Chellas, B. F. (1980). Modal Logic. Cambridge, U.K.: Cambridge University Press.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....least one situation where X is false and Y is false. The second condition has to do with the causal or implicative relevance of a premise with respect to a given conclusion. We will call the condition (a) a modal condition, since it amounts to use the standard necessary implication of modal logic (Chellas 1980). If donc is modally stronger than alors, we should relax Iatridou s condition on then for alors, for instance by substituting (a ) for (a) a ) The speaker asserts that Y holds in some situations where X holds. If we adopt condition (b) it is still possible to distinguish alors from et ....

Chellas, Brian F. 1980. Modal Logic. An Introduction. Cambridge: Cambridge University Press.

....directed equivalence frames. We now establish that this is indeed the case. The proof will be by means of a standard technique for completeness proofs in modal logic, namely the construction of a canonical model. We now brie y review this technique to x the notation, but refer the reader to Chellas [1980] and [Hughes and Cresswell 1984] for details. A logic L consists of a derivability relation L typically de ned inductively using a basis of a set of axioms and closing under a set of inference rules. Given a logic L, a set of formulae is L inconsistent if there are formulae 1 ; m 2 , ....

Chellas, B. 1980. Modal Logic. Cambridge University Press, Cambridge.

....and ) infer (Modus ponens) Gen. From infer K (Knowledge Generalization) The system with axioms and rules Prop, K, MP, and Gen has been called K. If we add T to K, we get the axiom system T; if we add 4 to T, we get S4; if we add 5 to S4, we get S5. Many other systems can also be formed [Chellas 1980]; these are the four I focus on here. The following result is well known (see, for example, Chellas 1980; Fagin, Halpern, Moses, and Vardi 1995] for proofs. Theorem 5.1: For formulas in the language L K : a) K is a sound and complete axiomatization with respect to M, c) T is a sound and ....

....rules Prop, K, MP, and Gen has been called K. If we add T to K, we get the axiom system T; if we add 4 to T, we get S4; if we add 5 to S4, we get S5. Many other systems can also be formed [Chellas 1980] these are the four I focus on here. The following result is well known (see, for example, [Chellas 1980; Fagin, Halpern, Moses, and Vardi 1995] for proofs. Theorem 5.1: For formulas in the language L K : a) K is a sound and complete axiomatization with respect to M, c) T is a sound and complete axiomatization with respect to M r , c) S4 is a sound and complete axiomatization with ....

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Chellas, B. F. (1980). Modal Logic. Cambridge, U.K.: Cambridge University Press.

....and proving are considered. Further one should not confuse this modality believing , which is categorical, with the weighted beliefs encountered in the subjective probability theory and in the transferable belief model. Here, #p means the agent believes p. Let # satisfies the classical KD system (Chellas, 1980): D : #p # #p K : #(p # q) # (#p # #q) Axiom D states that if you believe p, you do not believe its negation. Axiom K states: if you believe an implication and its antecedent, then you believe its consequent. 11 Under KD, one deduces: AND : #(p # q) #p # #q that states that ....

Chellas, B. F. (1980). Modal logic. Cambridge Univ. Press, G.B.

....equivalence frames. We now establish that this is indeed the case. The proof will be by means of a standard technique for completeness proofs in modal logic, namely the construction of a canonical model. We now brie y review this technique to x the notation, but refer the reader to [Chellas 1980; Hughes and Cresswell 1984] for details. A logic L consists of a derivability relation L typically de ned inductively using a basis of a set of axioms and closing under a set of inference rules. Given a logic L, a set of formulae is L inconsistent if there are formulae 1 ; m 2 , such ....

Chellas, B. 1980. Modal Logic. Cambridge University Press, Cambridge.

....negotiators. As in our model, they also use explicit time structures and their modal operators G (for goal) Int (for intention) and Do have some similarities to our obligation, permission and do operators. However, their semantics is very di#erent from ours. They use a minimal structures Chellas (1980) style semantics for each of their modal operators which leads to a set of axioms that are not appropriate for our agents. In addition, they require a fully specified history and use a discrete point based representation of time, while we use an interval based representation of discrete time. An ....

Chellas, B. (1980). Modal Logic. Cambridge University Press.

