Showing papers on "Normal modal logic published in 2019"

Lattice logic as a fragment of (2-sorted) residuated modal logic

Abstract: Correspondence and Shalqvist theories for Modal Logics rely on the simple observation that a relational structure F=(W,R) is at the same time the basis for a model of modal logic and for a model of first-order logic with a binary predicate for the accessibility relation. If the underlying set of the frame is split into two components, W=X|Y, and R⊆X×Y, then frames G=(X,R,Y) are at the same time the basis for models of non-distributive lattice logic and of two-sorted, residuated modal logic. This suggests that a reduction of the first to the latter may be possible, encoding Positive Lattice Logic (PLL) as a fragment of Two-Sorted, Residuated Modal Logic. The reduction is analogous to the well-known Godel-McKinsey-Tarski translation of Intuitionistic Logic into the S4 system of normal modal logic. In this article, we carry out this reduction in detail and we derive some properties of PLL from corresponding properties of First-Order Logic. The reduction we present is extendible to the case of lattice...

The modal logic of Bayesian belief revision

Abstract: In Bayesian belief revision a Bayesian agent revises his prior belief by conditionalizing the prior on some evidence using Bayes’ rule. We define a hierarchy of modal logics that capture the logical features of Bayesian belief revision. Elements in the hierarchy are distinguished by the cardinality of the set of elementary propositions on which the agent’s prior is defined. Inclusions among the modal logics in the hierarchy are determined. By linking the modal logics in the hierarchy to the strongest modal companion of Medvedev’s logic of finite problems it is shown that the modal logic of belief revision determined by probabilities on a finite set of elementary propositions is not finitely axiomatizable.

Combining Monotone and Normal Modal Logic in Nested Sequents – with Countermodels

Abstract: 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.

Complete Additivity and Modal Incompleteness

Abstract: In this paper, we tell a story about incompleteness in modal logic. The story weaves together a paper of van Benthem [1979], “Syntactic aspects of modal incompleteness theorems,” and a longstanding open question: whether every normal modal logic can be characterized by a class of completely additive modal algebras, or as we call them, V-BAOs. Using a first-order reformulation of the property of complete additivity, we prove that the modal logic that starred in van Benthem’s paper resolves the open question in the negative. In addition, for the case of bimodal logic, we show that there is a naturally occurring logic that is incomplete with respect to V-BAOs, namely the provability logic GLB [Japaridze, 1988, Boolos, 1993]. We also show that even logics that are unsound with respect to such algebras do not have to be more complex than the classical propositional calculus. On the other hand, we observe that it is undecidable whether a syntactically defined logic is V-complete. After these results, we generalize the famed Blok Dichotomy [Blok, 1978] to degrees of V-incompleteness. In the end, we return to van Benthem’s theme of syntactic aspects of modal incompleteness.

A Note on Algebraic Semantics for S5 with Propositional Quantifiers

Abstract: In two of the earliest papers on extending modal logic with propositional quantifiers, R. A. Bull and K. Fine studied a modal logic S5Π extending S5 with axioms and rules for propositional quantification. Surprisingly, there seems to have been no proof in the literature of the completeness of S5Π with respect to its most natural algebraic semantics, with propositional quantifiers interpreted by meets and joins over all elements in a complete Boolean algebra. In this note, we give such a proof. This result raises the question: for which normal modal logics L can one axiomatize the quantified propositional modal logic determined by the complete modal algebras for L?

Remarks about the unification type of several non-symmetric non-transitive modal logics

Abstract: The problem of unification in a normal modal logic L can be defined as follows: given a formula F, determine whether there exists a substitution s such that s(F) is in L. In this paper, we prove that for several non-symmetric non-transitive modal logics, there exists unifiable formulas which possess no minimal complete set of unifiers.

First-order possibility models and finitary completeness proofs

Abstract: This article builds on Humberstone’s idea of defining models of propositional modal logic where total possible worlds are replaced by partial possibilities. We follow a suggestion of Humberstone by introducing possibility models for quantified modal logic. We show that a simple quantified modal logic is sound and complete for our semantics. Although Holliday showed that for many propositional modal logics, it is possible to give a completeness proof using a canonical model construction where every possibility consists of finitely many formulas, we show that this is impossible to do in the first-order case. However, one can still construct a canonical model where every possibility consists of a computable set of formulas and thus still of finitely much information.

