Frames and Theories III: Motivating the Semantic Clauses

Ok, so I had to take a break to grade finals and then to do the whole Christmas thing. Now I'm back on the wagon.

Let's dive right in with some intuition pumping for clauses 4 and 5 from the definition of truth we covered last time.

To begin, I need to point out that only prime theories *have* complements. I don't think I said that before. I'm saying it now. The reason is simple: the complement of a theory needs to be a theory. And, intuitively, the complement of the theory t is the theory containing every sentence A whose negation is not in t. In a motto, the complement of t agrees to all the things t doesn't reject. The problem this gives rise to is that if 'A or B' is in t, but neither A nor B is, then 'not A' and 'not B' must both be in the complement of t. But then, on minimal assumptions, 'not A and not B' will be in the complement of t. So, 'not (not A and not B)' must not be in t. But (again on minimal assumptions) this just is 'A or B', which we already said *was* in t.

That's a contradiction. To avoid it, we restrict complementation to being a function from primes to primes. Consider this officially added to the description of frames and models from here on.

Now take a prime extension u of a theory t. Since u is prime, u decides every disjunction in t. The complement of u accepts everything u doesn't reject. So if the complement of u doesn't accept A, then u rejects A.

So the basic idea behind the negation clause is that a theory t makes the sentence A false just when each way, u, of settling all the matters that t raises (that is, each prime extension u of t) rejects A. Such a t has effectively made A impossible. So it makes sense to say that this theory makes A false.

So much for clause 4. Clause 5 is a bit easier to motivate. It says that the conditional `if A, then B' is true in a theory t just when every time we close a theory where A is true under t, the result is a theory where B is true. This is a nonstandard, nontruthfunctional, but nonetheless plausible and fairly simple story about what it takes for a certain sort of conditional to be true.