6.3 Misplace parenthesis

When I wrote the definitions of the rules, I used the letters $A$ and $B$, but these can represent any expression.

For instance, here we do negation introduction, in which -following the rule- we suppose some formula $A$, attain a contradiction, and we conclude $\neg A$, so, the original formula, but negated. Let's see:

$$\begin{fitch*} \par 1 & \fh P \Rightarrow Q & H \\ \par \ldo... ...nd{fitch*} {\textcolor{red}{\bigotimes INCORRECTO \bigotimes}}$$

I think it's clear that the $A$ which appears in the rule represents to $P\Rightarrow Q$ in this example. The problem comes when we do the $\neg A$. The negation of $P\Rightarrow Q$ is not $\neg P\Rightarrow Q$, but $\neg(P\Rightarrow Q)$. It's necessary that parenthesis because if not present, the negation affects only $P$.

If you don't know when to put parenthesis, always put them, and then try to remove the unneeded ones. For instance, if you must write that $\neg P\vee R$ implies $R\wedge Q$, put parenthesis around each expression and thus write $(\neg P\vee R)\Rightarrow(R\wedge Q)$. This way, there are absolutely no errors. Now learn when is it possible to remove parenthesis, and take away all that you can. In this case, both can be suppressed and it remains $\neg P\vee R\Rightarrow R\wedge Q$.