Fuzzy Set Structure with Strong Implication

Zhang Jinwen
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Zhang Jinwen: Academia Sinica, Institute of Computing Technology

A chapter in Advances in Fuzzy Sets, Possibility Theory, and Applications, 1983, pp 107-136 from Springer

Abstract: Abstract Fuzzy set structure with strong implication has some very interesting logical characteristics. It is weaker than the normal fuzzy set structures, but a little stronger than some fuzzy set structures. We have the following results: Theorem 1. Every axiom of weak logic calculus WL is true in this structure and every rule of inference of WL is also valid in it. Theorem 2. Let G’ be any complete quasi-Boolean sublattice. Then for any Σ0-formula, A(xl,…,xn) not including the negative ר, and any cl,…, cnεF(G’), we have $${{\left[\kern-0.15em\left[ {A\left( {{{C}_{1}}, \ldots ,{{C}_{n}}} \right)} \right]\kern-0.15em\right]}^{{G'}}} = {{\left[\kern-0.15em\left[ {A\left( {{{C}_{1}}, \ldots ,{{C}_{n}}} \right)} \right]\kern-0.15em\right]}^{G}}$$ Theorem 3. Every axiom of a weak axiom system ZF1U of set theory is true in our structure; that is, our structure is a model of ZF1U.

Date: 1983
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DOI: 10.1007/978-1-4613-3754-6_10

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