What is Logical Paradox?

A paradox is a statement that, despite apparently sound reasoning from true premises, leads to a self-contradictory or a logically unacceptable conclusion. As an example we can use the egg and chicken paradox.The paradoxes concerning the notion of a set are called logical paradoxes.

Paradoxes have been in Mathematics since a long time. The earliest one, which I know is “Parallel Postulate by Euclid”. For two thousand years, many attempts were made to prove the parallel postulate using Euclid’s first four postulates. Many of them being accepted as proofs for long periods until the mistake was found. Invariably the mistake was assuming some ‘obvious’ property which turned out to be equivalent to the fifth postulate! For more information check https://www.youtube.com/watch?v=nsZsd5qtbo4

In this blog post , I will be specifically looking at three paradoxes :

1 — Barber Paradox

The barber is the “one who shaves all those, and those only, who do not shave themselves.” The question is, does the barber shave himself?

Answering this question results in a contradiction. The barber cannot shave himself as he only shaves those who do not shave themselves. As such, if he shaves himself he ceases to be a barber. Conversely, if the barber does not shave himself, then he fits into the group of people who would be shaved by the barber, and thus, as the barber, he must shave himself.

This Paradox is actually derived from Russell’s paradox — “Consider the set A of all those sets X such that X is not a member of X. Clearly, by definition, A is a member of A if and only if A is not a member of A. So, if A is a member of A, the A is also not a member of A; and if A is not a member of A, then A is a member of A. In any case, A is a member of A and A is not a member of A.” Check this out for a better understanding.

2 — Jourdain’s Card Paradox

Suppose there is a card with statements printed on both sides:

Front:The sentence on the other side of this card is TRUE.

Back:The sentence on the other side of this card is FALSE.If the first statement is true, then so is the second. But if the second statement is true, then the first statement is false. It follows that if the first statement is true, then the first statement is false.

If the first statement is false, then the second is false, too. But if the second statement is false, then the first statement is true. It follows that if the first statement is false, then the first statement is true.

Hence trying to assign a truth value to either of them leads to a paradox!

3— The Paradox of Epimenides the Cretan

According to Thomas Fowler (1869) the paradox is as follows: “Epimenides the Cretan says, ‘that all the Cretans are liars,’ but Epimenides is himself a Cretan; therefore he is himself a liar. But if he be a liar, what he says is untrue, and consequently the Cretans are veracious; but Epimenides is a Cretan, and therefore what he says is true; saying the Cretans are liars, Epimenides is himself a liar, and what he says is untrue. Thus we may go on alternately proving that Epimenides and the Cretans are truthful and untruthful.”

This paradox can actually be solved just by adjusting our assumptions! In the above paradox, we are assuming that a liar will always tell a lie which may not be true in all the cases. If we suspend it, then the Epimenides will tell the truth at some point.

But actually there is an ambiguity with the paradox. Those of you who are familiar with Propositional Logic would have noticed it. The mistake made by Thomas Fowler above is to think that the negation of “all Cretans are liars” is “all Cretans are honest” (a paradox) when in fact the negation is “there exists a Cretan who is honest”, or “not all Cretans are liars”!

Lastly, I would like to point out one important observation/property about these paradoxes. All of these either have a sense of self-reference or circular definitions which actually make it undecidable!

Πηγή: medium