scholarly journals Admissible Inference Rules and Semantic Property of Modal Logics

2021 ◽  
Vol 37 ◽  
pp. 104-117
Author(s):  
V.V. Rimatskiy ◽  
Keyword(s):  
Modal Logic ◽  
Inference Rule ◽  
Expressive Power ◽  
Modal Logics ◽  
Human Reasoning ◽  
Inference Rules ◽  
Semantic Property ◽  
Admissible Rule ◽  
Admissible Inference Rules ◽  
Deductive Power

Firstly semantic property of nonstandart logics were described by formulas which are peculiar to studied a models in general, and do not take to consideration a variable conditions and a changing assumptions. Evidently the notion of inference rule generalizes the notion of formulas and brings us more flexibility and more expressive power to model human reasoning and computing. In 2000-2010 a few results on describing of explicit bases for admissible inference rules for nonstandard logics (S4, K4, H etc.) appeared. The key property of these logics was weak co-cover property. Beside the improvement of deductive power in logic, an admissible rule are able to describe some semantic property of given logic. We describe a semantic property of modal logics in term of admissibility of given set of inference rules. We prove that modal logic over logic 𝐺𝐿 enjoys weak co-cover property iff all given rules are admissible for logic.

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.

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.

Noncompactness in propositional modal logic

1972 ◽  
Vol 37 (4) ◽  
pp. 716-720 ◽  
Author(s):  
S. K. Thomason
Keyword(s):  
Modal Logic ◽  
Predicate Logic ◽  
Expressive Power ◽  
Second Order ◽  
Order Logic ◽  
Relational Models ◽  
Propositional Modal Logic ◽  
Bimodal Logic ◽  
Deductive Power ◽  
Second Order Logic

We have come to believe that propositional modal logic (with the usual relational semantics) must be understood as a rather strong fragment of classical second-order predicate logic. (The interpretation of propositional modal logic in second-order predicate logic is well known; see e.g. [2, §1].) “Strong” refers of course to the expressive power of the languages, not to the deductive power of formal systems. By “rather strong” we mean sufficiently strong that theorems about first-order logic which fail for second-order logic usually fail even for propositional modal logic. Some evidence for this belief is contained in [2] and [3]. In the former is exhibited a finitely axiomatized consistent tense logic having no relational models, and the latter presents a finitely axiomatized modal logic between T and S4, such that □p → □2p is valid in all relational models of the logic but is not a thesis of the logic. The result of [2] is strong evidence that bimodal logic is essentially second-order, but that of [3] does not eliminate the possibility that unimodal logic only appears to be incomplete because we have not adopted sufficiently powerful rules of inference. In the present paper we present stronger evidence of the essentially second-order nature of unimodal logic.

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