I was reading this (older) post on the Law of Non-Contradiction over at Philosophy, et cetera. Is it really the case that it is not rational to question LNC? One might think that (P v -P) is an analytic truth. Indeed, I think that it is. But this is true only if one is operating within the confines of a two-valued logic. If one takes a many-valued logic then -(P & -P) is not analytic at all! So then the issue is whether or not classical two-valued logic is The One True Logic or not. How would we know this? I take it that someone like Richard would say that classical logic is what a maximally rational ideal agent would subscribe to. But how could we possibly know what kind of logic such an agent would subscribe to?
In an earlier post (Why Does 1+1=2?) I argued that there is nothing that could possibly decide between whether or not mathematics is an empirical science or an a priori one. The same seems true of logic. The basic laws of logic only seem rational to us because of the kind of experience that we have in this world filled with ‘medium sized dry-goods’ and this means that coherence as a criterion of truth is only valuable because of the world we live in and the kind of experience that we have. So the fact that the LNC seems to us to be denied at our own peril cannot possibly be evidence that it cannot be denied. We have no way of knowing whether it is actually ideally rational or not, since what we can concive of is limited to the kind of experience that we have.
Richard Brown 
Hmm, I could be mistaken, but I thought a many-valued logic merely allowed P’s truth value to be other than 0 or 1 [thus rejecting (P v -P)], but still not both 0 and 1 at once!
“how could we possibly know what kind of logic such an agent would subscribe to?”
How do we know anything? Some positions seem more coherent and defensible than others. We’re not totally irrational ourselves, so our own best judgments constitute (defeasible) evidence of what we would conclude on ideal reflection.
And don’t forget the strategy described in my post: although I deny their view, the logical skeptic has no reason to disagree with me, because my contradictory assertion only threatens their position on the assumption of LNC, which they claim not to believe in any case!
Wittgenstein might argue that logical truths (and others) are not of a sort that we can possibly doubt. All knowing and doubting takes place within a system, and principles such as the law of contradiction are sufficiently basic that nothing can undercut them. Indeed, our very manner of arguing presumes the integrity of certain modes of inference, a certain epistemic system, so there is something self-stultifying about trying to use that system against itself; it is like sawing off the branch on which one is sitting.
Hi Richard, interesting post. You write: “We have no way of knowing whether it is actually ideally rational or not, since what we can concive of is limited to the kind of experience that we have.”
As it turns out, Graham Priest argues that certain logical and semantic paradoxes, such as the Liar or Grelling’s paradox, are true contradictions. Thus, “This statement is false” (and its more sophisticated variants) are both true and false, and ‘heterological’ is both heterological and non-heterological. If his case can be borne out, then we can come to know that LNC is not ideally rational. He calls his position “dialetheism”. There’s a good introductory article here:
http://plato.stanford.edu/entries/dialetheism/
There’s also a good anthology on the subject:
http://www.amazon.com/Law-Non-Contradiction-Graham-Priest/dp/0199204195/ref=sr_1_7/102-7573754-0496959?ie=UTF8&s=books&qid=1187581464&sr=1-7
Hope you find this interesting.
Hi Richard,
As Jason points out below, dialetheists do allow (some) sentences to be both true and false at the same time (this is the third truth value!). This is because if one rejects (p v -p) then one also rejects the logically equivelent -(p & -p)…unless one has a form of many-valued logic that rejects De Morgan’s laws…but not all do…
“Some positions seem more coherent and defensible than others.”
Of course they do, to us! The point is that this may be just because of the kind of experience we have had in this kind of world. So intuitions about coherency cannot possibly be of any help here!
Finally, your strategy doesn’t work against the kind of skepticism I argue for. I of course believe in LNC…given the kind of world that we inhabit (at the macro level) it would be irrational to deny it. What I am questioning is whether an ideally rational agent would have to believe it.
Hi Andy,
Thanks for the comment!!
The problem with that line of argument is that it assumes that basic first-order logic is the One true Logic, but there is no evidence for that! There are many, many kinds of logic…paraconsistent, dialietheists, many-valued logics, relevent logics etc…all of which cherry pick which logical truths to preserve and which to jettison…so we never saw off the branch we are sitting on, just the ones that are next door…
Hi Jason,
Thanks for the links. I am aware of Priest’s work (we almost hired him at the Graduate center, but went with Stephan Neale instead…)
I guess I don’t see why, if Priest’s programme can be borne out, that shows that LNC is not ideally rational? That would seem to assume that Priest’s is? I just don’t see any way of deciding which is ideally rational…
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