Topic
Heyting arithmetic
About: Heyting arithmetic is a research topic. Over the lifetime, 317 publications have been published within this topic receiving 3767 citations.
Papers published on a yearly basis
Papers
TL;DR: It is shown that the variety of Heyting algebras has finitary unification type and that the subvariety obtained by adding it De Morgan law is the biggest variety ofHeyting alGEbras having unitary unification type.
Abstract: We show that the variety of Heyting algebras has finitary unification type. We also show that the subvariety obtained by adding it De Morgan law is the biggest variety of Heyting algebras having unitary unification type. Proofs make essential use of suitable characterizations (both from the semantic and the syntactic side) of finitely presented projective algebras. ?0. Introduction. Unification under equational conditions (briefly, E-unification) is an important tool in equational reasoning whithin the point of view of automated deduction. Research on this field mainly concentates in two directions: general algorithms enumerating all E-unifiers of an E-unification problem (see, e.g., [20]) and specific algorithms for relevant special theories [1], [19]. One of the main results obtained within the second direction is the unitarity of Boolean unification [14]. This result, once translated in logical terms, says that for every formula A in classical propositional calculus, if there is a substitution making A a theorem in this calculus, then there is also 'the best' substitution with this property, i.e., there is a substitution a such that av(A) is provable and any -c such that T(A) is provable is, up to provable equivalence, an instantiation of a. We wonder whether the same property holds for other logical calculi. In the case of intuitionistic propositional calculus (IPC), the answer is negative, as the following simple counterexample shows. The formula x V ix has unifiers (by this we mean substitutions making it a theorem in IPC) oI:x y-T, o2:xH -I (where T is 'syntactic truth' and I is 'syntactic false') and there is no unifier more general than both because of the disjunction property of IPC. In fact, if
209 citations
TL;DR: This paper concerns the question of determining all formulas which are valid in every linearly ordered Heyting algebra, a particularly simple case intermediate between the intuitionist and classical logics.
Abstract: It is known that the theorems of the intuitionist predicate calculus are exactly those formulas which are valid in every Heyting algebra (that is, pseudo-Boolean algebra). The simplest kind of Heyting algebra is a linearly ordered set. This paper concerns the question of determining all formulas which are valid in every linearly ordered Heyting algebra. The question is of interest because it is a particularly simple case intermediate between the intuitionist and classical logics. Also the interpretation of implication is such that in general there exists no nondiscrete Hausdorff topology for which this operation is continuous.
160 citations
Journal Article•
138 citations
1 Jan 2016
TL;DR: In this article, it was shown that the set of formulas valid in every linearly ordered Heyting algebra can be axiomatized by adding Dummett's axiom and one additional axiom to the intuitionist predicate calculus.
Abstract: It is known that the theorems of the intuitionist predicate calculus are exactly those formulas which are valid in every Heyting algebra (that is, pseudo-Boolean algebra). The simplest kind of Heyting algebra is a linearly ordered set. This paper concerns the question of determining all formulas which are valid in every linearly ordered Heyting algebra. The question is of interest because it is a particularly simple case intermediate between the intuitionist and classical logics. Also the interpretation of implication is such that in general there exists no nondiscrete Hausdorff topology for which this operation is continuous. For the case of propositional calculus, the question was settled by M. Dummett [1] who showed that the set of formulas in question can be axiomatized by adding the axiom schema (a v P) v (P : a) to the intuitionist propositional calculus. A simple proof of Dummett's result will be given. It will be proved below that the set of formulas valid in every linearly ordered Heyting algebra can be axiomatized by adding Dummett's axiom and one additional axiom to the intuitionist predicate calculus. We conclude with a compactness theorem for this intermediate predicate calculus. If L is any lattice, we shall use + and - for the operations of least upper bound and greatest lower bound of two elements. The smallest and largest elements of L (if they exist) will be denoted by 0 and 1 respectively. A Heyting algebra is a lattice with 0 such that for any two elements x, y there exists a largest element z such that xz < y. z is called the relative pseudo-complement of x relative to y and is denoted
84 citations
TL;DR: In this note, a sequence of intermediate logics D is constructed, with the following properties, which are interested in extensions of the intuitionistic logic which are both decidable and have the disjunction property.
Abstract: 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).
77 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 6 |
| 2020 | 7 |
| 2019 | 5 |
| 2018 | 5 |
| 2017 | 14 |
| 2016 | 8 |