Modal Logic Research Articles

New paraconsistent modal logics based on rough modus ponens rules and their interrelations

Labelled transitions systems can be studied in terms of modal logic and in terms of bisimulation. These two notions are connected by Hennessy-Milner theorems, that show that two states are bisimilar precisely when they satisfy the same modal logic formulas. Recently, apartness has been studied as a dual to bisimulation, which also gives rise to a dual version of the Hennessy-Milner theorem: two states are apart precisely when there is a modal formula that distinguishes them. In this paper, we introduce "directed" versions of Hennessy-Milner theorems that characterize when the theory of one state is included in the other. For this we introduce "positive modal logics" that only allow a limited use of negation. Furthermore, we introduce directed notions of bisimulation and apartness, and then show that, for this positive modal logic, the theory of $s$ is included in the theory of $t$ precisely when $s$ is directed bisimilar to $t$. Or, in terms of apartness, we show that $s$ is directed apart from $t$ precisely when the theory of $s$ is not included in the theory of $t$. From the directed version of the Hennessy-Milner theorem, the original result follows. In particular, we study the case of branching bisimulation and Hennessy-Milner Logic with Until (HMLU) as a modal logic. We introduce "directed branching bisimulation" (and directed branching apartness) and "Positive Hennessy-Milner Logic with Until" (PHMLU) and we show the directed version of the Hennessy-Milner theorems. In the process, we show that every HMLU formula is equivalent to a Boolean combination of Positive HMLU formulas, which is a very non-trivial result. This gives rise to a sublogic of HMLU that is equally expressive but easier to reason about.

We investigate the set-theoretic strength of several maximality principles that play an important role in the study of modal and intuitionistic logics. We focus on well-known Fine’s and Esakia’s Maximality Principles, present two formulations of each, and show that the stronger formulations are equivalent to the Axiom of Choice (AC), while the weaker ones to the Boolean Prime Ideal Theorem (BPI).

Abstract Explainability is emerging as a key requirement for autonomous systems. While many works have focused on what constitutes a valid explanation, few have considered formalizing explainability as a system property. In this work, we approach this problem from the perspective of hyperproperties. We start with a combination of three prominent flavors of modal logic and show how they can be used for specifying and verifying counterfactual explainability in multi-agent systems: With Lewis’ counterfactuals, linear-time temporal logic, and a knowledge modality, we can reason about whether agents know why a specific observation occurs, i.e., whether that observation is explainable to them. We use this logic to formalize multiple notions of explainability on the system level. We then show how this logic can be embedded into a hyperlogic. Notably, from this analysis we conclude that the model-checking problem of our logic is decidable, which paves the way for the automated verification of explainability requirements.

Abstract In this paper we explore the construction of canonical models for K45 and weaker systems by extending Wittgenstein’s picture theory of language to a family of locally tabular modal logics. After discussing the philosophical implications of revisiting Wittgenstein’s concept of logical space, especially for K45, we turn to the mathematical side, establishing soundness and completeness results for K$4^{k}!5^{-}$. These theorems pave the way for the description of canonical models, which we illustrate for the systems K45, K5, and K$4^{2}!5^{-}$ in the case where a single proposition is considered. The purpose of these illustrations is to gain insight into these systems through the combinatorial structure of their canonical models.

None of the first-order modal logics between $\mathsf{K}$ and $\mathsf{S5}$ under the constant domain semantics enjoys Craig interpolation or projective Beth definability, even in the language restricted to a single individual variable. It follows that the existence of a Craig interpolant for a given implication or of an explicit definition for a given predicate cannot be directly reduced to validity as in classical first-order and many other logics. Our concern here is the decidability and computational complexity of the interpolant and definition existence problems. We first consider two decidable fragments of first-order modal logic $\mathsf{S5}$: the one-variable fragment $\mathsf{Q^1S5}$ and its extension $\mathsf{S5}_{\mathcal{ALC}^u}$ that combines $\mathsf{S5}$ and the description logic$\mathcal{ALC}$ with the universal role. We prove that interpolant and definition existence in $\mathsf{Q^1S5}$ and $\mathsf{S5}_{\mathcal{ALC}^u}$ is decidable in coN2ExpTime, being 2ExpTime-hard, while uniform interpolant existence is undecidable. These results transfer to the two-variable fragment $\mathsf{FO^2}$ of classical first-order logic without equality. We also show that interpolant and definition existence in the one-variable fragment $\mathsf{Q^1K}$ of first-order modal logic $\mathsf{K}$ is non-elementary decidable, while uniform interpolant existence is again undecidable.

A Function-Set Framework: General Properties and Applications to Modal Logic

Abstract This paper presents a comprehensive proof-theoretic analysis of Jaśkowski’s discussive (or discursive) logic, working with a set of connectives including classical negation and disjunction, as well as so-called (right-)discussive conjunction and discussive implication. By employing established techniques two labelled frameworks are introduced: sequent and natural deduction systems. The paper explores the ability of the proposed calculi to accurately represent Jaśkowski’s discussive logic, particularly in light of its paraconsistent nature, and establishes cut-admissibility and normalization theorems. Additionally, the introduced sequent calculus – shown to allow terminating proof search – is employed to prove the embedding of discussive logic within modal logic $$\textbf{S5}$$ S 5 . Finally, it is proved that the natural deduction calculus translates into the corresponding sequent system, with soundness and completeness established for both calculi. Concluding remarks highlight the potential for expanding this study and suggest directions for future research.

