# What does the fact that “~P is equivalent to (P --> ~P) ” tell us about the nature of logical falsity?

After all, what does " false" mean in logic?

Does this fact: "~P is equivalent to (P --> ~P) " deliver the essence of logical falsity?

I mean , does this formula express the idea that being false, in logic, is not simply to disagree with facts/ reality, but something deeper: to be self-negating, or, so to say to be "self-destroying"? For this formula says: a proposition is false if and only if this proposition implies it's own negation.

However, is not the essence of logical falsity contained, so to speak, in the constant F, "falsum" , the antilogy per excellence? If (P--> ~P) were the key to logical falsity, it should be equivalent to " falsum"; which is not the case.

I can see that these reflections are rather confused and might originate from some conceptual blurrring, but the question remains to me: what does the fact that "(P --> ~P) is equivalent to ~P" tell us about logical falsity?

• $P\to\lnot P$ informs us that $P$ is not true; not that falsum is true. Why should it? – Graham Kemp Apr 25 at 22:47
• Re. your thoughts about the role of falsum: $\neg P$ is also equivalent to $P \to \bot$. In fact, $P \to \bot$ is often used as the definition of the $\neg$ symbol, if one decides (as often done for reasons of syntactic minalism) to treat $\neg$ only as an abbreviation in a language that formally consists only of a limited set of logical connectives like $\to, \land,\bot$ as primitive. So I certainly wouldn't say that in the tradition of logic, falsum is less of a key to negation than what you expressed as "self-negation" is, rather the contrary. – lemontree Apr 25 at 22:52
• @GrahamKemp. I understand what you mean about the second argument. – Eleonore Saint James Apr 25 at 22:52

If (P--> ~P) were the key to logical falsity, it should be equivalent to " falsum"; which is not the case.

No. Why should it? $$P\to\lnot P$$ informs us that $$P$$ is not true; not that falsum is true.

$$P\to \lnot P$$ states: If $$P$$ were true it would imply the contradiction, that $$P$$ is also not true. Thus we infer that $$P$$ is false, ie that $$\lnot P$$ is true.

Another way to state this is to define the falsum constant, $$\bot$$, such that the negation of a predicate is equivalent to stating that predictate implies falsum. (Or to take falsum as the primative and use this to define the negation symbol.) $$\lnot P~~\iff~~ P\to \bot$$

Extending my comment:

Not every formula which is logically equivalent - of which there are, by the way, infinitely many for any given formula - must be the "key" to understand or define the connective. Actually, among the possible logical equivalences for $$\neg P$$, $$P \to \neg P$$ is a rather useless one, as it describes negation in terms of itself ($$\neg$$ occurs again in the re-formulation of $$\neg P$$, so you can't understand $$P \to \neg P$$ without also understanding $$\neg P$$, which creates a circularity).

As an example, take implication:
A reasonable "key" to understand the meaning and use of $$P \to Q$$ is the equivalence $$\neg P \lor Q$$, since that is the equivalence that is used in a good portion of proofs of statements of the form $$P \to Q$$. If we know what $$\neg$$ is, and we know what $$\lor$$ is, then we can use this to grasp the behavior if $$\to$$. (The other big portion would be proofs that start with assuming P and derive Q from that hypothesis, which more directly reflects the meaning of implication taken as a primitive connective.)
Another possible "key" to capturing the meaning of implication is the equilvance $$\neg (P \land \neg Q)$$, since $$P \land \neg Q$$ is what would be used to refute a claim $$P \to Q$$. We can fall back to the connectives $$\neg$$ and $$\land$$ to reason about the truth of $$\to$$.
And then there is contraposition, $$\neg Q \to \neg P$$, which is also logically equilvalent to $$P \to Q$$, and sometimes makes things easier to prove than the forward direction. But contraposition doesn't don't tell us that much more about what implication means, or what is at the heart of the proof of an implicative statement, as the contraposition consists itself of an implication - so we could hardly regard it as a "key" to grapsing implication, just because it is logically equivalent to it. A similar claim could be made about $$P \to \neg P$$ w.r.t. explaining $$\neg P$$.

Re. your thoughts about the role of falsum: $$\neg P$$ is also equivalent to $$P \to \bot$$. It is obviously not equilvalent to $$\bot$$, but there is no reason why it should be. Instead, relating $$P$$ with $$\bot$$ in the form of "If $$P$$ is true, then falsity is the case" is what explains the meaning of $$\neg$$ without reference to itself.
In fact, $$P \to \bot$$ is often used as the definition of the $$\neg$$ symbol, if one decides (as often done for reasons of syntactic minalism) to treat $$\neg$$ only as an abbreviation in a language that formally takes only of a limited set of logical connectives like {\to,\land,\bot} as primitive. This is also an imporant equilvance from the perspective of proof-writing, as they key to proving a statement of the form $$\neg P$$ is to assume $$P$$ and derive a contradiction - that is, falsum.

So I certainly wouldn't say that in the tradition of logic, falsum is any less of a key to negation than what you expressed as "self-negation" is, rather the contrary. $$P \to \neg P$$ may be a logically equivalent formula, but that doesn't mean it's a useful one that gives us much insight about the way we interpret $$\neg$$ or use it in proofs.