Modal Logic S4 Research Articles

Wanted dead or alive: epistemic logic for impure simplicial complexes

Abstract We propose a logic of knowledge for impure simplicial complexes. Impure simplicial complexes represent synchronous distributed systems under uncertainty over which processes are still active (are alive) and which processes have failed or crashed (are dead). Our work generalizes the logic of knowledge for pure simplicial complexes, where all processes are alive, by Goubault et al. In our semantics, given a designated face in a complex, a formula can only be true or false there if it is defined. The following are undefined: dead processes cannot know or be ignorant of any proposition, and live processes cannot know or be ignorant of factual propositions involving processes they know to be dead. The semantics are therefore three-valued, with undefined as the third value. We propose an axiomatization that is a version of the modal logic S5. We also show that impure simplicial complexes correspond to certain Kripke models where agents’ accessibility relations are equivalence relations on a subset of the domain only.

On Combining Intuitionistic and S4 Modal Logic

We address the problem of combining intuitionistic and S4 modal logic in a non-collapsing way inspired by the recent works in combining intuitionistic and classical logic. The combined language includes the shared constructors of both logics namely conjunction, disjunction and falsum as well as the intuitionistic implication, the classical implication and the necessity modality. We present a Gentzen calculus for the combined logic defined over a Gentzen calculus for the host S4 modal logic. The semantics is provided by Kripke structures. The calculus is proved to be sound and complete with respect to this semantics. We also show that the combined logic is a conservative extension of each component. Finally we establish that the Gentzen calculus for the combined logic enjoys cut elimination.

Satisfiability Threshold of Random Propositional S5 Theories

Modal logic S5, which isan important knowledge representation and reasoning paradigm, has been successfully applied in various artificial-intelligence-related domains. Similar to the random propositional theories in conjunctive clause form, the phase transition plays an important role in designing efficient algorithms for computing models of propositional S5 theories. In this paper, a new form of S5 formula is proposed, which fixes the number of modal operators and literals in the clauses of the formula. This form consists of reduced 3-3-S5 clauses of the form l1∨l2∨ξ, where ξ takes the form □(l3∨l4∨l5), ◊(l3∧l4∧l5), or a propositional literal, and li(1≤i≤5) is a classical literal. Moreover, it is demonstrated that any S5 formula can be translated into a set of reduced 3-3-S5 clauses while preserving its satisfiability. This work further investigates the probability of a random 3-c-S5 formula with c=1,2,3 being satisfied by random assignment. In particular, we show that the satisfiability threshold of random 3-1-S5 clauses is −ln2(1−Pd−Ps)ln78+Ps·ln34, where Ps and Pd denote the probabilities of different modal operators appearing in a clause. Preliminary experimental results on random 3-1-S5 formulas confirm this theoretical threshold.

On Paracomplete Versions of Jaśkowski's Discussive Logic

Jaśkowski's discussive (discursive) logic D2 is historically one of the first paraconsistent logics, i.e., logics which 'tolerate' contradictions. Following Jaśkowski's idea to define his discussive logic by means of the modal logic S5 via special translation functions between discussive and modal languages, and supporting at the same time the tradition of paracomplete logics being the counterpart of paraconsistent ones, we present a paracomplete discussive logic D2p.

On some classes of formulas in S5 which are pre-complete relative to existential expressibility

Existential expressibility for all k-valued functions was proposed by A. V. Kuznetsov and later was investigated in more details by S. S. Marchenkov. In the present paper, we consider existential expressibility in the case of formulas defined by a logical calculus and find out some conditions for a system of formulas to be closed relative to existential expressibility. As a consequence, it has been established some pre-complete as to existential expressibility classes of formulas in some finite extensions of the paraconsistent modal logic S5.

An elementary belief function logic

ABSTRACT Non-additive uncertainty theories, typically possibility theory, belief functions and imprecise probabilities share a common feature with modal logic: the duality properties between possibility and necessity measures, belief and plausibility functions as well as between upper and lower probabilities extend the duality between possibility and necessity modalities to the graded environment. It has been shown that the all-or-nothing version of possibility theory can be exactly captured by a minimal epistemic logic (MEL) that uses a very small fragment of the KD modal logic, without resorting to relational semantics. Independently, a belief function logic has been obtained by extending the modal logic S5 to probabilistic graded modalities using Łukasiewicz logic, albeit using relational semantics. This paper shows that a simpler belief function logic can be devised by adding Łukasiewicz logic on top of MEL. It allows for a more natural semantics in terms of Shafer basic probability assignments.

Įrodymų specializacija modalumo logikai S5

Loop-check-free decidable specialization of sequent calculus for modal logic S5 is presented. Soundness and completness of this calculus is proved.

