scholarly journals Pooling Modalities and Pointwise Intersection: Semantics, Expressivity, and Dynamics

2022 ◽  
Author(s):  
Frederik Van De Putte ◽  
Dominik Klein
Keyword(s):  
Expressive Power ◽  
Modal Logics ◽  
Relational Semantics ◽  
Modal Operators

AbstractWe study classical modal logics with pooling modalities, i.e. unary modal operators that allow one to express properties of sets obtained by the pointwise intersection of neighbourhoods. We discuss salient properties of these modalities, situate the logics in the broader area of modal logics (with a particular focus on relational semantics), establish key properties concerning their expressive power, discuss dynamic extensions of these logics and provide reduction axioms for the latter.

Canonical modal logics and ultrafilter extensions

1979 ◽  
Vol 44 (1) ◽  
pp. 1-8 ◽  
Author(s):  
J. F. A. K. van Benthem
Keyword(s):  
Modal Logic ◽  
Algebraic Theory ◽  
Main Tool ◽  
Modal Logics ◽  
Modal Operators ◽  
Modal Algebras ◽  
Lower Case ◽  
Elementary Classes ◽  
Preceding Discussion ◽  
Ultrafilter Extensions

In this paper thecanonicalmodal logics, a kind of complete modal logics introduced in K. Fine [4] and R. I. Goldblatt [5], will be characterized semantically using the concept of anultrafilter extension, an operation on frames inspired by the algebraic theory of modal logic. Theorem 8 of R. I. Goldblatt and S. K. Thomason [6] characterizing the modally definable Σ⊿-elementary classes of frames will follow as a corollary. A second corollary is Theorem 2 of [4] which states that any complete modal logic defining a Σ⊿-elementary class of frames is canonical.The main tool in obtaining these results is the duality between modal algebras and general frames developed in R. I. Goldblatt [5]. The relevant notions and results from this theory will be stated in §2. The concept of a canonical modal logic is introduced and motivated in §3, which also contains the above-mentioned theorems. In §4, a kind of appendix to the preceding discussion, preservation of first-order sentences under ultrafilter extensions (and some other relevant operations on frames) is discussed.The modal language to be considered here has an infinite supply of proposition letters (p, q, r, …), a propositional constant ⊥ (the so-calledfalsum, standing for a fixed contradiction), the usual Boolean operators ¬ (not), ∨ (or), ∨ (and), → (if … then …), and ↔ (if and only if)—with ¬ and ∨ regarded as primitives—and the two unary modal operators ◇ (possibly) and □ (necessarily)— ◇ being regarded as primitive. Modal formulas will be denoted by lower case Greek letters, sets of formulas by Greek capitals.

Probability as a modal operator: the possibilities of its combination with other modalities

2020 ◽  
Author(s):  
Nino Guallart
Keyword(s):  
Subjective Probability ◽  
Modal Operator ◽  
Relational Semantics ◽  
Modal Operators ◽  
Probabilistic Semantics ◽  
Probability Operator

Abstract In this work we examine some of the possibilities of combining a simple probability operator with other modal operators, in particular with a belief operator. We will examine the semantics of two possible situations for expressing probabilistic belief or the lack of it, a simple subjective probability operator (SPO) versus the composition of a belief operator, plus an objective modal operator (BOP). We will study their interpretations in two probabilistic semantics: a relational Kripkean one and a variation of neighbourhood semantics, showing that the latter is able to represent the lack of probabilistic belief more directly, just with the SPO, whereas relational semantics needs the combination of BOP probability to represent lack of belief.

scholarly journals THE EXPRESSIVE POWER OF MEMORY LOGICS

2011 ◽  
Vol 4 (2) ◽  
pp. 290-318 ◽  
Author(s):  
CARLOS ARECES ◽  
DIEGO FIGUEIRA ◽  
SANTIAGO FIGUEIRA ◽  
SERGIO MERA
Keyword(s):  
Modal Logic ◽  
Expressive Power ◽  
Modal Logics ◽  
Hybrid Logic ◽  
Satisfiability Problem ◽  
Current Node

We investigate the expressive power of memory logics. These are modal logics extended with the possibility to store (or remove) the current node of evaluation in (or from) a memory, and to perform membership tests on the current memory. From this perspective, the hybrid logic ℋℒ (↓), for example, can be thought of as a particular case of a memory logic where the memory is an indexed list of elements of the domain.This work focuses in the case where the memory is a set, and we can test whether the current node belongs to the set or not. We prove that, in terms of expressive power, the memory logics we discuss here lie between the basic modal logic ${\cal K}$ and ℋℒ (↓). We show that the satisfiability problem of most of the logics we cover is undecidable. The only logic with a decidable satisfiability problem is obtained by imposing strong constraints on which elements can be memorized.

