Propositional quantification in the monadic fragment of intuitionistic logic

1998 ◽  
Vol 63 (1) ◽  
pp. 269-300 ◽  
Author(s):  
Tomasz Połacik
Keyword(s):  
Intuitionistic Logic ◽  
Metric Space ◽  
Propositional Logic ◽  
Second Order ◽  
Intuitionistic Propositional Logic ◽  
Topological Interpretation ◽  
Propositional Quantifiers ◽  
Propositional Quantification

AbstractWe study the monadic fragment of second order intuitionistic propositional logic in the language containing the standard propositional connectives and propositional quantifiers. It is proved that under the topological interpretation over any dense-in-itself metric space, the considered fragment collapses to Heyting calculus. Moreover, we prove that the topological interpretation over any dense-in-itself metric space of fragment in question coincides with the so-called Pitts' interpretation. We also prove that all the nonstandard propositional operators of the form q ↦ ∃p (q ↔ F(p)), where F is an arbitrary monadic formula of the variable p, are definable in the language of Heyting calculus under the topological interpretation of intuitionistic logic over sufficiently regular spaces.

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 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.

On second order intuitionistic propositional logic without a universal quantifier

2009 ◽  
Vol 74 (1) ◽  
pp. 157-167 ◽  
Author(s):  
Konrad Zdanowski
Keyword(s):  
Propositional Logic ◽  
Second Order ◽  
Universal Quantifier ◽  
Intuitionistic Propositional Logic ◽  
Glivenko’S Theorem ◽  
Free Variables ◽  
Universal Quantification

AbstractWe examine second order intuitionistic propositional logic, IPC2. Let ℱ∃ a be the set of formulas with no universal quantification. We prove Glivenko's theorem for formulas in ℱ∃ that is, for φ ∈ ℱ∃, φ is a classical tautology if and only if ┐┐φ is a tautology of IPC2. We show that for each sentence φ ∈ ℱ∃ (without free variables), φ is a classical tautology if and only if φ is an intuitionistic tautology. As a corollary we obtain a semantic argument that the quantifier ∀ is not definable in IPC2 from ⊥, ⋁, ⋀, →, ∃.

The decidability of the Kreisel-Putnam system

1970 ◽  
Vol 35 (3) ◽  
pp. 431-437 ◽  
Author(s):  
Dov M. Gabbay
Keyword(s):  
Intuitionistic Logic ◽  
Propositional Logic ◽  
Finite Model ◽  
Model Property ◽  
Intuitionistic Propositional Logic ◽  
Disjunction Property ◽  
Finite Model Property ◽  
Image Position

The intuitionistic propositional logic I has the following disjunction property This property does not characterize intuitionistic logic. For example Kreisel and Putnam [5] showed that the extension of I with the axiomhas the disjunction property. Another known system with this propery is due to Scott [5], and is obtained by adding to I the following axiom:In the present paper we shall prove, using methods originally introduced by Segerberg [10], that the Kreisel-Putnam logic is decidable. In fact we shall show that it has the finite model property, and since it is finitely axiomatizable, it is decidable by [4]. The decidability of Scott's system was proved by J. G. Anderson in his thesis in 1966.

scholarly journals COMPLETENESS OF SECOND-ORDER PROPOSITIONAL S4 AND H IN TOPOLOGICAL SEMANTICS

2018 ◽  
Vol 11 (3) ◽  
pp. 507-518
Author(s):  
PHILIP KREMER
Keyword(s):  
Modal Logic ◽  
Intuitionistic Logic ◽  
Second Order ◽  
Topological Semantics ◽  
First Order ◽  
Propositional Modal Logic ◽  
Propositional Quantifiers ◽  
Modal Logic S4 ◽  
Natural Way

AbstractWe add propositional quantifiers to the propositional modal logic S4 and to the propositional intuitionistic logic H, introducing axiom schemes that are the natural analogs to axiom schemes typically used for first-order quantifiers in classical and intuitionistic logic. We show that the resulting logics are sound and complete for a topological semantics extending, in a natural way, the topological semantics for S4 and for H.

ON FLATTENING ELIMINATION RULES

2014 ◽  
Vol 7 (1) ◽  
pp. 60-72 ◽  
Author(s):  
GRIGORY K. OLKHOVIKOV ◽  
PETER SCHROEDER-HEISTER
Keyword(s):  
Intuitionistic Logic ◽  
Propositional Logic ◽  
Second Order ◽  
Kripke Semantics ◽  
Elimination Rule

AbstractIn proof-theoretic semantics of intuitionistic logic it is well known that elimination rules can be generated from introduction rules in a uniform way. If introduction rules discharge assumptions, the corresponding elimination rule is a rule of higher level, which allows one to discharge rules occurring as assumptions. In some cases, these uniformly generated elimination rules can be equivalently replaced with elimination rules that only discharge formulas or do not discharge any assumption at all—they can be flattened in a terminology proposed by Read. We show by an example from propositional logic that not all introduction rules have flat elimination rules. We translate the general form of flat elimination rules into a formula of second-order propositional logic and demonstrate that our example is not equivalent to any such formula. The proof uses elementary techniques from propositional logic and Kripke semantics.

Logical connectives for intuitionistic propositional logic

1971 ◽  
Vol 36 (1) ◽  
pp. 15-20 ◽  
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.

scholarly journals An Alternative Natural Deduction for the Intuitionistic Propositional Logic

2016 ◽  
Vol 45 (1) ◽  
Author(s):  
Mirjana Ilić
Keyword(s):  
Intuitionistic Logic ◽  
Propositional Logic ◽  
Natural Deduction ◽  
Sequent Calculus ◽  
Intuitionistic Propositional Logic ◽  
Deduction System ◽  
Natural Deduction System

A natural deduction system NI, for the full propositional intuitionistic logic, is proposed. The operational rules of NI are obtained by the translation from Gentzen’s calculus LJ and the normalization is proved, via translations from sequent calculus derivations to natural deduction derivations and back.

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