Rooted Hypersequent Calculus for Modal Logic S5

Rooted Hypersequent Calculus for Modal Logic S5

Fitch-Style Modal Necessity as a Substructural Sequent-Style System

The question of defining a Jaskowski-Fitch natural deduction system for modal logic has been settled since the very introduction of such formalisms in the middle of the last century. In contrast, a sequent-style formulation of this approach has only been discussed since the turn of this century but exclusively from the point of view of type theories. In this paper we propose a substructural sequent-style deductive system, based on previous ideas by Borghuis and Clouston, which captures Fitch-style for modal logic in a faithful way, meaning that the features of the original diagrammatic proofs are enforced by the sequent rules. This answers the question of what is a sequent-style version of Fitch-style natural deduction in the case of the necessity fragment of minimal modal logic S4.

A reduction-based cut-free Gentzen calculus for dynamic epistemic logic

Abstract Dynamic epistemic logic (DEL) is a multi-modal logic for reasoning about the change of knowledge in multi-agent systems. It extends epistemic logic by a modal operator for actions which announce logical formulas to other agents. In Hilbert-style proof calculi for DEL, modal action formulas are reduced to epistemic logic, whereas current sequent calculi for DEL are labelled systems which internalize the semantic accessibility relation of the modal operators, as well as the accessibility relation underlying the semantics of the actions. We present a novel cut-free ordinary sequent calculus, called $ \textbf{G4}_{P,A}[] $, for propositional DEL. In contrast to the known sequent calculi, our calculus does not internalize the accessibility relations, but—similar to Hilbert style proof calculi—action formulas are reduced to epistemic formulas. Since no ordinary sequent calculus for full S5 modal logic is known, the proof rules for the knowledge operator and the Boolean operators are those of an underlying S4 modal calculus. We show the soundness and completeness of $ \textbf{G4}_{P,A}[] $ and prove also the admissibility of the cut-rule and of several other rules for introducing the action modality.

An algebraic study of the logic S5’(BL)

Abstract P. Hájek introduced an S5-like modal fuzzy logic S5(BL) and showed that is equivalent to the monadic basic predicate logic mBL∀ . Inspired by the above important results, D. Castaño et al. introduced monadic BL-algebras and their corresponding propositional logic S5’(BL), which is a simplified set of axioms of S5(BL). In this paper, we review the algebraic semantics of S5’(BL) and obtain some new results regarding to monadic BL-algebras. First we recall that S5’(BL) is completeness with respect to the variety 𝕄𝔹𝕃 of monadic BL-algebras and obtain a necessary and sufficient condition for the logic S5’(BL) to be semilinear. Then we study some further algebraic properties of monadic BL-algebras and discuss the relationship between monadic MV-algebras and monadic BL-algebras. Finally we give some characterizations of representable, simple, semisimple and directly indecomposable monadic BL-algebras, which are important members of the variety 𝕄𝔹𝕃. These results also constitute a crucial first step for providing an equivalent algebraic foundation for mBL∀ .

A dual-context sequent calculus for the constructive modal logic S4

AbstractThe proof theory of the constructive modal logic S4 (hereafter $\mathsf{CS4}$ ) has been settled since the beginning of this century by means of either standard natural deduction and sequent calculi or by the reconstruction of modal logic through hypothetical and categorical judgments à la Martin-Löf, an approach carried out by using a special kind of sequents, which keeps two separated contexts representing ordinary and enhanced hypotheses, intuitively interpreted as true and valid assumptions. These so-called dual-context sequents, originated in linear logic, are used to define a natural deduction system handling judgments of validity, truth, and possibility, resulting in a formalism equivalent to an axiomatic system for $\mathsf{CS4}$ . However, this proof-theoretical study of $\mathsf{CS4}$ lacks, to the best of our knowledge, its third fundamental constituent, namely a sequent calculus. In this paper, we define such a dual-context formalism, called ${\bf DG_{CS4}}$ , and provide detailed proofs of the admissibility for the ordinary cut rule as well as the elimination of a second cut rule, which manipulates enhanced hypotheses. Furthermore, we make available a formal verification of the equivalence of this proposal with the previously defined axiomatic and dual-context natural deduction systems for $\mathsf{CS4}$ , using the Coq proof-assistant.

Potentialism and S5

AbstractModal potentialism as proposed by Barbara Vetter (2015) is the view that every possibility is grounded in something having a potentiality. Drawing from work by Jessica Leech (2017), Samuel Kimpton-Nye (2021) argues that potentialists can have an S5 modal logic. I present a novel argument to the conclusion that the most straightforward way of spelling out modal potentialism cannot validate an S5 modal logic. Then I will propose a slightly tweaked version of modal potentialism that can validate an S5 modal logic and still does justice to the core claim of potentialism.

