G. Hughes, M. Cresswell: "A Companion to Modal Logic", Methuen, London, 1984.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....true for every w 2 W . Given a frame F a formula p is F valid if it is M valid for every model M on F . Finally, a formula p is valid if it is F valid for every frame F . The axiom system in table 3. 1 has been proved sound and complete with respect to the set of all frames F (see, for instance, [71]) Any theorem in it is valid and, conversely, a valid sentence is provable in this system. A set of formulae including these properties is called normal modal logic. The smallest normal modal logic is named K or minimal normal logic. It is the most basic system we can form. Other modal logics ....

G. E. Hughes and M. J. Cresswell. A Companion to Modal Logic. Methuen, 1968.

....and autoepistemic logics in terms of possible world structures. A possible world structure is a collection of two valued propositional interpretations each representing a state of the world that is possible according to the agent. Possible world structures can be seen as special Kripke structures [HC84]. They are of fundamental importance in semantic studies of the modalities of knowledge and belief. Possible world structures were used in the study of autoepistemic logic. Moore described a possible world characterization of expansions in [Moo84] They also figured prominently in Levesque s ....

....values, together with the ordering f t, form the standard Boolean lattice of truth values. The set of all two valued interpretations of At will be denoted by A. Any set Q A is called a possible world structure and can be viewed as a universal Kripke model with a total accessibility relation [Che80, HC84] . Possible world structures constitute a basic tool in semantic studies of modal logics. As we stated earlier, they represent the agent s knowledge about the world. Possible world structures were used by Moore [Moo84] and later by Levesque [Lev90] in the investigations of autoepistemic logic. ....

G.E. Hughes and M.J. Cresswell. A companion to modal logic. Methuen and Co., 1984.

....and autoepistemic logics in terms of possible world structures. A possible world structure is a collection of two valued propositional interpretations each representing a state of the world that is possible according to the agent. Possible world structures can be seen as special Kripke structures [HC84]. They are of fundamental importance in semantic studies of the modalities of knowledge and belief. Possible world structures were used in the study of autoepistemic logic. Moore described a possible world characterization of expansions in [Moo84] They also gured prominently in Levesque s ....

....values, together with the ordering f t, form the standard Boolean lattice of truth values. The set of all two valued interpretations of At will be denoted by A. Any set Q A is called a possible world structure and can be viewed as a universal Kripke model with a total accessibility relation [Che80, HC84] . Possible world structures constitute a basic tool in semantic studies of modal logics. As we stated earlier, they represent the agent s knowledge about the world. Possible world structures were used by Moore [Moo84] and later by Levesque [Lev90] in the investigations of autoepistemic logic. ....

G.E. Hughes and M.J. Cresswell. A companion to modal logic. Methuen and Co., 1984.

.... McDermott and Doyle style nonmonotonic modal logics because of the following property of nonmonotonic modal logics: for any modal logics S 1 ; S 2 contained in S5, if S 1 is contained in S 2 , then any S 1 expansion is a S 2 expansion [80] For example, Chellas [14] and Hughes and Cresswell [46] provide thorough introductions to modal logics. In Example 4.3 the weakly grounded expansion seems to hinge on the fact that :L:Lp is obtained by direct negative introspection. A straightforward technique to eliminate such a possibility is to restrict the set of formulae subject to direct ....

....by adding some new axioms and inference rules, i.e. by strengthening the underlying derivability relation used in the fixed point equation (7.35) McDermott [95] considered modal logics T, S4, and S5 as the basis of nonmonotonic reasoning. For example, Chellas [14] and Hughes and Cresswell [46] are good sources on modal logics. The work was extented first by Shvarts [135] and then by Marek et al. 80] Using Equation (7.35) nonmonotonic modal logics based on monotonic modal logics are defined in the following way as discussed already in Chapter 4. Given a classical monotonic modal ....

G.E. Hughes and M.J. Cresswell. A Companion to Modal Logic. Methuen and Co., London, 1984. -- 153 --

....D (PiM(p M(p) 1.2 THEOREM. The system S5AP(n ) is sound and complete with respect to S AP(n ) PROOF. Soundness is evident. Completeness is proved in the usual way by showing that any (maximally) consistent set of formulas is satisfiable in a canonical modal 6 structure in S5AP(n ) cf. e.g. [HC2]) using the usual modal correspondences of S5 for L M and K45 for Pi, and furthermore the fact that (8) corresponds to the property goi G go; 9) corresponds to s,t,u go(s,t) goi(t,u) goi(s,u) 10) corresponds to: if goi is locally serial at s, i.e. if there exists a t with goi(s,t) ....

G.E. Hughes & M.J. Cresswell, A Companion to Modal Logic, Methuen, London, 1984.

