This is a follow-up to Is "+" a two-place predicate? If the function symbol "+" is not a "two-place predicate", then what part of mathematical speech is it?

The context is the "Logical Analysis of Propositions". I will sometimes use function sign and the signified function interchangeably.

The authors have only introduced "subject", "predicate" and "function sign" as parts of propositions or propositional forms. I accept that "+" is a token signifying a function which adjoined with two subjects (arguments) forms a meaningful syntactic unit. The only place which that composite unit seems to have in a proposition or propositional form is that of subject.

The token "+" fits into the category of "function sign". In general, function signs are n-place syntactic structures which either evaluate concretely, or signify to a dependent variable.

So, even though the concept of mapping has not yet been introduced, I will say that a function signified by a function sign is a mapping of a set of subjects into a, possibly different set of subjects.

I use the term expression to categorize a composite syntactic unit that evaluates to a concrete entity, either immediately, or when its subject variables are instantiated.

By my way of thinking, expressions include propositions, propositional forms and n-place functions adjoined with n conforming subjects. Since a proposition is an expression consisting of syntactic form with adjoined subjects, and has a value, a proposition (or a propositional form) is a function.

So, I guess my question is: what categorical designation should identify syntactic forms which may serve as subjects in propositions or propositional forms?

Note that, by the language above, propositions and propositional forms are functions, but not all functions are propositions or propositional forms.

  • $\begingroup$ Hermes calls "expression" what almost everybody else calls "formula." At least, this is what he does in his *Introduction to Mathematical Logic." $\endgroup$ – Fabio Somenzi Jun 1 '18 at 5:23

In my experience the most common term for things to which you can apply a predicate, i.e. a predicate's arguments, is "term". "Expression" isn't uncommon, but I usually see that in a programming context, while "term" seems more common in a formal logic context. The former is the terminology that is used, e.g., on the Wikipedia page for predicate logic.

To be precise, $+$ by itself is not a term. It is usually called a "function symbol" or "operator"/"operation". However, an expression like $x+y$ would be a term. $x+y$ is something you could apply a predicate to, e.g. $P(x+y)$; $+$ is not (i.e. $P(+)$ is not a well-formed formula).

Predicates and formulas in general are usually not terms/expressions unless you are working in some kind of higher order logic. In first-order logic, you can't write $P(Q(x))$ where $P$ and $Q$ are (unary) predicate symbols, for example.

Certainly other words are used upon occasion, and you'll have to see how any particular text you're reading defines things, but these are fairly common words to use.

  • $\begingroup$ Ah! The formal definition of the terms used in formal definitions! Perhaps this is why the authors have been somewhat inexact in their definitions. $\endgroup$ – Steven Thomas Hatton Jun 1 '18 at 2:42

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