just getting my start in proofs and had a little difficulty with the following problem in the book How to Prove it by Velleman.
Which of the following expressions are well-formed formulas?
$1. \neg(\neg P\vee \neg\neg R)\\
2. \neg(P,Q,\wedge R)\\
3. P\wedge \neg P\\
4. (P\vee Q)(P\vee R)$
I did some research here and found that this question was asked before here well formed formulas?, but I didn't find the answer too satisfactory and it contradicted many of the independent solutions posted out there that don't justify their reasoning.
So I understand why 1 and 3 are well-formed formulas, and my reasoning for 2. not being a well-formed formula is that $\neg(P,Q,\wedge R) \iff \neg P, \neg Q, \neg\wedge R$, and it's that last part $\neg\wedge R$ that is nonsensical and leads the expression to not being a well-formed formula.
As for 4, my thoughts are that there needs to be conjunction or disjunction between the two statements for it to be a well-formed formula as it reads to me either P or Q either P or V, which sounds nonsensical to me without the "and" or "or".
Is my reasoning for arriving at these conclusions correct? Thanks.
Best Answer
You need to define the syntactic rules you are checking against. They should have things like if $a$ is a wff then $\lnot a$ is a wff and if $a$ and $b$ are wffs then $(a \vee b)$ is a wff. You justify something is a wff by citing the rules necessary to build it up. $P\wedge \lnot P$ may well not be a wff because it does not have parentheses around it. For $2$, you may not have any rules that involve a comma, so it cannot be well formed. For $4$ you should not have any rules that result in neighboring parentheses in that configuration.