....each of these axioms may appear. Indeed, fl has a flavour of both possibility and of necessity without being one or the other. Axioms fl R and fl M are typical of a modality of possibility 3 while fl S is typical for necessity 2. On the other hand, in standard systems, say Lewis modal system S4 [Chellas, 1980], the axiom fl R is never adopted for necessity while fl S never for possibility. In fact, if we add the axiom of the Excluded Middle (EM) and : fl false (which is valid for both 3 and 2) to the modal system fl R, fl M , fl S then fl becomes trivial. We can derive both fl M oe M and M oe ....

....PLL M oe N . Proof: The statement follows immediately from the deduction theorem for IPC (see e.g. Dummett, 1977] and the fact that PLL is an axiomatic extension of IPC. The deduction theorem does not hold for the standard Hilbert presentation of ordinary modal logics. For instance in K, T, S4 [Chellas, 1980] we have M 2M but 6 M oe 2M , and M oe N 3M oe 3N but 6 (M oe N) oe (3M oe 3N ) The Gentzen style calculus for PLL is presented in terms of ordinary sequents Gamma Delta, where Gamma is a finite, possibly empty, list of hypotheses and Delta a finite list of assertions with length 0 or ....

Chellas, B. (1980). Modal Logic. Cambridge University Press.

....approach of this paper is model theoretic in the standard sense of logic, as applied in AI or elsewhere. Statements of fact (including statements of what a given agent intends) are evaluated with respect to a formal model in which different possible states of the world are involved (e.g. see [Chellas, 1980, pp. 34 35] Whether the statement it is raining is true in the model or not depends only on the state of the world relative to which this statement is evaluated, not the beliefs of any agent. Similarly, whether an agent intends something is to be differentiated from the question of whether ....

....is a standard claim of pragmatism, the philosophy behind the formal logic and semantics that is based on possible worlds models [Stalnaker, 1984, pp. 15 19] Such models are used in a number of formal theories, e.g. those of [Fischer Ladner, 1979] Halpern Moses, 1987] and this paper (see [Chellas, 1980] for a textbook level introduction) This idea is also given importance in other naturalist frameworks, e.g. the one of Barwise and Perry [1983] From the technical logical point of view, what distinguishes this approach from an internal approach is that all the evidence we may have for ....

Chellas, Brian F.; 1980. Modal Logic. Cambridge University Press, New York, NY.

....prove the PSPACE lower bound in row 2 of the table. In Section 4, we discuss the effects of bounding the nesting of modal operators. We conclude in Section 5. 2 A brief review of modal logic We briefly review some standard notions of modal logic here. Further details can be found in, for example, [Chellas 1980; Halpern and Moses 1992; Hughes and Cresswell 1968] In this paper we focus on six logics known as K n , T n , S4 n , K45 n , KD45 n , and S5 n . The subscript n in all these logics is meant to emphasize the fact that we are considering the n agent version of the logic. We omit it when ....

....It is well known that there is a close connection between conditions placed on K and the axioms. In particular, T corresponds to the K i s being reflexive, 4 to the K i s being transitive, 5 to the K i s being Euclidean, and D to the K i s being serial. Thus, we get the following result (see [Chellas 1980; Hughes and Cresswell 1968; Halpern and Moses 1992] for proofs) Theorem 2.1: K n (resp. T n , S4 n , KD45 n , K45 n , S5 n ) is a sound and complete axiomatization for the language L n ( Phi) with respect to M n ( Phi) resp. M r n ( Phi) M rt n ( Phi) M est n ( Phi) M st n ( Phi) M ....

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Chellas, B. F. (1980). Modal Logic. Cambridge, U.K.: Cambridge University Press.

....follows that Gamma Delta 2 (A) A, as desired. Thus, A is a fixed point of Gamma Delta 2 . D A Closer Look at Stable Expansions We start with a review of a few of the basic notions of modal logic that we shall need. We refer the reader to one of the standard texts on modal logic, such as [Chellas 1980] or [Hughes and Cresswell 1968] for further details. The modal logic K45 is characterized by the following axiom system, consisting of four axioms: P. All instances of axioms of propositional logic K. L L( L 4. L ) LL 5. L ) L:L and two rules of inference: R1. From and ....

Chellas, B. F. (1980). Modal Logic. Cambridge, U.K.: Cambridge University Press.

....combination results in a rather strange modality. Indeed, fl has a flavour of both possibility and of necessity without being one or the other. Axioms fl R and fl M are typical of possibility while fl F is typical for necessity. On the other hand, in standard systems, say Lewis modal system S4 [Chellas, 1980], the axiom fl R is never adopted for necessity while fl F never for possibility, and in fact they would trivialize the modalities. The second noteworthy feature of PLL is that it is an intuitionistic rather than classical logic which so far has been the basis of the great majority of approaches ....

