Topic
Modal operator
About: Modal operator is a research topic. Over the lifetime, 1151 publications have been published within this topic receiving 22865 citations. The topic is also known as: modal connective.
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TL;DR: A family of noncongruential modal logics obtained by restricting the smallest classical modal logic E and some of its extensions are considered, and determination results proved.
Abstract: We consider a family of noncongruential modal logics obtained by restricting the smallest classical modal logic E and some of its extensions. We show that these logics are also properly contained in Lemmon's S0.5; semantics for them are adapted from Cresswell's semantics for S0.5 and from neighborhood semantics for classical modal logics. Some extensions of these logics by means of usual modal logical axioms are also considered, and determination results proved. As a further example, we also show how to obtain restricted versions of some of Chellas and Segerberg's prenormal modal logics.
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TL;DR: In this paper, the authors present a reformulation in topos logic of a safety result arising in an abstract presentation of blockchain consensus protocols, where it is shown that a proposition and its negation cannot both be safe in protocol states that have executions to some common state.
Abstract: This paper presents a reformulation in topos logic of a safety result arising in an abstract presentation of blockchain consensus protocols. That is, in a high-level template for "correct-by-construction" consensus protocols, it is shown that a proposition and its negation cannot both be safe in protocol states that have executions to some common state. This is in fact true for any inconsistent propositions and the proof requires only intuitionistic reasoning. This opens the door for work on consensus protocols in the internal language of a topos. As a first pass on such a program, the main contribution of this paper is the formulation of estimate safety in abstract correct-by-construction protocols as a forcing statement in the internal logic of a given topos. This is illustrated first in the setting of copresheaf toposes. It is also seen there that safety can be viewed as a modal statement. For these interpretations, some extensions and adaptations of results in the literature on modal operators in toposes are presented. The final reformulation of estimate safety is a completely elementary version in the language of an arbitrary topos where it is seen that estimate safety is equivalent to a certain forcing statement.
TL;DR: In this article , it is shown that certain statements discussed in the metaphysics of modality de re, such as the sufficiency condition for essential properties, cannot be faithfully formalized.
Abstract: It is commonplace to formalize propositions involving essential properties of objects in a language containing modal operators and quantifiers. Assuming David Lewis’s counterpart theory as a semantic framework for quantified modal logic, I will show that certain statements discussed in the metaphysics of modality de re, such as the sufficiency condition for essential properties, cannot be faithfully formalized. A natural modification of Lewis’s translation scheme seems to be an obvious solution but is not acceptable for various reasons. Consequently, the only safe way to express some intuitions regarding essential properties is to use directly the language of counterpart theory without modal operators.
TL;DR: In this paper , the authors show that a single operation can account for three seemingly distinct properties of the semantics of cause and because: comparative nature, asymmetry in logical strength between the two conditions, and the duality between universal and existential quantification.
Abstract: We show that a single operation can account for three seemingly distinct properties of the semantics of cause and because. The properties are, firstly, their comparative nature: interpreting cause and because involves comparing what would happen in the presence of the cause (a positive condition) with what would happen in the absence of the cause (a negative condition). Secondly, there is an asymmetry in logical strength between the two conditions: the positive condition involves a universal modal while the negative condition involves an existential modal. Thirdly, the positive and negative conditions have the same modal base, i.e. are interpreted while assuming the same set of background facts.Despite their apparent dissimilarity, we show that these three properties are predicted by a single operation: exhaustification. The comparative nature of cause and because follows from the comparative nature of exhaustification, which compares a sentence with its alternatives. The asymmetry in strength arises because exhaustification negates alternatives: given the duality between universal and existential quantification, negation flips a necessity modal into a possibility modal, producing the observed strength asymmetry. Finally, the positive and negative condition have the same modal base since, rather cause and because having two modals in their semantics—one for the positive condition and one for the negative condition—their semantics contains a single modal which is copied by exhaustification.We conclude by showing that this exhaustification operator violates Economy constraints, suggesting that it is not subject to licensing conditions but part of lexical semantics of cause and because.