I have some doubts about the "natural" interpretation of $\bot$ in Natural Deduction and sequent calculus.
In Prawitz (1965) $\bot$ (falsehood or absurdity) is called a sentential constant [page 14]
Chiswell & Hodges (2007) list $\bot$ (absurdity) between the truth function symbols [page 32], and this is not very clear for me; then, in the formal definition of formula of a language [page 33] they say :
$\bot$ is a formula.
Negri & von Plato (2001) [page 2] list $\bot$ (falsity) between the prime formulas, specifying that :
Often $\bot$ is counted among the atomic formulas, but this will not work in proof theory. It is best viewed as a zero-place connective.
I think that the last comment is contra D.van Dalen, Logic and Structure (5th ed, 2013) [page 7] where $\bot$ is defined as a connective and :
The proposition symbols and $\bot$ stand for the indecomposable propositions, which we call atoms, or atomic propositions.
I'm wondering if all the above definitions are equivalent.
A (propositional) connective is an "operator" that maps one or more propositional variables into a formula; e.g.
$\land$ : <$P,Q$> $\quad \rightarrow \quad P \land Q$.
This means that the zero-place connective $\bot$ is a mapping
$\bot$ : $\emptyset \quad \rightarrow \quad \bot$.
If so, may we say that, being at the same time the mapping and the output of the mapping, it is both a connective and a formula ?