, such as Bradley, sometimes spoke of the “judgment” as something distinct from a sentence, but this is not quite the same. As a further confusion the word “sentence” derives from the Latin, meaning an opinion or judgment, and so is equivalent to “proposition”. The logical quality of a proposition is whether it is aﬃrmative (the predicate is aﬃrmed of the subject) or negative (the predicate is denied of the subject). Thus every philosopher is mortal is aﬃrmative, since the mortality of philosophers is aﬃrmed universally, whereas no philosopher is mortal is negative by denying such mortality in particular. The quantity of a proposition is whether it is universal (the predicate is aﬃrmed or denied of all subjects or of “the whole”) or particular (the predicate is aﬃrmed or denied of some subject or a “part” thereof). In case where existential import is assumed, quantiﬁcation implies the existence of at least one subject, unless disclaimed. 13.5 Singular terms For Aristotle, the distinction between singular and universal is a fundamental metaphysical one, and not merely grammatical. A singular term for Aristotle is primary substance, which can only be predicated of itself: (this) “Callias” or (this) “Socrates” are not predicable of any other thing, thus one does not say every Socrates one says every human (De Int. 7; Meta. Δ9, 1018a4). It may feature as a grammatical predicate, as in the sentence “the person coming this way is Callias”. But it is still a logical subject. He contrasts “universal” (katholou, “whole”) secondary substance, genera, with primary substance, particular spec- imens. The formal nature of universals, in so far as they can be generalized “always, or for the most part”, are the subject matter of both scientiﬁc study and formal logic.[4] The essential feature of the syllogistic is that, of the four terms in the two premises, one must occur twice. Thus All Greeks are men
- 94 CHAPTER 13. TERM LOGIC All men are mortal. The subject of one premise, must be the predicate of the other, and so it is necessary to eliminate from the logic any terms which cannot function both as subject and predicate, namely singular terms. However, in a popular 17th century version of the syllogistic, Port-Royal Logic, singular terms were treated as universals:[5] All men are mortals All Socrates are men All Socrates are mortals This is clearly awkward, a weakness exploited by Frege in his devastating attack on the system (from which, ultimately, it never recovered, see concept and object). The famous syllogism “Socrates is a man ...”, is frequently quoted as though from Aristotle,[6] but fact, it is nowhere in the Organon. It is ﬁrst mentioned by Sextus Empiricus in his Hyp. Pyrrh. ii. 164. 13.6 Inﬂuence on philosophy 13.7 Decline of term logic Term logic began to decline in Europe during the Renaissance, when logicians like Rodolphus Agricola Phrisius (1444–1485) and Ramus (1515-1572) began to promote place logics. The logical tradition called Port-Royal Logic, or sometimes “traditional logic”, saw propositions as combinations of ideas rather than of terms, but otherwise followed many of the conventions of term logic. It remained inﬂuential, especially in England, until the 19th century. Leibniz created a distinctive logical calculus, but nearly all of his work on logic remained unpublished and unremarked until Louis Couturat went through the Leibniz Nachlass around 1900, publishing his pioneering studies in logic. 19th-century attempts to algebraize logic, such as the work of Boole (1815–1864) and Venn (1834–1923), typically yielded systems highly inﬂuenced by the term-logic tradition. The ﬁrst predicate logic was that of Frege's landmark Begriﬀsschrift (1879), little read before 1950, in part because of its eccentric notation. Modern predicate logic as we know it began in the 1880s with the writings of Charles Sanders Peirce, who inﬂuenced Peano (1858–1932) and even more, Ernst Schröder (1841–1902). It reached fruition in the hands of Bertrand Russell and A. N. Whitehead, whose Principia Mathematica (1910–13) made use of a variant of Peano’s predicate logic. Term logic also survived to some extent in traditional Roman Catholic education, especially in seminaries. Medieval Catholic theology, especially the writings of Thomas Aquinas, had a powerfully Aristotelean cast, and thus term logic became a part of Catholic theological reasoning. For example, Joyce’s Principles of Logic (1908; 3rd edition 1949), written for use in Catholic seminaries, made no mention of Frege or of Bertrand Russell.[7] 13.8 Revival Some philosophers have complained that predicate logic: � Is unnatural in a sense, in that its syntax does not follow the syntax of the sentences that ﬁgure in our every- day reasoning. It is, as Quine acknowledged, “Procrustean,” employing an artiﬁcial language of function and argument, quantiﬁer, and bound variable. � Suﬀers from theoretical problems, probably the most serious being empty names and identity statements. Even academic philosophers entirely in the mainstream, such as Gareth Evans, have written as follows: “I come to semantic investigations with a preference for homophonic theories; theories which try to take serious account of the syntactic and semantic devices which actually exist in the language ...I would
- 13.9. SEE ALSO 95 prefer [such] a theory ... over a theory which is only able to deal with [sentences of the form “all A’s are B’s"] by “discovering” hidden logical constants ... The objection would not be that such [Fregean] truth conditions are not correct, but that, in a sense which we would all dearly love to have more exactly explained, the syntactic shape of the sentence is treated as so much misleading surface structure” (Evans 1977) 13.9 See also 13.10 Notes [1] Degnan, M. 1994. Recent Work in Aristotle’s Logic. Philosophical Books 35.2 (April, 1994): 81-89. [2] � Review of “Aristotle, Prior Analytics: Book I, Gisela Striker (translation and commentary), Oxford UP, 2009, 268pp., $39.95 (pbk), ISBN 978-0-19-925041-7.” in the Notre Dame Philosophical Reviews, 2010.02.02. [3] Parsons, Terence (2012). “The Traditional Square of Opposition”. In Edward N. Zalta. The Stanford Encyclopedia of Philosophy (Fall 2012 ed.). 3-4. [4] They are mentioned brieﬂy in the De Interpretatione. Afterwards, in the chapters of the Prior Analytics where Aristotle methodically sets out his theory of the syllogism, they are entirely ignored. [5] Arnauld, Antoine and Nicole, Pierre; (1662) La logique, ou l'art de penser. Part 2, chapter 3 [6] For example: Kapp, Greek Foundations of Traditional Logic, New York 1942, p. 17, Copleston A History of Philosophy Vol. I., p. 277, Russell, A History of Western Philosophy London 1946 p. 218. [7] Copleston's A History of Philosophy 13.11 References � Bocheński, I. M., 1951. Ancient Formal Logic. North-Holland. � Louis Couturat, 1961 (1901). La Logique de Leibniz. Hildesheim: Georg Olms Verlagsbuchhandlung. � Gareth Evans, 1977, “Pronouns, Quantiﬁers and Relative Clauses,” Canadian Journal of Philosophy. � Peter Geach, 1976. Reason and Argument. University of California Press. � Hammond and Scullard, 1992. The Oxford Classical Dictionary. Oxford University Press, ISBN 0-19-869117- 3. � Joyce, George Hayward, 1949 (1908). Principles of Logic, 3rd ed. Longmans. A manual written for use in Catholic seminaries. Authoritative on traditional logic, with many references to medieval and ancient sources. Contains no hint of modern formal logic. The author lived 1864-1943. � Jan Łukasiewicz, 1951. Aristotle’s Syllogistic, from the Standpoint of Modern Formal Logic. Oxford Univ. Press. � John Stuart Mill, 1904. A System of Logic, 8th ed. London. � Parry and Hacker, 1991. Aristotelian Logic. State University of New York Press. � Arthur Prior 1962: Formal Logic, 2nd ed. Oxford Univ. Press. While primarily devoted to modern formal logic, contains much on term and medieval logic. 1976: The Doctrine of Propositions and Terms. Peter Geach and A. J. P. Kenny, eds. London: Duckworth. � Willard Quine, 1986. Philosophy of Logic 2nd ed. Harvard Univ. Press. � Rose, Lynn E., 1968. Aristotle’s Syllogistic. Springﬁeld: Clarence C. Thomas.
- 96 CHAPTER 13. TERM LOGIC � Sommers, Fred 1970: “The Calculus of Terms,” Mind 79: 1-39. Reprinted in Englebretsen, G., ed., 1987. The new syllogistic New York: Peter Lang. ISBN 0-8204-0448-9 1982: The logic of natural language. Oxford University Press. 1990: "Predication in the Logic of Terms," Notre Dame Journal of Formal Logic 31: 106-26. and Englebretsen, George, 2000: An invitation to formal reasoning. The logic of terms. Aldershot UK: Ashgate. ISBN 0-7546-1366-6. � Szabolcsi Lorne, 2008. Numerical Term Logic. Lewiston: Edwin Mellen Press. 13.12 External links � Term logic at PhilPapers � Aristotle’s Logic entry by Robin Smith in the Stanford Encyclopedia of Philosophy � Term logic entry in the Internet Encyclopedia of Philosophy � Aristotle’s term logic online—This online program provides a platform for experimentation and research on Aristotelian logic. � Annotated bibliographies: Fred Sommers. George Englebretsen. � PlanetMath: Aristotelian Logic. � Interactive Syllogistic Machine for Term Logic A web based syllogistic machine for exploring fallacies, ﬁgures, terms, and modes of syllogisms.
