Strong noncontingency: on the modal logics of an operator expressively weaker than necessity.

*(English)*Zbl 1441.03019The author provides a detailed study of propositional modal logic with unary strong noncontingency operator. It has intuitive meaning to the effect that if a formula with this operator is true, then it is necessarily true, and if it is false, then it is necessarily false. It may be applied as a formal way of expressing the content of question-embedding verbs like “knowing whether” or “knowing who”, as analysed by Hintikka. For example “\(a\) knows whether \(p\)” may be understood as “if \(p\), then \(a\) knows that, and if not-\(p\), then \(a\) knows that not-\(p\)”, and such understanding can be formalised by means of the new operator. However, the paper is not concerned much with linguistic or philosophical motivations and applications. It is a formal study of logical properties of the new operator.

This new operator is compared with respect to its expressive power with operators of standard necessity, noncontingency and essence/accident. In particular, it is shown that logic of strong noncontingency is less expressive than standard logic of necessity, incomparable with the logic of noncontingency, and expressively equivalent to the logic of essence and accident. In contrast to necessity, the new operator is rather weak tool for frame definability; although symmetry is definable, the properties of seriality, reflexivity, transitivity and Euclideaness are not definable. Two sections (4, 5) are devoted to characterization results based on suitable notion of bisimulation. It is shown that this logic is a bisimulation-invariant fragment of standard modal logic and of the first-order logic. Axiomatization and completeness results for the basic logic and some extensions occupy Section 6 and 7. Finally a comparison with Humberstone’s work on the logic of agreement operator is provided.

The paper is interesting and readable. Despite the philosophical motivations beyond the new operator, it is adressed rather to formal logicians interested in technical development of propositional modal logic.

This new operator is compared with respect to its expressive power with operators of standard necessity, noncontingency and essence/accident. In particular, it is shown that logic of strong noncontingency is less expressive than standard logic of necessity, incomparable with the logic of noncontingency, and expressively equivalent to the logic of essence and accident. In contrast to necessity, the new operator is rather weak tool for frame definability; although symmetry is definable, the properties of seriality, reflexivity, transitivity and Euclideaness are not definable. Two sections (4, 5) are devoted to characterization results based on suitable notion of bisimulation. It is shown that this logic is a bisimulation-invariant fragment of standard modal logic and of the first-order logic. Axiomatization and completeness results for the basic logic and some extensions occupy Section 6 and 7. Finally a comparison with Humberstone’s work on the logic of agreement operator is provided.

The paper is interesting and readable. Despite the philosophical motivations beyond the new operator, it is adressed rather to formal logicians interested in technical development of propositional modal logic.

Reviewer: Andrzej Indrzejczak (Łódź)

##### MSC:

03B45 | Modal logic (including the logic of norms) |

03B42 | Logics of knowledge and belief (including belief change) |

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