Modal Logic Research Articles

Uniform lyndon interpolation for basic non-normal modal and conditional logics

Abstract In this paper, a proof-theoretic method to prove uniform Lyndon interpolation (ULIP) for non-normal modal and conditional logics is introduced and applied to show that the logics, $\textsf{E}$, $\textsf{M}$, $\textsf{EN}$, $\textsf{MN}$, $\textsf{MC}$, $\textsf{K}$, and their conditional versions, $\textsf{CE}$, $\textsf{CM}$, $\textsf{CEN}$, $\textsf{CMN}$, $\textsf{CMC}$, $\textsf{CK}$, in addition to $\textsf{CKID}$ have that property. In particular, it implies that these logics have uniform interpolation (UIP). Although for some of them the latter is known, the fact that they have uniform LIP is new. Also, the proof-theoretic proofs of these facts are new, as well as the constructive way to explicitly compute the interpolants that they provide. On the negative side, it is shown that the logics $\textsf{CKCEM}$ and $\textsf{CKCEMID}$ enjoy UIP but not uniform LIP. Moreover, it is proved that the non-normal modal logics, $\textsf{EC}$ and $\textsf{ECN}$, and their conditional versions, $\textsf{CEC}$ and $\textsf{CECN}$, do not have Craig interpolation, and whence no uniform (Lyndon) interpolation.

On structural proof theory of the modal logic K+ extended with infinitary derivations

Abstract We consider an extension of the modal logic of transitive closure $\textsf{K}^{+}$ with certain infinitary derivations and present a sequent calculus for this extension, which allows non-well-founded proofs. We establish continuous cut-elimination for the given calculus using fixed-point theorems for contractive mappings. The infinitary derivations mentioned above are well founded and countably branching, while the non-well-founded proofs of the sequent calculus can only be finitely branching. The ordinary derivations in $\textsf{K}^{+}$, as we show additionally, correspond to the non-well-founded proofs of the calculus that are regular and cut-free. Therefore, in this article, we explore the relationship between deductive systems for $\textsf{K}^{+}$ with well-founded infinitely branching derivations and (regular) non-well-founded finitely branching proofs.

Quantum modal logic

Abstract A modal logic based on quantum logic is formalized in its simplest possible form. Specifically, a relational semantics and a sequent calculus are provided, and the soundness and the completeness theorems connecting both notions are demonstrated. This framework is intended to serve as a basis for formalizing various modal logics over quantum logic, such as quantum alethic logic, quantum temporal logic, quantum epistemic logic, and quantum dynamic logic.

A boolean-algebraic approach to completeness for normal modal predicate logics

This paper introduces an innovative methodology for demonstrating completeness for normal modal predicate logics. Traditional proofs typically involve constructing canonical models, wherein possible worlds are defined as maximal consistent sets possessing specific properties, with a heavy reliance on the Barcan Formula to affirm the existence of these worlds. Our approach deviates from the classical method by utilizing Boolean algebras and ultrafilters to construct models. Unlike conventional methods, our construction of possible worlds does not depend on the Barcan Formula; rather, these properties are ensured by Tarski’s Lemma. Furthermore, our proof distinguishes itself from other Boolean-algebraic completeness proofs in two key respects: it employs Kripke semantics instead of algebraic semantics and exclusively relies on ultrafilters, thereby offering a more concise approach. This methodology facilitates a natural extension from modal propositional logic to modal predicate logics and circumvents the added complexity of Q-filters. In our model, the equivalence class of each theorem of a normal modal predicate logic is a member of all worlds, while the equivalence class of each non-theorem is a member of some worlds. Consequently, the structure of these worlds ensures that non-theorems are false in the model.

Axiomatization of Boolean Connexive Logics with syncategorematic negation and modalities

Abstract In the article we investigate three classes of extended Boolean Connexive Logics. Two of them are extensions of Modal and non-Modal Boolean Connexive Logics with a property of closure under an arbitrary number of negations. The remaining one is an extension of Modal Boolean Connexive Logic with a property of closure under the function of demodalization. In our work we provide a formal presentation of mentioned properties and axiom schemata that allow us to incorporate them into Hilbert-style calculi. The presented axiomatic systems are provided with proofs of soundness, completeness and decidability. The properties of closure under negation and demodalization are motivated by the syncategorematic view on the connectives of negation and modalities, which is discussed in the paper.

Synergistic knowledge

Simplicial complexes are a successful model for distributed computing. They have recently been observed to provide an interesting model for epistemic multi-agent logic where the agents' local states are the main building blocks (instead of the global states). A natural generalization is to study epistemic logic on semi-simplicial sets. However, finding the appropriate modal logic for semi-simplicial models has been an open question. We answer this by introducing the logic of synergistic knowledge and establishing its soundness and completeness.

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.

Games for hybrid logic from semantic games to analytic calculi

Abstract Game semantics and winning strategies offer a potential conceptual bridge between semantics and proof systems of logics. We illustrate this link for hybrid logic—an extension of modal logic that allows for explicit reference to worlds within the language. We lift a Hintikka-style semantic game to a disjunctive game. The disjunctive game adequately models entailment and validity over classes of frames characterizable in the hybrid language. The search for winning strategies in this game can be reformulated as a sequent-style proof system.

