Intermediate logics with the same disjunctionless fragment as intuitionistic logic

Studia Logica ◽  
1986 ◽  
Vol 45 (2) ◽  
pp. 207-222 ◽  
Author(s):  
Plerluigi Minari
Keyword(s):  
Intuitionistic Logic ◽  
Intermediate Logics

scholarly journals Tableau Calculus for Dummett Logic Based on Present and Next State of Knowledge

10.29007/dhz5 ◽  
2018 ◽  
Author(s):  
Guido Fiorino
Keyword(s):  
Intuitionistic Logic ◽  
Search Space ◽  
Kripke Semantics ◽  
Proof Systems ◽  
Dummett Logic ◽  
Intermediate Logics ◽  
Tableau System

In this paper we use the Kripke semantics characterization of Dummett logic to introduce a new way of handling non-forced formulas in tableau proof systems. We pursue the aim of reducing the search space by strictly increasing the number of forced propositional variables after the application of non-invertible rules. The focus of the paper is on a new tableau system for Dummett logic, for which we have an implementation. The ideas presented can be extended to intuitionistic logic and intermediate logics as well.

scholarly journals Applications of intuitionistic logic in Answer Set Programming

2004 ◽  
Vol 4 (3) ◽  
pp. 325-354 ◽  
Author(s):  
MAURICIO OSORIO ◽  
JUAN A. NAVARRO ◽  
JOSÉ ARRAZOLA
Keyword(s):  
Recent Result ◽  
Intuitionistic Logic ◽  
Answer Set Programming ◽  
Minimal Models ◽  
Logic Programs ◽  
Intermediate Logics ◽  
Answer Set Semantics ◽  
Answer Sets ◽  
Answer Set

We present some applications of intermediate logics in the field of Answer Set Programming (ASP). A brief, but comprehensive introduction to the answer set semantics, intuitionistic and other intermediate logics is given. Some equivalence notions and their applications are discussed. Some results on intermediate logics are shown, and applied later to prove properties of answer sets. A characterization of answer sets for logic programs with nested expressions is provided in terms of intuitionistic provability, generalizing a recent result given by Pearce. It is known that the answer set semantics for logic programs with nested expressions may select non-minimal models. Minimal models can be very important in some applications, therefore we studied them; in particular we obtain a characterization, in terms of intuitionistic logic, of answer sets which are also minimal models. We show that the logic G3 characterizes the notion of strong equivalence between programs under the semantic induced by these models. Finally we discuss possible applications and consequences of our results. They clearly state interesting links between ASP and intermediate logics, which might bring research in these two areas together.

scholarly journals On an Application of Intermediate Logics

1960 ◽  
Vol 16 ◽  
pp. 119-133
Author(s):  
Toshio Umezawa
Keyword(s):  
Intuitionistic Logic ◽  
Classical Logic ◽  
Logical System ◽  
Intermediate Logics ◽  
Individual Variables ◽  
Classical Predicate

In [1] I investigated some logics intermediate between intuitionistic and classical predicate logics. The purpose of this paper is to show the possibility of applying some intermediate logics to mathematics namely, to show that some mathematical theorems which are provable in the classical logic but not provable in the intuitionistic logic are provable in some intermediate logics. Let LZ be the logical system obtained from LJ′ a variant of Gentzen’s LJ [2], by adding as axioms all those sequents which can be obtained from a sequent scheme Z by substitution for propositional, predicate, or individual variables.

A sequence of decidable finitely axiomatizable intermediate logics with the disjunction property

1974 ◽  
Vol 39 (1) ◽  
pp. 67-78 ◽  
Author(s):  
D. M. Gabbay ◽  
D. H. J. De Jongh
Keyword(s):  
Intuitionistic Logic ◽  
Finite Model ◽  
Ordered Sets ◽  
Disjunction Property ◽  
University Of Illinois ◽  
Finite Model Property ◽  
Intermediate Logics ◽  
Heyting Arithmetic ◽  
Image Position ◽  
System 1

