scholarly journals Failure of Interpolation in Constant Domain Intuitionistic Logic

2013 ◽  
Vol 78 (3) ◽  
pp. 937-950 ◽  
Author(s):  
Grigori Mints ◽  
Grigory Olkhovikov ◽  
Alasdair Urquhart
Keyword(s):  
Intuitionistic Logic ◽  
Interpolation Property ◽  
Interpolation Theorem ◽  
Constant Domain ◽  
Constant Domains

AbstractThis paper shows that the interpolation theorem fails in the intuitionistic logic of constant domains. This result refutes two previously published claims that the interpolation property holds.

Craig interpolation theorem for intuitionistic logic and extensions Part III

1977 ◽  
Vol 42 (2) ◽  
pp. 269-271 ◽  
Author(s):  
Dov M. Gabbay
Keyword(s):  
Possible Worlds ◽  
Predicate Logic ◽  
Interpolation Property ◽  
Interpolation Theorem ◽  
Kripke Structure ◽  
Unary Predicate ◽  
The World ◽  
Image Position ◽  
Constant Domains ◽  
Positive Integers

This is a continuation of two previous papers by the same title [2] and examines mainly the interpolation property for the logic CD with constant domains, i.e., the extension of the intuitionistic predicate logic with the schemaIt is known [3], [4] that this logic is complete for the class of all Kripke structures with constant domains.Theorem 47. The strong Robinson consistency theorem is not true for CD.Proof. Consider the following Kripke structure with constant domains. The set S of possible worlds is ω0, the set of positive integers. R is the natural ordering ≤. Let ω0 0 = , Bn, is a sequence of pairwise disjoint infinite sets. Let L0 be a language with the unary predicates P, P1 and consider the following extensions for P,P1 at the world m.(a) P is true on ⋃i≤2nBi, and P1 is true on ⋃i≤2n+1Bi for m = 2n.(b) P is true on ⋃i≤2nBi, and P1 for ⋃i≤2n+1Bi for m = 2n.Let (Δ,Θ) be the complete theory of this structure. Consider another unary predicate Q. Let L be the language with P, Q and let M be the language with P1, Q.

scholarly journals On the correspondence between nested calculi and semantic systems for intuitionistic logics

2020 ◽  
Author(s):  
Tim Lyon
Keyword(s):  
Intuitionistic Logic ◽  
Structural Rules ◽  
First Order ◽  
Constant Domains ◽  
The Relationship ◽  
Intuitionistic Logics

Abstract This paper studies the relationship between labelled and nested calculi for propositional intuitionistic logic, first-order intuitionistic logic with non-constant domains and first-order intuitionistic logic with constant domains. It is shown that Fitting’s nested calculi naturally arise from their corresponding labelled calculi—for each of the aforementioned logics—via the elimination of structural rules in labelled derivations. The translational correspondence between the two types of systems is leveraged to show that the nested calculi inherit proof-theoretic properties from their associated labelled calculi, such as completeness, invertibility of rules and cut admissibility. Since labelled calculi are easily obtained via a logic’s semantics, the method presented in this paper can be seen as one whereby refined versions of labelled calculi (containing nested calculi as fragments) with favourable properties are derived directly from a logic’s semantics.

A new version of Beth semantics for intuitionistic logic

1977 ◽  
Vol 42 (2) ◽  
pp. 306-308 ◽  
Author(s):  
Dov M. Gabbay
Keyword(s):  
Intuitionistic Logic ◽  
Predicate Logic ◽  
Tree Structure ◽  
Truth Conditions ◽  
Truth Definition ◽  
Image Position ◽  
Constant Domains ◽  
Condition B

We use the notation of Kripke's paper [1]. Let M = (G, K, R) be a tree structure and D a domain and η a Beth model on M. The truth conditions of the Beth semantics for ∨ and ∃ are (see [1]):(a) η (A ∨ B, H) = T iff for some B ⊆ K, B bars H and for each H′ ∈ B, either η(A, H′) = T or η(B, H′) = T.(b) η(∃xA(x), H) = T iff for some B ⊆ K, B bars H and for each H′ ∈ B there exists an a ∈ D such that η(A (a), H′) = T.Suppose we change the truth definition η to η* by replacing condition (b) by the condition (b*) (well known from the Kripke interpretation):Call this type of interpretation the new version of Beth semantics. We proveTheorem 1. Intuitionistic predicate logic is complete for the new version of the Beth semantics.Since Beth structures are of constant domains, and in the new version of Beth semantics the truth conditions for ∧, →, ∃, ∀, ¬ are the same as for the Kripke interpretation, we get:Corollary 2. The fragment without disjunction of the logic CD of constant domains (i.e. with the additional schema ∀x(A ∨ B(x))→ A ∨ ∀xB(x), x not free in A) equals the fragment without disjunction of intuitionistice logic.