- Chapter 14 Three-valued logic In logic, a three-valued logic (also trinary logic, trivalent, ternary, or trilean, sometimes abbreviated 3VL) is any of several many-valued logic systems in which there are three truth values indicating true, false and some indeter- minate third value. This is contrasted with the more commonly known bivalent logics (such as classical sentential or Boolean logic) which provide only for true and false. Conceptual form and basic ideas were initially created by Jan Łukasiewicz and C. I. Lewis. These were then re-formulated by Grigore Moisil in an axiomatic algebraic form, and also extended to n-valued logics in 1945. 14.1 Representation of values As with bivalent logic, truth values in ternary logic may be represented numerically using various representations of the ternary numeral system. A few of the more common examples are: � in balanced ternary, each digit has one of 3 values: −1, 0, or +1; these values may also be simpliﬁed to −, 0, +, respectively.[1] � in the redundant binary representation, each digit can have a value of �1, 0, 0, or 1 (the value 0 has two diﬀerent representations) � in the ternary numeral system, each digit is a trit (trinary digit) having a value of: 0, 1, or 2 � in the skew binary number system, only most-signiﬁcant non-zero digit has a value 2, and the remaining digits have a value of 0 or 1 � 1 for true, 2 for false, and 0 for unknown, unknowable/undecidable, irrelevant, or both.[2] � 0 for false, 1 for true, and a third non-integer “maybe” symbol such as ?, #, ½,[3] or xy. Inside a ternary computer, ternary values are represented by ternary signals. This article mainly illustrates a system of ternary propositional logic using the truth values {false, unknown, and true}, and extends conventional Boolean connectives to a trivalent context. Ternary predicate logics exist as well; these may have readings of the quantiﬁer diﬀerent from classical (binary) predicate logic, and may include alternative quantiﬁers as well. 14.2 Logics Where Boolean Logic has 4 monadic operators, the addition of a third value in ternary logic leads to a total of 27 distinct operators on a single input value. Similarly, where Boolean logic has 16 distinct diadic operators (operators with 2 inputs), ternary logic has 19,683 such operators. Where we can easily name a signiﬁcant fraction of the Boolean operators (not, and, or, nand, nor, exclusive or), it is unreasonable to attempt to name all but a small fraction of the possible ternary operators.[4] 97
- 98 CHAPTER 14. THREE-VALUED LOGIC 14.2.1 Kleene and Priest logics See also: Kleene algebra (with involution) Below is a set of truth tables showing the logic operations for Kleene's “strong logic of indeterminacy” and Priest’s “logic of paradox”. In these truth tables, the unknown state can be thought of as neither true nor false in Kleene logic, or thought of as both true and false in Priest logic. The diﬀerence lies in the deﬁnition of tautologies. Where Kleene logic’s only designated truth value is T, Priest logic’s designated truth values are both T and U. In Kleene logic, the knowledge of whether any particular unknown state secretly represents true or false at any moment in time is not available. However, certain logical operations can yield an unambiguous result, even if they involve at least one unknown operand. For example, since true OR true equals true, and true OR false also equals true, one can infer that true OR unknown equals true, as well. In this example, since either bivalent state could be underlying the unknown state, but either state also yields the same result, a deﬁnitive true results in all three cases. If numeric values, e.g. balanced ternary values, are assigned to false, unknown and true such that false is less than unknown and unknown is less than true, then A AND B AND C... = MIN(A, B, C ...) and A OR B OR C ... = MAX(A, B, C...). Material implication for Kleene logic can be deﬁned as: A! B def= NOT(A) OR B , and its truth table is which diﬀers from that for Łukasiewicz logic (described below). Kleene logic has no tautologies (valid formulas) because whenever all of the atomic components of a well-formed formula are assigned the value Unknown, the formula itself must also have the value Unknown. (And the only designated truth value for Kleene logic is True.) However, the lack of valid formulas does not mean that it lacks valid arguments and/or inference rules. An argument is semantically valid in Kleene logic if, whenever (for any interpretation/model) all of its premises are True, the conclusion must also be True. (Note that the Logic of Paradox (LP) has the same truth tables as Kleene logic, but it has two designated truth values instead of one; these are: True and Both (the analogue of Unknown), so that LP does have tautologies but it has fewer valid inference rules.)[5] 14.2.2 Łukasiewicz logic Further information: Łukasiewicz logic The Łukasiewicz Ł3 has the same tables for AND, OR, and NOT as the Kleene logic given above, but diﬀers in its deﬁnition of implication. This section follows the presentation from Malinowski’s chapter of the Handbook of the History of Logic, vol 8.[6] In fact, using Łukasiewicz’s implication and negation, the other usual connectives may be derived as: � A ∨ B = (A → B) → B � A ∧ B = ¬(¬A ∨ ¬ B) � A ↔ B = (A → B) ∧ (B → A) It’s also possible to derive a few other useful unary operators (ﬁrst derived by Tarski in 1921): � MA = ¬A → A � LA = ¬M¬A � IA = MA ∧ ¬LA They have the following truth tables: M is read as “it is not false that...” or in the (unsuccessful) Tarski–Łukasiewicz attempt to axiomatize modal logic using a three-valued logic, “it is possible that...” L is read “it is true that...” or “it is necessary that...” Finally I is read “it is unknown that...” or “it is contingent that...”
- 14.3. APPLICATION IN SQL 99 In Łukasiewicz’s Ł3 the designated value is True, meaning that only a proposition having this value everywhere is considered a tautology. For example A → A and A ↔ A are tautologies in Ł3 and also in classical logic. Not all tautologies of classical logic lift to Ł3 “as is”. For example, the law of excluded middle, A ∨ ¬A, and the law of non-contradiction, ¬(A ∧ ¬A) are not tautologies in Ł3. However, using the operator I deﬁned above, it is possible to state tautologies that are their analogues: � A ∨ IA ∨ ¬A [law of excluded fourth] � ¬(A ∧ ¬IA ∧ ¬A) [extended contradiction principle]. 14.2.3 Bochvar logic Main article: Many-valued_logic § Bochvar.27s_internal_three-valued_logic_.28also_known_as_Kleene.27s_weak_three- valued_logic.29 14.2.4 ternary Post logic 14.2.5 Modular algebras Some 3VL modular algebras have been introduced more recently, motivated by circuit problems rather than philo- sophical issues:[7] � Cohn algebra � Pradhan algebra � Dubrova and Muzio algebra 14.3 Application in SQL Main article: Null (SQL) The database structural query language SQL implements ternary logic as a means of handling comparisons with NULL ﬁeld content. The original intent of NULL in SQL was to represent missing data in a database, i.e. the assumption that an actual value exists, but that the value is not currently recorded in the database. SQL uses a common fragment of the Kleene K3 logic, restricted to AND, OR, and NOT tables. In SQL, the intermediate value is intended to be interpreted as UNKNOWN. Explicit comparisons with NULL, including that of another NULL yields UNKNOWN. However this choice of semantics is abandoned for some set operations, e.g. UNION or INTERSECT, where NULLs are treated as equal with each other. Critics assert that this inconsistency deprives SQL of intuitive semantics in its treatment of NULLs.[8] The SQL standard deﬁnes an optional feature called F571, which adds some unary operators, among which IS UNKNOWN corresponding to the Łukasiewicz I in this article. The addition of IS UNKNOWN to the other operators of SQL’s three-valued logic makes the SQL three-valued logic functionally complete,[9] meaning its logical operators can express (in combination) any conceivable three-valued logical function. 14.4 See also � Aymara language – a Bolivian language famous for using ternary rather than binary logic[10] � Binary logic (disambiguation) � Boolean algebra (structure) � Boolean function
- 100 CHAPTER 14. THREE-VALUED LOGIC � Digital circuit � Four-valued logic � Setun - an experimental Russian computer which was based on ternary logic � Ternary numeral system (and Balanced ternary) � Three-state logic 14.5 References [1] Knuth, Donald E. (1981). The Art of Computer Programming Vol. 2. Reading, Mass.: Addison-Wesley Publishing Com- pany. p. 190. [2] Hayes, Brian (November–December 2001). “Third Base”. American Scientist (Sigma Xi, the Scientiﬁc Research Society) 89 (6): 490–494. doi:10.1511/2001.6.490. [3] The Penguin Dictionary of Mathematics. 2nd Edition. London, England: Penguin Books. 1998. p. 417. [4] Douglas W. Jones, Standard Ternary Logic, Feb. 11, 2013 [5] http://www.uky.edu/~{}look/Phi520-Lecture7.pdf [6] Grzegorz Malinowski, “Many-valued Logic and its Philosophy” in Dov M. Gabbay, John Woods (eds.) Handbook of the History of Logic Volume 8. The Many Valued and Nonmonotonic Turn in Logic, Elsevier, 2009 [7] Miller, D. Michael; Thornton, Mitchell A. (2008). Multiple valued logic: concepts and representations. Synthesis lectures on digital circuits and systems 12. Morgan & Claypool Publishers. pp. 41–42. ISBN 978-1-59829-190-2. [8] Ron van der Meyden, "Logical approaches to incomplete information: a survey" in Chomicki, Jan; Saake, Gunter (Eds.) Logics for Databases and Information Systems, Kluwer Academic Publishers ISBN 978-0-7923-8129-7, p. 344; PS preprint (note: page numbering diﬀers in preprint from the published version) [9] C. J. Date, Relational database writings, 1991-1994, Addison-Wesley, 1995, p. 371 [10] “El idioma de los aymaras” (in Spanish). Aymara Uta. Retrieved 2013-08-20. 14.6 Further reading � Bergmann, Merrie (2008). An Introduction to Many-Valued and Fuzzy Logic: Semantics, Algebras, and Deriva- tion Systems. Cambridge University Press. ISBN 978-0-521-88128-9. Retrieved 24 August 2013., chapters 5-9 � Mundici, D. The C*-Algebras of Three-Valued Logic. Logic Colloquium ’88, Proceedings of the Colloquium held in Padova 61–77 (1989). doi:10.1016/s0049-237x(08)70262-3 14.7 External links � Introduction to Many-Valued Logics by Bertram Fronhöfer. Handout from a Technische Universität Dresden 2011 summer class. (Despite the title, this is almost entirely about three-valued logics.)
