Are there rules for dealing with self-reference "paradoxes" in logic?

Question

My favorite paradox that leads to an endless regress, and also leads to a question:

The sentence after this is true.

The sentence before this is false.

When contradictions appear in proofs, we have rules to finish out the proof. I believe that the upside-down T symbol is used to signify a contradiction. Then the rule was that from a contradiction anything could be derived. I always found that rule quite interesting.

However, a paradox to me does not seem to be quite equal to a contradiction. Maybe they get treated as such, but are there rules for dealing with paradoxes in formal proofs? Much like they do with contradictions?

Answer 1

The same effect can be achieved with a single sentence:"This sentence is false". It is known as the Liar paradox and goes back to an ancient sophist Epimenides. Your two sentences simply split the Liar in two. There is no endless regress though, it ends in one step. We accept both sentences as "axioms", i.e. "true", but the second sentence implies that the first one is false, a contradiction. The problem is that in the usual proofs by contradiction there is an underlying premise that leads to it, and can be rejected, but there seems to be no such extra premise here. This means that the Liar's sentence, or your two sentences, form an inconsistent "theory": a statement and its negation can both be derived in them. By the logical law of explosion, then any sentence whatsoever can be derived in them, in other words they are uninteresting.

Detecting such self-referential paradoxes is easy enough, see e.g. Wen's Semantic Paradoxes as Equations. Your two sentences can be coded as equations x=y and y=¬x, implying x=¬x, which codes the Liar sentence x, interpreting "=" as "refers to", and ¬ as "not". The Boolean variable x can take only two values, 0 or 1, and neither fits. A collection of sentences produces a paradox if the system of equations coding it has no solution. The real question is how to interpret "no solution". Several approaches exist, all of them controversial, see Paradoxes of Self-Reference on SEP.

One way is not to interpret it at all, but to ban inconsistent theories altogether using syntax. Mathematical and logical theories go to great lengths to make sure that paradoxical "collections of sentences" are always a syntax error. This is what makes proofs by contradiction work in them. Nothing like Liar can be expressed in set theory language, for example, or in Russell's Principia, the problem is defined out of existence there.

Another approach is to declare that there is a hidden premise in the Liar after all. We implicitly assume that the sentence is either true or false, that it has a truth value. The equational interpretation shows us that this assumption is false, and we have to accept that some sentences have no truth value. We already accept this in natural language, "similar moon slowly" isn't true or false, it is gibberish, and "electron is a green dignity" is meaningless too, although it is grammatically correct. So there are different ways to be gibberish, and the Liar sentence and its cousins are "logical gibberish", still neither true nor false.

The most popular version of logic and semantics, where some syntactically correct sentences have no truth value, was developed by Kripke and is called truth value gap logic. In addition to true and false it introduces a third truth value: undefined. But such a logic creates many technical complications in assessing truth values of compound expressions and in manipulating them, so it is rarely used in mathematics or applications.

Answer 2

(1) Not "sentences", but only statements, can be true or false. Machines can generate sentences, but only speakers can make statements, and to make a statement is implicitly to ask that attention be paid to it and that it not be contradicted without a reason being given, which implicitly commits the speaker to not making any other statement which contradicts it either explicitly or implicitly.

(2) The reference of the word "this" in the statement, "This statement is false", is indeterminate, so that nothing obliges an interlocutor to assume that the speaker intends the statement to refer to itself, and is therefore entitled to ask "Which statement do you mean?"

(3) To make explicit for his interlocutor which statement "this statement" refers to, the speaker would have to say, "This statement, - 'This statement is false', - is false," which brackets it in a way that removes the logical problem, because the speaker is no longer asserting anything paradoxical, and, since the bracketed sentence is no longer asserted by anyone as a statement, but merely quoted, we have no speaker's authority for supposing it to be self-referential.

(4) The same principle applies also in the other version: "The following statement is false. The previous statement is true." Here the references of the expressions, "the following" and "the previous", are logically indeterminate, so there can be no certainty that they refer to each other. After all, if these sentences are written down, other sentences might be interpolated, which would remove the logical problem.

(5) In conclusion, if we aren't obliged to take a statement as self-referential, then we can't be obliged to take seriously any problems that might arise from its being so.

Answer 3

There's no general rule. My favourite example is what happens when an irresistable force meets an unmoveable object?

It's actual an inversion and recapitulation of Aristotles definition of force and the original definition is far more important in the history of thought than the eye-catching rephrasing above.

In fact one can draw a line between Aristotles definition to Newtons and then to Einsteins; whereas the paradoxical statement merely sits as an isolated fragment of thought, twisted into thought-catching and thought-threatening, paradoxical terms which achieves nothing of any substance.