The Lure of Paraconsistency
A paraconsistent logic is a logic that allows some contradictions. To make this somewhat bearable, paraconsistent logics find various ways of stopping the rule of explosion: "from a contradiction, anything follows". The rule of explosion holds in both classical logic and intuitionistic logic, and makes inconsistent theories very uninteresting: there is only one of them, and in it all things are both true and false. Any theory that turns out to be inconsistent is logically equivalent to this boring theory.
So, contradictions are not as bad in paraconsistent logics-- the explosion is contained, so it's not like everything falls apart. We might still be irked, and ask how this counts as "logic" if it is explicitly contradictory, but at least we can do interesting formal work without worry.
For example, it would be possible to construct paraconsistant versions of naive set theory, my naive lambda calculus, and naive theories of truth. In the context of my most recent posts, it is interesting to consider a naive theory of logical implication: collapse all of the different levels of implication (the systems I named one, two, infinity, A(infinity, infinity)...) into a single implication relation that can talk about any level.
Now, the temptation: these "naive" foundational theories appear to be referentially complete! Frege was able to use naive set theory to construct basic mathematics, and indeed it doesn't seem as if there is any bound to what can be constructed in it. On my end, a naive theory of implication obviously contains all the levels I might construct: the system one is the fragment in which implication is only wrapped around first-order statements, two is the fragment in which implications are additionally allowed to be wrapped around statements in one, infinity is the fragment in which we can use any of one, two, three.... infinity^infinity allows us to use one, two, three,... infinity+1, infinity+2, infinity+3...
The problem, of course, is that in addition to allowing all meaningful statements to enter, we apparently let some meaningless statements through as well. But, at the moment, I know of no other way to let in all the meaningful ones! So paraconsistent logic is looking pretty good at the moment.
Essentially, it seems, a meaningful statement is any statement constructable in a naive theory that happens to have a truth value that relies on other meaningful statements. Those other meaningful statements gain their meaning from yet other meaningful statements, and so on down the line until we reach good solid first-order facts that everything relies on. But, the path down to basic statements can be arbitrarily complicated; so, it is impossible to construct a logic that contains all of the meaningful statements and none of the meaningless ones, because we can't know ahead of time which is which for every single case.
I found an argument that paraconsistent logic isn't the only way to preserve naive set theory (free version), but it apparently is only hinting at the possibility, not providing a concrete proposal. Actually, I've made some relevant speculations myself... in this post, towards the end, I talk about a "logic of definitions". Such a logic would be four-valued: true, false, both, neither. A definition can easily be incomplete (rendering some statements neither true nor false), but it can just as easily be inconsistant (rendering some statements both true and false). This is suited particularly well to the way the comprehension principle works in naive set theory; essentially, we can read the naive comprehension principle as stating that any set that has a definition, exists. The trouble comes from the fact that some such definitions are contradictory!
This seems nice; just alter naive set theory to use a four-valued logic and there you go, you've got your foundational logic that can do anything we might want to do. But I'm not about to claim that... first off, I haven't even begun to define how the four-valued logic would work. Second, that scheme omits the extra grounding that nonmonotonic methods seem somewhat capable of providing; I would want to look into that omission... Third, the non-classical manipulations provided by the four-valued logic may not be sufficiently similar to classical manipulations to justify much of classical mathematics. That would be a big warning sign. So, generally, the idea needs to be worked out in much more detail before it can be judged. (But, it looks like there is some work on four-valued logics... here, here, here... haven't read them yet.)
But, anyway, the naive theories (and the idea of using paraconsistant logic to make them workable) are quite valuable in that they provide a glimpse into the possible, showing that it is not utterly crazy to ask for a logic that can define any infinity a human can define... we just might have to give up consistency.