Download PDF by G. E. Hughes: A Companion to Modal Logic

By G. E. Hughes

ISBN-10: 0416375103

ISBN-13: 9780416375107

Word: This ebook was once later changed via "A New advent to Modal good judgment" (1996).

An past publication of ours, entitled An advent to Modal good judgment (IML), was once released in 1968. after we wrote it, we have been in a position to provide a fairly complete survey of the kingdom of modal common sense at the moment. We greatly doubt, even though, no matter if any similar survey will be attainable this present day, for, considering that 1968, the topic has built vigorously in a wide selection of directions.

The current publication is as a result now not an try to replace IML within the kind of that paintings, however it is in a few experience a sequel to it. the majority of IML was once fascinated by the outline of a number specific modal platforms. we've made no test the following to survey the very huge variety of platforms present in the new literature. stable surveys of those can be present in Lemmon and Scott (1977), Segerberg (1971) and Chellas (1980), and we've not wanted to replicate the fabric present in those works. Our target has been relatively to be aware of convinced contemporary advancements which predicament questions about normal homes of modal structures and that have, we think, ended in a real deepening of our realizing of modal good judgment. lots of the appropriate fabric is, even though, at this time to be had purely in magazine articles, after which usually in a sort that is available basically to a pretty skilled employee within the box. we've attempted to make those very important advancements obtainable to all scholars of modal logic,as we think they need to be.

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If cz). g. ) Now consider the following relation that might obtain between a system and a class of models: (2) For every wif if is S-consistent then there is some model in which for some weW, V(cx,w)= I. It is not difficult to show that if (2) holds, so does ( 1 ). For if this means that is S-consistent ; so if (2) holds, there is a OC, w) = 1. But by [V model in which for some weW, V we then have V (oc, w) = 0, and so (1) holds. ) What the canonical model method directly proves about a system S and a class of models is (2).

FlSOdOAd 1HM 'r)A (M i ill 24 A COMPANION TO MODAL LOGIC clause (3) in the definition of a canonical model. We then prove (a) that if the theorem holds for a wif oc, it also holds for cc, (b) that if it holds for each of a pair of wif ci and fi, it holds for cc v and (c) that if it holds for a wif oc, it also holds for Lcc. Since , and L are our only primitive operators, this will show that the theorem holds for every wif. We now prove each of (a) — (c) in turn. (a) Consider a wif cx and any weW.

Then (a) F is S-consistent. For if it were not, then some finite subset of F would be S-inconsistent. But clearly every finite and we have shown that subset of F is a subset of some no F is maximal. For consider any wif ;. By the construction of F1 , either cc e F1 or e F1 and so, since F1 c F, either ;eFor ; (c) AcF,sinceAisF0andF0cF. 2. All the results we have proved so far depend only on the fact that S contains PC. They therefore hold for any system, whether modal or not, which contains PC.

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A Companion to Modal Logic by G. E. Hughes


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