# Proof that: given any falsehood, any other falsehood can be proven

I saw this once on a puzzle page in Discovery or Science or some such.

They gave an instance:

Given 1 = 2, prove that I am the pope.
Solution: the pope and I are two men. And since 1 = 2, the pope and I are one man. Therefore, I am the pope. Q.E.D.

The answer appeared in the back, but I was too lazy to look at it.

Does someone know how to prove that any falsehood can be proven, given any other falsehood?

I’m not up on what constitutes a formal “proof” in logic, but the premise cited seems based almost entirely on an assertion: “The Pope and I are two men”. Without taking that as a given, there doesn’t seem to be any way to logically link the two falsehoods. I’m just guessing but I would presume that ANY two statements of truth or falsehood could “prove” each other given a customized assertion that links the two.

In formal logic, the premise must be true if any conclusion is to be drawn from it.

The problem with a false premise is not that the conclusion must be also therefore false, but that given a false premise any conclusion, true, false, or nonsensical, can logically be derived from it.

I’ll bet that the structure of the problem in the OP is not quoted properly, because it doesn’t make much sense as given. But the fundamental principle is there. False premises can lead to illogical conclusions, which is why you must take so many pains to ensure that your premises are true.

1. Exactly 1 of the following must be true: 1=2; 1!=2.

2. I am the Pope OR 1 != 2.

3. Assume that 1 = 2.

4. From step (1), it follows that 1 != 2 is false.

5. By process of elimination, I must be the Pope.

QED.

This is often attributed to Bertrand Russell.

If you are given a falsehood, and then you can also prove that it is in fact false, you have a contradiction, from which, in conventional formal logic, anything can be proven (this fact is referred to as ex contradictione quodlibet, among a million other names). Nothing more really needs to be said about this other than that this is the rule in conventional formal logic: “A implies B” is considered, by conventional definition*, to hold in every case except where A is true and B is false; therefore, if A is actually false, then it implies everything. So if A is provably false, then everything can be proven from it. That’s just the conventional way one sets up a formal logic.

But one needn’t use such a convention in one’s formalization. There are systems called paraconsistent logics, where not everything is provable from a contradiction. Of particular importance are the relevance logics, which try to only ascribe truth to “A implies B” when there is some sort of relevant connection between the statements A and B to carry the force of the implication, so to speak. These logics aren’t the standard one (classical logic), but they definitely have their uses.

*: As for this convention regarding “A implies B” (the convention of material implication), why would one want to adopt it? The best motivation is probably out of a desire for truth-functional semantics. Suppose you want the truth value of “A implies B” to be solely determined by the truth values of A and of B, with all truth values as either True or False. Well, the statements “Bush is a husband implies Bush is a male”, “Bush is a Parisian implies Bush is a Frenchman”, and “Bush is a Californian implies Bush is an American” are all pretty clearly true, while the statement “Bush is an American implies Bush is a Frenchman” is pretty clearly false. Thus, we see that “True implies True”, “False implies False”, and “False implies True” all come out to True, while “True implies False” comes out to False, fixing for us the definition of the material conditional. [Note that, as a result, one can’t reasonably give a truth-functional semantics to paraconsistent/relevance logics].

If one of your premises is contradictory, then from that you can prove anything true:

1. Assume P and not P.
2. P is true from 1.
3. P is false from 2.
4. P or Q is true from 2.
5. Thus, Q is true from 3 and 4.

The fancy latin name for this is ex falso quodibet. More about it here: http://tinyurl.com/ywoezl

More jargon: A deduction is valid if the conclusion must be true whenever the premises are true, and sound if it is valid and the premises are true. Thus, the argument given by totoismomo is valid but not sound.

I remember that we had a problem in my introductory symbolic logic class with the following argument:
[ol][li]If it rains, we’ll go to the movies.[/li][li]If it doesn’t rain, we’ll go to the movies.[/li][li]We’re not going to the movies.[/li][li]Therefore, my uncle Bernie will die of a heart attack next week.[/li][/ol]
For whatever reason, people didn’t seem to like it.

Let just emphasize as a matter of pure nitpickery, although I sort of mentioned it in my above post as well: if you want to prove arbitrary things from a premise A, then it’s key that A not merely be false, but actually be provably false. (Of course, naturally, if you can prove arbitrary things (like obvious falsehoods) from A, then you can prove that A is itself false). The statement from the OP shouldn’t be “Given a falsehood, any other statement can be proven (even false ones)” but “Given a falsehood which can be proven false, any other statement can be proven” or perhaps “A falsehood materially implies every other statement, although a proof of the implication is only guaranteed to exist in the case where the premise is provably false”.

If merely being false was enough to guarantee that a premise proved everything, then we’d have the silliness of all true things being provable, a dubious assumption on many natural interpretations (even without Goedelian interjections). [Want to know whether Lee Harvey Oswald shot J.F.K.? Just keep looking for a proof of a falsehood from either the assumption “He did it” or “He didn’t do it”, till you figure out which one is itself false…]