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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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Book ChapterDOI
01 Jan 2015
TL;DR: In this article, a modal logic over a finitely-valued Łukasiewicz logic is defined to capture possibilistic reasoning, and a completeness proof is provided.
Abstract: In this chapter we revisit a 1994 chapter by Hajek et al. where a modal logic over a finitely-valued Łukasiewicz logic is defined to capture possibilistic reasoning. In this chapter we go further in two aspects: first, we generalize the approach in the sense of considering modal logics over an arbitrary finite MTL-chain, and second, we consider a different possibilistic semantics for the necessity and possibility modal operators. The main result is a completeness proof that exploits similar techniques to the ones involved in Hajek et al.’s previous work.

8 citations

Journal ArticleDOI
TL;DR: Recherche d'une interpretation fidele de la theorie intuitionniste des ensembles dans le programme d'integration des mathematiques classiques and intuitionnistes.
Abstract: In [6] Godel observed that intuitionistic propositional logic can be interpreted in Lewis's modal logic (S4). The idea behind this interpretation is to regard the modal operator □ as expressing the epistemic notion of “informal provability”. With the work of Shapiro [12], Myhill [10], Goodman [7], [8], and Scedrov [11] this simple idea has developed into a successful program of integrating classical and intuitionistic mathematics.There is one question quite central to the above program that has remained open. Namely:Does Scedrov's extension of the Godel translation to set theory provide a faithful interpretation of intuitionistic set theory into epistemic set theory?In the present paper we give an affirmative answer to this question.The main ingredient in our proof is the construction of an interpretation of epistemic set theory into intuitionistic set theory which is inverse to the Godel translation. This is accomplished in two steps. First we observe that Funayama's theorem is constructively provable and apply it to the power set of 1. This provides an embedding of the set of propositions into a complete topological Boolean algebra . Second, in a fashion completely analogous to the construction of Boolean-valued models of classical set theory, we define the -valued universe V(). V() gives a model of epistemic set theory and, since we use a constructive metatheory, this provides an interpretation of epistemic set theory into intuitionistic set theory.

8 citations

01 Jan 1995
TL;DR: It is shown that reasoning in ground logics is p 3-hard, and it is proved that p 3 is also an upper bound for reasoning in the major ground logic, as well as identifying some special cases where the complexity of reasoning in Ground Logics is lower than in the general case.
Abstract: In this paper we discuss ground logics, a family of nonmonotonic modal logics, with the goal of using them in knowledge representation. Ground logics are based on the idea of minimizing the knowledge expressed by non-modal formulae. The nonmonotonic character of the logics can be described either by a x-point equation or by means of a preference relation on possible-worlds models. We address both the epistemological and the computational properties of ground logics. We discuss their representational features and provide a thorough comparison with McDermott and Doyle logics. Then, we show that reasoning in ground logics is p 3-hard, and prove that p 3 is also an upper bound for reasoning in the major ground logics. Moreover, we identify some special cases where the complexity of reasoning in ground logics is lower than in the general case.

8 citations

Book
24 Mar 2011
TL;DR: In this paper, the authors introduce the notion of negation in the context of Quantificational Logic (QL) and Model Sets for QL (MSQL), which is a generalization of the concept of truth-functionality and negation.
Abstract: 1. Introduction / Sentences / Truth and Falsity / Defense and Refutation / Inference, Form and Implication / Formally Valid Inference / Conjunctions / Inference with Conjunctions / Negation / Inference with Negation / Truth-Functionality and Negation / Grouping / 2. Sentential Logic / Simple Sentences / Sentences / Derivations: A First Look / A Note on Sets / Lines / Derivations Again / Theorems / Truth Sets / Soundness / Completeness / Extensions of SL / Conditionalization / Model Sets / Syntax and Semantics / 3. Quantificational Logic / Singular Terms / Predicates / Some Symbolic Conventions / Some / The Language QL / Derivations / Truth Sets / All / Further Extensions of QL / Model Sets / Identity / Model Sets for QL / 4. Sentential Modal Logic / Non-Truth-Functional Sentential Operators / Sentential Modal Operators / Derivations / S5, S4, T, and B / Possible Worlds / At a World and In a World / Model Sets and Model Systems / Deontic Logic and Model Sets / 5. Quantification and Modality / Some Derivations / Model Sets and Systems / An Alternative / 6. Set Theory / The Axiom of Extensionality / Axioms of Separation / Pairing Axiom and Rule U / The Restriction on the A2 Axiom / The Null Set / An Interpretation / More Axioms / General Intersection Operation / Order and Relations / Functions / Sizes of Sets / The Power Set Axiom / A Basic Theorem / 7. Incompleteness / The Language of Arithmetic / Three Key Concepts / Three Key Theorems / The Core Argument / Concluding Observations / 8. An Introduction to Term Logic / Syllogistic / The Limits of Syllogistic / Term Functor Logic / Singular Terms and Identity in TFL / Relationals in TFL / The Logic of Sentences in TFL / Rules of Inference for Derivations in TFL / Derivation in TFL / The Bridge to TFL / 9. Modal Term Logic / Modal Operators on Terms / Modal Operators on Sentences / Rules of Derivation for Modal TFL / Modal Inference in TFL / Rules, Axioms and Principles / List of Symbols / Glossary / Index.

8 citations

Book ChapterDOI
08 Oct 2015
TL;DR: A machine is defined that, for (at least) a specific case, knows its own code and knows to be sound and the definition of decidability by some Turing machine is extended.
Abstract: Church-Turing Thesis, mechanistic project, and Godelian Arguments offer different perspectives of informal intuitions behind the relationship existing between the notion of intuitively provable and the definition of decidability by some Turing machine. One of the most formal lines of research in this setting is represented by the theory of knowing machines, based on an extension of Peano Arithmetic, encompassing an epistemic notion of knowledge formalized through a modal operator denoting intuitive provability. In this framework, variants of the Church-Turing Thesis can be constructed and interpreted to characterize the knowledge that can be acquired by machines. In this paper, we survey such a theory of knowing machines and extend some recent results proving that a machine can know its own code exactly but cannot know its own correctness (despite actually being sound). In particular, we define a machine that, for (at least) a specific case, knows its own code and knows to be sound.

8 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202316
202222
202138
202035
201946
201844