Adding an implication to logics of perfect paradefinite algebras

Abstract Perfect paradefinite algebras are De Morgan algebras expanded with an operation that allows for the full behavior of classical negation to be restored. They form a variety that is term-equivalent to the variety of involutive Stone algebras. Their associated multiple-conclusion (Set-Set) and single-conclusion () order-preserving logics are non-algebraizable self-extensional logics of formal inconsistency and undeterminedness determined by a six-valued matrix. We studied these logics extensively in Gomes et al. ((2022). Electronic Proceedings in Theoretical Computer Science357 56–76.) from both the algebraic and the proof-theoretical perspectives. In the present paper, we continue that study by investigating directions for conservatively expanding these logics with an implication connective (essentially, one that admits the deduction-detachment theorem). We first consider logics given by very simple and manageable non-deterministic semantics whose implication (in isolation) is classical. These, nevertheless, fail to be self-extensional. We then consider the implication realized by the relative pseudo-complement over the six-valued perfect paradefinite algebra. Our strategy is to expand the language of the latter algebra with this connective and study the (self-extensional) Set-Set and order-preserving and $\top$ -assertional logics of the variety induced by the resulting algebra. We provide axiomatizations for such new variety and for such logics, drawing parallels with the class of symmetric Heyting algebras and with Moisil’s “symmetric modal logic.” For the order-preserving Set-Set logic, in particular, we obtain a Set-Set axiomatization that is analytic. We close by studying interpolation properties for these logics and concluding that the new variety has the Maehara amalgamation property.

Universal proof theory: Feasible admissibility in intuitionistic modal logics

We introduce a general and syntactically defined family of sequent-style calculi over the propositional language with the modalities {□,◇} and its fragments as a formalization for constructively acceptable systems. Calling these calculi constructive, we show that any strong enough constructive sequent calculus, satisfying a mild technical condition, feasibly admits all Visser's rules. This means that there exists a polynomial-time algorithm that, given a proof of the premise of a Visser's rule, provides a proof for its conclusion. As a positive application, we establish the feasible admissibility of Visser's rules in sequent calculi for several intuitionistic modal logics, including CK, IK, their extensions by the modal axioms T, B, 4, 5, and the axioms for bounded width and depth and their fragments CK□, propositional lax logic and IPC. On the negative side, we show that if a strong enough intuitionistic modal logic (satisfying a mild technical condition) does not admit at least one of Visser's rules, it cannot have a constructive sequent calculus. Consequently, no intermediate logic other than IPC has a constructive sequent calculus.

Formalized soundness and completeness of epistemic and public announcement logic

Abstract I strengthen the foundations of epistemic logic by formalizing the family of normal modal logics in the proof assistant Isabelle/HOL. I define an abstract canonical model over any set of axioms and formalize completeness-via-canonicity: when the canonical model for the chosen axioms belongs to a certain class of frames, strong completeness over that class follows immediately. I instantiate the result with logics based on various epistemic principles to obtain completeness results for systems from K to S5. I then move to a family of public announcement logics (PAL) and prove abstract results for strong soundness and completeness. I lift the completeness results from epistemic logic to the setting with public announcements in a modular way. This work formulates the completeness-via-canonicity technique as a proper theorem and demonstrates its applicability. Additionally, it succinctly formalizes the requirements for lifting completeness from bare epistemic logic to the addition of public announcements.

Propositional logic and modal logic—A connection via relational semantics

Abstract In this paper, by slightly generalizing an observation of Dalla Chiara and Giuntini in their chapter on quantum logic in Handbook of Philosophical Logic, we propose a relational semantics for propositional language with negation and conjunction, which unifies the relational semantics of intuitionistic logic and that of ortho-logic. We study the semantic and syntactic consequence relations and prove the soundness and completeness theorems for five propositional logics: $\textbf{BL}$, $\textbf{PL}$, $\textbf{IL}$, $\textbf{OL}$ and $\textbf{CL}$. Moreover, we prove that they can be translated into the modal logics $\textbf{K}$, $\textbf{T}$, $\textbf{S4}$, $\textbf{KTB}$ and $\textbf{S5}$, respectively, and thus establish a systematic connection between propositional logics and modal logics. The paper ends with a discussion about the possibility and difficulty of incorporating disjunction into our framework.

Conditional normative reasoning as a fragment of HOL

We report on the mechanisation of (preference-based) conditional normative reasoning. Our focus is on Åqvist's system E for conditional obligation, and its extensions. Our mechanisation is achieved via a shallow semantical embedding in Isabelle/HOL. We consider two possible uses of the framework. The first one is as a tool for meta-reasoning about the considered logic. We employ it for the automated verification of deontic correspondences (broadly conceived) and related matters, analogous to what has been previously achieved for the modal logic cube. The equivalence is automatically verified in one direction, leading from the property to the axiom. The second use is as a tool for assessing ethical arguments. We provide a computer encoding of a well-known paradox (or impossibility theorem) in population ethics, Parfit's repugnant conclusion. While some have proposed overcoming the impossibility theorem by abandoning the presupposed transitivity of ‘better than’, our formalisation unveils a less extreme approach, suggesting among other things the option of weakening transitivity suitably rather than discarding it entirely. Whether the presented encoding increases or decreases the attractiveness and persuasiveness of the repugnant conclusion is a question we would like to pass on to philosophy and ethics.

