But the very possibility of applying negation to a negated statement presupposes the analysis of contradictory negation as an iterative operator (one capable of applying to its own output), or as a function whose domain is identical to its range.
"Wherever we can construct concepts that, through their determinations (i.e., their limitations), do not merely negate, but they simultaneously determine positively the negated component that is missing from the whole, we can think in terms of alternatives.
The inquirers assume that a negation specifies the opposite of the item negated (Sophist 240b, 240d).
Nor, incidentally, could the intuitionist acknowledge the existence of a right inverse for negation, a # with arbitrary φ equivalent to ¬ # φ, as this makes every formula equivalent to a negated formula and hence to its own double negation.
The negation designates something within a wider kind, and the predicate negated (say "large") indicates that wider kind.
The analysis of negative predication is complex: the item negated in "Simmias is not large" indicates a division of the genus size, and the negation designates something different (non-identical, distinct) from largeness within that genus.
Volume II, a study of the calculus of classes using both equations and negated equations, attempted to cover the same topics covered in Vol. I, in particular there was considerable effort devoted to finding an Elimination Theorem.
There is, however, no clear syntactic restriction on speech acts of denial, as denials can be realized not only by assertions of negated sentences but, for example, also by means of irony.
The preceding typology of negated implications and co-implications has been developed in Wansing 2008, and one might add to this list the equivalences $\osim (A \rightarrow B) \leftrightarrow (B \rightarrow \osim A)$ and $\osim (A \coimp B) \leftrightarrow (\osim B \coimp A)$.
In the faithful embedding, the modal translation of negated implications is as expected: γ(~(A → B)) = □(γ(A) ⊃ γ ~B)).
