The Size of the World of Logic Jan Woleński Jagiellonian University, Krakow, Poland

Talk Outline What is the world of logic; Different accounts; Other logics; T, (BI) and propositional calculus; The general form of the world of logic; Argument for bivalence; Other.

What is the World of 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’s semantics of sentences: the True and the False as references (senses) of sentences.

What is the World of 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 wtw w(A) = w(B) = 0; otherwise w(A  B) = 0. A  B  A. 1 as the distinguished values.

Different Accounts 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.

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

Different Accounts 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’ = .

Different Accounts Truth and falsehood as modalities:    

Different Accounts  – 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.

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

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

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

Other Logics .        

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

T, (BI) and Propositional Calculus. (17) w(T(A)) = 1 iff w(A) = 1; otherwise w(A) = 0. The formula (17) is not generalized to predicate calculus.

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

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

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.