A Companion to Modal Logic by G. E. Hughes

By G. E. Hughes

Observe: This e-book used to be later changed by means of "A New creation to Modal good judgment" (1996).

An past booklet of ours, entitled An advent to Modal common sense (IML), was once released in 1968. after we wrote it, we have been in a position to provide a pretty finished survey of the country of modal good judgment at the moment. We a great deal doubt, in spite of the fact that, no matter if any similar survey will be attainable this day, for, given that 1968, the topic has built vigorously in a large choice of directions.

The current ebook is accordingly no longer an try to replace IML within the sort of that paintings, however it is in a few feel a sequel to it. the majority of IML was once involved in the outline of a number of specific modal platforms. we now have made no try the following to survey the very huge variety of platforms present in the new literature. sturdy surveys of those could be present in Lemmon and Scott (1977), Segerberg (1971) and Chellas (1980), and we haven't wanted to replicate the fabric present in those works. Our objective has been relatively to be aware of sure contemporary advancements which main issue questions about normal homes of modal structures and that have, we think, ended in a real deepening of our realizing of modal good judgment. many of the proper fabric is, although, at this time to be had in simple terms in magazine articles, after which usually in a kind that is obtainable merely to a pretty skilled employee within the box. we now have attempted to make those vital advancements obtainable to all scholars of modal logic,as we think they need to be.

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PROOF It is a straightforward consequence of the definition of R in a canonical model that if wWw' then {cx c w'. For the converse we have to show that, for every n 0, (A) For any w and w'eW, if {cz :L"cxew} c w', then wR"w'. We prove this inductively, by showing that (A) holds when n = 0, and then that, on the hypothesis that it holds for an arbitrary natural number n, it also holds for n + 1. e. w itself. So if {x: Thxew} w', then w c w'. e. that wR°w'. We now take as our induction hypothesis that (A) holds for n.

The proof is a generalization of the proof given on pp. 32f. 2, and the reader may find it helpful to look back at that proof. Suppose, then, that for some w1, w2, w3 eW in the canonical model for a system S in which (1) MmL"p We want to show that there is also in W some w4 such that both and w3 R'w4. 8 (p. 26), it is sufficient for this purpose to show that is a theorem, we have w1 Rmw2 and w1 (A) is S-consistent. Now suppose it is not. 8), since . Therefore by (1) we have must have we have MkcLEW3 .

Condition: If w1Rw2 and w1Rw3, then w2Rw3. (c) 54 + MLp z (p n Lp). Conditions : (i) Reflexiveness; (ii) transitivity ; (iii) if w1 Rw2 and w1 w2 , then for every w3, if w1 Rw3 , w3 Rw2 . ) (d) T + Lp D LMLp. Conditions : (i) Reflexiveness ; (ii) if w1Rw2 then there is some w3 such that both (1) w2Rw3 and (2) for any w4 , if w3 Rw4 then w1 Rw4 . (Hint : for condition (ii) in the completeness proof, assume that w1 Rw2 , show that L (w2) u { Lcc :Lcsew1} is consistent, and let w3 be a world that includes this set.

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