Download Logica Universalis: Towards a General Theory of Logic by Jean-Yves Beziau PDF

By Jean-Yves Beziau

Common good judgment isn't really a brand new common sense, yet a common thought of logics, regarded as mathematical buildings. The identify was once brought approximately ten years in the past, however the topic is as previous because the starting of recent good judgment: Alfred Tarski and different Polish logicians similar to Adolf Lindenbaum constructed a normal conception of logics on the finish of the Nineteen Twenties in line with final result operations and logical matrices. the topic was once revived after the flowering of millions of latest logics over the past thirty years: there has been a necessity for a scientific idea of logics to place a few order during this chaotic multiplicity. This e-book comprises fresh works on common common sense through top quality researchers from all over the international. The booklet is stuffed with new and tough rules that would consultant the way forward for this intriguing topic. it will likely be of curiosity for those who are looking to higher comprehend what common sense is. instruments and ideas are supplied the following in case you are looking to examine periods of already current logics or are looking to layout and construct new ones.

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Hence, M od(L) is a subbasis of this topology. In of the topology σL 2 a similar way follows that VL is a basis of the topology σL on (IntL =L ) and therefore (M od(L) =L ) is a subbasis of this topology. We have proved the first assertion of the Proposition. Now suppose that L has conjunction. , an ). Hence, M od(L) is closed under finite intersections and therefore M od(L) = UL . Similarly, (M od(L) =L ) is closed under finite intersections and (M od(L) =L ) = VL . Thus, if L has conjunction, then M od(L) and (M od(L) =L ) are bases of the 7 Note that if a basis of a topology is closed under complement, then, in particular, it is a basis of clopen sets.

1. A model-theoretical abstract logic is a structure L = (ExprL , IntL , L ). ) (i) ExprL is a set of formulas over a given alphabet. We call the elements of ExprL expressions or L-expressions. (ii) IntL is a class of interpretations in which the L-expressions receive their denotations. The element of IntL are also called models. (iii) L is a relation, called satisfaction relation, on IntL ×ExprL . The satisfaction relation determines the truth values (“true” or “false”) of the expressions in the interpretations.

F is regular, if for all A, B ∈ IntL holds: A =L B =⇒ f (A) =L f (B). g is regular, if for all a, b ∈ ExprL holds: a =L b =⇒ g(a) =L g(b). (ii) A logic-homomorphism from L to L is a pair (f, g) of functions f : IntL → IntL and g : ExprL → ExprL that satisfy the following condition: For every a ∈ ExprL there exists an Aa ⊆ ExprL such that the following holds: • f −1 (M odL (a )) = {(M odL (a)) | a ∈ Aa } and • g −1 (a ∗L ) = {a∗L | a ∈ Aa }. We write (f, g) : L → L . We say that a logic-homomorphism (f, g) is injective (surjective, bijective), if f and g are injective (surjective, bijective) functions, respectively.

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