I have a problem understanding the Completeness Theorem in Sentential Calculus, which appears in the Merrie Bergmann's The Logic Book 4th edition. The theorem states that if P is a logical consequence of a consistent set of sentences S, then there is a derivation in SD such that P is derivable from S. Here is the sketch of the proof.
To prove the theorem, first, the authors prove the following statement: If S is consistent in SD, then S is truth-functionally consistent. The theorem follows from this statement, so the task is reduced to prove that statement.
Here is the sketch of the proof of the statement. Supposing S is a consistent set of sentences in SD, our goal is to build a truth assignment A* such that all members of S is TRUE. First, we extend it to a maximally consistent set S* in SD. Then we can build a truth assignment A* such that all members of S* are TRUE under this assignment. This will prove that S is truth-functionally consistent under A*, for S* is the superset of S. To prove that there is such a truth assignment, the proof in the book assumes that there exists a truth assignment A* such that it assigns TRUE to all atomic sentences of S* and FALSE to all other atomic sentences. Why it is legitimate to assume the existence of such a truth assignment?
Thank you very much.
"The proof in the book assumes that there exists a truth assignment A* such that it assigns TRUE to all atomic sentences of S* and FALSE to all other atomic sentences. Why it is legitimate to assume the existence of such a truth assignment?"
Why would it not be legitimate?
To talk of an assignment of truth values to the atomic sentences of a propositional language L is just to talk of a function which assigns to each of those atomic sentences one and only one of the values T and F (or 1 and 0 if you prefer). Any such function counts as an assignment function.
Suppose then that we partition the atomic sentences of L in some way or other into two classes (so every atom is in one and only one of these two classes). Then consider the function that sends all the atomic sentences in the first class to T and all the atomic sentence in the other class to F. Then lo and behold, you have defined a valuation of L -- that is all it takes.
So in particular, if we consider the function that sends all the atoms is S* to T and all the other atoms to F, then yes, we have a perfectly kosher valuation of the relevant language.