Classical propositional logic and decidability of variables in intuitionistic propositional logic

AbstractA 1-ary sentential context is aggregative (according to a consequence relation) if the result of putting the conjunction of two formulas into the context is a consequence (by that relation) of the results of putting first the one formula and then the other into that context. All 1-ary contexts are aggregative according to the consequence relation of classical propositional logic (though not, for example, according to the consequence relation of intuitionistic propositional logic), and here we explore the extent of this phenomenon, generalized to having arbitrary connectives playing the role of conjunction; among intermediate logics, LC, shows itself to occupy a crucial position in this regard, and to suggest a characterization, applicable to a broader range of consequence relations, in terms of a variant of the notion of idempotence we shall call componentiality. This is an analogue, for the consequence relations of propositional logic, of the notion of a conservative operation in universal algebra.

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Logical connectives for intuitionistic propositional logic

Journal of Symbolic Logic ◽

10.2307/2271511 ◽

1971 ◽

Vol 36 (1) ◽

pp. 15-20 ◽

Cited By ~ 11

Author(s):

Dean P. McCullough

Keyword(s):

Intuitionistic Logic ◽

Propositional Logic ◽

Classical Type ◽

Classical Propositional Logic ◽

Intuitionistic Propositional Logic ◽

Truth Function ◽

Logical Connectives ◽

Complete Set ◽

Complete Sets

In classical propositional logic it is well known that {7, ⊃ } is a functionally complete set with respect to a two-valued truth function modeling. I.e. all definable logical connectives are definable from 7 and ⊃. Other modelings of classical type propositional logics may have different functionally complete sets; for example, multivalued truth function modelings.This paper examines the question of a functionally complete set of logical connectives for intuitionistic propositional logic with respect to S. Kripke's modeling for intuitionistic logic.

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Cardinalities of proper ideals in some lattices of strengthenings of the intuitionistic propositional logic

Studia Logica ◽

10.1007/bf01063837 ◽

1983 ◽

Vol 42 (2-3) ◽

pp. 173-177

Author(s):

Wies?aw Dziobiak

Keyword(s):

Propositional Logic ◽

Intuitionistic Propositional Logic

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Deduction in Non-Fregean Propositional Logic SCI

Axioms ◽

10.3390/axioms8040115 ◽

2019 ◽

Vol 8 (4) ◽

pp. 115 ◽

Cited By ~ 1

Author(s):

Joanna Golińska-Pilarek ◽

Magdalena Welle

Keyword(s):

Propositional Logic ◽

Sequent Calculus ◽

Classical Propositional Logic ◽

Tableau System ◽

Soundness And Completeness

We study deduction systems for the weakest, extensional and two-valued non-Fregean propositional logic SCI . The language of SCI is obtained by expanding the language of classical propositional logic with a new binary connective ≡ that expresses the identity of two statements; that is, it connects two statements and forms a new one, which is true whenever the semantic correlates of the arguments are the same. On the formal side, SCI is an extension of classical propositional logic with axioms characterizing the identity connective, postulating that identity must be an equivalence and obey an extensionality principle. First, we present and discuss two types of systems for SCI known from the literature, namely sequent calculus and a dual tableau-like system. Then, we present a new dual tableau system for SCI and prove its soundness and completeness. Finally, we discuss and compare the systems presented in the paper.

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RELEVANCE LOGIC AND THE CALCULUS OF RELATIONS

The Review of Symbolic Logic ◽

10.1017/s1755020309990293 ◽

2010 ◽

Vol 3 (1) ◽

pp. 41-70 ◽

Cited By ~ 6

Author(s):

ROGER D. MADDUX

Keyword(s):

Propositional Logic ◽

Relevance Logic ◽

Binary Relations ◽

Classical Propositional Logic

Sound and complete semantics for classical propositional logic can be obtained by interpreting sentences as sets. Replacing sets with commuting dense binary relations produces an interpretation that turns out to be sound but not complete for R. Adding transitivity yields sound and complete semantics for RM, because all normal Sugihara matrices are representable as algebras of binary relations.

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