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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.

user1153980
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    Often in logic things like AND are not defined. Instead, they are logical primitives, and there are rules of inference about how they can be used. So it does matter so much what a conjunction "is" as what it "does". For example, from A AND B you are allowed to conclude A (and you are also allowed to conclude B). See rules of inference. – Jair Taylor Mar 10 '22 at 21:24
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    Just like it doesn't matter what a pawn in chess really "is", but how it can move. You could use a bottlecap or a nickle just as well as a pawn piece as long as it follows the rules for a pawn. – Jair Taylor Mar 10 '22 at 21:26
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    @JairTaylor The OP's question has more force with the other direction - how do you make sense of "From $\alpha$ and $\beta$, we can infer $\alpha\wedge\beta$" without something like a meta-theoretic notion of "AND"? (OK, we can talk instead about "${\alpha,\beta}\vdash\alpha\wedge\beta$," but adjunction isn't any less complicated than conjunction in my opinion.) This is different from the "rules-versus-nature" issue, and ultimately not something we can fully get around - see also the story What the tortoise said to Achilles. – Noah Schweber Mar 11 '22 at 00:34
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    Well, the point is we can program a computer to formally manipulate such rules even though the computer has no concept of "and", it's just a lot of electrons moving around and such. Any human-understandable definition is ultimately going to be circular. – Jair Taylor Mar 11 '22 at 00:58
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    Tortoise: But we must be careful in combining sentences. For instance, you'd grant that "Politicians lie" is true, wouldn't you? Achilles: Who could deny it? Tortoise: Good. Likewise, "Cast-iron sinks" is a valid utterance, isn't it? Achilles: Indubitably. Tortoise: Then, putting them together, we get "Politicians lie in cast-iron sinks" — Douglas R. Hofstadter, Godel, Escher, Bach: an Eternal Golden Braid – PM 2Ring Mar 11 '22 at 04:53

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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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