Semantic characterization of intuitionistic logical connectives
type: Item
instance of: scholarly article
title: Semantic characterization of intuitionistic logical connectives
zbMATH Open document ID: 0658.03006
DOI: 10.1007/BF01137476
published in: Mathematical Notes
publication date: 1985-01-01T00:00:00Z
review text: In classical propositional logic any \(2^ n\)-distribution of 0 and 1 can be interpreted as a suitable combination of propositional variables connected only by \& and \(\neg\). So, any clssical connective given semantically can be expressed by \& and \(\neg\); or, the pair (\&,\(\neg)\) is functionally complete. The author investigates the analogous question for intuitionistic propositional logic: do the four ``standard'' intuitionistic connectives (\&,\(\vee,\Rightarrow,\neg)\) suffice to express any semantically given connective? And, in general, what is an intuitionistic connective when speaking of functional completeness? It is natural to agree that the analogue of classical 0-1-tableau will be a one-free-variable predicate formula (evaluating the connective in Kripke-models) which links the validity of the operation in an arbitrary possible world with validity of the operands in some other worlds. E.g. the evaluation of the intuitionistic implication, \(x\Vdash A\Rightarrow B\), can be regarded as the predicate \((A\Rightarrow B)(x)\) defined by the formula (\(\forall y\geq x)\) (if \(A(y)\) then \(B(y)\)). Furthermore, the author postulates the following three principles for this predicate (they also seem to be natural from the intuitionistic point of view): 1) monotonicity (if C(x) and \(x\leq y\) then C(y)), i.e. what has been proved remains true forever; 2) relativization of quantifiers (\(\forall z\geq w\), \(\exists z\geq w\), etc.), i.e. the truth value at given vertex is determined only by the situation at this vertex and in the future, but not in the past; 3) similarity (intuitively: a proposition constructed by means of logical connectives should not separate models which are equivalent in a precise sense). Then the fundamental theorem affirms that the set of all semantically expressed intuitionistic connectives is exactly covered by the set of all propositional formulas. In such a way the sufficiency of the four standard connectives obtains some sema…
Mathematics Subject Classification ID: 03B20
zbMATH DE Number: 4075018
zbMATH Keywords: Kripke-semantics
zbMATH Keywords: intuitionistic propositional logic
zbMATH Keywords: intuitionistic connectives
zbMATH Keywords: functional completeness
author: Aleksandr D. Yashin
MaRDI profile type: Publication
cites work: Q3880841
cites work: On some new intuitionistic propositional connectives. I
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description: scientific article; zbMATH DE number 4075018