Definition
A truth function is a logical connective whose output truth value is fixed entirely by the truth values of the statements it joins — and by nothing else. Once you know whether the component statements are true or false, the truth value of the whole compound is settled automatically.
Classical logic treats and, or, not, and if...then as truth functions. "It is raining and it is cold" is true exactly when both parts are true; no further information is needed. The meaning of the connective is given completely by this input-to-output mapping.
Why it matters
How it works
Because a truth function ignores everything except input truth values, it can be specified completely by a finite table listing every combination of inputs and the resulting output. This is what makes classical propositional logic so tractable: validity becomes a matter of checking finitely many cases.
Priest highlights both the strength and the controversy here. Truth-functionality makes logic clean and computable, but it forces every connective into a purely extensional mould. The if...then connective resists this treatment — its ordinary meaning seems to depend on a connection between the parts, not just their truth values — and that tension drives much of the book's later discussion.