On the interpolation theorem for the logic of constant domains

1981 ◽  
Vol 46 (1) ◽  
pp. 87-88 ◽  
Author(s):  
E. G. K. López-Escobar
Keyword(s):  
Simple Proof ◽  
Open Problem ◽  
Formal System ◽  
Kripke Model ◽  
Interpolation Theorem ◽  
Predicate Calculus ◽  
Weak Consistency ◽  
Simple Matter ◽  
Constant Domains

In Gabbay [1] it is stated as an open problem whether or not Craig's Theorem holds for the logic of constant domains CD, i.e. for the extension of the intuitionistic predicate calculus, IPC, obtained by adding the schema; . Then in the later article, [2], Gabbay gives a proof of it. The proof given in [2] is via Robinson's (weak) consistency theorem and depends on relatively complicated (Kripke-) model-theoretical constructions developed in [1] (see p. 392 of [1] for a brief sketch of the method). The aim of this note is to show that the interpolation theorem for CD can also be obtained, by simple proof-theoretic methods, from §80 of Kleene's Introduction to Metamathematics [3].GI is the classical formal system whose postulates are given on p. 442 of [3]. Let GD be the system obtained from GI by the following modifications: (1) the sequents of GD are to have at most two formulas in their succedents and (2) the intuitionistic restriction that Θ be empty is required for the succedent rules (→ ¬) and (→ ⊃). It is a simple matter to show that: , x not free in . It then follows that, using Theorem 46 of [3], if then .

Representability and amalgamation for Heyting polyadic algebras

2011 ◽  
Vol 48 (4) ◽  
pp. 509-539 ◽  
Author(s):  
Tarek Ahmed
Keyword(s):  
Intuitionistic Logic ◽  
Heyting Algebra ◽  
Interpolation Property ◽  
Amalgamation Property ◽  
Infinite Dimension ◽  
Locally Finite ◽  
Craig Interpolation ◽  
Polyadic Algebras ◽  
Craig Interpolation Property

We prove that every (not necessarily locally finite) polyadic Heyting algebra of infinite dimension is representable in some concrete sense. We also show that this class has the super amalgamation property. As a byproduct we infer that a certain infinitary extension of predicate intuitionistic logic, or equivalently, the intuitionistic fragment of Keisler’s infinitary logics, is complete and enjoys the Craig interpolation property.

scholarly journals First-order justification logic with constant domain semantics

2020 ◽  
Vol 30 (1) ◽  
pp. 193-216
Author(s):  
Melvin Fitting ◽  
Felipe Salvatore
Keyword(s):  
Justification Logic ◽  
Propositional Justification ◽  
Logic Of Proofs ◽  
First Order ◽  
Barcan Formula ◽  
Constant Domain ◽  
New Family ◽  
Domain Semantics ◽  
Constant Domains ◽  
Completeness Theorems

Abstract Justification logic is a term used to identify a relatively new family of modal-like logics. There is an established literature about propositional justification logic, but incursions on the first-order case are scarce. In this paper we present a constant domain semantics for the first-order logic of proofs with the Barcan Formula (FOLPb); then we prove Soundness and Completeness Theorems. A monotonic semantics for a version of this logic without the Barcan Formula is already in the literature, but constant domains require substantial new machinery, which may prove useful in other contexts as well. Although we work mainly with one system, we also indicate how to generalize these results for the quantified version of JT45, the justification counterpart of the modal logic S5. We believe our methods are more generally applicable, but initially examining specific cases should make the work easier to follow.

ANALYTIC CUT AND INTERPOLATION FOR BI-INTUITIONISTIC LOGIC

2016 ◽  
Vol 10 (2) ◽  
pp. 259-283 ◽  
Author(s):  
TOMASZ KOWALSKI ◽  
HIROAKIRA ONO
Keyword(s):  
Intuitionistic Logic ◽  
Interpolation Property ◽  
Separation Property ◽  
Variable Separation ◽  
Craig Interpolation ◽  
Craig Interpolation Property ◽  
Sequent Systems ◽  
Subformula Property

AbstractWe prove that certain natural sequent systems for bi-intuitionistic logic have the analytic cut property. In the process we show that the (global) subformula property implies the (local) analytic cut property, thereby demonstrating their equivalence. Applying a version of Maehara technique modified in several ways, we prove that bi-intuitionistic logic enjoys the classical Craig interpolation property and Maximova variable separation property; its Halldén completeness follows.

scholarly journals Calderon's Complex Interpolation of Morrey Spaces

2020 ◽  
Vol 26 (1) ◽  
pp. 137-164
Author(s):  
Denny Hakim
Keyword(s):  
Interpolation Method ◽  
Interpolation Property ◽  
Interpolation Theorem ◽  
Morrey Spaces ◽  
Complex Interpolation ◽  
Generalized Morrey Spaces ◽  
Definition Of

In this note we will discuss some results related to complex interpolation of Morreyspaces. We first recall the Riesz-Thorin interpolation theorem in Section 1.After that, we discuss a partial generalization of this theorem in Morrey spaces proved in \cite{St}.We also discuss non-interpolation property of Morrey spaces given in \cite{BRV99, RV}.In Section 3, we recall the definition of Calder\'on's complex interpolation method andthe description of complex interpolation of Lebesgue spaces.In Section 4, we discuss the description of complex interpolation of Morrey spaces given in\cite{CPP98, HS2, Lemarie, LYY}. Finally, we discuss the description of complex interpolationof subspaces of Morrey spaces in the last section.This note is a summary of the current research about interpolation of Morrey spaces,generalized Morrey spaces, and their subspaces in\cite{CPP98, HS, HS2, H, H4, Lemarie, LYY}.

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