Kripke Semantics Research Articles

Kripke semantics for the logic of problems and propositions

In this paper we study the propositional fragment of the joint logic of problems and propositions introduced by Melikhov. We provide Kripke semantics for this logic and show that is complete with respect to those models and has the finite model property. We consider examples of the use of -models usage. In particular, we prove that is a conservative extension of the logic . We also show that the logic is complete with respect to Kripke frames with sets of audit worlds introduced by Artemov and Protopopescu (who called them audit set models). Bibliography: 31 titles.

Belief Changes and Cognitive Development: Doxastic Logic {mathsf {LCB}}

We present the logic {mathsf {LCB}} which is expressed in a propositional language constantly enriched by new atomic expressions. Our formal framework is the propositional doxastic logic {mathsf {KD45}} with the belief operator {mathcal {B}}, extended by the {mathcal {C}} operator, to be read it changes that ... . We describe the changing beliefs of an agent who uses progressively expanding language. The approach presented here allows us to weaken pragmatic objections to the so-called principle of negative retrospection accepted in {mathsf {KD45}} and the problem of logical omniscience. In what follows, we present the expanding propositional language used in our formalism, interpreted using an epistemic version of Kripke semantics. Next, we give a syntactic characterization of logic {mathsf {LCB}} and prove the soundness and completeness of {mathsf {LCB}} in respect to our semantics. Finally, we compare the idea of expanding language with the notion of agent awareness and we relate our formalism to two epistemic temporal logics.

Kripke semantics for the logic of problems and propositions

Рассматривается пропозициональный фрагмент $\mathrm{HC}$ объединенной логики задач и высказываний, введенной C. A. Мелиховым. Строятся модели типа Крипке для этой логики, доказывается полнота логики $\mathrm{HC}$ относительно таких моделей, а также свойство конечных моделей. Рассмотрены примеры применения моделей типа Крипке логики $\mathrm{HC}$ для решения некоторых вопросов (в частности, доказательство того, что $\mathrm{HC}$ является консервативным расширением логики $\mathrm{H4}$). Также показано, что логика $\mathrm{HC}$ полна относительно шкал Крипке с проверяющими мирами, введенных С. Н. Артeмовым и Т. Протопопеску. Библиография: 31 название.

Kripke Semantics for Intuitionistic \u0141ukasiewicz Logic

This paper proposes a generalization of the Kripke semantics of intuitionistic logic IL appropriate for intuitionistic Łukasiewicz logicIŁL —a logic in the intersection between IL and (classical) Łukasiewicz logic. This generalised Kripke semantics is based on the poset sum construction, used in Bova and Montagna (Theoret Comput Sci 410(12):1143–1158, 2009). to show the decidability (and PSPACE completeness) of the quasiequational theory of commutative, integral and bounded GBL algebras. The main idea is that w Vdash psi —which for ILis a relation between worlds w and formulas psi , and can be seen as a function taking values in the booleans (w Vdash psi ) in {{mathbb {B}}}—becomes a function taking values in the unit interval (w Vdash psi ) in [0,1]. An appropriate monotonicity restriction (which we call sloping functions) needs to be put on such functions in order to ensure soundness and completeness of the semantics.

Introducing H, an Institution-Based Formal Specification and Verification Language

This is a short survey on the development of the formal specification and verification language H with emphasis on the scientific part. H is a modern highly expressive language solidly based upon advanced mathematical theories such as the internalisation of Kripke semantics within institution theory.

Recapturing Dynamic Logic of Relation Changers via Bounded Morphisms

The present contribution shows that a Hilbert-style axiomatization for dynamic logic of relation changers is complete for the standard Kripke semantics not by a well-known rewriting technique but by the idea of an auxiliary semantics studied by van Benthem and Wang et al. A key insight of our auxiliary semantics for dynamic logic of relation changers can be described as: “relation changers are bounded morphisms.” Moreover, we demonstrate that this semantic insight can be used to provide a modular cut-free labelled sequent calculus for the logic in the sense that our calculus can be regarded as a natural expansion of a labelled sequent calculus of iteration-free propositional dynamic logic.

