Modal Logics and Topological Semantics for Hybrid Systems

STRONG COMPLETENESS OF MODAL LOGICS OVER 0-DIMENSIONAL METRIC SPACES

The Review of Symbolic Logic ◽

10.1017/s1755020319000534 ◽

2019 ◽

Vol 13 (3) ◽

pp. 611-632

Author(s):

ROBERT GOLDBLATT ◽

IAN HODKINSON

Keyword(s):

Metric Spaces ◽

Deductive System ◽

Modal Logics ◽

Topological Semantics ◽

Strong Completeness

AbstractWe prove strong completeness results for some modal logics with the universal modality, with respect to their topological semantics over 0-dimensional dense-in-themselves metric spaces. We also use failure of compactness to show that, for some languages and spaces, no standard modal deductive system is strongly complete.

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Modifier Logics Based on Graded Modalities

Journal of Advanced Computational Intelligence and Intelligent Informatics ◽

10.20965/jaciii.2003.p0072 ◽

2003 ◽

Vol 7 (2) ◽

pp. 72-78 ◽

Cited By ~ 6

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.

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