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Additional info for Logic: Form and function - The Mechanization of Deductive Reasoning
Sample text
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. So if we are given merely that some rule is a rule of S and that S’ is an extension of S, this does not by itself guarantee that it is also a rule of S’. This is just one of the pitfalls one may encounter in studying axiomatic systems and which should put us on our guard against jumping to conclusions too easily.
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. We shall prove this in a moment, and we shall also prove that T is not a theorem of D. D, however is a theorem of T, since it follows straightforwardly from T and Tl by Syll.
We noted earlier on that the rules DRl-DR3 and Eq which we derived in K still hold in all extensions of K. But consider the rule we discussed above and showed not to be a rule of T: Now K, as we observed, has no theorems at all of the form Ma; so P is (trivially) a rule of K. Less trivially, it is also a rule of D. 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.