Question Since one's own reasoning is a basically set of rules of inference operating on a set of axiomatic beliefs, can one reliably prove one's own reasoning to be logically consistent from within one's own reasoning? Might not such reasoning itself be inconsistent? If our own reasoning were inconsistent, might not the logical consistency (validity) of such "proofs" as those of Godel's Incompleteness Theorems, be merely a mirage? How could we ever hope to prove otherwise? How could we ever trust our own "perception" of "implication" or even of "self-contradiction"?

Accepted:
August 21, 2008

Comments

This question raises a number of issues it is worth disentangling.

It is far from clear that we should think of our reasoning as "operating on a set of axiomatic beliefs". That makes it sound as if there's a foundational level of beliefs (which are kept fixed as "axioms"), and then our other beliefs are all inferentially derived from those axioms. There are all sorts of problems with this kind of crude foundationalist epistemology, and it is pretty much agreed on all sides that -- to say the least -- we can't just take it for granted. Maybe, to use an image of Quine's, we should instead think of our web of belief not like a pyramid built up from foundations, but like an inter-connected net, with beliefs linked to each other and our more "observational" beliefs towards the edges of the net linked to perceptual experiences, but with no beliefs held immune from revision as "axioms". Sometimes we revise our more theoretical beliefs in the light of newly acquired, more observationally-based, beliefs: but also sometimes we revise some observational beliefs (e.g. diagnosing perceptual error) on the basis of well-confirmed theory. It's not a one-way process, always reasoning from "axioms" to our remaining beliefs, but a complex balancing act, trading off considerations of theoretical organization against the observational appearances to keep the net both useful and coherent.
The notions of consistency and validity are indeed connected. Roughly, an inference is logically valid just if the premisses taken together are not consistent with the negation of the conclusion. But note that this connection ties establishing validity with establishing inconsistency. And establishing inconsistency is easy compared with establishing consistency. For to establish an inconsistency, we just need to find one compelling argument that leads to a contradiction: while to establish consistency, we have to show that, take any argument you like, it won't lead to contradiction. That means that, even if we are not sure that our repertoire of logical apparatus is overall consistent (we are not sure that there's no lurking contradictions anywhere), we can still be sure that one particular argument is valid (i.e. sure that one particular collection of propositions does lead to contradiction). That's why, contrary to the implication in the question, doubts about consistency claims don't automatically spread into doubts about validity claims.
But set those two points aside. There remains the following issue (which I take to be the key intended question). Suppose that we regiment our principles of logical reasoning into One Big Theory. Suppose we worry whether the One Big Theory is consistent. Then how can we show that the One Big Theory is in good order after all, except by using reasoning of the kind embodied in the One Big Theory whose status is in question? It looks as if we are doomed to go round in circles. To make this more pointed: it is arguable that our logical/mathematical reasoning can be regimented inside a set theory like ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice). What if we worried whether ZFC is consistent? Then using ZFC to argue that it is consistent would hardly be convincing. Indeed it's worse that: in fact, if ZFC "proved" its own consistency, not only would that not convince a skeptic of its consistency (for an inconsistent theory would "prove" anything, including its own consistency), but it would actually demonstrate the theory's inconsistency, by Gödel's second incompleteness theorem. So doesn't this show that we shouldn't trust ZFC?
No! For note, we just haven't any good reason to suppose it is inconsistent. It isn't that we have grounded worries about ZFC's consistency and so we are casting around for some independent proof to quiet those doubts. So we don't need a "consistency proof". Decades of intensive study have turned up no inconsistencies; and that's no surprise, as ZFC is custom-designed to avoid the problems that lead to inconsistencies in more "naive" set theories. The theory is in great shape, and by this stage in the game it seems to be mere idle skepticism to imagine things might be otherwise. And while idle skepticisms might be fun for half-an-hour in a philosophy classroom (though the game soon palls), they are no reason to stop trusting our mathematics!