A strong completeness theorem in intuitionistic quantified modal logic

2000 ◽  
Vol 43 (1) ◽  
pp. 60-70 ◽  
Author(s):  
Hengshan Gao
Keyword(s):  
Modal Logic ◽  
Completeness Theorem ◽  
Strong Completeness ◽  
Quantified Modal Logic

A unified completeness theorem for quantified modal logics

2002 ◽  
Vol 67 (4) ◽  
pp. 1483-1510 ◽  
Author(s):  
Giovanna Corsi
Keyword(s):  
Modal Logic ◽  
Completeness Theorem ◽  
Modal Logics ◽  
General Strategy ◽  
Quantified Modal Logic ◽  
Barcan Formula ◽  
Completeness Theorems

AbstractA general strategy for proving completeness theorems for quantified modal logics is provided. Starting from free quantified modal logic K. with or without identity, extensions obtained either by adding the principle of universal instantiation or the converse of the Barcan formula or the Barcan formula are considered and proved complete in a uniform way. Completeness theorems are also shown for systems with the extended Barcan rule as well as for some quantified extensions of the modal logic B. The incompleteness of Q°.B + BF is also proved.

Some results in modal model theory

1974 ◽  
Vol 39 (3) ◽  
pp. 496-508 ◽  
Author(s):  
Michael Mortimer
Keyword(s):  
Modal Logic ◽  
Model Theory ◽  
Classical Logic ◽  
Classical Model ◽  
Completeness Theorem ◽  
Strong Completeness ◽  
Barcan Formula ◽  
Weak Completeness ◽  
Modal Model ◽  
Omitting Types

This paper is concerned with extending some basic results from classical model theory to modal logic.In §1, we define the majority of terms used in the paper, and explain our notation. A full catalogue would be excessive, and we cite [3] and [7] as general references.Many papers on modal logic that have appeared are concerned with (i) introducing a new modal logic, and (ii) proving a weak completeness theorem for it. Theorem 1, in §2, in many cases allows us to conclude immediately that a strong completeness theorem holds for such a logic in languages of arbitrary cardinality. In particular, this is true of S4 with the Barcan formula.In §3 we strengthen Theorem 1 for a number of modal logics to deal with the satisfaction of several sets of sentences, and so obtain a realizing types theorem. Finally, an omitting types theorem, generalizing the result for classical logic (see [5]) is proved in §4.Several consequences of Theorem 1 are already to be found in the literature. [2] gives a proof of strong completeness in languages of arbitrary cardinality of various logics without the Barcan formula, and [8] for some logics in countable languages with it. In the latter case, the result for uncountable languages is cited, without proof, in [1], and there credited to Montague. Our proof was found independently.

scholarly journals Interpretability over peano arithmetic

1999 ◽  
Vol 64 (4) ◽  
pp. 1407-1425
Author(s):  
Claes Strannegård
Keyword(s):  
Modal Logic ◽  
Completeness Theorem ◽  
Peano Arithmetic ◽  
Embedding Theorem ◽  
Compactness Theorem ◽  
Partial Answer ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Interpretability Logic

AbstractWe investigate the modal logic of interpretability over Peano arithmetic. Our main result is a compactness theorem that extends the arithmetical completeness theorem for the interpretability logic ILMω. This extension concerns recursively enumerable sets of formulas of interpretability logic (rather than single formulas). As corollaries we obtain a uniform arithmetical completeness theorem for the interpretability logic ILM and a partial answer to a question of Orey from 1961. After some simplifications, we also obtain Shavrukov's embedding theorem for Magari algebras (a.k.a. diagonalizable algebras).

Quantified Modal Logic With Rigid Terms

1988 ◽  
Vol 34 (3) ◽  
pp. 251-259 ◽  
Author(s):  
Giovanna Corsi
Keyword(s):  
Modal Logic ◽  
Quantified Modal Logic

QUANTIFIED MODAL LOGIC ON THE RATIONAL LINE

2014 ◽  
Vol 7 (3) ◽  
pp. 439-454 ◽  
Author(s):  
PHILIP KREMER
Keyword(s):  
Modal Logic ◽  
Real Line ◽  
Topological Spaces ◽  
Topological Semantics ◽  
Cantor Space ◽  
Quantified Modal Logic ◽  
Propositional Modal Logic ◽  
The Family ◽  
Modal Logic S4 ◽  
Rational Line

AbstractIn the topological semantics for propositional modal logic, S4 is known to be complete for the class of all topological spaces, for the rational line, for Cantor space, and for the real line. In the topological semantics for quantified modal logic, QS4 is known to be complete for the class of all topological spaces, and for the family of subspaces of the irrational line. The main result of the current paper is that QS4 is complete, indeed strongly complete, for the rational line.

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