Sentences and Non-Sentences

The previous handout gave the definitions of a language, of terms, and of formulas. Now we need to distinguish between two kinds of formulas, sentences and non-sentences. (The non-sentences are sometimes called "open sentences.")

We begin by defining a free variable. The following is (almost) Leary's Definition 1.5.2, p. 23.

Definition: Suppose that $v$ is a variable and $\phi$ is a formula. We will say that $v$ is free in $\phi$ if and only if:

1. $\phi$ is atomic and $v$ occurs in (is a symbol in) $\phi$, or
2. $\phi$ is $(\neg \alpha)$ and d$v$ is free in $\alpha$, or
3. $\phi$ is $(\alpha \vee \beta)$ and $v$ is free in at least one of $\alpha$ or $\beta$, or
4. $\phi$ is $(\forall u)(\alpha)$ and $v$ is not $u$ and $v$ is free in $\alpha$.

Leary's definition has "if" where the one above has "if and only if," but the latter seems clearly to be his intention. (Mathematicians seem to sometimes use simply "if" where logicians with a background in philosophy would write "if and only if" or its abbreviation "iff".)

The basic idea here is that an occurrence of a variable $v$ is free if it is not bound by a quantifier, i.e. if it is not within the scope of a quantifier $(\forall v)$ (or a quantifier $(\exists v)$, but that is not part of our official language $\mathcal{L}$). (Of course, being within the scope of a different quantifier $(\forall u)$, where $u \not= v$, does not prevent $v$ from being free.) Another way to say this is that $v$ is free if it is not bound by a quantifier.

The text mostly discusses "free variables" rather than free occurrences of a variable, but the latter term is a little more accurate, since the same variable might be free in one occurrence and bound in another (as exercise 6 on p. 25 makes clear).

Now we can define sentence very simply:

Definition: A sentence in a language $\mathcal{L}$ is a formula of $\mathcal{L}$ that contains no free variables.

A bit more precisely, a sentence is a formula that contains no free occurrences of any variable.