Careless discussion tends to ignore important philosophical criteria, as noted in previous notes and articles in this blog. One of these is the distinction between two basic types of negative statements: __true negations, and complements of the predicate__.

Daniel J. Castellano, a mathematician and historian from MIT and Boston University covers these points in his web pages dedicated to logic and language – http://www.arcaneknowledge.org/philtheo/logiclang/logiclang2.htm

Castellano writes:

“Although we regard negative predicates, i.e., complements, as having real conceptual content, the same is not as obvious in the case of true negations, i.e., negative propositions or statements. The Greek sophists famously claimed that sentences of the form ‘X is not Y’ have no conceptual content, since we cannot speak of that which is not. To this we may respond that the sentence, ‘Socrates is not tall,’ does have content. Although it does not articulate a state of affairs, it allows us to make judgments about states of affairs given certain conditions. For example, if we stipulate that Socrates exists and has the dimension of height, we know from the sentence ‘Socrates is not tall,’ that Socrates must be short or of medium build. Negation does not say anything by itself, but in combination with other statements it does convey conceptual content.

“The distinction between the two kinds of negative statement (one is a true negation; the other is a complement predicate) can be represented in symbolic logic if we retain a copula. ‘~(x is P)’ and ‘x is ~P’ are semantically distinct sentences, and the negative symbol ‘~’ actually represents different things in the two sentences. In the first, it expresses negation of the proposition “x is P.” In the second, it does not negate any proposition, but signifies the logical complement of the predicate P, or “non-P.” The scope of this complement depends on the scope of allowable predicates of *x*. The distinction between negation and complement is lost when we drop the copula from our symbology. The sentence ‘~P(x)’ is syntactically ambiguous, and could correspond to either of the semantically distinct sentences ‘~(x is P)’ and ‘x is ~P’.

“For complex sentences, consisting of propositions coupled by logical connectors (AND, OR, etc.), it is important to distinguish which propositions are being negated. If we use a particle such as ‘~’, we can enclose the negated propositions in parentheses. In this way, we can distinguish ‘~(p AND q)’ from ‘~p AND ~q’. Another approach is to use a bar over that part of the sentence that represents what is negated. With this notation, we would write these same two sentences as ‘p AND q’ and ‘p AND q’ respectively. The first system of notation is analogous to the use of the adverb ‘not’ in ordinary language, while the second system resembles the negative mood. We can express both the negative mood and the complement predicate without a copula if we combine these notations, so that ‘x is not non-P’ would be rendered ‘~P(x)’, using the particle ‘~’ to signify the predicate complement, and the overline to signify negative mood.

“In discussions of symbolic logic, it is common to characterize negation as an operator that reverses the truth value of a proposition. This is a purely syntactic definition of negation, motivated by the grammar of Western languages, which use the adverb ‘not’ to negate the expression it modifies, rather than have a negative grammatical mood. The effect of negation on a proposition’s truth value is not the essence of what it means to deny something, though it is a necessary consequence of negation, as we shall see presently.”

The negation of a statement and the **complement** of the statement are two logical figures which have to be distinguished and –accordingly will have different roles in philosophical discourse. In my own approach, negation and complement and direct negation (or negation of the statement) are the two basic inputs into the “logical square of opposition.”