A cut-free labelled sequent calculus for dynamic epistemic logic

Abstract Dynamic epistemic logic is a logic that is aimed at formally expressing how a person’s knowledge changes. We provide a cut-free labelled sequent calculus ($\textbf{GDEL}$) on the background of existing studies of Hilbert-style axiomatization $\textbf{HDEL}$ of dynamic epistemic logic and labelled calculi for public announcement logic. We first show that the $cut$ rule is admissible in $\textbf{GDEL}$ and show that $\textbf{GDEL}$ is sound and complete for Kripke semantics. Moreover, we show that the basis of $\textbf{GDEL}$ is extended from modal logic K to other familiar modal logics including S5 with keeping the admissibility of cut, soundness and completeness.

A Logic for Multi-Agent Conformant Planning Over Transition Systems

Reasoning about actions and information is one of the most active areas of research in artificial intelligence. In this article, we study the reasoning about actions and information in multi-agent conformant planning, which is an essential branch of artificial intelligence. We introduce a semantic-driven formalism to capture the information change during the execution of actions. We then propose a dynamic epistemic logic for multi-agent conformant planning. Moreover, we give an alternative Kripke semantics and provide a complete axiomatization.

An Investigation into Intuitionistic Logic with Identity

We define Kripke semantics for propositional intuitionistic logic with Suszko’s identity (ISCI). We propose sequent calculus for ISCI along with cut-elimination theorem. We sketch a constructive interpretation of Suszko’s propositional identity connective.

Recursive enumerability and elementary frame definability in predicate modal logic

Abstract We investigate the relationship between recursive enumerability and elementary frame definability in first-order predicate modal logic. On one hand, it is well known that every first-order predicate modal logic complete with respect to an elementary class of Kripke frames, i.e. a class of frames definable by a classical first-order formula, is recursively enumerable. On the other, numerous examples are known of predicate modal logics, based on ‘natural’ propositional modal logics with essentially second-order Kripke semantics, that are either not recursively enumerable or Kripke incomplete. This raises the question of whether every Kripke complete, recursively enumerable predicate modal logic can be characterized by an elementary class of Kripke frames. We answer this question in the negative, by constructing a normal predicate modal logic which is Kripke complete, recursively enumerable, but not complete with respect to an elementary class of frames. We also present an example of a normal predicate modal logic that is recursively enumerable, Kripke complete, and not complete with respect to an elementary class of rooted frames, but is complete with respect to an elementary class of frames that are not rooted.

On the multi-dimensional modal logic of substitutions

Abstract We prove completeness, interpolation, decidability and an omitting types theorem for certain multi-dimensional modal logics where the states are not abstract entities but have an inner structure. The states will be sequences. Our approach is algebraic addressing varieties generated by complex algebras of Kripke semantics for such logics. The algebras dealt with are common cylindrification free reducts of cylindric and polyadic algebras. For finite dimensions, we show that such varieties are finitely axiomatizable, have the super amalgamation property, and that the subclasses consisting of only completely representable algebras are elementary, and are also finitely axiomatizable in first order logic. Also their modal logics have an N P complete satisfiability problem. Analogous results are obtained for infinite dimensions by replacing finite axiomatizability by finite schema axiomatizability.

