If $A$ is decidable, then $A$ is true.

Question

Given is a statement $A$. Now I know the following for this particular statement:

If $A$ is decidable, then $A$ is true.

What can you conclude about the truth value of $A$? Obviously, if $A$ is true, then $A$ is decidable.

It seems like the decidability of $A$ does not really change the outcome, so I would assume that the truth value of $A$ is actually already determined before I decide what it is (with a proof).

My problem with this question probably comes from the fact that I do not know much about logic, (un)decidability or the mathematical axioms mathematicians are working with.

Answer 1

First, to clear up some possibly confusing terminology: "decidable" is usually a term applied to sets of formulas, and not individual formulas themselves. In this usage, it means that there exists an algorithm to decide whether a formula is a member of the set. However, there is another usage -- typically older -- which I will assume you are referring to in this question. By the older usage, we say that $A$ is decidable in a logical system $\mathcal{L}$ (roughly, a set of axioms) if either $A$ or its negation $\lnot A$ is provable in $\mathcal{L}$. I will assume this usage for the remainder of the answer.

With this usage in mind for decidable, there are still two ambiguities left in your question. First,

Now I know the following for this particular statement:

(1) Do you know it in the sense you have a proof, or we are just supposing it is true? And second,

If $A$ is decidable, then $A$ is true

(2) when you say "$A$ is decidable" -- decidable in what logical system $\mathcal{L}$?

Depending on the answers to (1) and (2), there are some things we can say. A famous result is: if the answer to (2) is ZFC, and the answer to (1) is that not only do you claim this is true -- you have a proof of in ZFC! -- then we can actually conclude from your statement ("if $A$ is decidable, then $A$ is true)" that $A$ is itself true! This is a consequence of Löb's theorem, which is closely related to Gödel's second incompleteness theorem. Löb's theorem states:

• If there is proof that "$A$ is provable" implies $A$, then there is a proof of $A$.

To apply it to your case: your statement is that if $A$ is decidable, then $A$ is true. It follows that if $A$ is provable, then $A$ is true (since decidability means either $A$ or $\lnot A$ is provable). So what Löb's theorem states is that the only way to know your statement (in a fixed logical system) is for $A$ itself to be known.