Abstract What happens if we drop the axiom (K) and the necessitation rule from the usual axiomatic presentation of modal logic $\mathbf{T}$? This system was first introduced by Ivlev (1988, Bull. Sect. Log., 17, 114–121). We show that this logic is a syntactical variant of the well-known paraconsistent logic $\mathbf{BK}$ (also known as mbCciw), which belongs to a large family of Brazilian paraconsistent logics known as Logics of Formal Inconsistency (LFIs). Our approach is semantical, as $\mathbf{BK}$ can be characterized by a 3-valued non-deterministic matrix (Nmatrix); we show that this Nmatrix can alternatively be axiomatized in a modal vocabulary by more convenient axioms. This result links this family of paraconsistent logics to another line of research aimed at characterizing modal logics without possible worlds, initiated by the works of Kearns (1981, J. Symb. Log., 46, 77–86) and Ivlev (1988, Bull. Sect. Log., 17, 114–121). We will see that the modal logics $\mathbf{T}$ and $\mathbf{S4}$, as well as their corresponding versions without the necessitation rule, can be characterized with the help of simple refinements1 of this single Nmatrix. All of this shows that this logic can serve as a base for both modal logics and LFIs.

In this paper, we introduce and study the variety of algebras ( A , ∧ , ∨ , → , ◻ , ◊ , 0 , 1 ) of type ( 2 , 2 , 2 , 1 , 1 , 0 , 0 ) whose { ∧ , ∨ , → , 0 , 1 } -reduct is a weak Heyting algebra and the following identities are satisfied: (a) ◻ 1 = 1 , (b) ◻ ( a ∧ b ) = ◻a ∧ ◻b , (c) ◊ 0 = 0 and (d) ◊ ( a ∨ b ) = ◊a ∨ ◊b . This variety, which is denoted by MWH , contains several varieties of Heyting algebras with modal operators, which are the algebraic semantics of well-known modal intuitionistic logics. The main goal of this paper is to study certain subvarieties of MWH , which are the algebraic semantics of particular subintuitionistic modal logics. We give representation theorems for these subvarieties, we prove that they are canonical and we also show that their associated deductive systems are frame complete with respect to certain class of relational frames.

Abstract We investigate a system of modal semantics in which $\Box \phi $ is true if and only if $\phi $ is entailed by a designated set of formulas by a designated logics. We prove some strong completeness results as well as a natural connection to normal modal logics via an application of some lattice-theoretic fixpoint theorems. We raise a difficult problem that arises naturally in this setting about logics which are identical with their own ‘meta-logic’, and draw a surprising connection to recent work by Andrew Bacon and Kit Fine on McKinsey’s substitutional modal semantics.

Abstract We study the Lyndon interpolation property (LIP) and the uniform LIP (ULIP) for extensions of $\mathbf {S4}$ and intermediate propositional logics. We prove that among the 18 consistent normal modal logics of finite height extending $\mathbf {S4}$ known to have CIP, 11 logics have LIP and 7 logics do not. We also prove that for intermediate propositional logics, the Craig interpolation property, LIP, and ULIP are equivalent.

Boxing Some: Axiomatizations of the $$\Box \exists $$-Bundled Fragment of First-Order Modal Logic

ABSTRACTThis paper argues that Avicenna was both a necessitarian and a realist about contingency. The two aspects of his modal metaphysics are reconciled by arguing that Avicenna's modal metaphysics is founded on realism about essences: strictly speaking, an individual has no contingent properties, but a modal distinction can be made between the properties that it has by virtue of its essence (and that are thus necessary by virtue of its identity) and those that it has by virtue of extrinsic causes (and that are thus contingent with respect to its identity). Consequently, despite its realism about contingency, Avicennian modal metaphysics is not committed to the existence of counterfactual scenarios. This may seem to be in tension with the fact that Avicenna takes counterfactuals seriously in his logic. The paper argues, however, that the logical discussion of counterfactuals must be set in the framework of Avicenna's conception of the relation between modal logic and modal metaphysics: the metaphysical account of the relation between essences and modalities should provide the foundation for logic, and the purely logical kind of possibility should be understood as a case of epistemic modality. The paper concludes by claiming that such a division of labour is in potentially productive contrast to the contemporary mainstream in modal metaphysics.

