What are some true paradoxes in philosophy?

Question

Some logical paradoxes are known to be invalid arguments, so I want to know what are some of the paradoxes based on valid logic in philosophy. So could you identify some of them?

Answer 1

Here is a non-comprehensive list of classical paradoxa that are not only surprising (Banach-Tarski-Paradox) or based on misunderstanding (Twin-Paradox, Achilles-Tortoise, Zeno's Arrow), but seem to constitute real problems with the notions involved. There is one paradox, the Skolem-Paradox, which is worth knowing. I purposefully omitted it from the list, since it is debatable whether it belongs there.

For all of these you can find formal analyses.

Semantic Paradoxa

Liar Paradox: Is "This sentence is false." a true or false sentence? (Or look up Revenge Liar if you have more than two truth values). Related: Grelling's Paradox (truth of sentence is replaced by the meaning of a word) and Yablo's Paradox (Replace the circle by an infinite chain).

Curry's Paradox: This one you best look up by yourself. It makes similar trouble as the Liar-Paradox, but contains no negation, and an implication instead.

Epistemic Paradoxa

Fitch's Paradox: If you accept the seemingly innocuous principle that truths are in principle knowable, you somehow can conclude that all truths are already known.

Surprise Paradox: If you announce a surprise test next week, you can use backward induction to show that the test won't happen at all, otherwise it wouldn't been surprising.

Preface Paradox: Rational belief that every sentence in a text/theory is true seems to be different from rational belief in each sentence. (Note the quantifier change).

Vagueness

Sorites Paradox: Removing one straw from a heap leaves you still with a heap. So by backwards induction a single straw is heap.

Answer 2

I would call Zeno’s motion paradoxes true paradoxes. Rather sensible assumptions lead to rather absurd conclusions.

An arrow can never move because at any given instant it is motionless, and all the motionless instants add up to zero progress through the air.

Achilles can never outrace a tortoise if the tortoise is given a head start. When Achilles reaches the point where the tortoise started, the tortoise has already moved some distance forward. When Achilles reaches that second point, the plodding tortoise has moved forward yet again. And so forth.

For further reading, you can spend many happy hours reviewing attempts to resolve the motion paradoxes by viewing YouTube or reading the entries in the Stanford Encyclopedia of Philosophy.

Answer 3

What are some true paradoxes in philosophy?

Here’s one I came across recently in the context of a legal argument:

Neccesitas non habet legem

Necessity knows no law.

This of course is a law, in the sense that it lays down a law in the world of laws; but also, notably, in the natural world, amongst natural laws, one might say, necessity, is the only law, and the closer our understanding approaches that ideal, the closer we are to a true law of nature. This might be one reason as to why logic and mathematics is so often seen as true laws of nature, as they can’t be seen otherwise.