A $\implies$ ¬A. Is my reasoning on the following example correct?

Question

I already asked the question in the Philosophy forum, but I haven't gotten any answer, yet. Maybe the mathematicians will be faster:

Oh, you// anything that rhymes is not true.

If this statement is true, then it is false.

First, although it seems very logic to me, is this reasoning correct? If so, how do you call this kind of "situation" in logic? Could we say that it is a paradox in the strict definition of the word? Do you have other interesting examples?

So, I had further thoughts on this and discovered, that if you assume that the statement is false, you do not come to any contradiction, because if there exist sentences that rhyme and are true, then it is exactly equivalent to the hypothesis that we have assuming that it is false that "anything that rhymes is not true".

Answer 1

It's just the plain old liar paradox.

Typically nowadays the liar paradox is presented as a sentence that speaks only about its own truth:

This sentence is false.

but the name is also applied to a sentence that claims that a whole class of sentences which includes itself are all false. The classical example of that was when Epimenides (who was Cretan) declared, "all Cretans are liars".

This does not in itself create an impossible situation as long as there is at least one Cretan who is not a liar (or at least one rhyme that is true), but it would still be quite an iffy sort of reasoning to conclude that truthful Cretans exist just because Epimenides uttered that particular lie.

How about

This sentence as well as the Riemann Hypothesis are both false!

If we accepted Epimenides as proof of the existence of a truthful Cretan, then I've just proved the Riemann Hypothesis ...

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However, reasoning from $A\to\neg A$ to $\neg A$ is in itself completely standard and above-board -- it's even intuitionistically valid. What goes wrong here is not the particular kind of reasoning being applied, but the assumption that the original rhyme (which makes claims about its own truth value) can be reasoned about at all.

Answer 2

$A \to \lnot A$

is not a contradiction. If we assume f as truth value for $A$, then $\lnot A$ is f, and thus the conditional becomes :

f $\to$ t

which is t.

Thus, we have the conclusion that the "correct" answer is : $A$ must be false.

In fact :

$(A \to \lnot A) \to \lnot A$

is a tautology, as you can easily check via truth table.

Answer 3

Your statement has to be false. As you said, if the statement were true, this would imply that the statement is false which is a contradiction. If the statement false, no contradiction occurs.

More problematic are sentences like “This sentence is false.” This can be neither true nor false as always a contradiction will follow. For more information on this, search for Barber paradox.