Why Does Propositional Quantification Make Modal and Temporal Logics on Trees Robustly Hard?

Adding propositional quantification to the modal logics K, T or S4 is known to lead to undecidability but CTL with propositional quantification under the tree semantics (tQCTL) admits a non-elementary Tower-complete satisfiability problem. We investigate the complexity of strict fragments of tQCTL as well as of the modal logic K with propositional quantification under the tree semantics. More specifically, we show that tQCTL restricted to the temporal operator EX is already Tower-hard, which is unexpected as EX can only enforce local properties. When tQCTL restricted to EX is interpreted on N-bounded trees for some N >= 2, we prove that the satisfiability problem is AExpPol-complete; AExpPol-hardness is established by reduction from a recently introduced tiling problem, instrumental for studying the model-checking problem for interval temporal logics. As consequences of our proof method, we prove Tower-hardness of tQCTL restricted to EF or to EXEF and of the well-known modal logics such as K, KD, GL, K4 and S4 with propositional quantification under a semantics based on classes of trees.

Plotkin's call-by-value λ-calculus as a modal calculus

In the authors' previous analysis of the calling paradigms call-by-name and call-by-value through Girard's and Gödel's embeddings into the S4 modal logic, an asymmetry remains: the two paradigms are unified by the call-by-box paradigm of the modal target, but only for call-by-name can one say that the paradigm exists, up to isomorphism, inside the modal target. In this paper, we show that, by pushing further the modal analysis, a symmetric situation is revealed, in that ordinary and Plotkin's λ-calculi are shown to truly co-exist inside a simple modal calculus.

The actual challenge for the ontological argument

Abstract Many versions of the ontological argument have two shortcomings: First, despite the arguments put forward, for example, by Hugh Chandler and Nathan Salmon, they assume that S5 is the correct modal logic for metaphysical modality. Second, despite the classical arguments put forward, for example, by David Hume and Immanuel Kant or the more recent arguments put forward, for example, by John Findlay and Richard Swinburne, they assume that necessary existence is possible. The aim of the paper is to develop an alternative version of the ontological argument that, by an appeal to the logic of ‘actually’, avoids these two shortcomings. The version helps to leave aside unnecessary debates about the modal logic S5 or about the possibility of necessary existence and to focus on the actual challenge for the debate about the ontological argument: the challenge to evaluate the claim that it is possible that God exists.

Varieties of Relevant S5

In classically based modal logic, there are three common conceptions of necessity, the universal conception, the equivalence relation conception, and the axiomatic conception. They provide distinct presentations of the modal logic S5, all of which coincide in the basic modal language. We explore these different conceptions in the context of the relevant logic R, demonstrating where they come apart. This reveals that there are many options for being an S5-ish extension of R. It further reveals a divide between the universal conception of necessity on the one hand, and the axiomatic conception on the other: The latter is consistent with motivations for relevant logics while the former is not. For the committed relevant logician, necessity cannot be the truth in all possible worlds.

Logic of the ontological argument

In his ontological argument Gödel says nothing about its underlying logic. The argument is modal and at least of second-order and since S5 axiom is used so it is widely accepted that the logic of the argument is the S5 second-order modal logic. However, there is a step in the proof in which Gödel applies the necessitation rule on the assumptions of the argument (see [3]). This is repeated by all of his followers (see [1] and [5]). This application of the necessitation rule can seriously harm the consequence relation of the logic of the ontological argument. It seems that the only way to preserve the modal logic S5 for the ontological argument is to assume some of its axioms in the necessitated form.

Attainable Knowledge and Omniscience

The paper investigates an evidence-based semantics for epistemic logics. It is shown that the properties of knowledge obtained from a potentially infinite body of evidence are described by modal logic S5. At the same time, the properties of knowledge obtained from only a finite subset of this body are described by modal logic S4. The main technical result is a sound and complete bi-modal logical system that describes properties of these two modalities and their interplay.

Expressing discrete spatial relations under granularity

The logic UBiSKt is a bi-intuitionistic modal logic with universal modalities having a semantics in which formulae are interpreted as subgraphs. We use the logic to formalize the idea of looking at subgraphs at different levels of detail. We show how the logic can be used to define various spatial relations between subgraphs. These relations are considered with respect to change in level of detail. We use an existing formulation of generalized partitions on graphs and hypergraphs in terms of relations to give a novel generalization of the classical modal logic S5 to a bi-intuitionistic modal logic we call UBiSKtS5.