# Propositional Logic - Exactly what does ~ (negation) mean

Let's say p is a statement. Is ~p (negation of p) just opposite of p or is it anything but p.

For example, let's say p = "None of the basketball players are blond"

Without just adding a 'not' in front of the statement, what would ~p be?

Would ~p be: "All of the basketball players are blond" (exact opposite of p)

or would ~p be: "At least one of the basketball players is blond" (anything but p)

Another side question I have is what is the precedence of ^ (and), V (or), ~ (negation), --> (implies), and anything else relevant

Thanks! Help is appreciated!

One way to keep these two apart in your mind is to remember that $p$ and $\neg p$ (1) cannot both be true at the same time, and (2) cannot both be false at the same time (one of the two must be true in a given situation). Considering this, it is not possible to simultaneously have "all blond" and "none blond" (unless there are no players), but it is possible to have neither "all blond" nor "none blond" situations being the case, for example if there are some blond and others not blond. By contrast, "none blond" and "some blond" can not happen at the same time, and also one of the two has to be true (either there is a blond or there isn't). Thus "some blond" is the correct negation of "none blond".

Regarding precedence, usually $\neg$ has the highest precedence and $\lor,\land,\to$ compete for a lower predecence - I would not recommend relying on the precedence order of these and just bracket everything. But I think the most common convention puts $\land$ above $\lor$ above $\to$, so that $A\land B\lor C\to \neg D$ is $((A\land B)\lor C)\to (\neg D)$.

• I think I might be getting it. For example, if p = "You are always early", what would ~p be, other than "You are not always early" (not including 'not' in sentence) – Temp Jon Feb 4 '15 at 3:47
• @TempJon Well, presumably you know what "not always early" means, that is indeed the correct negation here. But if you want to avoid the word "not" here, you might also say "you are sometimes late". – Mario Carneiro Feb 4 '15 at 5:35

I think for your first question the best way to think about is $\neg P$ is "it is not the case that $P$." So in your example, if $P$ is "None of the basketball players are blond," then $\neg P$ is "it is not the case that none of the basketball players are blond" which is like saying "there is some basketball player with blond hair."