# Mendelson's Logic book “cheats” in the propositional calculus?

In Mendelson's book ("Introduction to mathematical logic") he defines truth values for sentences in the propositional calculus using truth tables. However, it seems to me he assumes implicitly that every well-formed sentence (what he calls "statement form") has a unique parsing; i.e. it is impossible for the same statement form to arise in two different ways.

This is of course correct, but it requires a proof. The omission of such proof (or even mentioning it is needed) is somewhat surprising for me as Mendelson's book is otherwise very explicit about everything. Am I missing something?

• Disclaimer: not expert on formal logic, but I find this question interesting. So you're saying that given a set of deductive rules defining well-formed formulas, prove that any WFF has a unique parsing? How would one define parsing, and unique parsing? By mapping to parse trees? or by defining a confluent mapping to, say, a canonical WFFs (e.g. infex)? – user2468 Mar 13 '12 at 18:21
• @J.D.: one example of a book that does do this rigorously is Enderton's book A Mathematical Introduction to Logic. He does this by defining an algorithm to perform the parsing and examining the algorithm in detail. A key step is that no proper initial segment of a well formed formula (as Enderton has defined them) can be a well formed formula. – Carl Mummert Mar 13 '12 at 18:54
• @J.D.: Another example of a text that proves "unique readability" is Shoenfield's "Mathematical Logic." In that case it is quite important because he introduces a syntax without parentheses in prefix (Polish) notation (so what is usually written "$(\exists x ) (\neg ( x = x ) \vee ( \exists y ) ( x = y ) )$" is rendered "$\exists v \vee \neg = x x \exists y = x y$") and so you cannot take advantage of a quick "unique pairing of parentheses" fact to get unique parsing. – user642796 Mar 13 '12 at 19:35
• @J.D.: Yes, you prove that every WFF has a unique parse tree, or generating sequence, or whatever; the important thing is it will have a unique "something you define the truth value by". – Gadi A Mar 14 '12 at 5:46