Index Aristotle 4–7, 45, 48, 166 artificial intelligence 62,

104–5 axioms 32, 43–7 Bacon, Francis 114 belief see web of belief brain as neural net 161–5 Brouwer, L.E. J. 92–4 Cantor, Georg 20 Carnap, Rudolf 30 Chomsky, Noam 146–57 Chrysippus of Soli 8–9, 19 classical logic 92, 102 computers 60, 62–4, 65, 78 connectives 8–9, 22, 36,

37–8, 47 consciousness 168 context principle, the 18 contradiction 5, 14, 24, 46

see also paradox Davidson, D. 54, 59, 142–4 deduction 115–21, 118, 126 Descartes, René 44–5, 114 disconfirmation theory 130–3 distributive law 107 Enigma code, the 42 falsity see paradox Feyerabend, Paul 141 formalism 52 Frege, G. 16, 36, 66, 68, 87 fuzzy logic 89–90, 102 Galileo 44, 112–14, 122 Gödel, Kurt 33

Incompleteness T h e o r e m

34, 74–9 intuitionistic logic 94 grammar 25, 50, 154

Chomsky 146–60 and mathematics 50–3 structures 156–7 see also language halting problem 78 Heap Paradox, see Sorites

Paradox Hempel, Karl 126–8 heterologies 73 Hilbert, David 32–3, 51, 74 Hilbert Spaces 77, 106

Hume, David 117–18, 121 Incompleteness Theorem

34, 74–9 induction 115–21 intuitionism 92–6 language 28–9, 54ff

Chomsky 146–57 law of excluded middle 93,

100 of identity 45 of non-contradiction 100 Leibniz, G. 10–16, 44–5, 68 liar paradox 67–92 logic classical 92, 102 and cognitive science 145 fuzzy 89–90 gates 40 and science 111 logical analysis 30 connectives 37–8 Lukasiewicz, Jan 97–102 mathematical logic 35 mathematics 20, 111, 122–5

Hilbert 32–3 incompleteness of 76–7 as language 52 metalanguage 71–2 Mill, John Stuart 119–24 neural nets 105, 161–5 Newton 113, 122, 130 nomological deduction

118, 126–9 organon, the 14 paradox 66–92

fuzzy logic 90 Gödel 74 heterologies 73 movement 80–3 Sorites 84–8 philosophy 28, 35 Popper, Karl 129–63 “possible” logic 98 predicate 4–5, 69 Predicate Calculus 49–50,

86, 91 proof theory 32, 36, 39, 51,

55, 66 and language 56

Propositional Calculus 19,

48–50, 86 Putnam, Hilary 109 quantifier 16, 20 quantum logic 107–10 mechanics 106 Quine, Willard 133–44 rational behaviour model

166 recursion 51 reductio ad absurdum

12–13, 33, 36, 95, 130 relativism 140–4 Russell, B. 23–7, 27–50

Gödel’s theorem 75 paradoxes 68–70 Russell’s system 26–7 science 129–40

and Carnap 30 and logic 111 and relativism 140–4 self-referential paradox 67 semantics 50, 60, 154–5 sentences 4–7

see also grammar set theory 20, 35, 69 Sorites Paradox 84–8, 90 square of oppositions 5–6 syllogisms 6–11, 48, 166–7 symbolic logic 35 Tarski, Alfred 57–8, 71–2 tautology 39 theory of types, the 69 tolerance principle 31 truth conditions 58–9 values as numbers 99 see also paradox Truth Tables 38–40, 43 Turing, Alan 42, 64–6, 145 Unger, Peter 85 universal grammar 146–60 viable theory 132 Vienna Circle, the 30, 118 web of belief 134–43 Wittgenstein, L. 28, 37–8,

170–2 Zeno of Elea 67, 80–3