3.1.s. Summary
Negation is an operation associated with the English word not. It generates a compound sentence from a single component, so it is a connective that serves to modify a sentence rather than to combine sentences. The not symbol ¬ is our notation for negation. As English notation for ¬ φ, we use not φ.
A sentence and its negation cannot be both true (they are mutually exclusive) and cannot be both false (they are jointly exhaustive); in short, they must have different truth values (they are contradictory). Each leaves open the possibilities the other rules out and rules out the possibilities the other leaves open. This means that negation, like conjunction, has a truth table; in other words it is a truth-functional connective. Not all connectives are truth-functional. Truth-functional logic is the branch of logic which studies those that are, but there are branches of logic—such as tense logic and modal logic—in which non-truth-functional connectives are studied.
Negation appears in English not only in connection with the word not but also with negative prefixes (though such a prefix does not always mark negation because it does not always produce a sentence that is contradictory to the original). Negation also appears with uses of no in phrases of the form no X, uses that can often be treated as the negation of at least one or some. The same sort of treatment is usually what is required when not appears along with the word any (which usually must be rephrased when not is removed). By negating a negation, we can produce a double negation, but this undoes the negation rather than generating a logical form with new properties.
The really new ideas come with the negation of conjunctions, but conjunctions whose components may involve negation also provide important forms of expression. A number of forms are shown below, with labels that suggest the sort of English sentences they serve to analyze:
| not-both form | ¬ (φ ∧ ψ) | not both φ and ψ |
| not-but form | ¬ φ ∧ ψ | both not φ and ψ |
| but-not form | φ ∧ ¬ ψ | both φ and not ψ |
| not-and-not form | ¬ φ ∧ ¬ ψ | both not φ and not ψ |
| not-without form | ¬ (φ ∧ ¬ ψ) | not both φ and not ψ |
That the last is the denial of the third reflects the fact that without can be used to express a but-not form. Also neither-nor can be used to express a not-and-not form. More generally, negation and conjunction form a truth-functionally complete set of connectives in the sense that any truth-functional compound can be expressed using them alone.