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# massey - 134 Axiomatization of Truth-Functional Logic 28...

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that the lemma holds for any wff A b in which b occurs. And if b does not occur in A b , we know that the lemma holds for A b, which then falls under the special case. This completes the proof of the substitution lemma. It follows as a corollary of the substitution lemma that substitution preserves validity (details are left to the reader). 28.1. Independence of Axioms and Rules Other things being equal, elegance is preferable to inelegance. Ac- cordingly, in logic and mathematics some emphasis is put on the creation of elegant primitive bases for axiomatic systems. One feature that contributes to the elegance of a primitive basis is the independence of its axioms and rules of inference. An axiom or rule of inference of a system L is dependent in L if the systemthat results by dropping the given axiom or rule from the primitive basis of L has the same set of theorems (actual output) that L has. So, if an axiom of a logistic system L is independent in L, then its omission from the primitive basis would diminish the set of theorems of L. The same holds for an independent rule of inference. We will show that all the axioms and rules of system P are in- dependent (in system P). Hence none of the axioms or rules of system P could be dropped from its primitive basis except on pain of denying theoremhood to some of the theorems (the valid wffs) of system P. First, the rules of inference. If the rule of substitution were dropped from the primitive basis of system P, no theorem of the resulting system would be longer than its longest axiom, axiom 2. The reason is that the conclusion of an application of modus ponens is always shorter than the longer premiss. But there are theorems of system P longer than axiom 2. So it follows from the definition of dependence that substitution is an independent rule in system P. Just as substitution is needed in system P to obtain theorems longer than the longest axiom, so modus ponens is needed to obtain theorems shorter than the shortest axiom. (Full details are left to the reader.) There are theorems of system P shorter than its short- est axiom, for example, '[p :) p]', so it follows that modus ponens is an independent rule in system P. To prove that an axiom of a logistic system L is independent 1 Any three objects would have done just as well as the numbers 0, 1, and 2. 135 28. Metatheory of System P (I) A ",A A B A:)B 0 1 0 0 0 1 1 0 1 2 2 1 0 2 2 1 0 2 1 1 2 1 2 0 2 0 0 2 1 0 2 2 0 We call 0 the designated truth value, and we refer to 1 and 2 as undesignated truth values. By a valid wff we now understand a wff that has a designated truth value under every assignment of values to its variables. For example, as the tables below show, axiom 2 and axiom 3 are valid in the sense just defined. (in L), it suffices to find some property which the given axiom lacks

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massey - 134 Axiomatization of Truth-Functional Logic 28...

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