- 14.8. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 101 14.8 Text and image sources, contributors, and licenses 14.8.1 Text � Alfred Tarski Source: https://en.wikipedia.org/wiki/Alfred_Tarski?oldid=670255173 Contributors: Kpjas, The Anome, Jan Hidders, Garyt, Michael Hardy, BoNoMoJo (old), Gabbe, Chinju, Eric119, Ahoerstemeier, Alexander, Snoyes, Jiang, Loren Rosen, Sociate, Charles Matthews, Selket, Markhurd, E23~enwiki, Hyacinth, Jose Ramos, Dimadick, Aleph4, Jaredwf, Fredrik, Vanden, MathMartin, Halibutt, Wikibot, JerryFriedman, Tobias Bergemann, Adam78, Tosha, Giftlite, Peruvianllama, Waltpohl, Carlo.Ierna, Fuzzy Logic, Gdr, Piotrus, Emax, Sam Hocevar, TonyW, Anirvan, Klemen Kocjancic, Lucidish, D6, Nathan Ladd, Naive cynic, Euthydemos, Paul August, Bender235, BACbKA, Rgdboer, Bill Thayer, Ruszewski, Lokifer, AshtonBenson, MPerel, Mdd, Mc6809e, Logologist, VivaEmilyDavies, Joaquin~enwiki, Kenyon, Japanese Searobin, Velho, Woohookitty, Kzollman, Wikiklrsc, Kbdank71, Chenxlee, Lockley, Nmegill, R.e.b., Truman Burbank, Tillmo, YurikBot, RussBot, Epolk, Trovatore, Rjensen, Varano, Tomisti, Avraham, Mike Dillon, Anclation~enwiki, Curpsbot-unicodify, Kaicarver, SmackBot, Roger Hui, Lestrade, Jacek Kendysz, Chris the speller, Zgyorﬁ~enwiki, RDBrown, Concerned cynic, Clconway, Josteinn, Oberwolfach, Valenciano, Jon Awbrey, Drunken Pirate, SashatoBot, Ser Amantio di Nicolao, DHR, Mathias- rex, Physis, Stampit, CRGreathouse, CmdrObot, CBM, Kowalmistrz~enwiki, Pierre de Lyon, Thomasmeeks, Myasuda, Gregbard, Cyde- bot, Poeticbent, Epbr123, King Bee, GentlemanGhost, Jd2718, Mibelz, Zickzack, Redmind, Danny lost, Hamaryns, Skomorokh, Matthew Fennell, Lurkman, .anacondabot, Glivi, Magioladitis, Swpb, Waacstats, David Eppstein, CommonsDelinker, Johnpacklambert, Stan J Klimas, LordAnubisBOT, Katharineamy, Policron, Kenneth M Burke, DavidCBryant, DorganBot, Ross Fraser, WOSlinker, TXiKiBoT, Ldonna, BertSen, Abtinb, Falcongl, Spaecious, Myscience, תמאה תויסחי, PeterBFZ, Romuald Wróblewski, SieBot, Nihil novi, Light- mouse, Likeminas, Jsqqq777, Kumioko (renamed), Vojvodaen, All Hallow’s Wraith, Niceguyedc, Altone, Masterpiece2000, Hans Adler, Thingg, Pichpich, Addbot, Atethnekos, Zarcadia, Feketekave, Dominidude, BOOLE1847, Lightbot, Luckas-bot, Yobot, Bunnyhop11, Denispir, Kilom691, Henryk Borawski, AnomieBOT, Citation bot, Xqbot, Mikołka, Omnipaedista, RibotBOT, January2009, Thehelp- fulbot, HJ Mitchell, Citation bot 1, Tkuvho, Kiefer.Wolfowitz, Skyerise, Gitana7, Trappist the monk, Doğu Kaan Eraslan, RjwilmsiBot, Spacejam2, DASHBot, EmausBot, Chimpionspeak, CrimsonBlue, WeijiBaikeBianji, Suslindisambiguator, El Roih, Satellizer, Citation- CleanerBot, Polmandc, BattyBot, Dirk101, VickiRedProject, Yoohoo234, Cerabot~enwiki, Epicuriousgeorge, Jochen Burghardt, Faizan, Oliszydlowski, A Boelen, POLY1956, Ice ax1940ice pick, GLG GLG, SoSivr, KasparBot, Heavenlyhermes and Anonymous: 111 � Aristotle Source: https://en.wikipedia.org/wiki/Aristotle?oldid=670301915 Contributors: Magnus Manske, Kpjas, General Wesc, Vicki Rosenzweig, Mav, Wesley, Bryan Derksen, Berek, Tarquin, Stephen Gilbert, Koyaanis Qatsi, Malcolm Farmer, DanKeshet, RK, Andre Engels, Eclecticology, Danny, XJaM, Deb, SimonP, Shii, Ben-Zin~enwiki, Glshadbolt, Camembert, Hirzel, Fonzy, Ezubaric, Hephaestos, Leandrod, Stevertigo, Spiﬀ~enwiki, Infrogmation, Pamplemousse, Michael Hardy, Llywrch, Fred Bauder, Owl, Aezram, BoNoMoJo (old), MartinHarper, Ixfd64, Bcrowell, Sannse, TakuyaMurata, Shoaler, GTBacchus, Nine Tail Fox, Paul A, Looxix~enwiki, Ellywa, Ahoerstemeier, Snoyes, Notheruser, Jniemenmaa, Angela, Darkwind, Александър, Cyan, Uri~enwiki, BenKovitz, LouI, Poor Yorick, Kwekubo, Andres, Evercat, John K, Ghewgill, Skyfaller, Schneelocke, Adam Conover, MichaelInskeep, Johs~enwiki, Renamed user 4, Alex S, Charles Matthews, Adam Bishop, EALacey, RickK, Jitse Niesen, Radgeek, Dandrake, The Anomebot, WhisperToMe, Wik, Dtgm, Zoicon5, Markhurd, Tpbradbury, Kaare, Hyacinth, Neiwai, Morwen, Itai, Populus, Mir Harven, Omegatron, Buridan, Phoebe, Joy, Prisonblues, Dpbsmith, Wetman, Pakaran, Johnleemk, Banno, Dimadick, Phil Boswell, Robbot, Jakohn, Fredrik, Alrasheedan, Goethean, Peak, Sam Spade, Lowellian, Mirv, Henrygb, Academic Challenger, Markewilliams, Flauto Dolce, Rursus, Paradox2, Rasmus Faber, Sun- ray, Rebrane, Hadal, Wikibot, Alba, Mushroom, Xanzzibar, Dina, Alan Liefting, Marc Venot, Sobelk, Giftlite, MPF, Awolf002, Andries, Tom harrison, Meursault2004, Aphaia, MSGJ, Obli, Rj, Peruvianllama, Everyking, Anville, Zmaj~enwiki, Carlo.Ierna, LarryGilbert, Beardo, Maarten van Vliet, Joshuapaquin, Node ue, Eequor, Rynelm, Solipsist, Matt Crypto, Chameleon, SWAdair, Deus Ex, Tagish- simon, Golbez, Gyrofrog, Utcursch, Gdr, Quadell, Antandrus, Williamb, Beland, OverlordQ, Cevlakohn, Anthony Mohen, Jossi, Euro- pracBHIT, 1297, Phil Sandifer, Rdsmith4, APH, Mikko Paananen, JimWae, Dmaftei, Tomruen, M.e, Pmanderson, Icairns, Tdombos, Kmweber, WpZurp, Joyous!, Herschelkrustofsky, Ukexpat, Trilobite, Adashiel, Trevor MacInnis, ELApro, Lacrimosus, Esperant, Zro, Hbmartin, Rﬂ, Simonides, Freakofnurture, Heegoop, Venu62, Poccil, Adambondy, Haiduc, DanielCD, Lectiodiﬃcilior, EugeneZelenko, Arcataroger, Noisy, Discospinster, Rich Farmbrough, Guanabot, Brutannica, FranksValli, Supercoop, Liso, Amicuspublilius, Vsmith, Parishan, Narsil, Ericamick, Xezbeth, Mal~enwiki, Number 0, Dbachmann, Pavel Vozenilek, TigerZukeX, Paul August, MarkS, Dcoet- zeeBot~enwiki, Lachatdelarue, Bender235, ESkog, PP Jewel, Ben Standeven, Brian0918, RJHall, Floorsheim, Livajo, Frankieist, El C, Chalst, Zenohockey, Mwanner, Kross, PhilHibbs, Shanes, AreJay, Art LaPella, Gershwinrb, Etz Haim, Spoon!, Wareh, Jpgordon, Causa sui, Thuresson, Bobo192, NetBot, Whosyourjudas, Ruszewski, Smalljim, Func, Evolauxia, John Vandenberg, BrokenSegue, Vor- texrealm, Cohesion, Arcadian, Oop, Urthogie, Rajah, PeterisP, Thewayforward, MPerel, Crust, Nsaa, Mdd, Batneil, Conny, Knucmo2, ADM, Jumbuck, Storm Rider, Alansohn, Gary, Anthony Appleyard, Jic, Mackinaw, Miranche, ChristopherWillis, Ben davison, Mr Adequate, Ricky81682, Verdlanco, Andrew Gray, D prime, Riana, Lectonar, SlimVirgin, WhiteC, Seans Potato Business, PAR, Eu- kesh, Mysdaao, Titanium Dragon, Jjhake, Snowolf, Pax~enwiki, Dkikizas, Wtmitchell, Binabik80, Kanodin, Andrew Norman, Suru- ena, Docboat, Evil Monkey, VivaEmilyDavies, RJFJR, RainbowOfLight, TenOfAllTrades, Sciurinæ, Sumergocognito, Pethr, LFaraone, Zereshk, HGB, Michael Ward, Ceyockey, Markaci, Phi beta, Oleg Alexandrov, Megan1967, Saeed, Snowmanmelting, Philthecow, Joriki, Velho, Mel Etitis, Woohookitty, FeanorStar7, TigerShark, Timo Laine, Etacar11, Daniel Case, Gruepig, DavidArthur, Benhocking, Deeahbz, Kzollman, Briangotts, Dodiad, Chochopk, MONGO, Schzmo, Wikiklrsc, KFan II, Prashanthns, G.W., Stefanomione, Palica, Tydaj, Dysepsion, Tslocum, Dpaking, SqueakBox, Graham87, Magister