Intuitionistic First-order Logic Research Articles

An Intuitionistically Complete System of Basic Intuitionistic Conditional Logic

We introduce a basic intuitionistic conditional logic IntCK\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{IntCK}$$\\end{document} that we show to be complete both relative to a special type of Kripke models and relative to a standard translation into first-order intuitionistic logic. We show that IntCK\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{IntCK}$$\\end{document} stands in a very natural relation to other similar logics, like the basic classical conditional logic CK\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{CK}$$\\end{document} and the basic intuitionistic modal logic IK\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{IK}$$\\end{document}. As for the basic intuitionistic conditional logic ICK\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{ICK}$$\\end{document} proposed in Weiss (Journal of Philosophical Logic, 48, 447–469, 2019), IntCK\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{IntCK}$$\\end{document} extends its language with a diamond-like conditional modality ◊→\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Diamond \\hspace{-4.0pt}\\rightarrow $$\\end{document}, but its (◊→\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Diamond \\hspace{-4.0pt}\\rightarrow $$\\end{document})-free fragment is also a proper extension of ICK\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{ICK}$$\\end{document}. We briefly discuss the resulting gap between the two candidate systems of basic intuitionistic conditional logic and the possible pros and cons of both candidates.

Constructive Logic is Connexive and Contradictory

It is widely accepted that there is a clear sense in which the first-order paraconsistent constructive logic with strong negation of Almukdad and Nelson, QN4, is more constructive than intuitionistic first-order logic, QInt. While QInt and QN4 both possess the disjunction property and the existence property as characteristics of constructiveness (or constructivity), QInt lacks certain features of constructiveness enjoyed by QN4, namely the constructible falsity property and the dual of the existence property. This paper deals with the constructiveness of the contra-classical, connexive, paraconsistent, and contradictory non-trivial first-order logic QC, which is a connexive variant of QN4. It is shown that there is a sense in which QC is even more constructive than QN4. The argument focuses on a problem that is mirror-inverted to Raymond Smullyan’s drinker paradox, namely the invalidity of what will be called the drinker truism and its dual in QN4 (and QInt), and on a version of the Brouwer-Heyting-Kolmogorov interpretation of the logical operations that treats proofs and disproofs on a par. The validity of the drinker truism and its dual together with the greater constructiveness of QC in comparison to QN4 may serve as further motivation for the study of connexive logics and suggests that constructive logic is connexive and contradictory (the latter understood as being negation inconsistent).

Completeness classes for intuitionistic first-order temporal logic with time gaps

There is not abstract.

Material dialogues for first-order logic in constructive type theory: extended version

Abstract Dialogues are turn-taking games which model debates about the satisfaction of logical formulas. A novel variant played over first-order structures gives rise to a notion of first-order satisfaction. We study the induced notion of validity for classical and intuitionistic first-order logic in the constructive setting of the calculus of inductive constructions. We prove that such material dialogue semantics for classical first-order logic admits constructive soundness and completeness proofs, setting it apart from standard model-theoretic semantics of first-order logic. Furthermore, we prove that completeness with regard to intuitionistic material dialogues fails in both constructive and classical settings. As an alternative, we propose material dialogues played over Kripke structures. These Kripke material dialogues exhibit constructive completeness when restricting to the negative fragment. The results concerning classical material dialogues have been mechanized using the Coq interactive theorem prover.

FIRST-ORDER HOMOTOPICAL LOGIC

Abstract We introduce a homotopy-theoretic interpretation of intuitionistic first-order logic based on ideas from Homotopy Type Theory. We provide a categorical formulation of this interpretation using the framework of Grothendieck fibrations. We then use this formulation to prove the central property of this interpretation, namely homotopy invariance. To do this, we use the result from [8] that any Grothendieck fibration of the kind being considered can automatically be upgraded to a two-dimensional fibration, after which the invariance property is reduced to an abstract theorem concerning pseudonatural transformations of morphisms into two-dimensional fibrations.

A Lindström theorem for intuitionistic first-order logic

We extend the main result of [1] to the first-order intuitionistic logic (with and without equality), showing that it is a maximal (with respect to expressive power) abstract logic satisfying a certain form of compactness, the Tarski union property and preservation under asimulations. A similar result is also shown for the intuitionistic logic of constant domains.

