Here's an odd thing. There seems, browsing along my shelves, to be no standard symbolic metalinguistic shorthand widely used in elementary books for assigning a truth-value to a wff (say, in the propositional calculus). You would have expected there to be some.
In the first edition of my Introduction to Formal Logic (CUP), I borrowed the symbol '$\Rightarrow$' to abbreviate 'takes the value ... [on some given valuation]' and wrote the likes of e.g.
If $\mathsf{P} \Rightarrow \textrm{T}$ and $\mathsf{Q} \Rightarrow \textrm{F}$ then $\mathsf{(P \land Q)} \Rightarrow \textrm{F}$.
But this on reflection was silly, given that the symbol '$\Rightarrow$' is already overloaded (not in my book, but elsewhere -- like on math.se! -- where, for a start, some use it for the conditional, some use it in place of a turnstile, and some get in a tangle by using it ambiguously for both!). It seems wiser not to add to possible confusion, especially when readers might well simultaneously get to see '$\Rightarrow$' being used in one of these different ways.
So for the upcoming second edition, I'm tentatively minded to use the \mapsto symbol for value-assignment, and write instead
If $\mathsf{P} \mapsto \textrm{T}$ and $\mathsf{Q} \mapsto \textrm{F}$ then $\mathsf{(P \land Q)} \mapsto \textrm{F}$.
(I suppose a colon could be another possibility, but I'd rather have something more distinctive. And something like $\textrm{T}(\mathsf{P})$ is not so pretty/easy to read in bulk, is usually part of an augmented object language -- and kinda misses the dynamics of some kind of arrow.) But am I missing a trick here? Is there a better symbolism?? Something now in more common/standard use in elementary contexts than I realize???
[Edited: I should add, that the book is baby-logic-for-philosophers -- so ease of readability/ease of blackboard use will trump e.g. portability to more sophisticated contexts.]
There are a variety of ways to handle this.
One option is to use some suitable notion of logical equivalence, for which I often use $\equiv$, but there are many other notations. So you would have "if $P\equiv\top$ and $Q\equiv \bot$ then $P\land Q \equiv \bot$". However, this would be more syntactic than I think you want. This also hides the dependence on some sort of context, though this could be recovered by adding a subscript to $\equiv$, unless you are only talking about tautologies.
Another is to use the satisfaction relation, e.g. "for a given structure $M$, if $M\vDash P$ and $M\vDash \neg Q$ then $M\vDash \neg(P\land Q)$". For classical propositional logic, this can also be written "for a given structure $M$, if $M\vDash P$ and $M\not\vDash Q$ then $M\not\vDash P\land Q$".
Finally, you could explicitly do what you are saying. I like using $[\![P]\!]$ for interpreting syntax into some mathematical object. This notation seems fairly common though hardly universal. Usually, the interpretation is with respect to some structure/valuation/assignment which will be added as a decoration, e.g. $[\![P]\!]^{\mathcal M}$. An alternate common notation is simply $P^{\mathcal M}$ as used e.g. on this Wikipedia page. For the context of classical propositional logic, I'd likely write $[\![P]\!]_\rho$ where $\rho$ is an assignment for the propositional variables. Your statement would then be: "for any propositional variable assignment $\rho$, if $[\![P]\!]_\rho = 1$ and $[\![Q]\!]_\rho = 0$ then $[\![P\land Q]\!]_\rho = 0$".
As a final option that's more akin to what you are currently doing, in many type/proof theories, there is a notion of reduction. For some, particularly simply typed lambda calculi used for higher-order logic, a formula is "true" if and only if its normal form with respect to this reduction is $\mathsf{tt}$, one of the two values of the Boolean type. You could then say "if $P\leadsto\mathsf{tt}$ and $Q\leadsto\mathsf{ff}$ then $P\land Q \leadsto \mathsf{ff}$", though, again, this would only be true of tautologies. However, in a higher-order logic, you can easily represent proposition variables as parameters, so this isn't as much of an issue. Note, that these concepts and this notation would be unusual in a traditional logic context.
My recommendation is to use the third approach, the $[\![-]\!]$ brackets or a variation of that idea at least.