Thursday, December 3, 2009

The Logical Foundations of Logic According to Wall-E

There is a prejudice abroad that an axiom is a certain sort of proposition or statement, which serves as one of a set of starting points to build the any science (construed as a set of interrelated propositions regarding a genus) through inferences from these axioms. For those who distinguish between axioms and principles--which we must if we are to countenance the notion of an empirical science (and let that be an axiom if you like)--this thesis may be refined to say that axioms serve as the most general starting points and function alongside the empirically disclosed and clarified principles of any given empirical science to govern the manner in which inference may and must proceed from those starting points.

Presumably Aristotle is somehow responsible for this state of affairs, since perhaps there are only a handful of prejudices of logic which cannot somehow be traced back to his work. I have not yet found anything of the sort asserted in the Analytics, but perhaps that will only tell you how little attention I have been able to give that text. What I do find is the fact that an axiom is never brought into a straightforward argument: e.g., "That it is not possible to affirm and deny at the same time is assumed by no demonstration--unless the conclusion too is to be proved in this form" (An Post 77a10). In other words, only if one wants to prove precisely that a certain true proposition is not to be denied, or that a certain false proposition is not to be asserted, do we appeal to the axiom known as the principle of non-contradiction. Yet, even in this case, if the principle of non-contradiction is truly an axiom in the sense of being indispensable for grasping anything whatsoever, the same principle which is made explicit as a premise must also undergird the connection of this premise through the other to the conclusion. And it cannot do this as a premise.

Okay, so what does this have to do with Wall-E? I guess all you robot-loving fools out there will just have to wait till next time to find out,