Model checking and validity in propositional and modal inclusion logics

Abstract: Propositional and modal inclusion logic are formalisms that belong to the family of logics based on team semantics. This article investigates the model checking and validity problems of these logics. We identify complexity bounds for both problems, covering both lax and strict team semantics. By doing so, we come close to finalising the programme that ultimately aims to classify the complexities of the central reasoning problems for modal and propositional dependence, independence, and inclusion logics.

Sequent Calculi for Global Modal Consequence Relations

Abstract: The global consequence relation of a normal modal logic $$\Lambda $$ is formulated as a global sequent calculus which extends the local sequent theory of $$\Lambda $$ with global sequent rules. All global sequent calculi of normal modal logics admits global cut elimination. This property is utilized to show that decidability is preserved from the local to global sequent theories of any normal modal logic over $$\mathsf {K4}$$ . The preservation of Craig interpolation property from local to global sequent theories of any normal modal logic is shown by proof-theoretic method.

A Modal Logic of Supervenience

Abstract: Inspired by the supervenience-determined consequence relation and the semantics of agreement operator, we introduce a modal logic of supervenience, which has a dyadic operator of supervenience as a sole modality. The semantics of supervenience modality very naturally correspond to the supervenience-determined consequence relation, in a quite similar way that the strict implication corresponds to the inference-determined consequence relation. We show that this new logic is more expressive than the modal logic of agreement, by proposing a notion of bisimulation for the latter. We provide a sound proof system for the new logic. We lift onto more general logics of supervenience. Related to this, we address an interesting open research direction listed in the literature, by comparing propositional logic of determinacy and noncontingency logic in expressive powers and axiomatizing propositional logic of determinacy over various classes of frames. We also obtain an alternative axiomatization for propositional logic of determinacy over universal models.

Modelling Sources of Inconsistent Information in Paraconsistent Modal Logic

Abstract: Epistemic logics based on normal modal logic are notoriously bad at handling inconsistent and yet non-trivial information. This fact motivates epistemic logics based on paraconsistent logic, examples of which can be traced back at least to the 1980s. These logics handle inconsistent and non-trivial information, but they usually do not articulate sources of the inconsistency. Yet, making the origin of an inconsistency present in a body of information explicit is important to assess the body—can we trace the mutually conflicting pieces of information to sources of information relevant to the body or is the inconsistency a result of an error unrelated to any outside sources? Is the inconsistency derived from various equally trustworthy sources or from a single inconsistent source? In this article we show that a paraconsistent modal logic, namely, the logic BK introduced by Odintsov and Wansing, is a first step toward a formalism capable of making these distinctions explicit. We interpret the accessibility relation between states in a model as a source relation—states accessible from a given state are seen as sources of potential justification of the information contained in the original state. This interpretation also motivates the study of a number of extensions of BK. We focus here on extensions of BK able to articulate the relation of compatibility between bodies of information and extensions working with labels explicitly differentiating between bodies of information. In the case of compatibility-based extensions a more detailed technical study including a completeness proof is provided; technical features of the simpler case of label-based extensions, on the other hand, are discussed without going into details.

Propositional quantifiers in labelled natural deduction for normal modal logic

Abstract: This article concerns the treatment of propositional quantification in a framework of labelled natural deduction for modal logic developed by Basin, Matthews and Viganò. We provide a detailed analysis of a basic calculus that can be used for a proof-theoretic rendering of minimal normal multimodal systems with quantification over stable domains of propositions. Furthermore, we consider variations of the basic calculus obtained via relational theories and domain theories allowing for quantification over possibly unstable domains of propositions. The main result of the article is that fragments of the labelled calculi not exploiting reductio ad absurdum enjoy the Church–Rosser property and the strong normalization property; such result is obtained by combining Girard’s method of reducibility candidates and labelled languages of lambda calculus codifying the structure of modal proofs.

About the unification type of simple symmetric modal logics

Abstract: The unification problem in a normal modal logic is to determine, given a formula F, whether there exists a substitution s such that s(F) is in that logic. In that case, s is a unifier of F. We shall say that a set of unifiers of a unifiable formula F is complete if for all unifiers s of F, there exists a unifier t of F in that set such that t is more general than s. When a unifiable formula has no minimal complete set of unifiers, the formula is nullary. In this paper, we prove that KB, KDB and KTB possess nullary formulas.