A New Introduction to Modal Logic

Hughes, G E; Cresswell, M. J.

doi:10.4324/9780203290644

Cited by 789 publications

(481 citation statements)

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“…The well-known modal logic S5 (see an introduction in [HC68]) shares with E the same set of models but the modal operators in S5 are denoted by 2 and 3. In the sequel, we write for 2 (φ 0 ) to denote the set of formulas for S5.…”

Section: Expressivity Of E With Respect To S5mentioning

confidence: 99%

A simple tableau system for the logic of elsewhere

Demri¹

1996

Theorem Proving With Analytic Tableaux and Related Methods

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Abstract. We analyze different features related to the mechanization of von Wright's logic of elsewhere E. First, we give a new proof of the NP-completeness of the satisfiability problem (giving a new bound for the size of models of the satisfiable formulae) and we show that this problem becomes linear-time when the number of propositional variables is bounded. Although E and the well-known propositional modal S5 share numerous common features we show that E is strictly more expressive than S5 (in a sense to be specified). Second, we present a prefixed tableau system for E and we prove both its soundness and completeness. Two extensions of this system are also defined, one related to the logical consequence relation and the other related to the addition of modal operators (without increasing the expressive power). An example of tableau proof is also presented. Different continuations of this work are proposed, one of them being to implement the defined tableau system, another one being to extend this system to richer logics that can be found in the literature.

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Section: Expressivity Of E With Respect To S5mentioning

confidence: 99%

A simple tableau system for the logic of elsewhere

Demri¹

1996

Theorem Proving With Analytic Tableaux and Related Methods

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“…Besides, we shall be all the more certain that X takes its value in some subset A as all the values outside A have a smaller degree of possibility; this can be estimated by means of a dual measure of necessity N(A) =1 -Π(A c ) (A c is the complement of A). This duality between possibility and necessity measures (Dubois and Prade, 1980) is a graded version of the one existing in modal logic (Hughes and Cresswell, 1968), which expresses a relation between the necessary and the possible, already advocated by Aristotle and his school; indeed some linkage between possibility theory and some modal logic systems have been pointed out (e.g., Fariñas del Cerro and Herzig, 1991). Necessity degrees N(A) express how certain the proposition "X is A" is, in the face of the possibility distribution induced by F. More precisely it expresses to what extent "X is A" is implied by "X is F", namely N(A) = I S (F, A), using (1.13) with a Kleene-Dienes implication.…”

Section: When a Is An Ordinary Subset π(A) Is Defined As The Maximummentioning

confidence: 99%

“…However, in propositional logic, uncertainty is three-valued: one can be sure about a proposition, sure about its negation, or unsure about both. Explicitly expressing and reasoning about these three situations requires the setting of modal logic (Hughes and Cresswell, 1968), where certainty is captured by the modal necessity and reflects provability, while the lack of certainty is captured by the modal possibility, and reflects logical consistency.…”

Section: World and Wordsmentioning

confidence: 99%

Fuzzy Sets: History and Basic Notions

Dubois¹,

Ostasiewicz

Prade³

2000

The Handbooks of Fuzzy Sets Series

138

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Abstract. This paper is an introduction to fuzzy set theory. It has several purposes. First, it tries to explain the emergence of fuzzy sets from an historical perspective. Looking back to the history of sciences, it seems that fuzzy sets were bound to appear at some point in the 20th century. Indeed, Zadeh's works have cristalized and popularized a concern that has appeared in the first half of the century, mainly in philosophical circles. Another purpose of the paper is to scan the basic definitions in the field, that are required for a proper reading of the rest of the volume, as well as the other volumes of the Handbooks of Fuzzy Sets Series. This Chapter also contains a discussion on operational semantics of the generally too abstract notion of membership function. Lastly, a survey of variants of fuzzy sets and related matters is provided.

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“…Completeness of K can be proved by e.g. the Lemmon-Scott extension of Henkin's technique of canonical models [29], [50]. Decidability of K follows from completeness and the collapse property mentioned above along with decidability of propositional calculus.…”

Section: Modal Logicmentioning

confidence: 99%

“…The derivation rules of modal propositional logic are: modus ponens M P and the necessity rule N which is the relation on formulas consisting of pairs of the form (α, [α]). This calculus is consistent [50]: to see it it suffices to collapse formulas of K onto formulas of propositional calculus by omitting all modal operator symbols. Then theorems of modal logic are in one-to-one correspondence with theorems of propositional calculus.…”

Section: Modal Logicmentioning

confidence: 99%

Rough Sets and Rough Logic: A KDD Perspective

Pawlak

Polkowski

Skowron

2000

Rough Set Methods and Applications

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Basic ideas of rough set theory were proposed by Zdzis law Pawlak [85,86] in the early 1980's. In the ensuing years, we have witnessed a systematic, world-wide growth of interest in rough sets and their applications.The main goal of rough set analysis is induction of approximations of concepts. This main goal is motivated by the basic fact, constituting also the main problem of KDD, that languages we may choose for knowledge description are incomplete. A fortiori, we have to describe concepts of interest (features, properties, relations etc.) not completely but by means of their reflections (i.e. approximations) in the chosen language. The most important issues in this induction process are:-construction of relevant primitive concepts from which approximations of more complex concepts are assembled, -measures of inclusion and similarity (closeness) on concepts, -construction of operations producing complex concepts from the primitive ones.Basic tools of rough set approach are related to concept approximations. They are defined by approximation spaces. For many applications, in particular for KDD problems, it is necessary to search for relevant approximation spaces in the large space of parameterized approximation spaces. Strategies for tuning parameters of approximation spaces are crucial for inducing concept approximations of high quality.Methods proposed in rough set approach are kin to general methods used to solve Knowledge Discovery and Data Mining (KDD) problems like feature selection, feature extraction (e.g. discretization or grouping of symbolic value),

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A New Introduction to Modal Logic

Cited by 789 publications

References 0 publications

A simple tableau system for the logic of elsewhere

A simple tableau system for the logic of elsewhere

Fuzzy Sets: History and Basic Notions

Rough Sets and Rough Logic: A KDD Perspective

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