The intuitionistic propositional logic I has the following (disjunction) property.We are interested in extensions of the intuitionistic logic which are both decidable and have the disjunction property. Systems with the disjunction property are known, for example the Kreisel-Putnam system [1] which is I + (∼ϕ → (ψ ∨ α))→ ((∼ϕ→ψ) ∨ (∼ϕ→α)) and Scott's system I + ((∼ ∼ϕ→ϕ)→(ϕ ∨ ∼ϕ))→ (∼∼ϕ ∨ ∼ϕ). It was shown in [3c] that the first system has the finite-model property.In this note we shall construct a sequence of intermediate logics Dn with the following properties:These systems are presented both semantically and syntactically, using the remarkable correspondence between properties of partially ordered sets and axiom schemata of intuitionistic logic. This correspondence, apart from being interesting in itself (for giving geometric meaning to intuitionistic axioms), is also useful in giving independence proofs and obtaining proof theoretic results for intuitionistic systems (see for example, C. Smorynski, Thesis, University of Illinois, 1972, for independence and proof theoretic results in Heyting arithmetic).

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.

scholarly journals On Correspondence between Selective CPS Transformation and Selective Double Negation Translation

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 385
Author(s):  
Hyeonseung Im
Keyword(s):  
Programming Languages ◽  
Intuitionistic Logic ◽  
Reference Point ◽  
Classical Logic ◽  
Proof System ◽  
Double Negation

A double negation translation (DNT) embeds classical logic into intuitionistic logic. Such translations correspond to continuation passing style (CPS) transformations in programming languages via the Curry-Howard isomorphism. A selective CPS transformation uses a type and effect system to selectively translate only nontrivial expressions possibly with computational effects into CPS functions. In this paper, we review the conventional call-by-value (CBV) CPS transformation and its corresponding DNT, and provide a logical account of a CBV selective CPS transformation by defining a selective DNT via the Curry-Howard isomorphism. By using an annotated proof system derived from the corresponding type and effect system, our selective DNT translates classical proofs into equivalent intuitionistic proofs, which are smaller than those obtained by the usual DNTs. We believe that our work can serve as a reference point for further study on the Curry-Howard isomorphism between CPS transformations and DNTs.

On the validity of hilbert's nullstellensatz, artin's theorem, and related results in grothendieck toposes

1988 ◽  
Vol 53 (4) ◽  
pp. 1177-1187
Author(s):  
W. A. MacCaull
Keyword(s):  
Intuitionistic Logic ◽  
Main Idea ◽  
Rational Functions ◽  
Completeness Theorem ◽  
Point Of View ◽  
Polynomial Rings ◽  
Von Neumann ◽  
Ordered Fields ◽  
Von Neumann Regular ◽  
Theoretic Point

Using formally intuitionistic logic coupled with infinitary logic and the completeness theorem for coherent logic, we establish the validity, in Grothendieck toposes, of a number of well-known, classically valid theorems about fields and ordered fields. Classically, these theorems have proofs by contradiction and most involve higher order notions. Here, the theorems are each given a first-order formulation, and this form of the theorem is then deduced using coherent or formally intuitionistic logic. This immediately implies their validity in arbitrary Grothendieck toposes. The main idea throughout is to use coherent theories and, whenever possible, find coherent formulations of formulas which then allow us to call upon the completeness theorem of coherent logic. In one place, the positive model-completeness of the relevant theory is used to find the necessary coherent formulas.The theorems here deal with polynomials or rational functions (in s indeterminates) over fields. A polynomial over a field can, of course, be represented by a finite string of field elements, and a rational function can be represented by a pair of strings of field elements. We chose the approach whereby results on polynomial rings are reduced to results about the base field, because the theory of polynomial rings in s indeterminates over fields, although coherent, is less desirable from a model-theoretic point of view. Ultimately we are interested in the models.This research was originally motivated by the works of Saracino and Weispfenning [SW], van den Dries [Dr], and Bunge [Bu], each of whom generalized some theorems from algebraic geometry or ordered fields to (commutative, von Neumann) regular rings (with unity).

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