TITLE: What *exactly* are truth values?
QUESTION [5 upvotes]: I know this question might sound stupid, but hear me out.
For a while I took truth values for granted, but recently I've gotten very confused about them. Wikipedia says that a truth value is a "value indicating the relation of a proposition to truth". Afterwards it goes on to say that in classical logic, there are two truth values, true and false. It also says that the set of those two values is the boolean domain $\mathbb{B}$. On the boolean domain page, it says that a boolean domain is a set of two elements whose interpretations include true and false.
Now, for my question.

Are truth values, true and false, specific mathematical objects, just like the number $5$ for example, so when someone says "logical truth", they refer to the specific mathematical object?
And is $\mathbb{B}$ a specific set, just like $\mathbb{R}$ but instead of numbers its elements are truth values?
If not, are truth values and $\mathbb{B}$ something abstract (for example, could the integers $0$ and $1$ be considered truth values)?

Any help is appreciated

REPLY [3 votes]: I think it is worth giving more detail to HallaSurvivor's brief remark about the distinction between using logic and studying logic.
When you use classical logical reasoning (because it works), and you make a statement $A$, you are saying that $A$ is true. Note that this truth-value is not an object in the intended domain/world for your reasoning. For example, if you work within ZFC set theory, the intended domain (whatever it is) does not include true and false. That is why predicate-symbols are not the same as function-symbols, and statements are not the same as terms!
On the other hand, when you study classical logic (or other logics), then you are using mathematical objects to represent all the bits of the logic, including truth-values. This is why an FOL interpretation for a given FOL language $L$ includes a function that maps each well-formed formula over $L$ to some truth-value (not the same as truth-values in the previous paragraph). This would not be possible if the truth-values here are not objects!
Note, however, that when you study a logic system $S$, you are still working within some foundational system MS, and you still make statements over MS. Frequently, the MS you would use is based on classical logic, and so these statements would have an 'external' truth-value that is not an object (in your intended domain for MS). At the same time, you might have some truth-values for $S$, which are objects in (the intended domain of) MS but not objects in $S$. Just make sure you are clear about what things are objects in which system.