....not hold, therefore #(A, g(i) S and #(A, g( k, i)# S4 ) S . However, according to lemma 3 g(i) #= #, which means that there is a world w n such that #(A, w n ) S and #(A, w n ) S , thus obtaining a contradiction. Theorem 6. ## # Proof. For the proof see, for example, [11]. Theorem 7. # #KES4 A. Proof. The characteristic axioms of S4 and modus ponens are provable in KES4 (see section 6, and [3] for a proof that modus ponens is a derived rule in KE, the propositional subsystem of KES4) We give a KES4 proof of the rule of necessitation. Let us assume that ....

G. E. Hughes and M. Creswell. A Companion to Modal Logic. London, Methuen, 1984.

....their own conception of these concepts, it is very important to unambiguously establish what is meant by these concepts when ascribed to some specific implemented agent. Formal specifications allow for such an unambiguous definition. The formal tool that we propose to model agency is modal logic [4, 20, 21]. Using modal logics offers a number of advantages. Firstly, using an intensional logic like modal logic allows one to come up with an intuitively acceptable formalisation of intensional notions with much less effort than it would take to do something similar using fully fledged first order logic. ....

....In Definition 3.3 we required these relations to be equivalence relations, and this demand indeed corresponds to knowledge being veridical and satisfying the properties of positive and negative introspection. The proof of the following proposition is standard and well known from the literature [21, 36]. 15 Proposition 4.7 The following correspondences hold. 1. T 8s( s; s) 2 R(i) i.e. R(i) is reflexive ) 2 R(i) s ) 2 R(i) s; s ) 2 R(i) i.e. R(i) is transitive ) 2 R(i) s; s ) 2 R(i) s ) 2 R(i) i.e. R(i) is Euclidean 4.2 Additional properties of ....

G.E. Hughes and M.J. Cresswell. A Companion to Modal Logic. Methuen & Co. Ltd., London, 1984.

....logic with equality. It is essential that we look at all models here; if we restrict ourselves to intitial models, completeness would only hold with respect to ground equations [9] We prove completeness of the modal part of DYN = by using a Henkin style proof that is standard in modal logic [24]. First note that IN] and [DL1] characterize Kripke structures, with accessibility relations R s for each Tprocess(X) Moreover, the inference rule [Sub2] corresponds to the property that 4Process 1 = 2 = R = 32 (13) Let us call the class of Kripke structures satisfying (13) Completeness ....

G.E. Hughes and M.J. Cresswell. A Companion to Modal Logic. Methuen, 1984.

....by Jennings and Schotch in the 80 s. 1 Introduction In the standard kripkean binary relational semantics, the truth condition for modal formulae is defined by j= M x A , 8y; Rxy )j= M y A where the notions of frame, model, and satisfaction are defined in the usual way (see [3] 7] [8], and [9] The minimal modal logic determined by the kripkean binary relational frame is the logic K, most economically axiomatized by adding to PL the single rule called Scott Rule: SR] Gamma A [ Gamma] A where [ Gamma] fB : B 2 Gammag. Alternatively, K can be axiomatized by adding ....

G. E. Hughes and M. J. Cresswell. A Companion To Modal Logic. Methuen & Co., 1984.

No context found.

G. Hughes, M. Cresswell: "A Companion to Modal Logic", Methuen, London, 1984.

No context found.

G.E. Hughes and M.J. Cresswell. A Companion to Modal Logic. Methuen, 1984.

No context found.

G. Hughes & M. Cresswell, A Companion to Modal Logic, Methuen, London, 1984.

No context found.

: Hughes, G.E., Cresswell, M.J., A Companion to Modal Logic, London, Methuen, 1984.

No context found.

G.E. Hughes, M.J. Cresswell, A companion to modal logic, Methuen, London, 1984.

No context found.

Hughes, G. E., and Cresswell, M. J. A Companion to Modal Logic. Methuen, London, 1984.

No context found.

H.G. Hughes and M.J. Creswell. A Companion to Modal Logic. Methuen, London, 1984.

No context found.

Hughes, G. and Creswell, M. (1984). A Companion to Modal Logic. Methuen.

No context found.

G. E. Hughes and M. J. Cresswell. A Companion to Modal Logic. Methuen, London, 1984. 31

No context found.

G.E. Hughes and M.J. Cresswell. A Companion to Modal Logic. Methuen, 1984.

No context found.

Hughes, G. and Cresswell, M.: 1984, A Companion to Modal Logic, Methuen & Co., Ltd.

No context found.

G. E. Hughes and M. J. Cresswell. A Companion to Modal Logic. Methuen, London, 1984.

No context found.

G.E. Hughes and M.J. Cresswell. A Companion to Modal Logic. Methuen, London, 1984.

No context found.

G.E. Hughes and M.J. Cresswell. A Companion to Modal Logic. Methuen, 1984.

No context found.

G. E. Hughes and M. J. Cresswell, A Companion to Modal Logic, Methuen, 1984.

No context found.

G.E. Hughes and M.J. Cresswell. A Companion To Modal Logic. Methuen, London and New York, 1984.

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC