Why is it possible to conclude everything from a false statement? [duplicate]

Solution 1:

Here's one for the non-mathematician:

Let's say that I promise you $1,000,000 on the condition that pigs fly. We could say something like:

If pigs fly, then I will give you $1,000,000

Now, if pigs don't fly and I don't give you $1,000,000, you have no reason to complain. I haven't lied. This is your case #2.

If pigs don't fly yet I still give you $1,000,000 , (case 1) then I am just very generous, but I haven't contradicted myself or broken my word.

Solution 2:

See http://www.nku.edu/~longa/classes/mat385_resources/docs/russellpope.html for Bertrand Russell's proof that 1=0 implies that he is the Pope.

Solution 3:

While the other answers have given mathematically adequate, I feel that perhaps I should explain why we define material implication this way.

First, let us consider the principle of modus ponens. If we know that $P$ is true, and we also know $P \to Q$, then it is reasonable that we may deduce $Q$ is also true. It is also reasonable to say that if it is possible to have $P$ true and $Q$ false at the same time, then we should have $P \to Q$ false. The question is how to define the truth value of $P \to Q$ so that the principle of modus ponens gives us sound deductions. It turns out there isn't a unique answer: the two ‘obvious’ properties of implication I just listed only force two of the entries in the truth table: $$\begin{array}{c|cc} & P \text{ false} & P \text{ true} \\ \hline \\ Q \text{ false} & ? & P \to Q \text{ false} \\ Q \text{ true} & ? & P \to Q \text{ true} \end{array}$$ It is possible to fill in the blanks in any way and still get an interpretation of $P \to Q$ which makes modus ponens valid. At least amongst mathematicians, it is conventional to choose the ‘weakest’ possible way to fill in the table, which is to say that $P \to Q$ is true whenever $P$ is false. This produces no logical inconsistencies and modus ponens is valid under this interpretation, so why not?

But actually, once we start adding other rules of inference, we rapidly find ourselves forced to conclude that $\to$ must be interpreted this way. Indeed, consider the following rules of inference:

1. Monotonicity: From $P$, we may deduce $P$.
2. Transitivity: If we may deduce $R$ from $Q$, and if we may deduce $Q$ from $P$, then from $P$ we may deduce $R$.
3. Conditional proof: If we may deduce $R$ from $P$ and $Q$, then from $P$ we may deduce $Q \to R$.
4. Negation introduction: If we may deduce $Q \to \bot$ from $P$, then from $P$ we may deduce $\lnot Q$. ($\bot$ is a proposition which is interpreted as ‘contradiction’. $\lnot$ is a logical operator which is interpreted as ‘not’.)
5. Disjunction introduction: From $P$, we may deduce $P \lor Q$. ($\lor$ is a logical connective which is interpreted as ‘or’.)
6. Modus tollendo ponens: From $P \lor Q$ and $\lnot P$, we may deduce $Q$.

I'm certain that everyone agrees that these rules of inference are entirely reasonable. But these already suffice to prove that ex falso quodlibet, that is, from $\bot$ we may deduce any proposition $Q$. Indeed:

1. By monotonicity, we may deduce $\bot$ from $\bot$.
2. By conditional proof, we may deduce $\bot \to \bot$ from $\bot$.
3. By negation introduction, we may deduce $\lnot \bot$ from $\bot \to \bot$.
4. By disjunction introduction, we may deduce $\bot \lor Q$ from $\bot$.
5. By transitivity, we may deduce $\lnot \bot$ from $\bot$.
6. By modus tollendo ponens and transitivity (a few times), we may deduce $Q$ from $\bot$.

We then may apply the principle of conditional proof again to deduce $\bot \to Q$ from any hypotheses whatsoever. So if we believe these rules of inference are valid, then we are forced to conclude that $P \to Q$ must be true whenever $P$ is false, no matter what $Q$ is.