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The Size of the World of Logic

The Size of the World of Logic. Jan Woleński Jagiellonian University , Krakow , Poland. Talk Outline. W hat is the world of logic ; Different a ccount s; Other logics ; T, (BI) and propositional calculus ; The general f orm of the wor l d of logic ; Argument for bivalence ;

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The Size of the World of Logic

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  1. The Size of the World of Logic Jan Woleński JagiellonianUniversity, Krakow, Poland

  2. Talk Outline • What is the world of logic; • Differentaccounts; • Other logics; • T, (BI) and propositional calculus; • The general formof the world of logic; • Argument for bivalence; • Other.

  3. What is the Worldof Logic The problem: what is the world of logic Russell: Logic is concerned with the real world just as truly as zoology, though with its more abstract and general features. But what are more abstract and general features of the world? Logic as consisting of tautologies. Frege: Logic is concerned with the predicate “true” Frege’ssemantics of sentences: the True and the False as references (senses) of sentences.

  4. What is the Worldof Logic Example: • p p q. • 1 (10), • 0 (01), 0 (00) and 0 (01). • p r q. • A A B as the way out. “Dual” logic • w(A B) = 1 wtww(A) = w(B) = 0; otherwisew(A B) = 0. • A B  A. • 1 as the distinguished values.

  5. DifferentAccounts The world of logic consists of logical value. (BI) every sentence is either true or false. 9. (BI)  (CN)  (EM). (BI) and the theorems of PC.

  6. DifferentAccounts Béziau on conditions for (BI): • Counter-domain of w is two-elements; • Domain of w - the set of sentences; • w is a total function.

  7. DifferentAccounts Other account: (a) card(V) = 0; no A is a theorem; (b) card(V) = 1; every A is a theorem; (c) card(V)  2; some A are theorems, some A are not theorems. D – distinguished values, D’ – non-distinguished values, L is consistent if card(V)  card(D) V = D  D’, D D’ = .

  8. DifferentAccounts Truth and falsehood as modalities:    

  9. DifferentAccounts  – 1A, – 1(A),  – 1(A),  – 1(A). (10)(a)  (b)  (c) () ; (d)  (e) ; (f)) . What about 0? Either  or . (11)(a) 0A 1A; (b) 0A 1A. (12) 1A 1A 0A.

  10. DifferentAccounts (11), (12) (BI) and more than 2 values.      

  11. DifferentAccounts  – 0,  – 1A0A,  – 1A 0A (11) and the triangle. (13) A(1A0A), Conclusion: (BI) is not a tautology.

  12. Other Logics Assumption: the only designated value. Is possible to save (BI)? (14) A(D(A) D(A), D’, DA (15) A(D(A)  DA(A)), DA(A) D(A) and its legitimization. T-scheme : TAA, DA and A

  13. OtherLogics .        

  14. Other Logics  , – A i A. (16) D(A) A, holds for every value, but reverse dependence not.

  15. T, (BI) and PropositionalCalculus. (17) w(T(A)) = 1iffw(A) = 1; otherwise w(A) = 0. The formula (17) is not generalized to predicate calculus.

  16. The GeneralForm of the World of Logic (WL) {w1, w2 ,…, wn, …}.

  17. Argument for Bivalence Argument for bivalence: metalogic (the role of classical logic, simplicity.

  18. Other • Truth – facts, falsehood – the lack of facts’; • Various oppositions, spatial, temporal; • Biological oppositions; • Passive- active; • Possession and its lack; Inner – outer; • Modal contrasts; • Biological rhythms are binary; • Perceptual contrasts; • Binary structure of the helix and genetic codes; • 0-1 nature of information; • Truth protects information, falsehood results in its dispersion; • Ordinary quantifiers are dual.

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