Intuitionistic modal logic made explicit

This paper characterizes the admissible rules for six interesting intuitionistic modal logics: iCK4, iCS4≡IPC, strong Löb logic iSL, modalized Heyting calculus mHC, Kuznetsov-Muravitsky logic KM, and propositional lax logic PLL. Admissible rules are rules that can be added to a logic without changing the set of theorems of the logic. We provide a Gentzen-style proof theory for admissibility that combines methods known for intuitionistic propositional logic and classical modal logic. From this proof theory, we extract bases for the admissible rules, i.e., sets of admissible rules that derive all other admissible rules. In addition, we show that admissibility is decidable for these logics.

The main results of the paper are the following: For each monadic prepositional formula φ which is classically true but not intuitionistically so, there is a continuum of intuitionistic monotone modal logics L such that L+φ is inconsistent. There exists a consistent intuitionistic monotone modal logic L such that for any formula φ of the kind mentioned above the logic L+φ is inconsistent. There exist at least countably many maximal intuitionistic monotone modal logics.

This paper presents sequent calculi in which proof search is terminating for two intuitionistic modal logics, the intuitionistic versions of the classical modal logics K and KD without a diamond operator. The calculi are extensions of the terminating sequent calculus G4ip for intuitionistic propositional logic that was discovered independently by Dyckhoff and Hudelmaier around 1990. It is shown by proof–theoretic means that these terminating calculi are equivalent to the cutfree extensions of G3ip that form some of the standard calculi for intuitionistic modal logics.

The purpose of the research presented in this paper is to extend the author's results on sequent forms of Herbrand theorems for classical and intuitionistic logics onto classical and intuitionistic modal sequent logics. It was found that the technique reported at the SYNASC 2008 symposium and based on the original notions of admissibility and compatibility can satisfactorily be applied for proving Herbrand theorems for the logics under consideration in the sequent form.

Intuitionistic logic encompasses the principles of reasoning which were used by L. Brouwer in developing his intuitionistic mathematics, beginning in [Bro07]. Intuitionistic logic can be succinctly described as classical logic without the law ϕ ∨ ¬ϕ of excluded middle. Brouwer observed that this law was abstracted from finite situations and its application to statements about infinite collections is not justified. One of the consequences of the rejection of the law of excluded middle is that every intuitionistic proof of an existential sentence can be effectively transformed into an intuitionistic proof of an instance of that sentence. More precisely, if a formula of the form ∃xϕ(x) without free variables is provable in the intuitionistic predicate logic, then there is a term t without free variables such that ϕ(t) is provable. In particular, if ϕ ∨ ψ is provable, then either ϕ or ψ is provable. In that sense intuitionistic logic may provide a logical basis for constructive reasoning. A formal system of intuitionistic logic was proposed in [Hey30]. A relationship between the classical propositional logic PC and intuitionistic propositional logic INT was proved by Glivenko in [Gli29], namely a PC-formula ϕ is PC-provable if and only if ¬ ¬ϕ is INT-provable. Kripke semantics for intuitionistic logic was developed in [Kri65].

Some modal logics based on logics weaker than the classical logic have been studied by Fitch [4], Prior [7], Bull [1], [2], [3], Prawitz [6] etc. Here we treat modal logics based on the intuitionistic propositional logic, which call intuitionistic modal logics (abbreviated as IML’s). Let H be the intuitionistic propositional logic formulated in the Hilbertstyle. The rules of inference of H are modus ponens and the rule of substitution. The IML L0 is obtained from H by adding the following three axioms,

We consider a general format for sequent rules for not necessarily normal modal logics based on classical or intuitionistic propositional logic and provide relatively simple local conditions ensuring cut elimination for such rule sets. The rule format encompasses e.g. rules for the boolean connectives and transitive modal logics such as S4 or its constructive version. We also adapt the method of constructing suitable rule sets by saturation to the intuitionistic setting and provide a criterium for translating axioms for intuitionistic modal logics into sequent rules. Examples include constructive modal logics and conditional logic $\mathbb{VA}$ .

