What is the nature of the difference between a contradiction and a paradox (as apparent contradiction)?

Question

Given the recent PhilSE question What framework or tool solves the Barber Paradox? it emerged from a number of answers including Bumble's and Mauro's that there is a technical distinction between Russell's paradox and the Barber paradox. (The former occurs within the context of set theory, and the latter does not.) The distinction is that the Barber paradox does not need the semantics of set theory in order to dissolve the paradox, and that the formal semantics of first-order logic is sufficient. There are also objections to the phrasing of the question itself that seem to hint that the question is not well-formed in some sense.

I was looking for clarification regarding contradictions and paradoxes. Specifically Bubmle states:

It is not really a paradox. It is just a contradiction. There can be no such barber. It seems paradoxical only because at first glance there does not seem anything odd in the expression, "person who shaves all and only those persons who do not shave themselves". It is only after a little reflection that the contradiction becomes apparent.

Clearly, a paradox is much more than contradiction because it is psychological involving expectation rather than logic which merely speaks to the logical structure. What is a succinct explanation of the difference between a paradox and a contradiction?

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Mauro's comment:

A paradox is counter-intuitive, not necessarily self-contradictory.

seems to me simply to move the problem and hide it behind the notions of intuition. As a clarificatory question: What is the difference between something which APPEARS to be contradictory, and something which is actually a contradiction?

Answer 1

A contradiction, traditionally, is a pair of propositions such that the truth of one implies the falsehood of the other and vice versa. In modern usage, it has come to refer to a single proposition that is a logical falsehood, or false under all interpretations.

The term paradox may be used to refer to several things.

1. Two propositions that cannot both be true, but where we appear to have a plausible argument in favour of each. One of the arguments must be wrong, and the paradox can be resolved by determining which. Or maybe both arguments are wrong.

2. A proposition that appears to be true, or at least possibly true, but where it turns out to be impossible and this is surprising and counterintuitive. Or equally, a proposition that appears to be false, but where it turns out that it can be proved to be true and this is counterintuitive.

3. A situation in which it appears to be impossible to determine that a proposition is true or false, but this is because of a lack of clarity in the wording, or a lack of understanding of the precise concepts involved.

4. Sometimes it is used for an antinomy. A contradiction that points up a fundamental incoherence in our understanding of some concept.