Mathematicae, Cuchullain, BD2412, Galwhaa, FreplySpang, DePiep, Jclemens, Porcher, Jorunn, Rjwilmsi, Mayumashu, Koavf, Vary, Tangotango, Sdornan, Salix alba, HandyAndy, ErikHaugen, SpNeo, Zizzybaluba, Crazynas, Tstockma, Blueskyboris, Boccobrock, Afterwriting, Kazak, The wub, DoubleBlue, Reinis, Dar-Ape, MartinC~enwiki, Sango123, Ev, Yamamoto Ichiro, Hanshans23, Miskin, FlaBot, CDThieme, Ian Pitchford, RobertG, Doc glasgow, Crazycomputers, TheMidnighters, Nivix, Andy85719, RexNL, Gurch, Wars, Str1977, TeaDrinker, Alphachimp, Langer, Tedder, Pinir- icc65, TheSun, Tofergregg, King of Hearts, CiaPan, CJLL Wright, Chobot, DTOx, Finnegar, Citizen Premier, Aethralis, Gdrbot, Bg- white, Gwernol, Uriah923, YurikBot, Split, Deeptrivia, Jimp, Mukkakukaku, RussBot, Jtkiefer, ThomistGuy, RJC, Pigman, Eupator, Chris Capoccia, CanadianCaesar, Al Capwned, Zuben, Subsurd, Akamad, Stephenb, Robert Turner, Gaius Cornelius, Pseudomonas, KSchutte, Cunado19, Tyugar, NawlinWiki, Matia.gr, Rick Norwood, Ben-T, Stephen Burnett, Wiki alf, Veledan, LaszloWalrus, Du- moren, Jaxl, Johann Wolfgang, Trovatore, Proyster, Cognition, SivaKumar, Milesbuckeridge, Eric Sellars, Shaun F, Ziel, BlackAndy, Yoninah, Ragesoss, Shinmawa, Brandon, Jpbowen, Pkrembs, Darcrist, Aldux, Moe Epsilon, Misza13, Alex43223, Xgu, Dbﬁrs, BOT- Superzerocool, Wangi, DeadEyeArrow, Darthkt, FestivalOfSouls, Dernhelm~enwiki, Jpeob, Tomisti, User27091, Wknight94, Jkelly, FF2010, Womble, Phgao, Lt-wiki-bot, Andrew Lancaster, Nikkimaria, Theda, Closedmouth, Skenmy, Oscurotrophic, Fang Aili, Moogsi, E Wing, Abune, Jogers, LordJumper, Canley, Beaker342, Sean Whitton, GraemeL, Rocketrye12, Kevin, Anjoe, Whobot, Mhenriday, Ethan Mitchell, Argos’Dad, Kungfuadam, Lowellplayer, Inﬁnity0, Zernhelt, DVD R W, CIreland, David Wahler, ���� robot, Sycthos,
- 102 CHAPTER 14. THREE-VALUED LOGIC VinceyB, Sardanaphalus, Crystallina, Havocrazy, Otheus, SmackBot, FocalPoint, Imz, Smitz, Lestrade, Temptinglip, KnowledgeOfSelf, Notaﬂy, Lagalag, SilverFox, Nikanako, Kimon, Lawrencekhoo, Jacek Kendysz, KocjoBot~enwiki, Davewild, AndreasJS, Chairman S., Delldot, Blackpower, Agentbla, Rachel Pearce, Kintetsubuﬀalo, Edgar181, Alsandro, Mary 23 mali, LonesomeDrifter, Sebesta, Xaosﬂux, Yamaguchi��, Vassyana, Aksi great, Gilliam, Portillo, ShalashaskaX, Hmains, ERcheck, Exlibris, DarkElf109, David Ludwig, Amatulic, Izehar, Chris the speller, Bluebot, Keegan, TimBentley, Jcc1, Persian Poet Gal, Ian13, Jordanhurley, Master of Puppets, Thumperward, Miquonranger03, MalafayaBot, Bethling, SchﬁftyThree, Jennneal1313, Interstate295revisited, KaptKos, Willardo, Viewﬁnder, Nbarth, Kasyapa, Go for it!, DHN-bot~enwiki, Tonica, Boﬀman, AdamSmithee, John Reaves, WikiPedant, Aﬂin, Can't sleep, clown will eat me, Vanished user llkd8wtiuawfhiuweuhncu3tr, John Hyams, Gamahucheur, Kelvin Case, Akhilleus, Onorem, Wisconjon, Yidisheryid, Matthew, EvelinaB, Jajhill, Clinkophonist, Addshore, Bardsandwarriors, Edivorce, Celarnor, Stevenmitchell, Junius~enwiki, WhereAmI, Iapetus, Downwards, Nibuod, Retinarow, Nakon, James McNally, RobinJ, Richard001, Alexandra lb, RandomP, Mini-Geek, Aniras, LoveMonkey, Hgilbert, Jan.Kamenicek, Weregerbil, Only, Lacatosias, Das Baz, Jon Awbrey, Illnab1024, Nathans, Jklin, Wybot, KeithB, Slotaa, Richard0612, ElizabethFong, Sadi Carnot, Vina-iwbot~enwiki, Aviron, Ck lostsword, Bejnar, Jwesalo, Kukini, Yevgeny Kats, Ohconfucius, Byelf2007, CIS, SashatoBot, Grommel~enwiki, Yannismarou, Clown in black and yellow, Rory096, Swatjester, Harry- boyles, Rklawton, Giovanni33, Rthefunkeymonkey, Dbtfz, Kuru, John, Scientizzle, Kipala, Ocanter, Disavian, VirtualDave, Sir Nicholas de Mimsy-Porpington, Shadowlynk, Merchbow, Hemmingsen, Mattbarton.exe, Mgiganteus1, Peterlewis, RedStar~enwiki, RomanSpa, PseudoSudo, KatToni, Aarandir, Kaewing, Bmistler, Defyn, Slakr, Special-T, Bfjs123, Stwalkerster, Apcbg, NJMauthor, Noah Salz- man, Mr Stephen, Waggers, Funnybunny, Ryulong, Risingpower, Pitman6787, RichardF, Texas Dervish, Zapvet, Jose77, LaMenta3, Ontoquantum, Inquisitus, Isokrates, Hectorian, Phuzion, Keitei, S t B, Hu12, Ginkgo100, BranStark, Azamat Abdoullaev, Mig77, OnBe- yondZebrax, Aursani, Fan-1967, Iridescent, K, Stangoldsmith, WGee, Shoeofdeath, AntonM~enwiki, J Di, Delta x, Gregtrueblood, MJO, Cbrown1023, Wwallacee, Blehfu, Musicmonk, Marysunshine, Amhboro1, Az1568, Tawkerbot2, Dave Runger, Daniel5127, Will Pit- tenger, Xcentaur, Cyrusc, JForget, Vaughan Pratt, CRGreathouse, Postmodern Beatnik, CmdrObot, Sir Vicious, Matthieu Houriet, Rigel1, Comrade42, CBM, KyraVixen, Ruslik0, N2e, OMGsplosion, Richaraj, MarsRover, Avillia, Casper2k3, Neelix, Andkore, Tim1988, Karenjc, Chicheley, Lookingforgroup, Gregbard, Seejyb, Slazenger, Michfan2123, Cydebot, Fluence, Gtxfrance, Steel, Aristophanes68, DrunkenSmurf, Astrochemist, Gogo Dodo, Corpx, ST47, Mvoltron, A Softer Answer, Jlpriestley, Pascal.Tesson, Scott14, Joegasper, Tawkerbot4, Doug Weller, Rlz, Christian75, Codetiger, DumbBOT, Chrislk02, In Defense of the Artist, Sirmylesnagopaleentheda, Vy- selink, IComputerSaysNo, Viridae, Briantw, SpK, SteveMcCluskey, Ebyabe, Omicronpersei8, JodyB, Zalgo, Daniel Olsen, Dimo414, Grubbiv, Gimmetrow, Nishidani, Bhvilar, FrancoGG, Thijs!bot, SnaX, Epbr123, Wikid77, CSvBibra, Ziggman93, Mime, Ucanlookitup, Vidor, N5iln, Andyjsmith, Headbomb, Victorlamp, John254, Tapir Terriﬁc, James086, Peter Gulutzan, Tellyaddict, BehnamFarid, Pavlo Moloshtan, Dfrg.msc, RichardVeryard, Philippe, CharlotteWebb, Deafchild, TangentCube, Klausness, WhaleyTim, SusanLesch, Na- talie Erin, CTZMSC3, Northumbrian, Escarbot, Oreo Priest, Hmrox, AntiVandalBot, Ais523, RobotG, Chaleyer61, Majorly, Abu-Fool Danyal ibn Amir al-Makhiri, Emeraldcityserendipity, Quintote, Prolog, Doc Tropics, DeanC, Sirol~enwiki, Neoptolemos, Julia Rossi, Mal4mac, Dr who1975, Jj137, Editor Emeritus, D. Webb, Modernist, Farosdaughter, Gdo01, MaXiMiUS, LéonTheCleaner, David auk- erman, Baskaransri, John Cho, JAnDbot, Denidoc@gmail.com, WANAX, Leuko, Husond, Athkalani~enwiki, Bobvila2, Smashman202, MER-C, Epeeﬂeche, Mcorazao, Matthew Fennell, Instinct, Janejellyroll, Tonyrocks922, Xeno, Hut 8.5, GurchBot, Chickyfuzz123, Tstrobaugh, Snowolfd4, Savant13, Beaumont, Cynwolfe, Dmacw6, LittleOldMe, Acroterion, Meeples, ΚΕΚΡΩΨ, Bibi Saint-Pol, Niko- laos Bakalis, Magioladitis, Connormah, Bongwarrior, Xwangtang, VoABot II, P64, Ishikawa Minoru, AuburnPilot, JNW, SHCarter, Careless hx, ZooTVPopmart, Sunﬂower at Dawn, Doug Coldwell, Avicennasis, Snowded, Bubba hotep, JaKoBay, Catgut, Ankitsingh83, Awwiki, Animum, Nposs, Ben Ram, MetsBot, User86654, Oldimagineer, 28421u2232nfenfcenc, Boﬀob, Allstarecho, Faded shado, SlamDiego, DerHexer, JaGa, Matt B., Megalodon99, CCS81, Debashish, Johnbrownsbody, TimidGuy, Erik.w.davis, Murraypaul, Gwern, Kitler0005, Gjd001, FisherQueen, GustavoDuarte, Ratherhaveaheart, Neonblak, Magnus