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.
3 comments:
What do you mean when you say that M* is just like M except wrt \phi? It will have to not only treat \phi differently, but also any formula \psi that has \phi as a subformula.
The interpretation functions M and M* are functions from proposition letters (not formulas generally) to {0,1}, so they don't map complex formulas anywhere.
Perhaps its confusing because it seems that I'm defining the extension of ~φ for any formula φ.
I should have said for any proposition letter φ, [[~φ]]^M = 1 iff [[φ]]^M* = 1, where M* is just like M except it assigns 1 - M(φ) to φ.
Does that clear it up or am I confused?
Hi Brian,
Looks OK to me. Looks like you could simplify to this:
[[~φ]]^M = [[φ]]^M*, where M* is just like M except it assigns 1 - M(φ) to φ.
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