Deductive Completeness

1996 ◽  
Vol 2 (3) ◽  
pp. 243-283 ◽  
Author(s):  
Kosta Došen
Keyword(s):  
Category Theory ◽  
Propositional Logic ◽  
Completeness Theorem ◽  
Substructural Logics ◽  
Combinatory Logic ◽  
Functional Completeness ◽  
Deduction Theorem ◽  
Intuitionistic Propositional Logic

AbstractThis is an exposition of Lambek's strengthening and generalization of the deduction theorem in categories related to intuitionistic propositional logic. Essential notions of category theory are introduced so as to yield a simple reformulation of Lambek's Functional Completeness Theorem, from which its main consequences can be readily drawn. The connections of the theorem with combinatory logic, and with modal and substructural logics, are briefly considered at the end.

Explicit Provability and Constructive Semantics

2001 ◽  
Vol 7 (1) ◽  
pp. 1-36 ◽  
Author(s):  
Sergei N. Artemov
Keyword(s):  
Modal Logic ◽  
Propositional Logic ◽  
Modal Logics ◽  
Combinatory Logic ◽  
Underlying Structure ◽  
Intuitionistic Propositional Logic ◽  
Intuitionistic Modal Logics

AbstractIn 1933 Gödel introduced a calculus of provability (also known as modal logicS4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logicLPof propositions and proofs and show that Gödel's provability calculus is nothing but the forgetful projection ofLP. This also achieves Gödel's objective of defining intuitionistic propositional logicIntvia classical proofs and provides a Brouwer-Heyting-Kolmogorov style provability semantics forIntwhich resisted formalization since the early 1930s.LPmay be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and λ-calculus.

scholarly journals On density of truth of the intuitionistic logic in one variable

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Zofia Kostrzycka
Keyword(s):  
Intuitionistic Logic ◽  
Classical Logic ◽  
Propositional Logic ◽  
Propositional Variable ◽  
Intuitionistic Propositional Logic ◽  
International Audience

International audience In this paper we focus on the intuitionistic propositional logic with one propositional variable. More precisely we consider the standard fragment $\{ \to ,\vee ,\bot \}$ of this logic and compute the proportion of tautologies among all formulas. It turns out that this proportion is different from the analog one in the classical logic case.

scholarly journals An ASP Approach to Generate Minimal Countermodels in Intuitionistic Propositional Logic

2019 ◽  
Author(s):  
Camillo Fiorentini
Keyword(s):  
Propositional Logic ◽  
Answer Set Programming ◽  
Kripke Model ◽  
Kripke Semantics ◽  
Minimal Models ◽  
Intuitionistic Propositional Logic ◽  
Answer Set

Intuitionistic Propositional Logic is complete w.r.t. Kripke semantics: if a formula is not intuitionistically valid, then there exists a finite Kripke model falsifying it. The problem of obtaining concise models has been scarcely investigated in the literature. We present a procedure to generate minimal models in the number of worlds relying on Answer Set Programming (ASP).

scholarly journals On an interpretation of second order quantification in first order intuitionistic propositional logic

1992 ◽  
Vol 57 (1) ◽  
pp. 33-52 ◽  
Author(s):  
Andrew M. Pitts
Keyword(s):  
Propositional Logic ◽  
Heyting Algebra ◽  
Propositional Calculus ◽  
Interpolation Theorem ◽  
Second Order ◽  
Heyting Algebras ◽  
Intuitionistic Propositional Logic ◽  
First Order ◽  
Intuitionistic Propositional Calculus ◽  
Algebraic Counterpart

AbstractWe prove the following surprising property of Heyting's intuitionistic propositional calculus, IpC. Consider the collection of formulas, ϕ, built up from propositional variables (p, q, r, …) and falsity (⊥) using conjunction (∧), disjunction (∨) and implication (→). Write ⊢ϕ to indicate that such a formula is intuitionistically valid. We show that for each variable p and formula ϕ there exists a formula Apϕ (effectively computable from ϕ), containing only variables not equal to p which occur in ϕ, and such that for all formulas ψ not involving p, ⊢ψ → Apϕ if and only if ⊢ψ → ϕ. Consequently quantification over propositional variables can be modelled in IpC, and there is an interpretation of the second order propositional calculus, IpC2, in IpC which restricts to the identity on first order propositions.An immediate corollary is the strengthening of the usual interpolation theorem for IpC to the statement that there are least and greatest interpolant formulas for any given pair of formulas. The result also has a number of interesting consequences for the algebraic counterpart of IpC, the theory of Heyting algebras. In particular we show that a model of IpC2 can be constructed whose algebra of truth-values is equal to any given Heyting algebra.

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