Wednesday, May 6, 2015

Fundamental logical relations

One might think the fundamental logical relations are between propositions. But I now think they are between relations (and propositions are a special case: 0-ary relations). Why? Well, in quantified logic (think: universal introduction and existential elimination) we need to talk about logical relationships between either (a) sentences with arbitrary names, (b) sentences with irrelevant names or (c) open formulas. Now ordinary sentences represent propositions, and the logical relations between sentences plausibly are grounded in the logical relations between the corresponding propositions. But sentences with arbitrary names don't represent anything, and so we have this unsatisfying grounding discontinuity: some logical relations between sentence-like entities in proofs are grounded in relations between proposition-like entities and others aren't. For somewhat similar reasons, if we want our logic to mirror the logical structure of reality, (b) isn't an option.

So we have philosophical reason to use a logic where there are logical relationships between open formulas. But an open formula represents a relation of arity equal to the number of free variables. It seems, thus, that some logical connections between propositions hold in virtue of logical connections between relations. Thus, the proposition that all humans are mortal follows from the propositions that all humans are animals and that all animals are mortal because the property (a property is a unary relation) of being mortal-if-human follows from the properties of being animal-if-human and being mortal-if-animal.

OK, time to stop procratinating grading the modal logic assignments!