How is it possible to have an axiom in logic that defines AND? Wouldn't you have to already understand what it means?  If you define A AND B by saying that it follows from A being True and B being True, aren't you using "and" to define AND. To be able to combine two or more premises to reach a conclusion, aren't you assuming the conjunction of the premises?  It would seem that some prior concept of conjunction is required for any system of axioms and theorems.
 A: 
One might again worry that something circular is going on. We defined the
symbols for disjunction and biconditionalization, $\lor$ and $\leftrightarrow$ in terms of $\lnot$ and $\to$ in Section 2.1, and now we've defined the valuation function in terms of disjunction and biconditionalization. So haven't we given a circular definition of disjunction and biconditionalization? No. When we define the valuation function, we're not trying to define logical concepts such as negation, conjunction, disjunction, conditionalization, biconditionalization, and so on, at all. Reductive definition of these very basic concepts is probably impossible (though one can define some of them
in terms of the others). What we are doing is starting with the assumption that we already understand the logical concepts, and then using those concepts to provide a semantics for a formal language. This can be put in terms of object language and metalanguage: we use metalanguage connectives, such as 'iff' and 'or', which we simply take ourselves to understand, to provide a semantics for the object-language connectives $\lnot$ and $\to$.
Sider, T., 2010. Logic for philosophy. Oxford: Oxford University Press, p.31.

While Sider is talking about other logical connectives, what he's saying holds for conjunction, $\land$. Basically, we sidestep the issue by differentiating between the object-language and the meta-language; in the meta-language we just assume we know what "and" means and leave it at that. This might not feel very satisfying, but it's both enough to avoid circularity and it works — "$v(\phi\land\psi)=1$ if, and only if, $v(\phi)=1$ and $v(\psi)=1$" is clearly understandable because we use it consistently.
There is another sense we give $\land$ meaning. To be clear, I don't mean "meaning" in a logical semantic sense. We supply meaning via syntactic rules. For example, $\land\text{I}=_{\tiny{Def}}\dfrac{\phi,~\psi}{\phi\land\psi}$ gives us a way to manipulate symbols. We've stated that $\phi$, $\psi$, and $\phi\land\psi$ are wffs in the object language so there's no circularity. The inference rule doesn't mean "and", it just happens to commonly coincide with that concept. All these symbols are are some squiggles on a page, and some rules for transforming those squiggles; we're free to assign any meaning to the squiggles we like. For instance, $\phi$ could mean $10$, $\land$ could mean "divides", and $\psi$ could mean $100$, hence, $\phi\land\psi$ means "$10$ divides $100$". While this wouldn't be very useful, it's not wrong because there's no right.
This phenomenon isn't just a quirk of logic by the way. Definitions are slippery, evasive things that do odd things when looked at closely.
A wild sorites paradox has appeared

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mountain

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