By G.E.Hughes, M.J.Cresswell
This long-awaited ebook replaces Hughes and Cresswell's vintage experiences of modal common sense: An creation to Modal common sense and A spouse to Modal Logic.A New creation to Modal common sense is a wholly new paintings, thoroughly re-written by way of the authors. they've got included the entire new advancements that experience taken position considering 1968 in either modal propositional common sense and modal predicate common sense, with no sacrificing tha readability of exposition and approachability that have been crucial gains in their previous works.The ebook takes readers from the main simple structures of modal propositional common sense correct as much as platforms of modal predicate with id. It covers either technical advancements corresponding to completeness and incompleteness, and finite and countless types, and their philosophical functions, particularly within the sector of modal predicate common sense.
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Extra info for A New Introduction to Modal Logic
So P is an example of a rule which holds in some systems but not in all their extensions, and this illustrates the care that must be taken with derived rules. If we look back at the way DRl-DR3 and Eq were proved to hold in K, we can easily see why they hold in all extensions of K: for they were derived by appealing only to elements in K (theorems and primitive transformation rules) which are still present in all its extensions. But P is a rule of K and D because of features of those systems which are not present in all their extensions - in the case of K because the system is too weak to have any theorem satisfying the antecedent of the rule.
The application of MP and N will be indicated by ‘ X MP’ and ‘ x N’ respectively. We shall first prove two theorems in full detail, and then describe some methods of abbreviating proofs. Theorems will be numbered using the name of the relevant system; thus Kl will be the first theorem we prove in K, and so on. ’ PC2 (6) @ A 4 1 g (6) x N (7) L(6-’ A 4) 1 4) (3)[P A &‘I (8) MP A d 1 q) 1 (UP A 4 1 W (9) L(p A q) 3 Lq (71, (8) x MP (10) @ 3 4) 1 (@ 1 9 1 @ 1 (4 A r))) PC3 (lO)[Lo? D. D. The proofs of these theorems satisfy exactly the requirements we listed 27 A NEW INTRODUCTION TO MODAL LOGIC for a proof in K.
E. D. In fact Dl would provide an alternative axiom for D, since if we add it alone to K we can derive D in the following way: (1) MP 1 P) = @P 1 MP) K7[P/q1 Dl, (1) x Eq. D. LP 1 MP It is worth noting that if any wff CYis a theorem of D, then so is Mol. For if CYis a theorem, N gives La as a theorem; and then by D[dp] and MP we obtain MCY. It is also worth noting that if any system which is an extension of K has any theorems of the form MCY,that system contains D. D. In introducing the system D we mentioned that its axiom D is not a theorem of K.