You are mixing up different aspects of logic, also some parts of your question are more philosophical than mathematical.
First headline: $\bot$ and $\top$ are wellformed formulas.
(On purpose I mention them both here because in this aspect they are the same)
Different authors have different formulations of this fact:
- $\bot$ and $\top$ are propositional constants
- $\bot$ and $\top$ are a zero-place connectives
- $\bot$ and $\top$ are atomic formula
They all point to the same thing $\bot$ and $\top$ can be part of a formula, it can be used like a normal propositional variable in all rules of the logic.
so if $ ( P \to (Q \to R )) \to ( (P \to Q )\to (P \to R )) $ is a theorem then so are
$ ( \top \to (Q \to \bot )) \to ( (\top \to Q )\to (\top \to \bot )) $ and
$ ( \bot \to (\bot \to R )) \to ( (\bot \to \bot )\to (\bot \to R )) $ and many more,
they are not very helpful but that is beside the point)
This is all about being wellformed and how you can use them in formulas , it has nothing about what $\bot$ means.
Some logics just don't define $\bot$ or $\top$ as a wellformed formula, so in those logics they just do not exist.
What does $\bot$ mean?
This is a philosophical question.
If you see logic just as symbol manipulation (the philosophy of mathematics known as formalism) , no symbol means anything and so questioning what a particular symbol means is meaningless.
The above is I guess not very helpful, so different logicians come up with different ideas.
$\bot$ means absurd: $ P \to \bot$ means that P leads to absurdity ( and we don't want that)
$\bot$ means refutability: $ P \to \bot$ means that P is refutable ( and so P is false)
$\bot$ means non-demonstrability $ P \to \bot$ means that P is not demonstable (so not provably true)
The above is a rewriting from "Foundations of Mathematical logic" Curry (1963), chapter 6 "negation" , the chapter goes much deeper in it, there is a dover edition of it, highly recomended, but negation is much more complex than it looks, in another article I saw, I think 7 different negations appeared, and i do doubt that article mentioned them all.
Wittgenstein came up with " meaning follows from use " so maybe the only way you can find the meaning is to look at how it is used.
If $ \bot \to P $ is a theorem then $\bot$ means absurdity, it is quite absurd that every formula is true.
If $ ((P \lor R) \to ((P \to\bot) \to R) $ is a theorem then $\bot$ means refutability, P is refuted (and therefore R is true)
If $ (P \lor (P \to\bot) ) $ you have classical logic.
so it all depends, but can you expect anything else with a philosophical question.
A propositional sentence, built out of propositional atoms, does indeed describe a set: namely, the set of valuations making it true. Remember that a valuation is just a map from the set of atoms to $\{$True, False$\}$ (such a map extends to a map from the set of all propositional sentences to $\{$True, False$\}$ via an appropriate recursion). Valuations in propositional logic serve the same semantic role as models in predicate logic: a propositional sentence is true, or false, with respect to a given valuation, and different valuations make different things true or false.
For $\varphi$ a propositional sentence, let $Set(\varphi)$ be the set of valuations making $\varphi$ true. Think of $Set(\varphi)$ as describing the conditions under which $\varphi$ is true: a smaller set corresponds to a more demanding sentence. For example, suppose I have two sentences, $\varphi$ and $\psi$. Then we have $$Set(\varphi\implies\psi)=Set(\neg\varphi)\cup Set(\psi),$$ since a valuation satisfies $\varphi\implies \psi$ iff it either satisfies $\neg\varphi$ or it satisfies $\psi$.
Note that this means that the Boolean propositional operations (conjunction, disjunction, negation, ...) correspond as desired to Boolean set operations (intersection, union, (relative) complementation, ...). So this does indeed give a set-theoretic interpretation of propositional logic.
Now let's look at Venn diagrams.
Venn diagrams are visual representations of the $Set(-)$ operation - or rather, they are a pictorial language for making assertions about the valuation sets assigned to propositional sentences. E.g. when we draw one circle wholly inside another, we're asserting that every valuation making the first circle's sentence true also makes the second circle's sentence true. Crucially, a "point on the plane" in a Venn diagram represents a valuation.
Keep in mind that the assertion made by a specific diagram may or may not be correct - it's always possible to write something meaningful but wrong.
But suppose I've gone ahead and done that: I have two propositional sentences $\varphi$ and $\psi$, and I've drawn circles $C_\varphi$ and $C_\psi$ representing them each respectively, and I've drawn $C_\varphi$ wholly inside $C_\psi$. That drawing is correct iff every valuation making $\varphi$ true also makes $\psi$ true. That statement in turn is true iff there are no valuations which make $\varphi$ true but $\psi$ false, and that statement's truth is equivalent to the claim that $(\neg\varphi)\vee\psi$ is true under every valuation - and that in turn is the assertion made by the Venn diagram which represents $(\neg\varphi)\vee\psi$ as everything.
Best Answer
We need some conventions on the terminology.
A sentence
A declarative sentence (stament, assertion) is a sentence stating a fact, like: "The rose is red".
In logic, there are declarative sentences and not e.g. questions or commands.
Unfortunately, we call the propositional calculus also sentential logic.
Thus, in propositional calculus we can replace a sentential variable $p_i$ with the declarative sentence: "The rose is red" and not with the question: "Which is the color of the rose?"
In propositional calculus, sentence and proposition are interchangeable, while in philosophical discourse, a proposition is usually an extra-linguistic entity: the content expressed by, the meaning of, the reference of a linguistc entity (a declarative sentence).
In predicate logic we have formulas with free variables (called open formulas), like: "$x$ is red".
The free variable acts as a pronoun; when we assert "it is red", the meaning of the assertion is disambiguated contextually: according to the context, "it" may stand for the book on the table, the car, the pen, and the truth value of the assertion may change accordingly.
In logic, we have to assign a denotation to the free variable $x$ (through e.g. a variable assigment fucntion) in order that the formula has a meaning (and a truth value).
In predicate logic, a formula without free variables (or closed formula), like "the rose in my hand is red" or "all the roses are red", is called a sentence.
Thus, in conclusion, either in propositional logic or in predicate one, a sentence is always meaningful, and it has always a definite truth value.