Modern perspectives in Proof Theory.

Modern perspectives in Proof Theory.

Another neighbourhood semantics for intuitionistic logic

Abstract In this paper we first introduce a new neighbourhood semantics for propositional intuitionistic logic. We then naturally extend this semantics to first-order intuitionistic logic. We also study bisimulation between neighbourhood models and prove some of their basic properties for both propositional and first-order intuitionistic logic.

THE FIRST-ORDER LOGIC OF CZF IS INTUITIONISTIC FIRST-ORDER LOGIC

AbstractWe prove that the first-order logic of CZF is intuitionistic first-order logic. To do so, we introduce a new model of transfinite computation (Set Register Machines) and combine the resulting notion of realisability with Beth semantics. On the way, we also show that the propositional admissible rules of CZF are exactly those of intuitionistic propositional logic.

Combining First-Order Classical and Intuitionistic Logic

This paper studies a first-order expansion of a combination C+J of intuitionistic and classical propositional logic, which was studied by Humberstone (1979) and del Cerro and Herzig (1996), from a proof-theoretic viewpoint. While C+J has both classical and intuitionistic implications, our first-order expansion adds classical and intuitionistic universal quantifiers and one existential quantifier to C+J. This paper provides a multi-succedent sequent calculus G(FOC+J) for our combination of the first-order intuitionistic and classical logic. Our sequent calculus G(FOC+J) restricts contexts of the right rules for intuitionistic implication and intuitionistic universal quantifier to particular forms of formulas. The cut-elimination theorem is established to ensure the subformula property. As a corollary, G(FOC+J) is conservative over both first-order intuitionistic and classical logic. Strong completeness of G(FOC+J) is proved via a canonical model argument.

Not all Kripke models of [formula omitted] are locally [formula omitted

Let K be an arbitrary Kripke model of Heyting Arithmetic, HA. For every node k in K, we can view the classical structure of k, Mk as a model of some classical theory of arithmetic. Let T be a classical theory in the language of arithmetic. We say K is locally T, iff for every k in K, Mk⊨T. One of the most important problems in the model theory of HA is the following question: Is every Kripke model ofHAlocallyPA? We answer this question negatively. We introduce two new Kripke model constructions to this end. The first construction actually characterizes the arithmetical structures that can be the root of a Kripke model K⊩HA+ECT0 (ECT0 stands for Extended Church Thesis). The characterization says that for every arithmetical structure M, there exists a rooted Kripke model K⊩HA+ECT0 with the root r such that Mr=M iff M⊨ThΠ2(PA). One of the consequences of this characterization is that there is a rooted Kripke model K⊩HA+ECT0 with the root r such that Mr⊭IΔ1 and hence K is not even locally IΔ1. The second Kripke model construction is an implicit way of doing the first construction which works for any reasonable consistent intuitionistic arithmetical theory T with a recursively enumerable set of axioms that has the existence property. We get a sufficient condition from this construction that describes when for an arithmetical structure M, there exists a rooted Kripke model K⊩T with the root r such that Mr=M. As applications of this sufficient condition, we construct two new Kripke models. The first one is a Kripke model K⊩HA+¬θ+MP (θ is an instance of ECT0 and MP is Markov's principle) which is not locally IΔ1. The second one is a Kripke model K⊩HA such that K forces exactly the sentences that are provable from HA, but it is not locally IΔ1. Also, we will prove that every countable Kripke model of intuitionistic first-order logic can be transformed into another Kripke model with the full infinite binary tree as the Kripke frame such that both Kripke models force the same sentences. So with the previous result, there is a binary Kripke model K of HA such that K is not locally IΔ1.

Automata theory approach to predicate intuitionistic logic

Abstract Predicate intuitionistic logic is a well-established fragment of dependent types. Proof construction in this logic, as the Curry–Howard isomorphism states, is the process of program synthesis. We present automata that can handle proof construction and program synthesis in full intuitionistic first-order logic. Given a formula, we can construct an automaton such that the formula is provable if and only if the automaton has an accepting run. As further research, this construction makes it possible to discuss formal languages of proofs or programs, the closure properties of the automata and their connections with the traditional logical connectives.

