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.
We can agree that there is some "variability" in the practice, regarding the definition (if any) of logical symbols in first-order logic.
According to the definition in Herbert Enderton, A Mathematical Introduction to Logic (2nd - 2001), of First-Order Languages [page 69], we have :
A. Logical symbols
$0$. Parentheses: $(,)$.
$1$. Sentential connective symbols: $\rightarrow, \lnot$.
$2$. Variables (one for each positive integer $n$):
$v_\, v_2, \ldots$.
$3$. Equality symbol (optional): $=$.
B. Parameters
$0$. Quantifier symbol: $\forall$.
$1$. Predicate symbols: For each positive integer $n$, some set (possibly empty) of symbols, called $n$-place predicate symbols.
[...]
Compare with Dirk van Dalen, Logic and Structure (5th ed - 2013), page 56 :
The alphabet consists of the following symbols:
Predicate symbols: $P_1, \ldots, P_n, =$
Function symbols: $f_1, \ldots, f_m$
Constant symbols: $c_i$ for $i \in I$
Variables: $x_0, x_1, x_2, \ldots$ (countably many)
Connectives: $∨,∧,→,¬,↔,⊥,∀,∃$
Auxiliary symbols: $(, )$.
Note that the expression "logical symbols" has been avoided.
The issue is related to that of Logical Form and Logical Constants. Traditionally :
The most venerable approach to demarcating the logical constants identifies them with the language's syncategorematic signs: signs that signify nothing by themselves, but serve to indicate how independently meaningful terms are combined.
Roughly speaking, syncategorematic signs are the logical constants or logical symbols, i.e. those symbols (like conncetives) which are not interpreted or, according to modern semantics of first-order logic, do not "change meaning" when we vary the interpretation of the language.
According to this criteria, variables are ambiguos, because they have no "fixed meaning"; but also, their meaning is not fixed by the interpretation.
They are placeholders; in principle (see Frege) we can dispense with them. We can index argument places writing, instead of $Q(x, y)$ :
$Q[( \quad )_i , ( \quad )_j]$.
Thus, we can say that variables are another category of auxiliary symbols.
Best Answer
First of all, $\top$ and $\bot$ are special constants whose valuations are always set to true and false respectively. In other words, we always have $\models \top$ and $\models \neg \bot$.
It is, however, convenient to consider them as connectives ($0$-ary connectives, hence) when one is dealing with sets of connectives. One never writes the braces, but you can imagine eg $\top$ as an operator $\top ()$.
As to why $X :=\{\rightarrow,\bot\}$ and $Y := \{\nrightarrow,\top\}$ are functionally complete sets of connectives, it is enough to show that you can express the operators $\wedge$ and $\neg$. I leave you as an exercise to show that:
(edited)