Weak subintuitionistic logics

We introduce nested sequent calculi for bimodal monotone modal logic, aka. Brown’s ability logic, a natural combination of non-normal monotone modal logic M and normal modal logic K. The calculus generalises in a natural way previously existing calculi for both mentioned logics, has syntactical cut elimination, and can be used to construct countermodels in the neighbourhood semantics. We then consider some extensions of interest for deontic logic. An implementation is also available.

In previous publications, it was shown that finite non-deterministic matrices are quite powerful in providing semantics for a large class of normal and non-normal modal logics. However, some modal logics, such as those whose axiom systems contained the Löb axiom or the McKinsey formula, were not analyzed via non-deterministic semantics. Furthermore, other modal rules than the rule of necessitation were not yet characterized in the framework.In this paper, we will overcome this shortcoming and present a novel approach for constructing semantics for normal and non-normal modal logics that is based on restricted non-deterministic matrices. This approach not only offers a uniform semantical framework for modal logics, while keeping the interpretation of the involved modal operators the same, and thus making different systems of modal logic comparable. It might also lead to a new understanding of the concept of modality.

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.

C. I. Lewis’ systems were the first axiomatisations of modal logics. However some of those systems are non-normal modal logics, since they do not admit a full rule of necessitation, but only a restricted version thereof. We provide G3-style labelled sequent calculi for Lewis’ non-normal propositional systems. The calculi enjoy good structural properties, namely admissibility of structural rules and admissibility of cut. Furthermore they allow for straightforward proofs of admissibility of the restricted versions of the necessitation rule. We establish completeness of the calculi and we discuss also related systems.

This investigation is concerned with weak subintuitionistic logics interpreted over neighborhood models introduced by the authors in 2016. The two types of neighborhood semantics introduced in that article are compared and their relationship is clarified. Thereby modal companions for various logics are recognized. Specifically, a logic is found which has basic monotonic logic with necessitation as its modal companion. Many of the extensions of the basic logics are discussed and characterized.

The main purpose of this paper is to define and study a particular variety of Montague-Scott neighborhood semantics for modal propositional logic. We call this variety the “first-order neighborhood semantics” because it consists of the neighborhood frames whose neighborhood operations are, in a certain sense, first-order definable. The paper consists of two parts. In Part I we begin by presenting a family of modal systems. We recall the Montague-Scott semantics and apply it to some of our systems that have hitherto be uncharacterized. Then, we define the notion of a first-order indefinite semantics, along with the more specific notion of a first-order uniform semantics, the latter containing as special cases the possible world semantics of Kripke. In Part II we prove consistency and completeness for a broad range of the systems considered, with respect to the first-order indefinite semantics, and for a selected list of systems, with respect to the first-order uniform semantics. The completeness proofs are algebraic in character and make essential use of the finite model property. A by-product of our investigations is a result relating provability in “S-systems” and provability in “T-systems”, which generalizes a known theorem relating provability in the systems S 2° and C 2.

We introduce Pretopologically-Neighborhoood Modal Logic (PNML), a formal framework for reasoning about local knowledge and uncertainty based on pretopological neighborhood semantics. Unlike classical Kripke or general neighborhood models assuming global structural properties or arbitrary accessibility, PNML restricts neighborhood systems to satisfy the axioms of pretopological spaces, i.e., upward closure and self-inclusion, without requiring intersection stability or closure under arbitrary unions. This enables a finer-grained representation of agents’ information in contexts where only partial or locally available knowledge is relevant. We define the truth conditions for modal operators in terms of pointwise neighborhood filters, introduce a basic axiomatic system and prove its soundness/completeness with respect to the full class of pretopological frames, ensuring that the syntactic and semantic components of the logic are aligned. Then, we examine the expressivity of PNML in relation to both normal and non-normal modal logics, arguing that pretopological constraints introduce structural distinctions not captured by standard neighborhood models, particularly under minimal closure conditions. We present examples illustrating the framework’s utility in modeling observer-dependent scenarios, including epistemic uncertainty, context-sensitive reasoning and localized inference. Notably, PNML may accommodate settings in which traditional modal logic either overgeneralizes or underrepresents the dynamics of localized information. By grounding modal reasoning in pretopological structure, PNML may apply to distributed computing with local inputs, quantum mechanics involving contextual observation, cellular signaling and ecological systems in biology, modal update operations under information change, connections to fixpoint and coalgebraic semantic frameworks, proof systems for local inference and real-world modeling of belief revision and protocol verification.