A Modal Defeasible Reasoner of Deontic Logic for the Semantic Web

2011 ◽  
Vol 7 (1) ◽  
pp. 18-43 ◽  
Author(s):  
Efstratios Kontopoulos ◽  
Nick Bassiliades ◽  
Guido Governatori ◽  
Grigoris Antoniou
Keyword(s):  
Modal Logic ◽  
Semantic Web ◽  
Deontic Logic ◽  
Expressive Power ◽  
Modal Logics ◽  
Cognitive State ◽  
Privacy Requirements ◽  
Defeasible Logic ◽  
Conflicting Information ◽  
Web Information

Defeasible logic is a non-monotonic formalism that deals with incomplete and conflicting information, whereas modal logic deals with the concepts of necessity and possibility. These types of logics play a significant role in the emerging Semantic Web, which enriches the available Web information with meaning, leading to better cooperation between end-users and applications. Defeasible and modal logics, in general, and, particularly, deontic logic provide means for modeling agent communities, where each agent is characterized by its cognitive profile and normative system, as well as policies, which define privacy requirements, access permissions, and individual rights. Toward this direction, this article discusses the extension of DR-DEVICE, a Semantic Web-aware defeasible reasoner, with a mechanism for expressing modal logic operators, while testing the implementation via deontic logic operators, concerned with obligations, permissions, and related concepts. The motivation behind this work is to develop a practical defeasible reasoner for the Semantic Web that takes advantage of the expressive power offered by modal logics, accompanied by the flexibility to define diverse agent behaviours. A further incentive is to study the various motivational notions of deontic logic and discuss the cognitive state of agents, as well as the interactions among them.

scholarly journals On the Mosaic Method for Many-Dimensional Modal Logics: A Case Study Combining Tense and Modal Operators

2012 ◽  
Vol 7 (1) ◽  
pp. 33-69 ◽  
Author(s):  
Carlos Caleiro ◽  
Luca Viganò ◽  
Marco Volpe
Keyword(s):  
Modal Logics ◽  
Modal Operators

A Modal Defeasible Reasoner of Deontic Logic for the Semantic Web

Semantic Web ◽  
2013 ◽  
pp. 140-167
Author(s):  
Efstratios Kontopoulos ◽  
Nick Bassiliades ◽  
Guido Governatori ◽  
Grigoris Antoniou
Keyword(s):  
Modal Logic ◽  
Semantic Web ◽  
Deontic Logic ◽  
Expressive Power ◽  
Modal Logics ◽  
Cognitive State ◽  
Privacy Requirements ◽  
Defeasible Logic ◽  
Conflicting Information ◽  
Web Information

Defeasible logic is a non-monotonic formalism that deals with incomplete and conflicting information, whereas modal logic deals with the concepts of necessity and possibility. These types of logics play a significant role in the emerging Semantic Web, which enriches the available Web information with meaning, leading to better cooperation between end-users and applications. Defeasible and modal logics, in general, and, particularly, deontic logic provide means for modeling agent communities, where each agent is characterized by its cognitive profile and normative system, as well as policies, which define privacy requirements, access permissions, and individual rights. Toward this direction, this article discusses the extension of DR-DEVICE, a Semantic Web-aware defeasible reasoner, with a mechanism for expressing modal logic operators, while testing the implementation via deontic logic operators, concerned with obligations, permissions, and related concepts. The motivation behind this work is to develop a practical defeasible reasoner for the Semantic Web that takes advantage of the expressive power offered by modal logics, accompanied by the flexibility to define diverse agent behaviours. A further incentive is to study the various motivational notions of deontic logic and discuss the cognitive state of agents, as well as the interactions among them.

