Why can't we just claim a paradox invalid, just as the way we treat contradiction in mathematical proof? (i.e. if we arrive at the proposition inconsistent with the assumption, then we can immediately claim the logical impossibility of the assumption).

• I will add to Hunan answer only the SEP entry regarding Paradoxes and Contemporary Logic. – Mauro ALLEGRANZA Mar 30 '14 at 14:57
• Given that an argument usually relies on more than one assumption, the difficulty is to tease out which assumption is problematic. – user5878 Mar 30 '14 at 18:05
• Do you regard the Banach-Tarski paradox as a paradox? Why or why not? If not, then what is your definition of a paradox? If so, then are you claiming it should be "claimed invalid?" Which means what, exactly? – user4894 Mar 30 '14 at 19:28
• @user4894 No, the Banach-Tarski paradox isn't a paradox: it's just a surprising consequence of a particular set of axioms. Most "paradoxes" in maths and physics are of this form. – David Richerby Mar 30 '14 at 21:04

Paradoxes are indeed invalid arguments, but what makes them special is that they rest on seemingly unproblematic assumptions. We know, for example, that Achilles will in fact outrun the tortoise (Zeno's Paradoxes), we know that the surprise exam will take place (Surprise Exam Paradox), and so on. Because we know the conclusions of those paradoxes are false, we know that something is wrong with the arguments. The task then is to identify the assumptions that lead to the false conclusion.

Paradoxes can be called 'invalid' and ignored, but if taken seriously they can help us diagnose and fix problems with existing logico-mathematical frameworks. Axiomatic set theory and type theory owe much to Russell's Paradox, for example. The above mentioned Surprise Exam Paradox has led to lots of interesting developments in epistemic, dynamic, and public announcement logics. There are, of course, the classical ones, like the Sorites, the Liar, and so on. Each has opened some interesting door.

Take paradoxes seriously, because they indicate that (at least) some thing is not as true as it seems.

• Russel's paradox or the Cretan paradox are not in any way invalid arguments. They are not even arguments. They are single sentences that must be both true and false simultaneously in the logical systems in which they were proposed. – user9166 Sep 1 '17 at 20:50
• Hang on. Yes, we know that Achilles will in fact outrun the tortoise, but we do not know that this disposes of Zeno's argument. He was not arguing that we will never see Achilles outrun the tortoise, but that what we see is some sort of illusion, and that if we assume otherwise paradoxes will arise. He argues for a change to our ideas of time and motion, not that motion is impossible. It seems to me that paradoxes always depend on contradictions. That is, paradoxes are caused by contradictions and I'm not sure they can have any other cause. – user20253 Sep 3 '17 at 12:48
• @jobermark - Not sure about your point. Russell's paradox may not be an argument but it allows us to make a very good argument against the idea that the Universe can be modeled as the 'set of all sets', and this argument has extensive consequences in metaphysics. The physicist Paul Davies is good on this one, in his book 'Mind of God'. – user20253 Sep 3 '17 at 12:52
• @PeterJ Or it is a good argument for intermediate truth values in general. I favor intuitionism over ZFC. The point is that real paradoxes, in general, are not invalid arguments. They are not really contradictory and that is the problem. In order to even be contradictory, they require moving outside of the logical system into which they are proposed. – user9166 Sep 3 '17 at 17:37
• @jobermark - Metaphysical dilemmas such as One-Many arise because both horns give rise to contradictions. Thus we are left with what seems to be a paradox. So many arise that philosophers often claim that the world is paradoxical. I'd agree that these are not real paradoxes and that they can be solved by a proper use of logic, but you'll find it difficult to find a professor who agrees. Darn it, there's too many interesting discussions here to keep up. – user20253 Sep 4 '17 at 10:32

Just to adress your question directly: We could treat paradoxes in the way suggested by you if the English predicates of truth and falsity were necessarily exclusive; that is if contradictions could not be true.

But whether this is so is a highly controversial issue. Proponents of weak paraconsistency say no, because (i) the English consequence relation is not explosive, i.e. contradictions do not entail everything (think of inconsistent fictions or theories, where apparently not everything holds) and (ii) consequence is truth preservation in all models (of a suitable non-classical logic) and (iii) models represent possibilities.

More boldly, propopnents of strong paraconsistency (aka dialetheism) hold that there actually are true contradictions and so true falsities are possible. Indeed dialetheists say that some paradoxes (such as the liar) are sound arguments. Non-trivial strong paraconsistency entails weak paraconsistency, but not the other way round.

If any of these positions can succeed depends on many intricate issues such as how best to treat paradoxes of self-reference (a strong justification for strong paraconsistency) and a quick settlement of the debate is not to be expected. So you're not justified in treating paradoxes by denying their validity.

A contradiction is something that cannot be true, because it refutes its premises.

In the strictest sense, a paradox is something that can be neither be true nor false, because refuting the premises provides an equally false set of premises.

Consider Russel's paradox: Does the collection of all collections that do not contain themselves contain itself? The question cannot be answered 'yes' or 'no'. Either answer implies the opposite. If it contains itself, then it does not meet the criteria for being admitted to itself. If it does not contain itself, then it does meet the criteria and must be included.

If you decide 'Russell's paradox is invalid.', what does that mean? It can only mean that the answer to that question has some truth value like 'Irrelevant' which is different from both True and False. This requires discarding the Law of the Excluded Middle -- because this falls flat into the 'middle' it is meant to 'exclude'.

This indicates that the basic naive intuitions involved in the statement are flawed. These can be worked around technically, but cannot really be resolved. Humans will still continue having the same flawed intuitions of logic and the paradox will continue to make a compelling case for the difficulty of reasoning correctly.

You can make up whatever special set of rules you want, but they will never be as compelling as naive logic, even though naive logic leads into plenty of paradoxes.

• Agree about ordinary logic being more compelling than any other. But I'd rather say that naive logic appears to lead to paradoxes, and only appears to do so. Even Russell's paradox is not a paradox if we apply Aristotle's rules as he instructs. – user20253 Sep 4 '17 at 10:40

To my understanding a paradox is a logical argument that fails because it is built on inconsistent axioms, e.g. a tower built on clay. In contrast, a fallacy is a logical argument that fails because of a faulty argumentation, e.g. a tower built without a plumb-line. A contradiction show that the axioms and the conclusion cannot be true simultaneously.

• paradoxes don't necessarily fail. – virmaior Sep 4 '17 at 2:55
• point taken, by fail I meant 'leads to a contradiction'. – Joel Malard Apr 9 '18 at 19:22