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On Fuzzy Logic III. Semantical completeness of some many‐valued propositional calculi
Author(s) -
Pavelka Jan
Publication year - 1979
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.19790252510
Subject(s) - completeness (order theory) , propositional calculus , computer science , citation , fuzzy logic , information retrieval , arithmetic , discrete mathematics , calculus (dental) , artificial intelligence , mathematics , programming language , medicine , world wide web , mathematical analysis , dentistry
Now we come to speak of the first non-trivial question which, in a way, puts to the test all the definitions presented so far, viz. the question of balance of syntax against semantics. Every choice of an enriched complete residuated lattice & = ((L, 8, +), S ) of a certain type ( A T : A + N,, Ex: A + N*), plus a set P of propositional variables, poses the problem of axiomatizability of the 8-valued propositional calculus over P. According to [3], M. 25, the question actually goes like this: (Q) Do there exist an L-fuzzy set of logical axioms A : F(P, L, A ) --f L and a set W of L-valued rules of inference in F(P, L, A ) such that for any L-fuzzy theory X: F(P, L, A ) + L over P and any formula e) E F(P, L, A ) the degree ( g y ( p , 8 ) X ) e), to which q~ follows from X in the L-semantical system ( F ( P , L , A ) , Y ( P , a)), equals exactly the degree ( q A , @ X ) q ~ , to which q~ is provable from X in the L-syntactical system ( F ( P , L, A ) , A, W ) ? l ) In this paper we investigate the case when the underlying lattice of B is a chain. an abstract set of formulas ([3]);