Abstract The quest of smoothly combining logics so that connectives from different logics can co-exist in peace has been a fascinating topic of research. In 2015, Dag Prawitz introduced a natural deduction system for an ecumenical first-order logic, unifying classical and intuitionistic logics within a shared language. Building upon this foundation, we introduced, in a series of works, sequent systems for ecumenical logics and modal extensions. In this work we propose a new pure sequent calculus version for Prawitz’s original system, where each rule features precisely one logical operator. This is achieved by extending sequents with an additional context, called stoup, and establishing the ecumenical concept of polarities. We smoothly extend these ideas for handling modalities, presenting a new pure labelled system for ecumenical modal logics. Finally, we show how this allows for naturally retrieving the ecumenical modal nested system proposed in a previous work.

Is it possible to write significantly smaller formulae when using Boolean operators other than those of the De Morgan basis (and, or, not, and the constants)? For propositional logic, a negative answer was given by Pratt: formulae over one set of operators can always be translated into an equivalent formula over any other complete set of operators with only polynomial increase in size. Surprisingly, for modal logic the picture is different: we show that elimination of bi-implication is only possible at the cost of an exponential number of occurrences of the modal operator $\lozenge$ and therefore of an exponential increase in formula size, i.e., the De Morgan basis and its extension by bi-implication differ in succinctness. Moreover, we prove that any complete set of Boolean operators agrees in succinctness with the De Morgan basis or with its extension by bi-implication. More precisely, these results are shown for the modal logic $\mathrm{T}$ (and therefore for $\mathrm{K}$). We complement them showing that the modal logic $\mathrm{S5}$ behaves as propositional logic: the choice of Boolean operators has no significant impact on the size of formulae.

The intersection of artificial intelligence (AI) and music is redefining the construction, preservation, and perception of musical memory. This study investigates how AI-generated compositions interact with human cognition and reshape our understanding of cultural continuity in music. Anchored in cognitive musicology and memory theory, it adopts a qualitative-computational framework to explore how algorithmic systems simulate and reinterpret traditional musical structures. Focusing on the Ottoman-Turkish makam tradition as a case study, the research compares AI-generated pieces with historically grounded compositions, analyzing their melodic contours, modal progressions, and formal architectures. The methodology combines structural music analysis, listener response studies, and computational profiling of AI models. Findings indicate that while AI can effectively reproduce surface-level features of traditional music, it often lacks the nuanced emotional and cultural depth embedded in human compositions. Listener responses reveal cognitive dissonance when AI-generated works deviate subtly from familiar modal logics, highlighting the complex interplay between form, memory, and authenticity. The study also engages with broader theoretical discourses in digital aesthetics and posthumanism, arguing that AI’s role in music extends beyond imitation. It positions AI as a co-author in the evolving ecology of musical memory an entity capable of both continuity and disruption. By articulating a model of hybrid authorship and distributed memory, the study challenges traditional notions of creativity, heritage, and authorship in the digital age. This research contributes to interdisciplinary discussions on the future of cultural heritage, offering critical insights into how emerging technologies reshape the way we remember, transmit, and reinterpret music.

We prove that for weakly transitive modal logics equipped with the universal modality whose satisfiability problem is already decidable in PSPACE, adding the connectivity axiom does not increase the complexity. Moreover, we present an algorithm that solves the satisfiability task within the same complexity class.

Ehrenfeucht-Fraïssé games provide means to characterize elementary equivalence for first-order logic, and by standard translation also for modal logics. We propose a novel generalization of Ehrenfeucht-Fraïssé games to hybrid-dynamic logics which is direct and fully modular: parameterized by the features of the hybrid language we wish to include, for instance, the modal and hybrid language operators as well as first-order existential quantification. We use these games to establish a new modular Fraïssé-Hintikka theorem for hybrid-dynamic propositional logic and its various fragments. We study the relationship between countable game equivalence (determined by countable Ehrenfeucht-Fraïssé games) and bisimulation (determined by countable back-and-forth systems). In general, the former turns out to be weaker than the latter, but under certain conditions on the language, the two coincide. As a corollary we obtain an analogue of the Hennessy-Milner theorem. We also prove that for reachable image-finite Kripke structures elementary equivalence implies isomorphism.

A formal context consists of objects, properties, and the incidence relation between them. Various notions of concepts defined with respect to formal contexts and their associated algebraic structures have been studied extensively, including formal concepts in formal concept analysis (FCA), rough concepts arising from rough set theory (RST), and semiconcepts and protoconcepts for dealing with negation. While all these kinds of concepts are associated with lattices, semiconcepts and protoconcepts additionally yield an ordered algebraic structure, called double Boolean algebras. As the name suggests, a double Boolean algebra contains two underlying Boolean algebras. In this article, we investigate logical and algebraic aspects of the representation and reasoning about different concepts with respect to formal contexts. We first review our previous work on two-sorted modal logic systems KB and KF for the representation and reasoning of rough concepts and formal concepts, respectively. Then, in order to represent and reason about both formal and rough concepts in a single framework, these two logics are unified into a two-sorted Boolean modal logic BM , in which semiconcepts and protoconcepts are also expressible. Based on the logical representation of semiconcepts and protoconcepts, we prove the characterization of double Boolean algebras in terms of their underlying Boolean algebras. Finally, we also discuss the possibilities of extending our logical systems for the representation and reasoning of more fine-grained quantitative information in formal contexts.