# A Companion to Modal Logic by G. E. Hughes

By G. E. Hughes

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

An past booklet of ours, entitled An creation to Modal good judgment (IML), used to be released in 1968. after we wrote it, we have been capable of provide a pretty finished survey of the kingdom of modal common sense at the moment. We greatly doubt, notwithstanding, no matter if any similar survey will be attainable at the present time, for, due to the fact that 1968, the topic has constructed vigorously in a wide selection of directions.

The current publication is for that reason now not an try and replace IML within the type of that paintings, however it is in a few experience a sequel to it. the majority of IML used to be fascinated by the outline of a number of specific modal structures. we now have made no test right here to survey the very huge variety of platforms present in the new literature. solid 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 particularly to be aware of convinced contemporary advancements which hindrance questions about common 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, despite the fact that, at the present on hand basically in magazine articles, after which usually in a sort that's obtainable in simple terms to a reasonably skilled employee within the box. we've attempted to make those vital advancements obtainable to all scholars of modal logic,as we think they need to be.

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**Sample text**

We say that A is S-consistent if;. e. if {;, If A is an infinite set, we cannot proceed quite so simply, since there is no such thing as the conjunction of all its members. What we say in this case is that A is S-consistent if every finite subset of A is S-consistent; or in other words, if there is no 18 A COMPANION TO MODAL LOGIC finite subset fr1 , . . , of A such that This indeed we shall take as our general definition of S-consistency, since it will clearly cover the case of a finite set, and even the case of a single formula, if we think of it as the set of which it is the only member.

Van Benthem (1980), p. 138, and Hughes and Cresswell (1975), p. 21. g. Segerberg (1971), p. 34) call a logic 'finitely axiomatizable' only if it can be axiomatized by a finite set of axioms with US and MP as the only transformation rules. In this sense, even some of the systems to be introduced in this chapter, such as K and T, are not finitely axiomatizable. 6 The name 'K' was given to this system in Lemmon and Scott (1977), p. 29, in honour of Saul Kripke, from whose work the model theory for normal modal systems is largely derived.

Hence 1 by [VL], V(Lcz, w) = 1. la, Locew. 3, L - (w) u { cc} is S-consistent. 2 and the definition of W, there is some w' e W such that L - (w) u { cx} w', and therefore such that (i) L (w) and (ii) '%' ctew'. Now (i) gives us wRw', by the definition of R. And since the theorem is assumed to hold for oc, and therefore, by (a) above, also for cx, (ii) gives us V( cx, w') = 1, and hence V(oc, w') # 1. So by [VL] we have V(Lcx, w) # 1. 4. 5 Any wif cx is valid in the canonical model for S if F—s PROOF