Bakken, Hdt83, MartinBot, Paoloster, Arjun01, Tekleni, Cadre99, CalendarWatcher, Kostisl, PGRandom, R'n'B, AlexiusHoratius, Johnpacklambert, Zygimantus, Irish2455, Kjmarino, Fconaway, LittleOldMe old, Mifa17, WelshMatt, Whale plane, Smokizzy, Jsmith86, Erkan Yilmaz, Artaxiad, RockMFR, Paranomia, J.delanoy, Captain panda, Pharaoh of the Wizards, Nev1, Rgoodermote, Atomic theorist, Ulyssesmsu, Silverxxx, Uncle Dick, Yonidebot, Jonpro, Ginsengbomb, StonedChipmunk, Zane2614, Fleiger, Extransit, Pajfarmor, TheTwiz, Alsandair, TomS TDotO, Kimedoncius, Cpiral, Katalaveno, Nsigniacorp, LordAnubisBOT, Ignatzmice, Keyblade12344, Janus Shadowsong, Ypetrachenko, Kelvin Knight, Sil- ver7scythe7, Gabr-el, Stevenw988, Masmas7, Hm john morse, Chiswick Chap, InspectorTiger, Richard D. LeCour, NewEnglandYankee, SJP, Cobi, Malerin, Phatius McBluﬀ, Mufka, Tanaats, Rumpelstiltskin223, Nrobin9, Madhava 1947, Sean0987, 1stBrigade, Juliancolton, Evb-wiki, RB972, Kolja21, DorganBot, Subtilior, Doctoroxenbriery, Lucaswennerholm, Bite Jones, Inwind, Useight, Adam Zivner, CJTweedy, Izno, RjCan, Millton2, Lilguys, Idontkknow, Levydav, The enemies of god, ThePointblank, RJASE1, Nailer123, Idarin, Jonas Mur~enwiki, Mastrchf91, Taquam, Dirak, Tigger99, X!, Cantdj, Deor, VolkovBot, TreasuryTag, Laurzor, Hersfold, Wrongkey- hole, Jeﬀ G., Nburden, Kwsn, Al.locke, Ryan032, Aesopos, Barneca, Philip Trueman, Elephantini4, TXiKiBoT, Hunter.krauch, Kww, Envee11, Antoni Barau, FitzColinGerald, Karynhuntting, Z.E.R.O., Anonymous Dissident, Ticketautomat, Aﬂuent Rider, Weikang526, Qxz, Someguy1221, Bdallen, HansMair, Rhrebs0913, Ocolon, Koranjem, Ontoraul, Melsaran, Corvus cornix, Xxdarkstar101xX, Sol- darnal, Broadbot, Manbss, Abdullais4u, Jcollins07, LeaveSleaves, Drappel, Seb az86556, Mmashark311, Domitius, Frogdoglogpog, Cre- mepuﬀ222, Actipolak, Ilyushka88, FrankSanMiguel, RadiantRay, Mwilso24, Eldredo, Ahmedoasis, Deneys, BobTheTomato, Tctwood, Mattmiller2, Graymornings, Falcon8765, Enviroboy, J Casanova, Floikas, Why Not A Duck, Brianga, HeirloomGardener, Symane, Cowlinator, NHRHS2010, EmxBot, Deconstructhis, Thony C., Is Mise, Macdonald-ross, Linguist1, SMC89, SieBot, Whiskey in the Jar, Tresiden, Fixer1234, Gprince007, Tiddly Tom, Nihil novi, Scarian, Euryalus, BotMultichill, Ghimboueils, ThePrince7, Adamoako221, Caltas, Crawfwil, Doesils13, Squelle, RJaguar3, Triwbe, Swaq, This, that and the other, The way, the truth, and the light, Santas back3, JabbaTheBot, Drknow2000, Cagnettaican, GrooveDog, Srushe, Eumix, Chinesearabs, The Unknown Hitchhiker, Likebox, SpitFire3129, Tiptoety, Radon210, Ako221, Arbor to SJ, Tuomas Parsio, Ferret, Richardcraig, JSpung, Shakko, Turtle123, Oxymoron83, Aelius28, Citador, Faradayplank, Linkpalmer, Steven Zhang, Phil Lu, Lightmouse, Poindexter Propellerhead, Hjelmerus, Hobartimus, Svm1 63, Da noob1, Homelessman123123123, Pediainsight, Vojvodaen, Calatayudboy, Datus, Vanished user ewﬁsn2348tui2f8n2ﬁo2utjfeoi210r39jf, Anchor Link Bot, Jacob.jose, Wolfgang84, Superbeecat, Nic bor, GirlyPanache, 3rdAlcove, Kanonkas, Gr8opinionater, Invertzoo, Leranedo, Loren.wilton, De728631, ClueBot, LAX, Snigbrook, Jgeortsis, Hippo99, Fyyer, The Thing That Should Not Be, Fadesga, DionysosProteus, EelkeSpaak, Taquito1, Herakles01, Arakunem, Steplin19, Cryptographic hash, J8079s, Migz Nexus, SuperHamster, Boing! said Zebedee, Poo9dle, Phahn7, CounterVandalismBot, Bryan.kromenacker, Waterfall117, Colentava, Jordoboy, Madman- luc, Singinglemon~enwiki, MrBosnia, Neverquick, Olivas ruben, Auntof6, DragonBot, Excirial, Universityuser, -Midorihana-, Soccer- punkrocker, Garrettissupercool, Macedonius, Big m0ma123, Gtstricky, Jedimaster121493, Vivio Testarossa, Lartoven, Alpha Ralpha Boulevard, Enochchan107067, Shadowrox, NuclearWarfare, Aristotle07, Jotterbot, Kcowluvr, Lilsaintdj, Afro Article, Zachmosher, Meardley, Hans Adler, Razorﬂame, Dekisugi, Jonathan316, Kieranlee999, Dwiddows, Krypton34, Askahrc, BOTarate, Thehelpfulone, Sprajah, Al-Andalusi, Aprock, Panos84, Catalographer, Thingg, Liquid Mercury~enwiki, Aitias, BVBede, Notanaccountname, Venera 7, Sunshinyness, Dana boomer, Akaszynski, MelonBot, Tsan2008, Bolchazy101, Apparition11, ImGladMyMomIsDead, Crazy Boris
- 14.8. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 103 with a red beard, JKeck, Seneca22, Chronicler~enwiki, XLinkBot, Jytdog, Rror, Gerhardvalentin, Onehundredbillion, Saeed.Veradi, AndreNatas, SilvonenBot, Badgernet, TomPointTwo, ZooFari, Good Olfactory, Xp54321, Wran, Willking1979, Ucdclassicscarty, Dude its nick, Uruk2008, Landon1980, Ramanujanredux, Atethnekos, Angel Alice, DougsTech, LightSpectra, MartinezMD, Guy of a place, Fieldday-sunday, CanadianLinuxUser, Leszek Jańczuk, Jpoelma13, Flame89~enwiki, Cst17, Footiemeister, Mentisock, 0measam, Berk- berk, Noobblack, CarsracBot, Moocow8696, Kurtcobain12345, LinkFA-Bot, Rtz-bot, Roman000, Austin pp, Meaty Weenies, Zqmdfg, Numbo3-bot, Sylvania w, F Notebook, DubaiTerminator, Nefnef, Tide rolls, BOOLE1847, Lightbot, Verazzano, Gail, MuZemike, Jarble, JEN9841, HerculeBot, Legobot, Rradulak, Ttasterul, Wcmead3, Luckas-bot, ZX81, Yobot, Ptbotgourou, Senator Palpatine, Kingkong77, Legobot II, Cow turdy, PMLawrence, Pitchneed, THEN WHO WAS PHONE?, AmeliorationBot, ChugiBear, KamikazeBot, Mar- bleofplaster, TheThomas, Keeratura, ��������, Creektheleftcheeksneak, Jacobisawesome, Licor, AnomieBOT, Rubinbot, Makedonas the Greek, Bomas Hawkins, Piano non troppo, AdjustShift, Bonﬁre of vanities, Mr pope man, Shadowmorph, Ulric1313, Owllovesux, Whatsupwiththat, Materialscientist, RobertEves92, Wandering Courier, Citation bot, Allen234, Paulatim, ArthurBot, CABlankenship, Xqbot, Timir2, Drilnoth, Sakaa, Br77rino, Koyos, GrouchoBot, Indeedous, Ute in DC, ProtectionTaggingBot, Omnipaedista, RibotBOT, PawtucketFacts, Amaury, Sabrebd, Auréola, Brayan Jaimes, White whirlwind, Cyberstrike3000X, Shadowjams, Griﬃnofwales, Green Cardamom, FrescoBot, Dolly1313, T of Locri, Tobby72, VI, Aghniyya, Steve Quinn, Craig Pemberton, NewEconomist, Machine Elf 1735, DivineAlpha, Cannolis, Rhalah, Citation bot 1, Kennyfsp, Glryutd, Rbh00, Pinethicket, Suman-kayastha, Kiefer.Wolfowitz, Dazed- bythebell, JuliaBaxter51, Connor269, MastiBot, SpaceFlight89, Cshaw100, Wikijos, Meaghan, Ava2083, RandomStringOfCharacters, Hiphive, Generalcommando, Gamewizard71, FoxBot, Greco22, Lpt101095, Trappist the monk, Ooh2009, Pollinosisss, Standardfact, Xlxfjh, GregKaye, Dinamik-bot, 777sms, Ninjasaursus, FrozenPencil, Brian the Editor, Sora3020, Satdeep Gill, DARTH SIDIOUS 2, TjBot, Ripchip Bot, Saruha, Salvio giuliano, Nyxaus, DASHBot, Steve03Mills, EmausBot, Icannothearyou, John of Reading, Wikitan- virBot, Gabby204, Ndkl, DuKu, S4city, Jake, Teiglykins123, Jean Alameda, Hpvpp, TomlinsonX21, AbhijayM., Limbero, Djembayz, Slawekb, AvicBot, Kkm010, ZéroBot, John Cline, Theirrulez, Imadjafar, Lateg, Resolver-Aphelion, Hazard-SJ, SporkBot, Christina Sil- verman, RaptureBot, Ventus55, Jsolorio14, OpenlibraryBot, L Kensington, Peace is contagious, Vistina101, Chrisdyer666, Bob