Modifier Logics Based on Graded Modalities

2003 ◽  
Vol 7 (2) ◽  
pp. 72-78 ◽  
Author(s):  
Jorma K. Mattila ◽  
Keyword(s):  
Modal Logics ◽  
Formal Semantic ◽  
Fuzzy Topology ◽  
Topological Semantics ◽  
Modal Operators ◽  
Operator Systems ◽  
Multimodal Logics ◽  
Interior Operator ◽  
Graded Modalities

Modifier logics are considered as generalizations of "classical" modal logics. Thus modifier logics are so-called multimodal logics. Multimodality means here that the basic logics are modal logics with graded modalities. The interpretation of modal operators is more general, too. Leibniz’s motivating semantical ideas (see [8], p. 20-21) give justification to these generalizations. Semantics of canonical frames forms the formal semantic base for modifier logics. Several modifier systems are given. A special modifier calculus is combined from some "pure" modifier logics. Creating a topological semantics to this special modifier logic may give a basis to some kind of fuzzy topology. Modifier logics of S4-type modifiers will give a graded topological interior operator systems, and thus we have a link to fuzzy topology.

An incomplete nonnormal extension of S3

1978 ◽  
Vol 43 (2) ◽  
pp. 211-212
Author(s):  
George F. Schumm
Keyword(s):  
Modus Ponens ◽  
Modal Logics ◽  
Relational Semantics ◽  
General Semantics ◽  
Normal Logic ◽  
Enormous Number

Fine [1] and Thomason [4] have recently shown that the familiar relational semantics of Kripke [2] is inadequate for certain normal extensions of T and S4. It is here shown that the more general semantics developed by Kripke in [3] to handle nonnormal modal logics is likewise inadequate for certain of those logics.The interest of incompleteness results, such as those of Fine and Thomason, is of course a function of one's expectations. Define a “normal” logic too broadly and it is not surprising that a given semantics is not adequate for all normal logics. In the case of relational semantics, for example, one would want to require at least that a normal logic contain T, the logic determined by the class of all normal frames, and that it be closed under certain (though perhaps not all) rules of inference which are validity preserving in such frames. The adequacy of that semantics will otherwise be ruled out at the outset.For Kripke a logic is normal if it contains all tautologies, □p→p and □ (p → q)→(□p → □q), and is closed under the rules of substitution, modus ponens and necessitation (from A infer □A). T is the smallest normal logic, and this fact, together with the “naturalness” of the definition and the enormous number of normal logics which have been shown to be complete, made it plausible to suppose that Kripke's original semantics was adequate for all normal logics. That it is not is indeed surprising and would seem to reveal a genuine shortcoming.

An extended joint consistency theorem for a family of free modal logics with equality

1984 ◽  
Vol 49 (1) ◽  
pp. 174-183 ◽  
Author(s):  
Raymond D. Gumb
Keyword(s):  
Infinite Family ◽  
Modal Logics ◽  
Relational Semantics ◽  
Accessibility Relation ◽  
Barcan Formula ◽  
Distinct Individual ◽  
Individual Variables

In this paper, we establish an extended joint consistency theorem for an infinite family of free modal logics with equality. The extended joint consistency theorem incorporates the Craig and Lyndon interpolation lemmas and the Robinson joint consistency theorem. In part, the theorem states that two theories which are jointly unsatisfiable are separated by a sentence in the vocabulary common to both theories.Our family of free modal logics includes the free versions of I, M, and S4 studied by Leblanc [5, Chapters 8 and 9], supplemented with equality as in [3]. In the relational semantics for these logics, there is no restriction on the accessibility relation in I, while in M(S4) the restriction is reflexivity (refiexivity and transitivity). We say that a restriction on the accessibility relation countenances backward-looping if it implies a sentence of the form ∀x1 …xn(x1Rx2 &…&xn ⊃ xkRxj) (1 ≤ j < k ≤ n ≥ 2), where the xi (1 ≤ i ≤ n) are distinct individual variables. Just as reflexivity and transitivity do not countenance backward-looping, neither do any of the restrictions in our family of free modal logics. (The above terminology is derived from the effect of such restrictions on Kripke tableaux constructions.) The Barcan formula, its converse, the Fitch formula, and the formula T ≠ T′ ⊃ □T ≠ T′ do not hold in our logics.

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