[Alta-Logic] Logic Talk today: reminder

[Gillman Payette]

Start: 10/28/2009 - 14:00
End: 10/28/2009 - 15:00

Speaker: Peter K. Schotch, Dalhousie University (Philosophy)

Place: SS 1253 (It is the seminar room in the Philosophy Department)

Abstract:

>From its earliest beginnings in the nineteenth Century, the algebra of
 logic has taken as its central focus, the notion of a finitary operation.
 These operations correspond in an obvious way with what the logicians call
 connectives. As advances have been made in what we now call abstract
 algebra, there has been no corresponding increase in the generality of the
 algebra of logic. Even when we consider algebra at its most abstract,
 category theory, we and that those who attempt to apply that lore to logic
 end up focusing on the operations no less than Boole himself did. This
 seems a shame in more ways than one. In the first place, by not attempting
 to do something a bit more general, practitioners of the algebra of logic
 seem to be committed to an anti-pluralist philosophy of logic whether or
 not they have ever given the matter much (or any) thought. This harsh
 criticism must be tempered a bit, when we consider the case of topos
 theory. In that particular branch of category theory, we can represent
 both intuitionistic logic and classical logic. This can be no better than
 a kind of limiting case of pluralism one cannot help but feel. One
 observes that the subobject classifier of a topos must be a Heyting
 algebra at which point it's back to operations again. Another reason to
 deplore the view which takes the operations as central, is that one misses
 entirely the chance to explain the connectives. The opportunity cost here
 is rather high, since the usual explanations in logic textbooks `explains'
 the connectives in terms of a supposed deep connection with certain words
 in natural language. It isn't hard to show that such an account is wrong,
 perhaps even wrong root and branch, or at least to show that logic has
 rather less to do with natural language than is dreamed of in most
 introductory logic texts.

In this essay we attempt to use category theory as the algebra of logic,
 but in a way that seems more helpful to logic as well as more in the
 categorical spirit. This means that instead of treating some fixed logic
 as if it were a category, we shall take the notion of a logic to be
 defined quite generally and then define, for each in a large class of
 logics, what we call the associated logical category. In this category we
 can define certain familiar limits which do duty for the connectives
 conjunction and disjunction. This much is a bit old hat. We also show that
 the familiar limits (and colimits) exist in a wider sense than is usually
 considered. In terms of these limits we may obtain the allusive limit
 definition of a generalized form of negation. Finally, we make some
 remarks about how modality might be introduced in these new categories,
 functorially.

-- 
Gillman Payette
Department of Philosophy
University of Calgary
2500 University Drive NW
Calgary, AB T2N 1N4, Canada
Ph 403.220.6463
Fax 403.289.5698



-- 
Gillman Payette
Department of Philosophy
University of Calgary
2500 University Drive NW
Calgary, AB T2N 1N4, Canada
Ph 403.220.6463
Fax 403.289.5698