duﬀy, Maximilianklein, Spicemix, GaleCarrLV, Miradre, Helpsome, ClueBot NG, Jean KemperNN, Jacksoncw, CocuBot, Macarenses, Aer- obicFox, Mccar408, Movses-bot, IfYouDoIfYouDon't, SilentResident, Two Wrongs, Bazuz, Frietjes, Kevin Gorman, CaroleHenson, EauLibrarian, Raoulis, Helpful Pixie Bot, Hagoth, Technical 13, BG19bot, MKar, Vagobot, Frog23, KateWishing, ElphiBot, Davidiad, Jordissim, Jahnavisatyan, Tyranitar Man, Brad7777, Aisteco, Vassto, Gundu1000, Mango845, ChrisGualtieri, Melenc, Generation zee, Dexbot, Belisariusgroup, Mr. Guye, Mogism, Binilmathew, VIAFbot, Neosmyrnian, Frosty, Juc123, Slurpy121, Condorcraft110, SomeF- reakOnTheInternet, Sɛvɪnti faɪv, MarcelBrandon, JPaestpreornJeolhlna, Nonsenseferret, Geofq, Msundqvist, Maria M Lopes, Shrikarsan, Dustin V. S., New worl, Recordstraight83, ����, Sol1, Prokaryotes, Bronx Discount Liquor, Fredmond4, Aubreybardo, Liz, Ryder- jalex, DraconiansUnleashed354, Meganesia, Sparlett, BillMoyers, AwesomeEvilGenius, Gts-tg, JMitchellUK, Uthorr, Tyro13, Monkbot, Stenskjaer, Trackteur, Amortias, Nimrainayat6290, MichelleSmith8, Akheller, I Love Adoption, SoSivr, Tetra quark, Hujt, KasparBot, Aliensyntax and Anonymous: 2174 � Emil Leon Post Source: https://en.wikipedia.org/wiki/Emil_Leon_Post?oldid=670203675 Contributors: Magnus Manske, Andre En- gels, Michael Hardy, IZAK, Ahoerstemeier, Docu, Ciphergoth, Charles Matthews, Robbot, Jaredwf, MathMartin, Pmineault, Cautious, Giftlite, Yekrats, Behnam, D3, Piotrus, Emax, PolishPoliticians, TonyW, Omassey, D6, EBL, KittySaturn, ZeroOne, Zaslav, EmilJ, Ruszewski, MPerel, TheParanoidOne, Darked~enwiki, Thivierr, Wikiklrsc, Kbdank71, Rjwilmsi, Koavf, Lockley, FlaBot, Tdoune, YurikBot, RussBot, R.e.s., Avraham, Petri Krohn, Curpsbot-unicodify, SmackBot, Vald, Chris the speller, Bluebot, Pax85, Jon Awbrey, Michael David, Wvbailey, Mathiasrex, CmdrObot, Drinibot, Gregbard, Cydebot, Alaibot, Biruitorul, Magioladitis, Connormah, JNW, Waacstats, Martin Davis, Lance6968, Alro, Christian Storm, TXiKiBoT, The Tetrast, SieBot, Nihil novi, Paradoctor, Monegasque, Justin W Smith, Universityuser, Addbot, Luckas-bot, Yobot, Delfort, AnomieBOT, Hairhorn, Gonzalcg, GrouchoBot, Skyerise, RedBot, Full- date unlinking bot, EmausBot, WikitanvirBot, CrimsonBlue, Ebrambot, Suslindisambiguator, ChuispastonBot, Carrotstacker, VIAFbot, Chaim1995, Lekoren, Oliszydlowski, Liz, Nagyjivad, KasparBot and Anonymous: 34 � Four-valued logic Source: https://en.wikipedia.org/wiki/Four-valued_logic?oldid=630213717 Contributors: Hyacinth, Jason Quinn, Cje~enwiki, Ruud Koot, Fresheneesz, Mikeblas, SmackBot, Oli Filth, BIL, “alyosha”, CRGreathouse, Amalas, Cydebot, Em3ryguy, R'n'B, Xenogene, Dekart, Paraconsistent, Paradoxe allemand, AnomieBOT, Erik9bot, AvicAWB, Tijfo098, Helpful Pixie Bot, Kahtar and Anonymous: 3 � Fuzzy logic Source: https://en.wikipedia.org/wiki/Fuzzy_logic?oldid=670685809 Contributors: Damian Yerrick, Tarquin, Ap, Rjstott, Christian List, Heron, Stevertigo, RTC, Michael Hardy, Pit~enwiki, Ixfd64, Eric119, Ahoerstemeier, Ronz, Harry Wood, AugPi, An- dres, Palfrey, EdH, Loren Rosen, Zoicon5, Markhurd, Furrykef, Hyacinth, Omegatron, Traroth, Robbot, Academic Challenger, Rursus, Blainster, Ruakh, Tobias Bergemann, Cedars, Giftlite, Zaphod Beeblebrox, Duniyadnd, Jason Quinn, Gyrofrog, Lawrennd, Quackor, Marcus Beyer, L353a1, Gauss, Icairns, Zfr, TreyHarris, Ohka-, Clemwang, Kadambarid, Xezbeth, Mani1, Paul August, Guard, El- wikipedista~enwiki, Mr. Billion, El C, Chalst, Moilleadóir, Causa sui, Smalljim, R. S. Shaw, Nortexoid, Adrian~enwiki, Abtin, Aronbeek- man, JesseHogan, Mdd, Denoir, Andrewpmk, Amram99, Samohyl Jan, Ajensen, Virtk0s, Oleg Alexandrov, Joriki, Velho, Woohookitty, Linas, Aperezbios, Olethros, Kzollman, Ruud Koot, WadeSimMiser, Brentdax, Smmurphy, BlaiseFEgan, Junes, Palica, Turnstep, MC MasterChef, Rjwilmsi, Koavf, Lese~enwiki, Arabani, Williamborg, Yamamoto Ichiro, FlaBot, Ultimatewisdom, Mathbot, Gurch, Intgr, Predictor, Scimitar, Chobot, YurikBot, Wavelength, Borgx, KSmrq, Manop, Ihope127, Trovatore, Srinivasasha, SAE1962, Expen- sivehat, Dhollm, Ndavies2, Dethomas, EverettColdwell, Dragonﬁend, Crasshopper, S. Neuman, Brat32, CLW, Andreasdr, Paul Mag- nussen, K.Nevelsteen, JimBrule, Closedmouth, Arthur Rubin, Scriber~enwiki, LanguidMandala, Mastercampbell, Acer, Peyna, Allens, Nekura, Jeﬀ Silvers, SmackBot, RedHouse18, Mneser, Slashme, Shervink, Eskimbot, Sebesta, Xaosﬂux, Ignacioerrico, Mhss, Sne- speca, Saros136, Catchpole, Thumperward, Oli Filth, Nbarth, DHN-bot~enwiki, Mladiﬁlozof, JonHarder, JustAnotherJoe, Cyberco- bra, Alca Isilon~enwiki, StephenReed, Ck lostsword, Evert Mouw, SashatoBot, Lambiam, Srikeit, Kuru, T3hZ10n, Jaganath, Bjanku- loski06en~enwiki, Ptroen, BenRayﬁeld, Hargle, Ace Frahm, Passino, Hu12, Iridescent, Igoldste, Bairam, George100, Megatronium, CRGreathouse, CmdrObot, Gbellocchi, Dgw, Requestion, Leujohn, Vizier, Gregbard, AndrewHowse, Rgheck, Peterdjones, Blackmet- albaz, Omicronpersei8, Jadorno, Letranova, Thijs!bot, Lord Hawk, Saibo, Amitauti, Klausness, Seaphoto, Mdotley, Vendettax, Gökhan, Kariteh, JAnDbot, Em3ryguy, MER-C, Dricherby, Typochimp, Magioladitis, Bongwarrior, Gerla314, Hkhandan~enwiki, Crunchy Num- bers, Boﬀob, Pkrecker, Oicumayberight, Oroso, EyeSerene, Arjun01, Rohan Ghatak, Honglyshin, Andreas Mueller, Sahelefarda, Ay- dos~enwiki, J.delanoy, Trusilver, Maurice Carbonaro, Gurchzilla, SuzanneKn, Jchernia, Jack and Mannequin, Gerla, DASonnenfeld, Spellcast, Babytoys, Philip Trueman, Mkcmkc, TXiKiBoT, Aylabug, Rei-bot, Atabəy, Anonymous Dissident, Fullofstars, Almadana, LBehounek, Swagato Barman Roy, Kilmer-san, Ululuca, VanishedUserABC, Sebastjanmm, Katzmik, GideonFubar, Hypertall, SieBot, Mathaddins, Malcolmxl5, BotMultichill, Phe-bot, Dawn Bard, Flyer22, Topher385, Panadero45, Allmightyduck, Ioverka, Cesarperma- nente, Vanished user oij8h435jweih3, Fratrep, OKBot, Melcombe, Rabend, Jcrada, Francvs, ClueBot, Fyyer, Drmies, Cryptographic hash, Ronaldloui, Excirial, Jbruck, Teutonic Tamer, Qwfp, Vansskater692, JHTaler, Cnoguera, Gerhardvalentin, PeterFisk, Avoided,
