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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01 Jan 1969
TL;DR: A system of modal logic is sketched in which modal operators are relativised to individuals or sets of individuals, suggested by certain expressions in ordinary language.
Abstract: In this paper I shall sketch a system of modal logic in which modal operators are relativised to individuals or sets of individuals. This extension of modal logic is suggested by certain expressions in ordinary language. For example, under certain circumstances we may utter the sentence (A) John can catch the trainwhich may be taken to be equivalent to (A’) It is possible for John to catch the train.
18 citations
TL;DR: The motivation for this work was to formally treat ordinary language counterfactual conditionals as global conditionals of a certain sort and to distinguish between two types of conditional operators.
Abstract: 1 Motivation In formal contexts, we may distinguish between two types of conditional operators.* The truth value of a local conditional is defined in terms of the truth values of its antecedent and consequent. The truth value of a global conditional is defined in terms of the possible truth-values of its antecedent and consequent. Global conditionals are usually formed by applying a modal operator to a local conditional. For example, strict implication is defined by applying the necessity operator to material implication. That is, \"p-$q\" is defined as \"L(p CO # ) \" , where CO is the standard two-valued material implication and the properties of the necessity operator \" L \" are determined by the particular modal logic being employed. There has been some move to formally treat ordinary language counterfactual conditionals as global conditionals of a certain sort. (See for example [1] and [3]). Viewed from this perspective, the local conditional involved is not the standard two-valued material implication, but is rather a three-valued operator. We may use Γ, F, and / for \" t r u e \" , \" fa lse\", and \"indeterminate\", respectively. We may then define the conditional Cl as follows (contrasting it with material implication CO):
18 citations
TL;DR: Provability logics with many modal operators for progressions of theories obtained by iterating their consistency statements are introduced and the corresponding arithmetical completeness theorem is proved.
Abstract: Provability logics with many modal operators for progressions of theories obtained by iterating their consistency statements are introduced. The corresponding arithmetical completeness theorem is proved.
18 citations
TL;DR: In this article, a modal language consisting of logic conjunctions extended with modal operators of knowledge, belief, and possibility is considered, where results of observations (reflections of objects) are treated as the source of data which induces internal meaning accessible to the agent.
Abstract: In this paper the language grounding problem is considered for a modal language consisting of logic conjunctions extended with modal operators of knowledge, belief and possibility. This language is grounded in an artificial cognitive agent which is located in a world built from objects that can exhibit particular sets of properties. This cognitive agent is equipped with sensors that make it possible to observe states of properties in external objects. Results of observations (reflections of objects) are collected and treated as the source of data which induces internal meaning accessible to the agent. This meaning is assigned to modal conjunctions. The triple of a modal conjunction, the embodied empirical content and an external object are a particular case of semiotic triangle concept within which the phenomenon of language grounding is modelled and studied. It is assumed that conditions for proper grounding of modal conjunctions are defined by the so called epistemic satisfaction relations which specify what empirical content should be collected in the cognitive agent in order to embody commonsense meaning of modal conjunctions. It is proved in this paper that the accepted definition for epistemic satisfaction results in desirable language behaviour corresponding to commonsense intentional semantics and pragmatics of natural language discourse. The desirable language expectations are considered for uniform sets of modal conjunctions.
18 citations
TL;DR: By developing an algebraic semantics, this work establishes the completeness of fifteen modal logics of gamma graphs and shows that, besides logical necessity and possibility, Peirce proposed an epistemic interpretation of the broken-cut modality, and was led to analyze constructions of knowledge in the style of epistemic logic.
Abstract: We describe Peirce’s 1903 system of modal gamma graphs, its transformation rules of inference, and the interpretation of the broken-cut modal operator. We show that Peirce proposed the normality rule in his gamma system. We then show how various normal modal logics arise from Peirce’s assumptions concerning the broken-cut notation. By developing an algebraic semantics we establish the completeness of fifteen modal logics of gamma graphs. We show that, besides logical necessity and possibility, Peirce proposed an epistemic interpretation of the broken-cut modality, and that he was led to analyze constructions of knowledge in the style of epistemic logic.
18 citations