Monday, March 28, 2011

Negation as an interpretation shifting operator

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:

Aaron Cotnoir said...

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.

Brian Rabern said...

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?

Tristan Haze said...

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.