The infinite, the indefinite and the critical turn: Kant via Kripke models

ABSTRACTThis paper aims to show that intuitionistic Kripke models are a powerful tool for interpreting Kant’s ‘Critical Philosophy’. Part I reviews some old work of mine that applies these models to provide a reading of Kant’s second antinomy about the divisibility of matter and to answer several attacks on Kant’s antinomies. But it also points out three shortcomings of that original application. First, the reading fails to account for Kant’s second antinomy claim that matter is divisible ‘ad infinitum’ and his first antinomy claim that empirical space extends only ‘ad indefinitum’. Secondly, in conformity with the ‘assertability’ heuristics for Kripke models, it attributes an assertability semantics to Kant. In fact, it attributes a pair of assertability semantics to Kant, but neither of these accords with modern assertabilism. And finally, one must wonder the propriety of using contemporary assertability and contemporary formal logic to save an eighteenth-century text. Part II goes historical to solve the problems about assertability. It outlines the Descartes–Leibniz dialectic about the infinite/indefinite distinction and demonstrates how each position reflects larger philosophical positions. It then demonstrates that Kant’s two versions of assertability and his version of the infinite/indefinite dichotomy emerge naturally from his attempts (pre-critical and then critical) to resolve a tension in Leibniz’s philosophy. In light of this demonstration, Part III adapts the Kripke model interpretation so that it does reflect that dichotomy. It then refines the Kripke semantics in order to model some of Kant’s signature critical doctrines, and it shows how Kant’s critical version of the infinite/indefinite distinction differs from all those of his predecessors and from his own pre-critical version. It also addresses the question of using contemporary logical machinery to interpret Kant. Part IV turns the tables and uses what we have learned about Kant and modern semantics in order to address a pair of issues in the contemporary critical foundations of logic.

Unicorn agnosticism

ABSTRACT Atheists and agnostics have a vexed relationship. Atheists often regard agnostics as timid, or perhaps as disguised apologists. Agnostics often regard atheists as dogmatic hypocrites: they proclaim something on insufficient evidence, while accusing theists of this. This dynamic is familiar from the academic and popular literature. Here, I consider a more radical conflict between the two, based on Kripkean semantics for empty terms applied to atheism. Sorensen (2018. “Unicorn Atheism.” Noûs 52 (2): 373–388) christened the Kripke-inspired formulation of atheism ‘Unicorn Atheism’ and argued from there to the incoherence of agnosticism. But, I argue, the objection fails and instead presents an opportunity to reformulate agnosticism. By appreciating the relevance of Kripkean semantics to the issue, a better understanding of the two positions, and their conflict, emerges.

Hybrid Logic as extension of Modal and Temporal Logic

La lógica temporal fue creada por Arthur Prior para representar información temporal en un sistema lógico mediante operadores modales-temporales como P, F, H o G. Intuitivamente tales operadores pueden entenderse respectivamente como “fue alguna vez en el pasado...”, “será alguna vez en el futuro...”, “ha sido siempre en el pasado...” y “será siempre en el futuro...”. La evaluación de las fórmulas construidas a partir de ellos se lleva a cabo en semánticas kripkeanas y, de este modo, la lógica modal y la temporal están relacionadas. Sin embargo, aunque sus mecanismos permiten formalizar la información modal-temporal con cierta precisión, ambas lógicas adolecen de un problema de expresividad que la lógica híbrida es capaz de solventar. En efecto, uno de los problemas de la lógica modal reside en su incapacidad para nombrar puntos concretos dentro de un modelo. La lógica temporal, al basarse en ella, tampoco puede hacerlo. Pero la lógica de primer orden sí es capaz gracias a las constantes y a la relación de identidad. La lógica híbrida, que resulta de combinar la lógica modal con la lógica de primer orden, sería una solución a este problema. El principal objetivo de este artículo consiste en explicar el origen de la lógica híbrida a partir de la modal-temporal para mostrar qué añade a ambos sistemas en la representación de información, porqué es más expresiva que ellos y qué relación guarda con el lenguaje de correspondencia de la lógica de primer orden.