Logics of intuitionistic Kripke-Platek set theory

We investigate the logical structure of intuitionistic Kripke-Platek set theory IKP, and show that the first-order logic of IKP is intuitionistic first-order logic IQC.

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

AbstractThis 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.

Intuitionistic fixed point logic

We study the system IFP of intuitionistic fixed point logic, an extension of intuitionistic first-order logic by strictly positive inductive and coinductive definitions. We define a realizability interpretation of IFP and use it to extract computational content from proofs about abstract structures specified by arbitrary classically true disjunction free formulas. The interpretation is shown to be sound with respect to a domain-theoretic denotational semantics and a corresponding lazy operational semantics of a functional language for extracted programs. We also show how extracted programs can be translated into Haskell. As an application we extract a program converting the signed digit representation of real numbers to infinite Gray code from a proof of inclusion of the corresponding coinductive predicates.

Anselme et Descartes, Deux Arguments Logico-Théologiques

Ce texte est une version profondément remaniée d’une conférence donnée sur ce sujet en février 2018 lors de la journée d’études à Nancy, dans le cadre des Archives Vuillemin, et en mai 2018, à l’Université de Brasilia. Contrairement à ce que j’ai affirmée dans ces conférences ainsi que dans une publication précédente, la formalisation de l’argument de Descartes (que je développe ici en déduction naturelle) montre que la logique minimale ne suffit pas à traduire l’argument de Descartes; la logique intuitionniste est requise pour que cet argument soit valide. Si la traduction de l’argument d’Anselme que je donne pour commencer contredit l’analyse et les conclusions de Vuillemin sur ce sujet, en revanche, l’interprétation que Vuillemin fait du système de Descartes comme philosophie intuitionniste est confirmée par la formalisation qui suit. Enfin, le pluralisme philosophique propre à l’esprit de la classification de Vuillemin est aussi éclairé par la conclusion de cet article.

Interpolation in Extensions of First-Order Logic

We prove a generalization of Maehara’s lemma to show that the extensions of classical and intuitionistic first-order logic with a special type of geometric axioms, called singular geometric axioms, have Craig’s interpolation property. As a corollary, we obtain a direct proof of interpolation for (classical and intuitionistic) first-order logic with identity, as well as interpolation for several mathematical theories, including the theory of equivalence relations, (strict) partial and linear orders, and various intuitionistic order theories such as apartness and positive partial and linear orders.

Intuitionistic ancestral logic

Abstract In this article we define pure intuitionistic Ancestral Logic ( iAL ), extending pure intuitionistic First-Order Logic ( iFOL ). This logic is a dependently typed abstract programming language with computational functionality beyond iFOL given by its realizer for the transitive closure, TC . We derive this operator from the natural type theoretic definition of TC using intersection. We show that provable formulas in iAL are uniformly realizable, thus iAL is sound with respect to constructive type theory. We further show that iAL subsumes Kleene Algebras with tests and thus serves as a natural programming logic for proving properties of program schemes. We also extract schemes from proofs that iAL specifications are solvable.

An intuitionistic logic for preference relations

AbstractWe investigate in intuitionistic first-order logic various principles of preference relations alternative to the standard ones based on the transitivity and completeness of weak preference. In particular, we suggest two ways in which completeness can be formulated while remaining faithful to the spirit of constructive reasoning, and we prove that the cotransitivity of the strict preference relation is a valid intuitionistic alternative to the transitivity of weak preference. Along the way, we also show that the acyclicity axiom is not finitely axiomatizable in first-order logic.

Undecidability of First-Order Modal and Intuitionistic Logics with Two Variables and One Monadic Predicate Letter

We prove that the positive fragment of first-order intuitionistic logic in the language with two variables and a single monadic predicate letter, without constants and equality, is undecidable. This holds true regardless of whether we consider semantics with expanding or constant domains. We then generalise this result to intervals [QBL, QKC] and [QBL, QFL], where QKC is the logic of the weak law of the excluded middle and QBL and QFL are first-order counterparts of Visser's basic and formal logics, respectively. We also show that, for most "natural" first-order modal logics, the two-variable fragment with a single monadic predicate letter, without constants and equality, is undecidable, regardless of whether we consider semantics with expanding or constant domains. These include all sublogics of QKTB, QGL, and QGrz -- among them, QK, QT, QKB, QD, QK4, and QS4.