Why is a predicate of 0-arity considered as a sentence letter?

Question

I often find that nullary functions are considered constants and 0-arity predicates are considered sentence letters. What is the intuition behind that notation?

Answer 1

A unary predicate symbol P(x) expresses a "property".

When we "instantiate" the variable (which is a place-holder) with a "name" a (a term of the language) denoting an onject in the domain of interpretation, what we gate is a sentence P(a) that express a fact.

It may be True or False, according to the fact that the property expressed by the predicate P holds or not of the object denoted by a .

Trivial example : let the domain of the interpretation the set N of natural numbers; let P(x) interpreted with "x is Even" and let the individual constant a interpreted with the number 2.

From an abstract point of view, an interpretation is a mapping from the set of sentences of the language into { T, F }.

If so, a 0-predicate symbol is a "degenerate" predicate symbol with no place-holders to be filled with "names".

Thus, in an interpretation it can have only one truth value, because there is no "parameter" on which the truth value of the formula can depend.

If so, having a specified truth value in an interpretation, it must be a sentence.