How to formally assign truth values to a proposition? What is the correct way of formally assigning a truth value to a proposition?
If $P$ is for instance true, can I just write $P = \mathbf{true}$? (Or 1 or T instead of $\textbf{true}$)
Should I use another sign than "$=$"?
What's the correct notation, what's the relation between a statement and its truth value?
 A: Main answer: There is no really standard notation.
At an elementary level, in my intro logic book I think I use ":=" for the assignment of a value to a wff (assuming some given valuation of relevant semantic primitives), and subscript "$:=_v$" when we need explicitly to relativize to some such valuation.
Note you are mentioning the wff and assigning a truth value to it. So, if you are being really pernickety you'd write

'$P$' takes the value T

or

'$P$' := T

for short. But that gets tedious to I'd just write

$P$  := T

with the convention that ':=' comes with hidden quotation marks on the left.
Another convention would simply be to write the likes of $v({P})$ = T.
Subsidiary comment. You move from talking of a "proposition" via (apparently) talk of a wff to talk of a "statement". That's rather careless, given that these are used in different ways by different logicians. Talking as you do in the question about "formal assignments", however, suggests that you mean to be talking about assignments to wffs.
A: A proposition is not literally the same thing as its truth value. This distinction is crucial when, for example, discussing the status of the same proposition in different contexts. Basically, in addition to the purely syntactic apparatus of the logical grammar itself there is also some semantic apparatus (e.g. structures in the case of first-order logic), and each "semantic object" will give rise to a function from propositions to truth values. So if you want to be very precise, you need to introduce some new notation: for example, maybe you introduce the notation $$\mathsf{tv}_\mathcal{M}(\varphi)=\mathsf{true}$$ to mean "the truth value of the sentence $\varphi$ in the context $\mathcal{M}$ is true."
Unfortunately there is not a standard notation here, and abuse of notation is actually quite common (e.g. "$\varphi=\mathsf{true}$"). Ultimately you should just follow whatever convention is used by your text, or clearly introduce a convention of your own.