Chapter 15 concerns a quite different semantics for intentional operators, neighbourhood semantics. This is similar to the neighbourhood semantics for modal logics. Such a semantics is presented, and its ramifications explored

We propose a new perspective on logics of computation by combining instantial neighborhood logic mathsf {INL} with bisimulation safe operations adapted from mathsf {PDL}. mathsf {INL} is a recent modal logic, based on an extended neighborhood semantics which permits quantification over individual neighborhoods plus their contents. This system has a natural interpretation as a logic of computation in open systems. Motivated by this interpretation, we show that a number of familiar program constructors can be adapted to instantial neighborhood semantics to preserve invariance for instantial neighborhood bisimulations, the appropriate bisimulation concept for mathsf {INL}. We also prove that our extended logic mathsf {IPDL} is a conservative extension of dual-free game logic, and its semantics generalizes the monotone neighborhood semantics of game logic. Finally, we provide a sound and complete system of axioms for mathsf {IPDL}, and establish its finite model property and decidability.

A number of significant contributions in the last four decades show that non-normal modal logics can be fruitfully employed in several applied fields. Well-known domains are epistemic logic, deontic logic, and systems capturing different aspects of action and agency such as the modal logic of agency, concurrent propositional dynamic logic, game logic, and coalition logic. Semantics for such logics are traditionally based on neighbourhood models. However, other model-theoretic semantics can be used for this purpose. Here, we systematically study multi-relational structures, whose investigation is still relatively underdeveloped. After a brief introduction to two different versions of multi-relational semantics – which we call strong and weak multi-relational semantics – we proceed to study several modal schemata. Special attention is paid to the schemata CON and D. Finally we offer completeness proofs for several systems using both strong and weak semantic tools: The proofs thus cover both classical systems and N-monotonic systems.

Historically, it was the interpretations of intuitionist logic in the modal logic S4 that inspired the standard Kripke semantics for intuitionist logic. The inspiration of this paper is the interpretation of intuitionist logic in the non-normal modal logic S3: an S3 model structure can be 'looked at' as an intuitionist model structure and the semantics for S3 can be 'cashed in' to obtain a non-normal semantics for intuitionist propositional logic. This non-normal semantics is then extended to intuitionist quantificational logic.

We develop a predicate logical extension of a subintuitionistic propositional logic. Therefore a Hilbert type calculus and a Kripke type model are given. The propositional logic is formulated to axiomatize the idea of strategic weakening of Kripke's semantic for intuitionistic logic: dropping the semantical condition of heredity or persistence leads to a nonmonotonic model. On the syntactic side this leads to a certain restriction imposed on the deduction theorem. By means of a Henkin argument strong completeness is proved making use of predicate logical principles, which are only classically acceptable. Semantic tableaux and an embedding into modal logic are defined straightforward.

Canonical models are of central importance in modal logic, in particular as they witness strong completeness and hence compactness. While the canonical model construction is well understood for Kripke semantics, non-normal modal logics often present subtle difficulties - up to the point that canonical models may fail to exist, as is the case e.g. in most probabilistic logics. Here, we present a generic canonical model construction in the semantic framework of coalgebraic modal logic, which pinpoints coherence conditions between syntax and semantics of modal logics that guarantee strong completeness. We apply this method to reconstruct canonical model theorems that are either known or folklore, and moreover instantiate our method to obtain new strong completeness results. In particular, we prove strong completeness of graded modal logic with finite multiplicities, and of the modal logic of exact probabilities.

In a recent paper, under the auspices of an unorthodox variety of bilateralism, we introduced a new kind of proof-theoretic semantics for the base modal logic \(\mathbf{K}\), whose values lie in the closed interval \([0,1]\) of rational numbers [14]. In this paper, after clarifying our conception of bilateralism – dubbed “soft bilateralism” – we generalize the fractional method to encompass extensions and weakenings of \(\mathbf{K}\). Specifically, we introduce well-behaved hypersequent calculi for the deontic logic \(\mathbf{D}\) and the non-normal modal logics \(\mathbf{E}\) and \(\mathbf{M}\) and thoroughly investigate their structural properties.

In this paper, we give an alternative semantics to the non-normal logic of knowing how proposed by Fervari et al. (Proc IJCAI 2017:1031–1038, 2017), based on a class of Kripke neighborhood models with both the epistemic relations and neighborhood structures. This alternative semantics is inspired by the same quantifier alternation pattern of $$\exists \forall $$ in the semantics of the know-how modality and the (monotonic) neighborhood semantics for the standard modality. We show that this new semantics is equivalent to the original Kripke semantics in terms of the validities. A key result is a representation theorem showing that the more abstract Kripke neighborhood models can be represented by the concrete Kripke models with action transitions modulo the valid formulas. We prove the completeness of the logic for the neighborhood semantics. The neighborhood semantics can be adapted to other variants of logics of knowing how. It provides us a powerful technical tool to study these logics while preserving the basic semantic intuition.