Medieval Type Theory


This post definitely falls under philosophical miscellanea, but is a nice example of a commonality between ancient and modern philosophy.

Sextus Empiricus, medieval skeptic, wrote a big (not Aquinas big, but definitely David Foster Wallace big) treatise on Pyrrhonism, a brand of radical skepticism which argues (more or less) that nothing is knowable. More specifically, Empiricus - aptly or inaptly named, I’m not sure - rejects dogma (roughly, scientific knowledge).

This runs into hot or at least lukewarm water soon enough, since, as the Pyrrhonists are quick to realise, that for P = “Dogma is unknowable.”, if one knows it, it’s false, thus refuting the whole thesis.

Empiricus’ solution is to offer two notions of dogma: narrow and broad. One must reject narrow dogma, but the rejection of narrow dogma (i.e. P=“Narrow dogma is unknowable.”) is a broad dogma, and that’s totally fine to keep.

Jump to Russell, and his famous paradox as to whether to set of all sets that don’t contain themselves contains itself. One solution, Russell’s, is to have a hierarchy of types. Type 1 sets can’t contain type 1 sets, but type 2 sets can (and so on for type 3 containing 2, etc). This eliminates the paradox.

Empiricus’ solution to self-refutation is Russell’s solution to Russell’s paradox, mutatis mutandis. Granted, you have to mutatis a little bit of mutandis, but underlyingly, it’s the same self-refuting paradox and the same type-based solution.

In actual fact, Empiricus doesn’t just give this one solution, but in a manner typical of rambling ancient philosophers, throws in a whole bunch to see what sticks, including the idea that the contradiction inherent in knowing that everything is unknowable is actually okay, and is like taking an emetic which purges the system. Nice and graphic.