soft question - Paradoxes in Logic

What is it that makes something a paradox? It seems to me that paradoxes are just, in many cases, misunderstandings about the properties some object can have and so misunderstandings about definitions. Is there something I might be missing? How is this kind of thought handled in logic?

Answer

"Paradox" is not a formally defined term. Many modern authors use "paradox" for all sorts of surprising or unexpected results; you can see examples by searching for "paradox" on Google News.

A more substantial use of the word "paradox" refers to a result that shows that a particular naive intuition is not sound. For example, the Banach-Tarski paradox shows that is it not possible to have a measure of volume for arbitrary subsets of Euclidean space, if that measure satisfies certain basic properties such as invariance under rigid motions and finite additivity. This goes against a certain naive thought that every subset of Euclidean space must have a well-defined volume (it might be 0 for strangely defined sets of points, but at least it should be defined, this intuition would say).

The classical paradoxes of logic (also called "antinomies") are somewhat different because they show that our intuition about logic itself is not valid. In particular, these show that the naive ways we talk about "truth" and "sets" in natural language lead to contradictions. An example is Curry's paradox, which shows that the naive way we prove "if/then" statements in normal mathematics can lead to false results when combined with self-referential sentences (even when there is no negation in the sentences).

The thing that makes the classical paradoxes more genuinely paradoxical is that it is hard to see where the problem comes from, or any straightforward way to resolve the issue. Consider the sentence of Curry's paradox

If this sentence is true, then 0 = 1

We can prove this sentence in the usual way: assume the hypothesis "this sentence is true" and prove that the conclusion must follow. That is how we prove many other implications in mathematics. But then, because the sentence is true, its hypothesis is true, so its conclusion must also be true: 0=1. It is very difficult to find a hidden assumption in this argument as with the Banach-Tarski "paradox".

One resolution in mathematics is to take refuge in formal logic, where the self-reference of the quoted sentence is impossible. But that does not resolve the issue that the sentence seems to be perfectly clear English, and yet applying the usual methods to it leads to a contradiction.