### NCM 09 (part 1)

As promised, I begin a series of posts about Non-Classical Mathematics 2009. (I've just started using this LaTeX editor for internet, so the formulas look kinda weird, I should get used to this system within a couple of weeks).

The conference started with Greg Restall's talk titled Theories, Co-Theories & Bi-Theories in Non-Classical Mathematics. In the non-classical setting the assertion of a negation of a formula and its denial are different things. Those who accept gluts will assert negations of certain formulas without denying the formulas themselves. Those who accept gaps will deny certain formulas without asserting their negations.

Now, in a setting of a mathematical theory we're dealing with a consequence operation such that for any A and B, if A entails B, then asserting A and denying B is a clash. This generalizes to sets of formulas.

$\small \mbox{For any } \Gamma \mbox{ and } \Delta, \mbox{ asserting each formula in } \Gamma \\ \mbox{ and denying each in } \Delta \\ \mbox{ is a clash.}$

The rules we buy into unconditionally are at least these:

$\mbox{\textsc{Identity:}}\,\, \Gamma, A \vdash A, \Delta \\ \mbox{\textsc{Cut:}}\,\, \Gamma \vdash A, \Delta \,\, \Gamma, A \vdash \Delta \Rightarrow \Gamma \vdash \Delta$

There are two interesting negation rules:

$\mbox{[]\neg L]}\,\,\,\,\,\,\,\,\, \Gamma \vdash A, \Delta \Rightarrow \Gamma, \neg A \vdash \Delta \\ \mbox{[]\neg R]}\,\,\,\,\, \,\,\,\,\, \Gamma,A \vdash \Delta \Rightarrow \Gamma \vdash \neg A, \Delta$

Both rules hold if there are no gaps and no gluts. If there are gaps, [~R] doesn't work. If there are gluts, [~L] doesn't hold, and if there are both gaps and gluts, none of the rules works. Perhaps, one might want to add other inference rules, but let's not be bothered by these issues.

Recall now that T is a theory iff for any A:

$T\vdash A \Rightarrow A \in T$.

In the non-classical setting, if you want to avoid clash (which you want to avoid even if you allow for gaps or gluts), you should assert whatever belongs to the theory you're committed to. A theory, however, doesn't tell you which formulas you should deny (for instance, ~A belonging to the theory only tells you that you should assert the negation of A, but from this, it still doesn't follow that you shouldn't assert A itself).

Greg then goes on to introducing theory-like notions that help one not only to tell what assertions one has to make, but also what has to be denied. The first one is the notion of a cotheory. U is a cotheory iff for every A,

$A\vdash U \Rightarrow A\in U.$

The intuition here is that U is a set of unassertable sentences, and the definition mirrors the fact that if something is not to be asserted, then nothing that entails it should be asserted either.

Now, combine these two notions to construct a thing that tells you what to accept and what to reject. is a bitheory iff for every A:

$T \vdash A, U \Rightarrow A\in T \\ T, A\vdash U \Rightarrow A \in U$

Again, the intuition is that T is what's to be accepted, and U is what's to be denied.

The rest of Greg's talk was devoted to applying this ideas to non-classical theories of numers, classes and truth.