Polynomial-time checking of generalized Sahlqvist syntactic shape

The best known modal logics are axiomatized by Sahlqvist axioms, i.e., axioms of a syntactic shape which guarantees these formulas to have such excellent properties as canonicity and elementarity. Recently, the definition of Sahlqvist formulas has been generalized and extended from formulas in classical modal logic to inequalities (sequents) in a wide family of logics known as LE-logics. We introduce an algorithm which checks if a given inequality is generalized Sahlqvist in polynomial time.

Epistemic Logics for Relevant Reasoners

We present a neighbourhood-style semantic framework for modal epistemic logic modelling agents who process information using relevant logic. The distinguishing feature of the framework in comparison to relevant modal logic is that the environment the agent is situated in is assumed to be a classical possible world. This framework generates two-layered logics combining classical logic on the propositional level with relevant logic in the scope of modal operators. Our main technical result is a general soundness and completeness theorem.

Modal weak Kleene logics: axiomatizations and relational semantics

Abstract Weak Kleene logics are three-valued logics characterized by the presence of an infectious truth-value. In their external versions, as they were originally introduced by Bochvar [4] and Halldén [30], these systems are equipped with an additional connective capable of expressing whether a formula is classically true. In this paper we further expand the expressive power of external weak Kleen logics by modalizing them with a unary operator. The addition of an alethic modality gives rise to the two systems $\textsf{B}_{\text{e}}^{\square }$ and $\textsf{PWK}^{\Box }_{\text{e}} $, which have two different readings of the modal operator. We provide these logics with a complete and decidable Hilbert-style axiomatization w.r.t. a three-valued possible worlds semantics. The starting point of these calculi are new axiomatizations for the non-modal bases $\textsf{B}_{\text{e}}$ and $\textsf{PWK}_{\text{e}}$, which we provide using the recent algebraization results about these two logics. In particular, we prove the algebraizability of $\textsf{PWK}_{\text{e}}$. Finally some standard extensions of the basic modal systems are provided with their completeness results w.r.t. special classes of frames.

Modal semantics for reasoning with probability and uncertainty

Abstract This paper belongs to the field of probabilistic modal logic, focusing on a comparative analysis of two distinct semantics: one rooted in Kripke semantics and the other in neighbourhood semantics. The primary distinction lies in the following: The latter allows us to adequately express belief functions (lower probabilities) over propositions, whereas the former does not. Thus, neighbourhood semantics is more expressive. The main part of the work is a section in which we study the modal equivalence between probabilistic Kripke models and a subclass of belief neighbourhood models, namely additive ones. We study how to obtain modally equivalent structures.

Universal proof theory: Semi-analytic rules and Craig interpolation

We provide a general and syntactically defined family of sequent calculi, called semi-analytic, to formalize the informal notion of a “nice” sequent calculus. We show that any sufficiently strong (multimodal) substructural logic with a semi-analytic sequent calculus enjoys the Craig Interpolation Property, CIP. As a positive application, our theorem provides a uniform and modular method to prove the CIP for several multimodal substructural logics, including many fragments and variants of linear logic. More interestingly, on the negative side, it employs the lack of the CIP in almost all substructural, superintuitionistic and modal logics to provide a formal proof for the well-known intuition that almost all logics do not have a “nice” sequent calculus. More precisely, we show that many substructural logics including UL−, MTL, R, Łn (for n⩾3), Gn (for n⩾4), and almost all extensions of IMTL, Ł, BL, RMe, IPC, S4, and Grz (except for at most 1, 1, 3, 8, 7, 37, and 6 of them, respectively) do not have a semi-analytic calculus.

Complexity results for modal logic with recursion via translations and tableaux

This paper studies the complexity of classical modal logics and of their extension with fixed-point operators, using translations to transfer results across logics. In particular, we show several complexity results for multi-agent logics via translations to and from the $\mu$-calculus and modal logic, which allow us to transfer known upper and lower bounds. We also use these translations to introduce terminating and non-terminating tableau systems for the logics we study, based on Kozen's tableau for the $\mu$-calculus and the one of Fitting and Massacci for modal logic. Finally, we describe these tableaux with $\mu$-calculus formulas, thus reducing the satisfiability of each of these logics to the satisfiability of the $\mu$-calculus, resulting in a general 2EXP upper bound for satisfiability testing.

Intuitionistic S4 as a logic of topological spaces

Abstract We design and study various topological semantics for the diamond-free intuitionistic modal logic $\textsf{iS4}$, an intuitionistic analogue of $\textsf{S4}$. Ultimately we prove that ordinary topological spaces can be used as semantics, using the specialization order to interpret intuitionistic implication and the interior for the modality. Some of our soundness and completeness results are mechanised in Coq.