What is your favourite paradox?

Jones asked:

What is your favourite paradox and why?

Answer by Craig Skinner

A paradox starts with acceptable assumptions, proceeds by apparently acceptable reasoning, and reaches an unacceptable conclusion.

Resolution therefore requires rejection of an assumption, finding a flaw in the reasoning, or accepting the conclusion after all.

The ‘paradoxes’ standardly studied in philosophy include:

* liar paradox
* set-theoretic paradoxes
* Zeno’s paradoxes of motion
* ravens (Hempel’s paradox)
* grue (Goodman’s paradox)
* prisoner’s dilemma
* Sorites
* unexpected hanging

Some are puzzles rather than genuine paradoxes (Zeno, ravens, grue, hanging, dilemma); some (Sorites) have suggested resolutions that require non-classical logic (3-valued; fuzzy; supervaluation).

The liar paradox (and other semantic paradoxes) and set-theoretic paradoxes are sometimes classed together, both involving self-reference, but some logicians think there are principled differences between them.

The liar paradox (‘All Cretans are liars’, said by a Cretan) is my favourite because it very old, simple to grasp, and (in my view) has no solution other than accepting that there are true contradictions.

A simple formulation is the statement ‘This sentence is false’.

Is it true or false?

If it is true, then what it says is correct. Hence it is false.

If, on the other hand, it is false, then what it says is incorrect. Hence it is not false, it is true.

So, if it’s true, it’s false, and if it’s false it’s true. Either way we have a contradiction.

Suggested solutions include:

1. Just ban self-referential sentences and say that comment about a sentence must be in a higher-level metalanguage. This is akin to Russell’s Theory of Types ‘solution’ to the set-of-sets-which-are-not-members-of-themself paradox. It seems to dodge the issue. In any case you can avoid the self-referring sentence by the following amendment:

* The sentence below is true.
* The sentence above is false.

2. Abandon true/false bivalence, admit a third truth value of neither-true-nor-false, or both-true-and-false, or a null value (truth gap).

3. Accept that there is a genuine contradiction. The sentence IS true, and the sentence IS not-true (not some new category embracing both, but a full-blooded contradiction) and the world therefore contains true contradictions.

My preference is for 3. Some people go wild at this, suggesting that if we accept a single contradiction, then ANYTHING can be proved (the ‘explosion’ problem). I don’t think this is so, but wont go into why. Those interested should read Graham Priest ‘In Contradiction’ 2nd ed OUP 2006.

Hegel, incidentally held that there are true contradictions, but based this on acceptance of Kant’s antinomies (which are fallacious) and on arguments of his own which are incomprehensible (to me at any rate). However, I think he was right.

Logic containing true contradictions is called dialetheic logic or paraconsistent logic, and its proponents say that dialetheic logic is to classical logic as Einstein’s theory of gravity is to Newton’s – both get it right in ordinary circumstances, but the newer view is more correct and also gets it right in extreme circumstances.


Answer by Peter Jones

I have two if that’s okay. The first would be the Something-Nothing problem, the problem of which came first. I like this because it is simple and yet if we can solve it we have solved metaphysics. I also like Russell’s paradox. This is because if we can solve it we have solved the Something-Nothing problem.

In my view, however, all metaphysical paradoxes would be the same problem, (just as the two I’ve mentioned are the same problem) and so it wouldn’t really matter which is our favourite. I cannot explain this idea in an answer here since it would take too long, but it is not a novel idea. All metaphysical paradoxes take the same form and would require the same logical resolution, so which one we decide to work on would be a matter of taste. I like the two I’ve mentioned here because they are very approachable, while some are pretty fiendish and difficult to clarify.

It may be helpful to add that the Something-Nothing problem became a favourite of mine thanks to Paul Davies’ book The Mind of God. I would recommend this to anyone interested in metaphysical paradoxes.


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