In propositional logic a formula φ of language L is only true (1) or false (0) relative to an interpretation M. The semantics of the negation symbol ~ is usually given as follows:
- [[~φ]]^M = 1 iff [[φ]]^M = 0.
But we could do it differently. We could conceive of negation as analogous to a modal operator (or quantifier). In this case it doesn't shift the world parameter or the assignment of values to individuals variables---instead it shifts the interpretation, i.e. the assignment of truth-values to propositional letters.
- [[~φ]]^M = 1 iff [[φ]]^M* = 1, where M* is just like M except it assigns 1 - M(φ) to φ.
As far as I can tell, that is a perfectly fine semantics for negation in propositional logic.