Does propositional logic have structures and domain(s)?

Question

I learned that "the alphabet of the language of propositional logic" has no function symbols, relation symbols, and constants.

Does propositional logic have no structure? So is an interpretation of propositional logic simply an assignment? (An interpretation for first order logic consists of a structure and an assignment.)
Does propositional logic have domain(s)? (A structure for first order logic consists of a domain set, and a mapping from relation symbols, function symbols and constants to relations, functions and elements in the domain set.) If yes, is its domain $\{T,F\}$?

Thanks.

Not exactly: one domain only ${ 0,1 }$ or ${ \top, \bot }$. An "interpretation" is a truth assignment. i.e. a list of propositional variables assumed as True. — Mauro ALLEGRANZA, Sep 02 '20 at 17:53
For model theory "applied" to propositional logic, see Chang & Keisler, Model Theory (3rd ed 1990) — Mauro ALLEGRANZA, Sep 02 '20 at 18:41

Natalie Clarius · Answer 1 · 2020-09-03T12:52:27.900

No, propositional logic does not have structures with a domain of objects in the way FOL does. Interpretations of propositional logic consist only of a valuation function which assigns truth values to the propositional variables.

You could take $\{True, False\}$ as the domain, as this is what the variables of the language are mapped to, but since this domain is constant across all PL interpretations, you wouldn't typically write it out as such and only mention the valuation function.

Analogous to the way Ebbinghaus et. al. write down an interpretation as

$\mathfrak{I} = ((A,a), \beta)$

interpretations of PL could be written as just

$\mathfrak{I} = (V)$

where $V$ is the valuation function.

Does propositional logic have structures and domain(s)?

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