well formed formulas? 
which of the following expressions are well formed formulas?
  $$\lnot(\lnot P\lor \lnot \lnot R)$$
  $$\lnot(P,Q, \land R)$$
  $$P \land \lnot P$$
  $$(P \lor Q)(P \lor R)$$

To me they all seem like they are not well formed formulas:
The first has a double negative
The second has ambiguous commas
The third contradicts itself
And the fourth is ambiguous as to what happens to both the brackets
However the way the question is phrased makes me think that at least one, maybe two, of these statements must be well formed, am i right? or am i missing something?
 A: Here's a possible grammar for well-formed formulae:
$$ \begin{align}\langle \text S\rangle&::= \langle \text V\rangle\mid\neg\langle S\rangle\mid (\langle \text S\rangle\langle \text{op}\rangle\langle \text S\rangle)\\
\langle\text V\rangle&::= P\mid Q\mid R\\
\langle\text{op}\rangle&::=\land\mid \lor\mid \to
\end{align}$$
According to this grammar, only the first of the given four strings is a wff, using the following derivation:
$$\begin{matrix}\langle \text S\rangle\\
\neg&&&\langle \text S\rangle\\
\neg&(&\langle \text S\rangle&\langle \text {op}\rangle&\langle \text S\rangle&)\\
\neg&(&\neg\langle \text S\rangle&\langle \text {op}\rangle&\langle \text S\rangle&)\\
\neg&(&\neg\langle \text V\rangle&\langle \text {op}\rangle&\langle \text S\rangle&)\\
\neg&(&\neg P&\langle \text {op}\rangle&\langle \text S\rangle&)\\
\neg&(&\neg P& \lor&\langle \text S\rangle&)\\
\neg&(&\neg P& \lor&\neg\langle \text S\rangle&)\\
\neg&(&\neg P& \lor&\neg\neg\langle \text S\rangle&)\\
\neg&(&\neg P& \lor&\neg\neg\langle \text V\rangle&)\\
\neg&(&\neg P& \lor&\neg\neg R&)
\end{matrix}
 $$
The second string is not a wff because the above grammar does not use "," at all -- but your text may use a grammar for wff that for example allows functions with more than one parameter and separate these with ",".
The third string is not a wff because any inroduction of "$\land$" necessarily also introduces "$($" and "$)$" -- but your text may employ a more complex grammar that manages to get rid of parentheses that look superfluous to the human reader (of all possibilities, I consider this the lest unlikely).
The fourth string is not wff because (as can be shown, if needed by structural induction) that "$)($" cannot occur as a substring in a wff -- but even here, your text may introduce juxtaposition as an alternative form of writing conjunction, for example.
Admittedly, all variants that I put into "your text may ..." are somewhat unlikely in a formal context.
