Why $(P \wedge ¬P) ⇒ Q$ means we can prove anything?

Question

I've been reading a Portuguese book on elementary logic. It gives the following definition:

$P(x_1,x_2, \dots)⇒ Q(x_1,x_2, \dots)$ if $Q$ is true whenever $P$ is true.

And establishes the following theorem:

$P(x_1,x_2, \dots)⇒ Q(x_1,x_2, \dots)$ iff $P(x_1,x_2, \dots)→ Q(x_1,x_2, \dots)$ is tautological.

A little bit further, they comment about the distinction between $→$ and $⇒$, but I think that given the theorem I just wrote, no further comment is needed. In the examples, it is given:

$$(P \wedge ¬P) → Q$$

And because this sentence is tautological, then:

$$(P \wedge ¬P) ⇒ Q$$

And it is said that from this, we could deduce anything. I don't see how, the last sentence means that we could prove anything. First, it says that (following the theorem):

• Q is true whenever $(P \wedge ¬P)$ is true.

But $(P \wedge ¬P)$ is never true. I know this is related to the principle of non contradiction, but we know that $(P \wedge ¬P)=0$, even assuming the principle of non-contradiction, I guess I still can write $0→ Q$, which leads me to $0⇒Q$. Wouldn't we face the same problem we had before?

Answer 1

Recall that for the logical implication

$$A\implies B$$

is equivalent to

$$\lnot A\lor B$$

and it is always true when proposition A is false.

Answer 2

This logical rule is a well-known and important principle called the principle of explosion. It is often stated as you have stated it in your question, but it can also be stated more succinctly as the logical rule $\boldsymbol{\text{F}} \rightarrow Q$, where $\boldsymbol{\text{F}} \equiv (P \land \neg P)$ is the falsehood proposition and $Q$ is any proposition. The tautological nature of this logical rule can be established easily with truth tables; it follows from the fact that logical implications make no claim in the case where the antecedent condition is false (i.e., the failure of the antecedent condition in an implication statement renders the statement vacuously true).

The principle of explosion operates within the domain of propositional logic and other logics that build on this. It establishes that if you hold a logical contradiction to be true then every proposition becomes provably true, which further means that every proposition is provably true and false (i.e., we can prove $Q$ and also $\neg Q$ for all propositions $Q$)! Yikes! In other words, propositional logic is extremely unforgiving of holding contradictions; if you have even a single logical contradiction in your system this ripples through the whole system and renders every statement provably true and false, giving rise to contradictions on every proposition!

This is the reason that it was such a big deal when Betrand Russell pointed out a contradiction in the axiom system of the logic of Gottlobb Frege. Both mathematicians were no-doubt familiar with the principle of explosion and so they both understood that this seemingly tiny contradiction in the axiom system implied that contradictions could be spread through the entire logical system.

Answer 3

Anything can be deduced from a contradiction, exactly because it is impossible for the contradiction to be true.

Remember that in general: $\varphi \Rightarrow \psi$ if and only if whenever $\varphi$ is true, $\psi$ is also true. Well, if $\varphi$ is a contradiction, then it is never true, and therefore it is automatically the case that whenever $\varphi$ is true, $\psi$ is also true.

Think about it this way. I must be the luckiest person in the world, because whenever I have played the lottery, I have won the jackpot! Incredible, right? Well, as it turns out, I have never played the lottery, and so also never won the jackpot. But it is therefore true that all zero times that I played the lottery, I won the jackpot, or: whenever I played the lottery, I won the jackpot! Of course, it is also true that whenever I played the lottery, I did not win the jackpot, and that whenever I played the lottery, pigs started to fly, and ... clearly I can say anything here.

By the same logic, we can say that all zero times that a contradiction is true, some other statement, whatever it is, is also true: whenever a contradicition is true, any statement is true. And thus, any statement logically follows from a contradiction.

A final way to think about this is: what would it take for some statement $\psi$ not to follow from some statement $\varphi$? It is when we can point to some possible scenario where $\varphi$ is true, but $\psi$ is false. Well, that is impossible when $\varphi$ is a contradiction, i.e. There is no counterexample to the claim that $\psi$ follows from $\varphi$ if $\varphi$ is a contradiction. And so, once again, anything follows from a contradiction.