Is there a technical reason to require relation symbols to have positive arity?

Question

I've noticed that many logic textbooks (oriented towards model theory) will require relation symbols to have positive arity. I cannot immediately think of a technical reason why one would require this condition.

What are the reasons for doing so? Historical?

We have also $0$-arity predicate (relation) symbols: they are propositions. — Mauro ALLEGRANZA, Mar 10 '23 at 13:38
@MauroALLEGRANZA I know that. My question was about the motivations to exclude them. — Alvaro Pintado, Mar 10 '23 at 13:48
"to exclude them"? See van Dalen, page 56-57: predicate symbols $P_i$ with arity $r_i \ge 0$. The set "FORM of formulas is the smallest set $X$ with the properties: (i) $⊥∈ X; P_i ∈ X \text { if } r_i = 0$ ... — Mauro ALLEGRANZA, Mar 10 '23 at 13:56
@MauroALLEGRANZA I said "many logic textbooks" not "all logic textbooks". See the body of the question. — Alvaro Pintado, Mar 10 '23 at 14:11
@Dan The question is about allowing arity $0$ or not, not about negative arity (which doesn't make sense). — Alex Kruckman, Mar 12 '23 at 02:56

score 9 · Accepted Answer · answered Mar 10 '23 at 17:04

Purely historical/conventional. There is no technical reason to exclude $0$-ary relation symbols (proposition symbols).

I can think of two reasons why model theorists tend to ignore proposition symbols. First, model theorists care about mathematical structures, like groups, graphs, ordered fields, etc, and propositions are typically not part of the data of these kinds of structures. Second, model theorists tend to be interested in models of complete theories, and a complete theory will decide the truth value of all proposition symbols. This means all models agree on the interpretations of all proposition symbols, so from this point of view they are irrelevant.

Is there a technical reason to require relation symbols to have positive arity?

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