What *exactly* are truth values? 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
 A: 
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?

In mathematical logic, yes, they are mathematical objects. And yes, 'True' and 'False' in this mathematical domain are just like the number $5$ in the domain of integers or reals.  It's just that in the domain of logic there are just these two objects, that's it. Statements/claims/propositions can be assigned a truth-value, just like variables can be assigned a number.

And is $\mathbb{B}$ a specific set, just like $\mathbb{R}$ but instead of numbers its elements are truth values?

Yep, that's it!

If not, are truth values and $\mathbb{B}$ something abstract (for example, could the integers $0$ and $1$ be considered truth values)?

People often use $0$ and $1$ when doing boolean logic, since they're a little easier to work with, and if you wanted to, you could define a boolean algebra with those two symbols .... but they wouldn't really be integers. Or, if they are, we would have an algebra defined over integers $0$ and $1$ but with some unusual operations (e.g. 'or' would not have a super intuitive counterpart in integer operations). So ... better to say that truth-values are one thing, integers something else. Hence the distinction between $\mathbb{B}$ and $\mathbb{N}$or $\mathbb{R}$
A: Welcome to MSE!
What "exactly" is a truth value is a tricky question to answer -- it depends on what you're doing. For everyday mathematics, a truth value is just true or false, and tells you something about your proposition/conjecture/etc.
In mathematical logic, we view logic itself as a mathematical object which we can study, and that means we can consider variants and see what happens. A lot of people get bogged down in questions like "well what are sets really" or "what is truth really", and (at least to me) these questions are analogous to learning that groups other than $\mathbb{Z}$ exist, then asking "but what is $\mathbb{Z}$ really"?
So, with my NB out of the way, let's talk about some ways that people have played with truth values. This (roughly) corresponds to the different kinds of semantics that people have used to interpret logical formulas.
First and foremost, truth values could be $\{ T, F \}$. Now if you see a formula like $p \lor q$, and you know whether $p$ and $q$ are true or false (that is, if you fix a model) you can bootstrap your way up to get the truth value of $p \lor q$, or any other (propositional) logical formula you're interested in.
Of course, there's a next kind of "obvious" step if you're abstraction-minded. We aren't using many properties of $\{ T, F \}$ here. All we really need is a way to interpret $\lor$, $\land$, and $\lnot$. There are also some rules we expect these symbols to satisfy, and it turns out we can do this with any boolean algebra! (There are actually ways to get by with only a heyting algebra, but you need to change your logic, and the semantics aren't different enough to warrant their own paragraph).
Concretely, maybe your lattice is the subsets of $\{a,b\}$, and we interpret $\lor$ by union, $\land$ by intersection, and $\lnot$ by complementation. Then now we have $4$ "truth values",
and if we think of $p$ as getting the truth value $\{a\}$ and $q$ as getting the truth value $\{b\}$, then $p \lor q$ must get the truth value $\{a,b\}$, $p \land q$ gets $\emptyset$, etc.
These semantics are useful in understanding forcing, and also have some application in computer science.
But possible answers get even wilder. This next section is a pretty big spike in complexity, so take care ye who enter here.
If we let $\mathcal{C}$ be a topos, then it has an internal logic whose truth values are the elements of $\Omega$, its subobject classifier. I want to emphasize that even though we're working in some abstract setting, we really haven't changed too much. If you unpack all the category theoretic language, it turns out the elements of $\Omega$ form a heyting algebra! Then we're really doing the truth values from the last paragraph, but the way we get those truth values is a bit more complicated (and lets us do more complicated things).

So again, when we as mathematicians talk daily about "truth values", we mean True or False. Logicians like to explore alternate systems, in which "truth values" are more interesting than that, but that doesn't mean truth values are "really" something else. The rabbit hole doesn't go that deep, there are just other rabbit holes to explore.

I hope this helps ^_^
A: *

*Some people study boolean algebras. See the definition in the article. In classical logic, the truth values are the elements of the set $A=\{0,1 \}$ or $A=\{F,T\}$ where the "meet" operation is the $AND$ and the join operation is the $OR$. But you can define other sets and other operations, see the section on examples in the article. These boolean algebras and set of truth values are the objects you're looking for.


*Other logics can have different truth values, for example: In fuzzy logic the truth values are in the interval $[0,1]$. Ie: There is a truth value $0.0652$.


*$0,1$ are not really integers in a boolean algebra. You can consider them as arithmetic $\text{mod}\; 2$ defining operations that look like the operations in the corresponding boolean algebra. For this, see Zhegalkin polynomials. It's easy to find the operations by trial and error, for example: You can define $a \wedge b$ as $ab \pmod2$. See if you can find the operation for $a\vee b$ and $ ¬ a$.
A: 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.