.... Gamma; M PLL N implies Gamma PLL M oe N . Proof: The statement follows immediately from the deduction theorem for IPC (see e.g. Dummett, 1977] and the fact that PLL is an axiomatic extension of IPC. The deduction theorem does not hold for ordinary modal logics. For instance in K, T, S4 [Chellas, 1980] we have M 2M but 6 M oe 2M , and M oe N 3M oe 3N but 6 (M oe N) oe (3M oe 3N ) The Gentzen style calculus for PLL is presented in terms of ordinary sequents Gamma Delta, where Gamma is a finite, possibly empty, list of hypotheses and Delta a finite list of assertions with length 0 ....

Chellas, B. (1980). Modal Logic. Cambridge University Press.

....interpretation of j y is that if j, then normally y. This conditional is interpreted in the manner of Stalnaker [1968] or Lewis [1973] using the device of possible worlds selection functions. Its truth conditions are completely standard except for the fact that the modal constraint centering (Chellas [1980] calls it mp) is not imposed on worlds selection functions. Such weak conditionals have been used to express pragmatic generalizations before, by Lascarides and Asher [1993] for example. What I will say about presuppositions and conversational implicatures dovetails with and complements their ....

Chellas, B.: 1980, Modal Logic, and Introduction, Cambridge University Press, Cambridge.

....logic, we cannot distinguish the statement the database db does not say that p is true from the statement the database db says that p is false ; both are represented by the formula :p. To solving this problem, a modality B db is usually defined (it generally corresponds to a KD logic (Chellas, 1980)) with formulae of the form B db p to be read: database db says (or believes) that p is true. This makes it possible to distinguish statements :B db p (database db does not say that p is true) from statement B db :p (database db says that p is false) We suggest using a similar approach by ....

....is a propositional formula without modality. ffl (Nec2) Fusion f = Fusion B r f if f is a normative formula 6 ffl (MP) Fusion f and Fusion (f g) Fusion g We can notice that, due to the axioms (A1) A5) the modality O and the modality B are ruled by the axioms of the KD logic (Chellas, 1980, Hughes Cresswell, 1972) In particular, this means that the deontic logic underlying this work is SDL; i.e, the logic used for reasoning with the norms expressed in the regulations, is SDL. A6) A8) are the axioms which rule the merging process. A6) means that if a normative formula f is ....

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Chellas, B. F. (1980). Modal logic, an introduction. Cambridge University Press.

....in the assumptions on which y : A (respectively z : B) depends, except those of the form x R y (respectively y : A and x R y) which are discharged by the inference. FIGURE 1 The rules of K count see (Basin et al. 1996a) We also assume that the reader is familiar with modal logics (see, e.g. (Chellas 1980)) and natural deduction (Prawitz 1965, Prawitz 1971) 0.2 A Hierarchy of Labelled Modal Logics 0.2.1 Labelled K Let W be a set of objects (called labels) representing possible worlds, and let R W Theta W be a binary relation. If x and y are labels, then x R y is a relational formula (rwff ....

....A is 33222A, and so on. A large and important class of modal logics falls under the generalized Geach axiom schema: 3 i 2 m A 2 j 3 n A (where i; j; m; and n are natural numbers) which corresponds to the semantic notion of (i; j; m;n) convergency (or incestuality in the terminology of (Chellas 1980); for correspondence theory see (van Benthem 1984) 8x8y8z(x R i y x R j z 9u(y R m u z R n u) where x R 0 y means x = y and x R i 1 y means 9v(x R v v R i y) There are instances of (i; j; m;n) convergency that explicitly require the equality predicate, e.g. 1; 0; 0; 0) ....

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Chellas, B. 1980. Modal Logic. New York: Cambridge University Press.

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Chellas, B. F. (1980). Modal Logic. Cambridge, U.K.: Cambridge University Press.

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B.F.Chellas (1980). Modal Logic. Cambridge University Press.

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B.F. Chellas, 1980, Modal logic, Cambridge University Press, Cambridge.

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Chellas, Brian F.; 1980. Modal Logic. Cambridge University Press, New York, NY.

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Manuscript. Chellas, B. F. (1980). Modal Logic. Cambridge, U.K.: Cambridge University Press.

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