Is "$x$ is a variable" a statement or a predicate?

Question

I just started learning about predicate logic, and I have a question: Suppose $x$ is a variable with the domain $D$. Is "$x$ is a variable" a statement or a predicate?

Answer 1

Usually, a term of the language "names" an object of the domain. Thus, for example, in the first-order language of Arithmetic the symbol $0$ names the number zero, when we interpret the language in the domain $\mathbb N$.

When we speak of the symbols and expressions of the language, what is usually done in a mathematical logic textbook, we need quotation marks to name the symbols themselves.

Thus, the statement: "$0$ is a number" will be symbolized with the following expression of the language: $0 \in \mathbb N$, while " '$0$' is a constant" is an expression of the meta-language.

Thus sum up, the expression " '$x$' is a variable" is a statement of the meta-language.

Assume now that we want to formalize the meta-theory, starting with a "theory of expressions". In this case we have that domain $D$ can be the collection of symbols of the language and the set of finite subsets of $D$ will be the collection of strings.

In this case, the expression "$x$ is a variable" can be a statement of the theory (of expressions).

---

The issue with the question is with its "too generic" aspect...

We have to define a language: if we want to speak of numbers, we have symbols (predicate, functions, etc) for numbers: $0,1,+,\times, <$.

With them, we define formulas expressing properties of numbers: $\text {Even}(n) \leftrightarrow \exists x (n=2 \times x)$.

If our theory is speaking of syntactical objects (variables, expression) instead of numbers, we will use predicate expressing properties of syntactical objects, like "$x$ is a variable" and "$\varphi$ is a formula".

About the question if a predicate variable can be a propositional variable, the answer is no. Propositional variables are used in propositional logic with connectives to produce compelx formulas.

In the case of predicate logic we have to start from atomic formulas (built up with predicate variables) like $x=x$ and $P(x)$ and use connectives and quantifier to build up formulas, like e.g. $\forall x (x=x \land P(x))$.

Answer 2

In ordinary predicate logic, "$x$ is a variable" is not a well formed sentence. So from the point of formal language it is as meaningless as "I love cats": it is simply not something the formal language can express. This does not imply the phrase does cannot have a meaning in a different context, just not a formal sentence of predicate logic. One context in which "$x$ is a variable" could be meaningful is when talking about a certain sentence of predicate logic. But sentences about a formal system are to be distinguished from sentences of the formal system.