Exactly how does it mean to negate a statement?

Question

Is $\lnot p,$ the negation of a statement $p,$ just the opposite of $p,$ or is it anything but $p\;?$

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

Is $\lnot p$: "All of the basketball players are blond" (exact opposite of $p$)?

Or is $\lnot p$: "At least one of the basketball players is blond" (anything but $p$)?

A side question: what is the precedence of ∧ (and), ∨ (or), ¬ (negation), $→$ (implies), etc.?

Answer 1

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."

Answer 2

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 $((A\land B)\lor C)\to (\neg D)$.

Answer 3

Consider any verbal statement. To negate it involves going back to its symbolic form, for example, $$\lnot P\lor(Q(x)\to\lnot R),$$ finding a symbolic sentence that has its opposite truth value no matter what meanings we assign to their atoms (here, there are three atoms: $P,Q(x)$ and $R$) and what set of objects we are discussing (here, $x$ varies over these objects), and finally translating the result to verbal form. In other words, negating a statement logically flips its truth value.

Is $\lnot P$ just the opposite of $P,$ or is it anything but $P\;?$

The logic formula $P$ has the opposite truth value as its negation $\lnot P.$

The set $S$ contains every element that is not in its complement $S^\complement.$

Consider this:

1. Let $T$ be the set of tautologies,
   and $P$ be the formula $x\in T,$ i.e., $x$ is a tautology.
2. Then $\lnot P$ means $x\in T^\complement;$ so, \begin{align}\lnot P\iff{}&x\text{ is either a contradiction or a contingency}\\\kern.6em\not\kern-.6em\implies{}&x\text{ is a contradiction (the negation/‘opposite’ of a tautology)}.\end{align}

$P$ = "None of the basketball players are blond"

Is $\lnot P$: "All the basketball players are blond" (exact opposite of $p$)?

Or is $\lnot P$: "At least one of the basketball players is blond" (anything but $p$)?

Following the procedure outlined above, we (use predicate instead of propositional logic to) symbolise “None of the basketball players are blond” as $$\forall x \;\lnot B(x);$$ no matter what the atom $B(x)$ means (e.g., $\text“x$ is blond”, $\text“x$ has four legs”, etc.) and what the universe of discourse is (e.g., the of basketball players in Beverly Hills High, the set of furniture at Pinewood Studios), this sentence has the opposite truth value as $$\exists x \; B(x);$$ thus, the required negation must be “Some basketball player is blond”, in other words, “At least one of the basketball players is blond” (the second option that you provided).

On the other hand, the first option, “All the basketball players are blond”, and “None of the basketball players are blond” are both true when the universe of discourse is the empty set, and are both false when the universe of discourse is {blond basketballer John, brunette basketballer Mary}, so do not always have opposite truth values, so are not negations of each other.

Answer 4

Is $\lnot p,$ the negation of a statement $p,$ just the opposite of $p,$ or is it anything but $p\;?$

It is anything but $p$. So in your example, "At least one of the basketball players is blond" .

A side question: what is the precedence of ∧ (and), ∨ (or), ¬ (negation), $→$ (implies), etc.?

Depends on the definition; usually, it is $\neg > \land > \lor > \rightarrow > \leftrightarrow$.