- 104 CHAPTER 14. THREE-VALUED LOGIC Addbot, Paper Luigi, DOI bot, Betterusername, LaaknorBot, Tide rolls, Zorrobot, Wireless friend, Luckas-bot, TheSuave, Yobot, Frag- gle81, H11, Legobot II, ArchonMagnus, SparkOfCreation, Gelbukh, AnomieBOT, DemocraticLuntz, Felipe Gonçalves Assis, Rubinbot, Jim1138, Riyad parvez, Lynxoid84, Flewis, Materialscientist, 90 Auto, Citation bot, Diegomonselice, ArthurBot, Pownuk, Obersachsebot, Xqbot, Jbbyiringiro, Grim23, Mechanic1c, Maddie!, J04n, Pickles8, False vacuum, Aiyasamy, Charvest, T2gurut2, Kingmu, Drwu82, Sector001, FrescoBot, Mark Renier, Spirographer, Citation bot 1, Pinethicket, Elockid, Tinton5, Skyerise, C2math, Lars Washing- ton, Alarichus, Gryllida, Serpentdove, Lbhales, Callanecc, ISEGeek, Chronulator, TankMiche, VernoWhitney, BertSeghers, Digichoron, EmausBot, Faolin42, ThornsCru, H3llBot, Carl Wivagg, Tolly4bolly, Labnoor, Donner60, Eulenreich, Tijfo098, ClueBot NG, Matthi- aspaul, Sfgrieco, Loopy48, ScottSteiner, Widr, Helpful Pixie Bot, Anidaane, Repep, Pacerier, Alex E. Clarke, Sqzx, Drift chambers, Sn1per, M.r.ebraahimi, WikiHannibal, Colbert Sesanker, Xca777, Flaminchimp, Diglio.simoni, ShashankSharma2511, Barakafrit, Illia Connell, Керен, Aklnih, Suraduttashandilya, Jochen Burghardt, Funnyperson22, Phmresearch, �, Eknigge, Pdecalculus, Jumpulse, Zsof- tua, Maple2013, Julaei, RudiSeising, Wangbo66653, Jptvgrey, Bilorv, Monkbot, Gregusmihai, ינלשב, Renates45, Dexalkaline, Sairp, Mrityunjaykr02, TranquilHope, Qzekrom, William Zachary Runyon, Brewstoo, Analplays, Aangell123, Mcconnellsc58, Sigma.4292, SocraticOath, Charlottecourtleeds and Anonymous: 451 � Hans Reichenbach Source: https://en.wikipedia.org/wiki/Hans_Reichenbach?oldid=670438932 Contributors: Dan~enwiki, Markhurd, Banno, Fredrik, Blainster, Peruvianllama, SPUI, Mailer diablo, Peter Wöllauer, Leondz, Ortcutt, Porcher, Rjwilmsi, FlaBot, KarlFrei, RussBot, KSchutte, Leutha, BOT-Superzerocool, Tomisti, Mike Dillon, Nikkimaria, Luk, Sardanaphalus, Attilios, SmackBot, Beta- command, Not Sure, Sistema13, Tsca.bot, OrphanBot, Vathek, CaAl, Stampit, Joseph Solis in Australia, Murzim, Vanisaac, Gregbard, Cydebot, Jdvelasc, Thijs!bot, Fayenatic london, Arch dude, Anthony Krupp, Waacstats, Cgingold, Arnold Reisman, CommonsDelinker, Inwind, TXiKiBoT, Just Jim Dandy, SieBot, MaynardClark, Vojvodaen, Alexbot, Addbot, Lightbot, JEN9841, Luckas-bot, Yobot, Om- nipaedista, D'ohBot, RandomStringOfCharacters, SaladDaisy23, Ceharanka, RjwilmsiBot, Beyond My Ken, ZéroBot, Chastra, Will- reed123, YFdyh-bot, VIAFbot, Jochen Burghardt, Monkbot, KasparBot and Anonymous: 27 � JanŁukasiewicz Source: https://en.wikipedia.org/wiki/Jan_%C5%81ukasiewicz?oldid=669968777Contributors: XJaM, Michael Hardy, Dominus, EdH, Bemoeial, Hyacinth, Aleph4, Robbot, Jaredwf, AdamReed, Altenmann, Vanden, Lzur, Adam78, Snobot, Giftlite, Pe- ruvianllama, Carlo.Ierna, Kpalion, Prosﬁlaes, Andycjp, MisﬁtToys, Piotrus, Emax, PolishPoliticians, Urhixidur, D6, Mindspillage, EBL, Djordjes, Kwamikagami, EmilJ, Ruszewski, ABCD, Logologist, Saga City, Gene Nygaard, Oleg Alexandrov, Japanese Searobin, Sburke, Wikiklrsc, Josh Parris, Sjakkalle, FlaBot, Witkacy, Vorpal Suds, Roboto de Ajvol, YurikBot, Leutha, Ziel, Scope creep, Nikkimaria, Curpsbot-unicodify, Appleseed, Neil Leslie, Eskimbot, Dr. Dan, Ligulembot, SashatoBot, Lambiam, Makyen, Myona, HennessyC, Kowalmistrz~enwiki, Cydebot, Julian Mendez, Al Lemos, Escarbot, Danny lost, Turgidson, Magioladitis, Waacstats, Gwern, STBot, PeterMSimons, EuTuga, Ontoraul, LBehounek, M0RD00R, Synthebot, Skarz, PeterBFZ, Laocoön11, Cirdan747, Nihil novi, Jack1956, Svick, Masterpiece2000, Lwyx, Laforgue, Addbot, Tassedethe, BOOLE1847, Lightbot, Yobot, Hohenloh, Analphabot, Xqbot, Om- nipaedista, 123unoduetre, Tkuvho, Martinvl, TobeBot, Podagrycznik, RjwilmsiBot, Neveln, EmausBot, Jllatimer, Hpvpp, HiW-Bot, Tijfo098, Memories of lost time, Snotbot, Helpful Pixie Bot, ChrisGualtieri, VIAFbot, UNOwenNYC, Liz, Tyro13, Ice ax1940ice pick, SoSivr, KasparBot and Anonymous: 41 � Many-valued logic Source: https://en.wikipedia.org/wiki/Many-valued_logic?oldid=670673489Contributors: Dan~enwiki, Bryan Derk- sen, Tarquin, Taw, B4hand, Michael Hardy, JakeVortex, MartinHarper, Justin Johnson, Eric119, Snoyes, Cyan, DesertSteve, Rzach, Reddi, Hyacinth, Hadal, Wikibot, Wile E. Heresiarch, Filemon, Snobot, Giftlite, Kim Bruning, Dissident, Muke, Jason Quinn, Gub- bubu, Lucidish, Mindspillage, Rich Farmbrough, Leibniz, EmilJ, Nortexoid, PWilkinson, Lysdexia, Oleg Alexandrov, Woohookitty, Mindmatrix, Kzollman, Ruud Koot, BD2412, Rjwilmsi, MWAK, David H Braun (1964), CiaPan, Urocyon, SmackBot, Mhss, Pwjb, Vina-iwbot~enwiki, MagnaMopus, Bjankuloski06en~enwiki, Makyen, Courcelles, JRSpriggs, Lahiru k, CRGreathouse, Giorgiomug- naini, Gregbard, ParmenidesII, Peterdjones, Quibik, Letranova, Escarbot, PChalmer, .anacondabot, STBot, Caregiver, Gurchzilla, Hey- itspeter, TXiKiBoT, Don4of4, LBehounek, Linguist1, Soler97, Cobalttempest, Francvs, Mild Bill Hiccup, Timberframe, Gerhard- valentin, Pgallert, Addbot, Rdanneskjold, SpBot, ChartreuseCat, Luckas-bot, Yobot, Legobot II, AnomieBOT, JackieBot, TheAMmol- lusc, Gilo1969, Oursipan, Argumzio, LittleWink, Trappist the monk, ZéroBot, Reasonable Excuse, Tijfo098, RockMagnetist, G8yingri, Helpful Pixie Bot, Repep, Sebrider, Jochen Burghardt, LvdT88, JMP EAX, Tecolotl 91, Phormicola and Anonymous: 50 � Principle of bivalence Source: https://en.wikipedia.org/wiki/Principle_of_bivalence?oldid=630181042Contributors: LC~enwiki, Bryan Derksen, Zundark, Ixfd64, Justin Johnson, Evercat, Charles Matthews, Hyacinth, NSash, Decoy, Guanabot, Paul August, Tsujigiri~enwiki, Chalst, Wareh, Beige Tangerine, Nortexoid, Lysdexia, Anthony Appleyard, Snowolf, RJFJR, Oleg Alexandrov, Velho, Linas, Pruss, Apokrif, Btyner, Graham87, Brighterorange, YurikBot, Hairy Dude, SmackBot, Rtc, Mhss, Nbarth, Frap, Jon Awbrey, Byelf2007, Wvbailey, Mets501, CBM, Gregbard, Letranova, R'n'B, WOSlinker, Don4of4, Modocc, Hugo Herbelin, RPHv, Legobot, Luckas-bot, Yobot, AnomieBOT, Hriber, Chharvey, Tijfo098, ClueBot NG, Helpful Pixie Bot, Ongepotchket, Harizotoh9, SoledadKabocha, Camila Cavalcanti Nery, Mathematical Truth and Anonymous: 40 � Probabilistic logic Source: https://en.wikipedia.org/wiki/Probabilistic_logic?oldid=670010734Contributors: Michael Hardy, Bender235, Oleg Alexandrov, Linas, Rjwilmsi, Joel7687, SmackBot, Mhss, Kripkenstein, Gregbard, Progicnet, FifthFloorLattimore, Jackbars, Quest for Truth, Valeria.depaiva, Melcombe, Classicalecon, Josang, DFRussia, ChuckEsterbrook, LeaW, Addbot, Tassedethe, Yobot, Cdrdata, Xqbot, PyonDude, Nkf31, ZéroBot, Egossvm, Donner60, बोधिचित्त, Widr, Scwarebang and Anonymous: 27 � Problem of future contingents Source: https://en.wikipedia.org/wiki/Problem_of_future_contingents?oldid=667681351 Contributors: Vadmium, Chalst, Miss Madeline, Koavf, Mercury McKinnon, SmackBot, Srnec, Monagz, Santa Sangre, O0pyromancer0o, CBM, Sdor- rance, Gregbard, Miguel de Servet, Barticus88, D. Webb, Arno Matthias, Stijn Vermeeren, R'n'B, Dionysiaca, Adavidb, Belovedfreak, Westfalr3, Ontoraul, Eletheia, Singinglemon~enwiki, CohesionBot, -Midorihana-, Spirals31, SchreiberBike, Qwfp, Addbot, Renamed user 5, Peter Damian (old), Yobot, AnomieBOT, Peter Damian, RjwilmsiBot, WikitanvirBot, Donner60, Tijfo098, Jack Greenmaven, Rezabot, Helpful Pixie Bot, Flosfa, CMDarling and Anonymous: 26 � Stephen Cole Kleene Source: https://en.wikipedia.org/wiki/Stephen_Cole_Kleene?oldid=668859670 Contributors: AxelBoldt, Vicki Rosenzweig, Gareth Owen, LA2, XJaM, Michael Hardy, Modster, Chinju, Karada, Snoyes, Emperorbma, Greenrd, Hyacinth, Phoebe, Jaredwf, MathMartin, Rebrane, Saforrest, Alan Liefting, Centrx, Giftlite, Yekrats, Stuuf, PDH, Beginning, D6, CALR, Rich Farm- brough, Guanabot, Leibniz, Ardonik, Kwamikagami, EmilJ, Duesentrieb, Nwerneck, Elipongo, Sligocki, Caesura, VivaEmilyDavies, Mcmillin24, Xiaoyanggu, Oleg Alexandrov, Velho, Linas, LOL, Sburke, Ruud Koot, Graham87, BD2412, Rjwilmsi, Lockley, Jive- cat, Bill37212, FlaBot, JYOuyang, YurikBot, RussBot, Harrisonmetz, Thnidu, Tim Parenti, Curpsbot-unicodify, SmackBot, TimBentley, RDBrown, AdamSmithee, Mhym, G716, Lambiam, Paul Foxworthy, HennessyC, Jonathan A Jones, CBM, Pierre de Lyon, Beeson, Greg- bard, Cydebot, Aanderson@amherst.edu, Master son, Omicronpersei8, Thijs!bot, Epbr123, Logomachon, Escarbot, Deﬂective, Postcard Cathy, .anacondabot, Waacstats, Mausy5043, DomBot, Senu, Salih, VolkovBot, Hotfeba, SieBot, DavisSta, MikeVitale, Niceguyedc, Addbot, Luckas-bot, AnomieBOT, Anne Bauval, Omnipaedista, Inerkor, DefaultsortBot, Foobarnix, FoxBot, RjwilmsiBot, BertSeghers,
