How to Prove Everything!
The Liars Paradox
By Donna L. McKereghan

Jack says "I am speaking falsely", referring to the words he is then uttering.
If Jack speaks truly when he says he is speaking falsely, he is speaking falsely.
If Jack speaks falsely when he says he is speaking falsely, he is speaking truly.

My simple question is:
How can Jack say this statement without lying?
I respond:
An interesting and mind bending fact, little known to most people (outside of the group of all logicians) is that from a contradiction, any and every conclusion follows. In symbolic logic, it looks like this (and I've matched the statements to the symbols):

I am speaking falsely assume this is a true statement, that A is true
If A then ~A If Jack speaks truly when he says he is speaking falsely, he is speaking falsely. If A is true, then A is not true (A is false)
If ~A then A  If Jack speaks falsely when he says he is speaking falsely, he s speaking truly. If A is not true, then A is true.
From ~A, A follows and from A, ~A follows, so:
A and ~A Jack is speaking truly and Jack is speaking falsely
Both of these are true!
A is true and A is not true
Thus, I can logically conclude that Jack is speaking truly! Of course, I can also logically conlude that Jack is speaking falsely, with equal validity, but that's not what you asked for!
But wait, there's more! Remember that any and every conclusion follows from a contradiction.
Knowing that A is true, (from above) then the statement "A or B" must also be true, because if either A or B is true, then that premise is true. Only one of them has to be true for the premise to be true.
A or B Jack is speaking truly OR I am the Queen of Sheba. Regardless of whether or not B is true, we know that A is true from the above argument, so this statement is true.
~A Jack is not speaking truly From the above argument, A is not true


I am the Queen of Sheba! B is true
From the first argument, we have it that A is true, so B doesn't need to be true for the first premise ("A" or "B") to be true. BUT, we also have it from the first argument that ~A (A is not true). This being the case, B MUST be true and I AM the Queen of Sheba! Of course, I can put any and every statement into the place of B and it will be true (or false, if you prefer).

Not only have I demonstrated how Jack can say this statement without lying, (if "not lying" means that what he says is true), I can put any statement I can imagine in place of the claim that I am the Queen of Sheba and it, too, will be true! I am god, you are the walrus, plants can think, aliens are invading.... But, on a more serious note, I wish people understood how this maneuver is used against them, unwittingly, to convince them of things that, were they to understand how this works, they would not be so vulnerable as to accept.

It is as important to examine the logical structure of an argument as the premises AND to check the line of reasoning being presented for contradictions. In the above arguments, the reasoning is sound. I can accept the premises as true. BUT, I cannot accept the conclusion. It's true, of course, but it demonstrates an important distinction between what is logically true and what is actually true.

So, the common apothegm "Everything is possible" is true...but it is only a logical truth. It is logically true, but it is not actually true. If it were, nothing else in our whole world would make sense and we couldn't even communicate sensibly with each other. Frankly, some of the nonsense out there can be traced back to a misunderstanding of this exact point.

© 1998

NOTE: I received an email from a gentleman who argues that the Liars Paradox is NOT a method by which we can prove everything. Apparently, I need to be explicit. This page was written to poke fun at the irrational assertion that "anything is possible." Of course, I don't believe that it proves everything!

He pointed out that when you reach an absurd conclusion, it indicates that something must be wrong with one of the premises from which you reached that conclusion. All I set out to demonstrate here is that, at best, the claim that "everything is possible" can mean only that "everything is logically possible."

He challenged that the argument can't even bring us to that conclusion because one of the premises violates a rule of logic - the law of non-contradiction. The whole of the above argument rests on the assumption of a and ~a, which is a illogical.

I agree. I only mean to demonstrate that, even if one allows that assumption (and there are philosophers who do, though I am certainly not one of them), the assertion that "anything is possible" is STILL ludicrous!

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