On a multilattice analogue of a hypersequent S5 calculus

In this paper, we present a logic MMLS5n which is a combination of multilattice logic and modal logic S5. MMLS5n is an extension of Kamide and Shramko’s modal multilattice logic which is a multilattice analogue of S4. We present a cut-free hypersequent calculus for MMLS5n in the spirit of Restall’s one for S5 and develop a Kripke semantics for MMLS5n, following Kamide and Shramko’s approach. Moreover, we prove theorems for embedding MMLS5n into S5 and vice versa. As a result, we obtain completeness, cut elimination, decidability, and interpolation theorems for MMLS5n. Besides, we show the duality principle for MMLS5n. Additionally, we introduce a modification of Kamide and Shramko’s sequent calculus for their multilattice version of S4 which (in contrast to Kamide and Shramko’s original one) proves the interdefinability of necessity and possibility operators. Last, but not least, we present Hilbert-style calculi for all the logics in question as well as for a larger class of modal multilattice logics.

Modality and folk psychology

The connection of the modern psychology and formal systems remains an important direction of research. This paper is centered on philosophical problems surrounding relations between mental and logic. Main attention is given to philosophy of logic but certain ideas are introduced that can be incorporated into the practical philosophical logic.
 The definition and properties of basic modal logic and descending ones which are used in study of mental activity are in view. The defining role of philosophical interpretation of modality for the particular formal system used for research in the field of psychological states of agents is postulated. Different semantics of modal logic are studied. The hypothesis about the connection of research in cognitive psychology (semantics of brain activity) and formal systems connected to research of psychological states is stated.

BELNAP–DUNN MODAL LOGICS: TRUTH CONSTANTS VS. TRUTH VALUES

AbstractWe shall be concerned with the modal logic BK—which is based on the Belnap–Dunn four-valued matrix, and can be viewed as being obtained from the least normal modal logic K by adding ‘strong negation’. Though all four values ‘truth’, ‘falsity’, ‘neither’ and ‘both’ are employed in its Kripke semantics, only the first two are expressible as terms. We show that expanding the original language of BK to include constants for ‘neither’ or/and ‘both’ leads to quite unexpected results. To be more precise, adding one of these constants has the effect of eliminating the respective value at the level of BK-extensions. In particular, if one adds both of these, then the corresponding lattice of extensions turns out to be isomorphic to that of ordinary normal modal logics.

MODAL LOGIC WITHOUT CONTRACTION IN A METATHEORY WITHOUT CONTRACTION

AbstractStandard reasoning about Kripke semantics for modal logic is almost always based on a background framework of classical logic. Can proofs for familiar definability theorems be carried out using anonclassical substructural logicas the metatheory? This article presents a semantics for positive substructural modal logic and studies the connection between frame conditions and formulas, via definability theorems. The novelty is that all the proofs are carried out with anoncontractive logicin the background. This sheds light on which modal principles are invariant under changes of metalogic, and provides (further) evidence for the general viability of nonclassical mathematics.

Generalized Update Semantics

Abstract This paper explores the relationship between dynamic and truth conditional semantics for epistemic modals. It provides a generalization of a standard dynamic update semantics for modals. This new semantics derives a Kripke semantics for modals and a standard dynamic semantics for modals as special cases. The semantics allows for new characterizations of a variety of principles in modal logic, including the inconsistency of ‘p and might not p’. Finally, the semantics provides a construction procedure for transforming any truth conditional semantics for modals into a dynamic semantics for modals with similar properties.

A semantic hierarchy for intuitionistic logic

Brouwer’s views on the foundations of mathematics have inspired the study of intuitionistic logic, including the study of the intuitionistic propositional calculus and its extensions. The theory of these systems has become an independent branch of logic with connections to lattice theory, topology, modal logic, and other areas. This paper aims to present a modern account of semantics for intuitionistic propositional systems. The guiding idea is that of a hierarchy of semantics, organized by increasing generality: from the least general Kripke semantics on through Beth semantics, topological semantics, Dragalin semantics, and finally to the most general algebraic semantics. While the Kripke, topological, and algebraic semantics have been extensively studied, the Beth and Dragalin semantics have received less attention. We bring Beth and Dragalin semantics to the fore, relating them to the concept of a nucleus from pointfree topology, which provides a unifying perspective on the semantic hierarchy.