- 14.8. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 105 Suslindisambiguator, ChuispastonBot, Helpful Pixie Bot, Helvitica Bold, Paolo Lipparini, VIAFbot, Jochen Burghardt, Kalyanam17, Jonarnold1985, JMP EAX, KasparBot and Anonymous: 46 � Term logic Source: https://en.wikipedia.org/wiki/Term_logic?oldid=669968687 Contributors: Ed Poor, Enchanter, Michael Hardy, AugPi, EdH, Renamed user 4, Charles Matthews, Timwi, Dysprosia, Wik, Markhurd, Maximus Rex, Hyacinth, Robbot, Fredrik, Stew- artadcock, Ruakh, Filemon, Giftlite, Siroxo, Gubbubu, Beland, Pmanderson, Deelkar, Paul August, Elwikipedista~enwiki, Chalst, Wood Thrush, BrokenSegue, Nortexoid, PWilkinson, Amerindianarts, Mark Dingemanse, Ricky81682, George Hernandez, Linas, Oriondown, BD2412, Grammarbot, Haya shiloh, Wavelength, Leuliett, SEWilcoBot, Cleared as ﬁled, Reyk, Tevildo, JoanneB, Bernd in Japan, GrinBot~enwiki, Sardanaphalus, SmackBot, Jagged 85, The great kawa, Yamaguchi��, Mhss, Oatmeal batman, Byelf2007, Anapraxic, CmdrObot, Gregbard, Cydebot, Gimmetrow, Barticus88, Bmorton3, DuncanHill, Gwern, Philcha, Jevansen, Djhmoore, Ontoraul, The Tetrast, Kumioko (renamed), Le vin blanc, JustinBlank, Andrewmlang, The Thing That Should Not Be, ImperfectlyInformed, Excirial, CohesionBot, PixelBot, Wordwright, MilesAgain, JDPhD, Palnot, Good Olfactory, Addbot, Markenrode, LightSpectra, Tassedethe, BOOLE1847, Lightbot, Vasiľ, Yobot, Ordre Nativel, AnomieBOT, LilHelpa, GrouchoBot, Peter Damian, Omnipaedista, Rb1205, Ma- chine Elf 1735, Winterst, Dhanyavaada, Dude1818, Pollinosisss, Wikielwikingo, EmausBot, Moswento, Rememberway, ClueBot NG, Jeraphine Gryphon, Regulov, JohnChrysostom, Hansen Sebastian, Hariket, Jochen Burghardt, Tyro13 and Anonymous: 50 � Three-valued logic Source: https://en.wikipedia.org/wiki/Three-valued_logic?oldid=666966799 Contributors: Ray Van De Walker, Booyabazooka, Shellreef, Cyp, AugPi, Dcoetzee, Furrykef, Hyacinth, AnonMoos, Saforrest, Ancheta Wis, Giftlite, Gwalla, DavidCary, Monedula, Jason Quinn, Jds, Nayuki, WhiteDragon, B.d.mills, Kate, Gazpacho, Guppyﬁnsoup, Mindspillage, Foolip, Ben Standeven, Kwamikagami, Nickj, EmilJ, Mairi, Spoon!, Telamon~enwiki, RJFJR, Alai, Klparrot, Forderud, Jörg Knappen~enwiki, Ruud Koot, Ash- moo, BD2412, Qwertyus, Rjwilmsi, Salix alba, Maxim Razin, Kakurady, YurikBot, Hillman, Trovatore, PrologFan, Vicarious, SmackBot, Tumbleman, Mhss, Bluebot, A Geek Tragedy, Cybercobra, Byelf2007, Bjankuloski06en~enwiki, Beard0, Norm mit, Judgesurreal777, Jason.grossman, Skapur, Ianji, SqlPac, ShelfSkewed, Shandris, Gregbard, Thijs!bot, Em3ryguy, Albmont, Loqi, Ssybesma, Nikpapag, Try0yrt, Ignat99, Peskydan, SparsityProblem, Robertgreer, Barraki, Dozen, Maghnus, Anonymous Dissident, Ruleof3, SieBot, Soler97, Svofski, Auntof6, Northernhenge, HumphreyW, Dekart, Addbot, DOI bot, Wireless friend, Luckas-bot, Yobot, AnomieBOT, Bci2, Fkereki, Omnipaedista, Douglas W. 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Original artist: Vectorized by Fvasconcellos (talk · contribs), based on original logo tossed together by Brion Vibber 14.8.3 Content license � Creative Commons Attribution-Share Alike 3.0 Alfred Tarski Life Mathematician Logician Truth in formalized languages Logical consequence What are logical notions? Works See also References Further reading External links Aristotle Life Thought Logic Aristotle’s epistemology Geology Physics Metaphysics Biology and medicine Psychology Practical philosophy Views on women Loss and preservation of his works Legacy Later Greek philosophers Influence on Byzantine scholars Influence on Islamic theologians Influence on Western Christian theologians Post-Enlightenment thinkers List of works Eponym See also Notes and references Further reading External links Emil Leon Post Early work Recursion theory Polyadic groups Selected papers See also Notes References Further reading External links Four-valued logic Applications Electronics Software Notes Fuzzy logic Overview Applying truth values Linguistic variables Early applications Example Hard science with IF-THEN rules Define with multiply Define with sigmoid Logical analysis Propositional fuzzy logics Predicate fuzzy logics Decidability issues for fuzzy logic Fuzzy databases Comparison to probability Relation to ecorithms Compensatory fuzzy logic See also References Bibliography External links Hans Reichenbach Life and work Selected publications See also References Sources External links Jan Łukasiewicz Life Work Recognition Chronology Selected works Books Papers See also Notes References Further reading External links Many-valued logic History Examples Kleene (strong) K3 and Priest logic P3 Bochvar’s internal three-valued logic (also known as Kleene’s weak three-valued logic) Belnap logic (B4) Gödel logics Gk and G∞ Łukasiewicz logics Lv and L∞ Product logic Π Post logics Pm Semantics Matrix semantics (logical matrices) Proof theory Relation to classical logic Suszko’s thesis Applications Research venues See also Notes References Further reading External links Principle of bivalence Relationship with the law of the excluded middle Classical logic Suszko’s thesis Criticisms Future contingents Vagueness See also References Further reading External links Probabilistic logic Historical context Modern proposals Possible application areas See also References Further reading External links Problem of future contingents Aristotle’s solution Leibniz 20th century See also Notes Further reading External links Stephen Cole Kleene Biography Important publications See also References External links Term logic Aristotle’s system Basics Term Proposition Singular terms Influence on philosophy Decline of term logic Revival See also Notes References External links Three-valued logic Representation of values Logics Kleene and Priest logics Łukasiewicz logic Bochvar logic ternary Post logic Modular algebras Application in SQL See also References Further reading External links Text and image sources, contributors, and licenses Text Images Content license