By C. Taylor
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Additional info for A Non-Mathematical Introduction to X-Ray Crystallography [short article]
We now present some connections between equivalent wffs, valuations, truth-values of swffs of a statement system, and true wffs. 46 Chap. 3 • Provable Wffs Lemma 4. If A == B, then vA = vB for each valuation v. Dem. Let v be a valuation such that vA = t and vB = f. Now, each provable wff is true, so A -+ B is true; thus v[A -+ B] = t. But v[A -+ B] = f since vA = t and vB = f. We conclude that vA = vB. Corollary 1. Let A == B; then both A and B are true, or neither is true. Although the converse of Lemma 4 is correct, we are not yet in a position to prove so.
Now, the leftmost LH parenthesis of B has a mate in B, namely the rightmosl RH parenthesis of B. C). In this case, the right most symbol of B is the rightmost symbol of (-,C); so B = (-,C), henet B is a component of C. (ii) The leftmost LH parenthesis of B is a symboi of C. But the mate of this LH parenthesis of C is a RH parenthesis of C, so B is a subwff of C. Since C E S, it follows that B is a component of C. Thus, by definition, B is a component of -,C. Sec. 1 • Subwft's, Components, and Wff-Builders 51 3.
Applying our Criterion for True Wffs, we conclude that A A B is true. We now present the important notion of a model of a set of wffs. A statement system I is said to be a model of K, a set of wffs, provided that there is a I -interpreter /-' such that /-,A is true for I for each A E K. In this case, we also say that I is a model of K under /-'. Z} under the I -interpreter /-' of that example. X), /-' Y, and /-,(--,. Z) are true swffs of I. Chap. 3 • Provable Wffs 36 Clearly, each statement system is a model of K if each member of K is a true wff.