This thesis is a study of intuitionistic semantics as presented by Beth [2] and Kripke [12], using the usual methods of investigation of classical informal logic. Beth models and Kripke models are presented in a manner which does not depend upon a prior definition of the notion of degree of a formula. It is shown that both classes of intuitionistic models are a generalization of the concept of classical models, i.e. they contain classical models as particular case, and that " branching " is a necessary condition in order that intuitionistic logic be complete with respect to them. Intuitionistic sentential calculus is complete with respect to the strong Beth model, the intersection of Beth models and Kripke models. But, if by analogy with the classical case, we extend them to first order logic we find that they are not adequate because, for example, the sentence [formula omitted]x(B(x) V C) [formula omitted] ([formula omitted]xB(x) V C), where C does not contain x free, is valid in these models but not intuitionistically provable. This observation helps to explain the formal differences between the two classes of models. The simplified Kripke models and the simplified Beth models are then introduced and their equivalence with the Kripke models and the Beth models, respectively, is proved. The first ones allow a better notation and a better understanding of the relation R occurring in the definition of Kripke models. The second ones have the important property that, if the domain is finite, any classically valid sentence is valid in them. Finally a semantic proof of most of the reduction theorems from classical to intuitionistic logic is given.

Justification logics were introduced by Artemov in 1995 to provide intuitionistic logic with a classical provability semantics, a problem originally posed by Gödel. Justification logics are refinements of modal logics and formally connected to them by so-called realization theorems. A constructive proof of a realization theorem typically relies on a cut-free sequent-style proof system for the corresponding modal logic. A uniform realization theorem for all the modal logics of the so-called modal cube, i.e., for the extensions of the basic modal logic K with any subset of the axioms d, t, b, 4, and 5, has been proven using nested sequents. However, the proof was not modular in that some realization theorems required postprocessing in the form of translation on the justification logic side. This translation relied on additional restrictions on the language of the justification logic in question, thus, narrowing the scope of realization theorems. We present a fully modular proof of the realization theorems for the modal cube that is based on modular nested sequents introduced by Marin and Straßburger.

Several justification logics have been created, starting with the logic LP, (Artemov, Bull Symbolic Logic 7(1):1---36, 2001). These can be thought of as explicit versions of modal logics, or of logics of knowledge or belief, in which the unanalyzed necessity (knowledge, belief) operator has been replaced with a family of explicit justification terms. We begin by sketching the basics of justification logics and their relations with modal logics. Then we move to new material. Modal logics come in various strengths. For their corresponding justification logics, differing strength is reflected in different vocabularies. What we show here is that for justification logics corresponding to modal logics extending T, various familiar extensions are actually conservative with respect to each other. Our method of proof is very simple, and general enough to handle several justification logics not directly corresponding to distinct modal logics. Our methods do not, however, allow us to prove comparable results for justification logics corresponding to modal logics that do not extend T. That is, we are able to handle explicit logics of knowledge, but not explicit logics of belief. This remains open.

The famous van Benthem theorem states that modal logic corresponds exactly to the fragment of first-order logic that is invariant under bisimulation. In this article we prove an exact analogue of this theorem in the framework of modal dependence logic (MDL) and team semantics. We show that Modal Team Logic (MTL) extending MDL by classical negation captures exactly the FO-definable bisimulation invariant properties of Kripke structures and teams. We also compare the expressive power of MTL to most of the variants and extensions of MDL recently studied in the area.

In this paper, intuitionistic modal logics which do not admit the law of the excluded middle are studied. The main result is that there exista a continuum of such logics.

We present a game semantics for modal propositional logic Grz. Since intuitionistic logic may be embedded into Grz, the semantics also covers intuitionistic propositional logic. That semantics is simpler than Lorenzen's ‘dialogue’ games, but lacks predicates.

Full classical S5 in natural deduction with weak normalization

The present paper studies two approaches to the expressiveness of propositional modal logics based on first-degree entailment logic, FDE. We first consider the basic FDE-based modal logic BK and certain systems in its vicinity, and then turn to some FDE-based modal logics in a richer vocabulary, including modal bilattice logic, MBL. On the one hand, model-theoretic proofs of the definability of connectives along the lines of [McCullough, “Logical connectives for intuitionistic propositional logic”, Journal of Symbolic Logic 36, 1 (1971): 15–20. DOI: 10.2307/2271511] and [[17] Wansing, “Logical connectives for constructive modal logic”, Synthese 150, 3 (2006): 459–482. DOI: 10.1007/s11229-005-5518-5] are given for various FDE-based modal logics. On the other hand, building on [Odintsov and Wansing, “Disentangling FDE-based paraconsistent modal logics, Studia Logica 105, 6 (2017): 1221–1254. DOI: 10.1007/s11225-017-9753-9], expressibility is considered in terms of mutual faithful embeddability of one logic into another logic. A distinction is drawn between definitional equivalence, which is defined with respect to a pair of structural translations between two languages, and weak definitional equivalence, which is defined with respect to a weaker notion of translations. Moreover, the definitional equivalence of some FDE-based modal logics is proven, especially the definitional equivalence of MBL and a conservative extension of the logic BK□×BK□, which underlines the central role played by BK among FDE-based modal logics.