Tuesday, September 29, 2009

What Calling The LNC "Metaphysical" Won't Get You A Free Pass On

[It's been recently forcefully pointed out to me that a regular, predictable schedule here would be good. Accordingly, I'm going to try to update every Monday and Wednesday for the foreseeable future. We'll see how that goes.]

So I just saw this extremely interesting article by Tuomas E. Tahko, "The Law Of Non-Contradiction As A Metaphysical Principle." (Thanks to William Shultz for the pointer.) From a quick skim, it looks like Tahko is making a useful distinction between the LNC understood just as a logical formula that says that for any sentence, the negation of the conjunction of that sentence and its negation is true, and something more like what I like to call "monaletheism" (a helpful term coined by Ryan Lake), a distinction often run together when the debate about dialetheism is casually referred to as a debate about the Law of Non-Contradiction. After all, the dialetheist can embrace the LNC in the former sense, as long as they're willing to commit to a secondary true contradiction, (P & ~P) & ~(P & ~P) for every simple contradiction (P & ~P) that they are committed to. Moreover, IIRC, the LNC is actually a logical truth in Graham Priest's favored logic LP (Logic of Paradox), and in a talk I saw in Melbourne, Koji Tanaka said that this was a feature of all of the paraconsistent logics that Koji liked. So this seems like a worthwhile distinction to make. As far as Tahko's favored way of expressing it....meh. I more or less agree with Quine on issues of ontological commitment, confirmational holism and the like, so I'm a bit skeptical about how much light is shed on the status of a given truth when we slap the "metaphysical" label on it, but Tahko's explication of what he's getting at with the label seems more or less unobjectionable to me:

"At its simplest, the metaphysical interpretation of LNC amounts to this: the entities of the mind-independent reality are plausibly governed by some sort of principles...as to what kind of properties a certain kind of entity can and cannot have, and further, some of these properties are mutually exclusive. For instance, a particle cannot both have and not have a charge at the same time, or an object cannot be both green and red all over at the same time. It seems that reality just is such that it conforms to the law of noncontradiction. For instance, a particle cannot both have and not have a charge at the same time, or an object cannot be both green and red all over at the same time."

The general spirit of this passage resonates nicely for me with what Frege seems to be getting at when he says that: "[l]ogic is concerned with the laws of truth, not of holding something to be true, not with the question of how men think, but with the question of how men must think if they are not to miss the truth." So far, so good.* What I have a problem with, in Tahko's presentation, comes shortly after that definition:

"Another thing to note before we proceed is that semantic paradoxes such as the Liar do not threaten LNC as a metaphysical principle. Any arbitrariness or vagueness over language has no bearing on LNC understood as a metaphysical principle. A counterexample to the metaphysical version of LNC could only be a true contradiction in the world."

I don't think this works. First of all, I'm deeply skeptical about the idea that the Liar Paradox has much of anything to do with vagueness. Take a typical Liar sentence:

# The sentence marked by the number sign is false.

No one, reading this sentence, has the slightest doubt about what sentence is being referred to, or about what it is to say of any sentence that it is false. Moreover, "true" and "false" certainly seem pre-philosophically to be mutually exclusive properties of sentences, certainly no less mutually exclusive than "red" and "green" as the colors of the entire visible surface of an object. So, unless sentences are not, for some reason, part of "the world" (which is certainly a strange thought, since they certainly *seem* to have an independent existence), then it looks like the LNC-as-a-"metaphysical"-principle as Tahko defines it should apply to the truth-and-falsehood of sentences as well as the redness-and-greenness of visible surfaces.

And what work is "arbitrariness" doing here? Of course, you could argue (as many people do) that any claim that "ungrounded" sentences like the Liar (or its non-paradoxical twin, the Truth-Teller) is either true or false is in some sense arbitrary, but if that line of thought gets you to the result that such sentences are "neither true nor false," you run into familiar problems with Strengthened Liar sentences, like:

$ The sentence marked with the dollar sign is not true.

...and the problem for the orthodox logical/metaphysical position on entities having some properties and not others, and some such combinations of properties being generally impossible, rears its ugly head once more. So, to sum up my objection to Tahko's move for side-stepping the semantic paradoxes, it seems to me that:

(a) Sentences are part of the world,
(b) Different semantic statuses of sentences seem to mutually exclude each other in the same way as e.g. different charges of particles or different colors of visible surfaces, and to equally be the sort of thing that the LNC-as-a-metaphysical-principle-as-defined-by-Tahko would pretty much have to apply to, &
(c) If the problem posed for logical orthodoxy by the Liar has anything to do with either vagueness or arbitrariness, it's far from obvious, and, at first blush, it's not clear how mention of either clarifies why the Liar isn't a problem for the-LNC-as-a-metaphysical-principle.

So, any thoughts out there? Is there a good way of glossing Tahko's discussion that makes his quick dismissal of the Liar more plausible than it sounds to me right now? Let me know what you think.

*Obviously, agreeing with Frege about *that* much is an entirely different matter from agreeing with him that logical truths are "analytic," or are about a metaphysically different sort of reality than other kinds of truths, or anything of the kind. Also, this description of what "logic is concerned with" on the relevant level of analysis shouldn't be taken as ruling out the possibility (unforeseen by Frege, as far as I know) that logical systems can fail to accurately model the way things really are but still be interesting and instrumentally useful tools for formal reasoning about various subjects.

Monday, September 14, 2009

A Quick Thought About Sylvan's Box, Paraconsistency and Truth-Value Gaps

Something occurred to me today and I'm curious to see what people who read this blog think about it.*

(1) Graham Priest has argued (convincingly, from my perspective) in many places that truth-value gaps are impossible. If there is no fact that makes P true, the fact that there is no such fact is sufficient to make it false, and that in any case gaps entail gluts--if a sentence is neither true nor false, then given that falsehood is best understood as truth of negation, by a single instance of the law of double negation, it follows that the sentence is both true and false, so (non-glutty) gaps are incoherent. Of course, there are many places in that extremely rough and condensed summary that a gap theorist could object and defend their position against these considerations, but for the moment, the important point is that it is Priest's position, and, since I don't tend to agree with him about much here, it seems worth noting that we're of the same mind on this.

(2) Priest once wrote a short story called Sylvan's Box in which the plot was explicitly inconsistent. (In it, the late Richard Sylvan (better known as Richard Routley) turns out to have been in a possession of a box that was simultaneously, observably empty and non-empty.) Of course, works of fiction frequently contain inconsistencies (the location of Watson's war wound is a famous stock example), but part of Priest's point in that story was to write something where the inconsistency was unambiguous and explicit, and no one could try to charitably interpret the inconsistency out, or break the story into maximal consistent chunks and look at them individually, or anything of the kind. We have to face up to the fact that there is a flat out contradiction (P & ~P) in the story, and that we are quite capable of (non-trivially) reasoning about it.** Over at Inconsistent Thoughts, Colin put this nicely, by saying that it shows that "we can navigate inconsistent bodies of information without triggering a “psychotic break."***

Now, in so far as this ability-to-maintain-our-sanity is supposed to show that the 'explosion' of inferences derivable from contradictions in classical logic is invalid, and we should instead adopt a weaker, inconsistency-tolerant (paraconsistent) account of the logical consequence relation, I think there's a tension here between that stance and Priest's position on truth-value gaps.

To see what I mean, let's think about a different possible moral you could take from Sylvan's Box (or, for that matter, from Watson's wandering war wound), which is not that P and ~P don't jointly entail any Q, but simply that F(P) and F(~P) don't jointly entail any F(Q). Just as it's false--but true in fiction--that Watson existed, it's true--but false in fiction that explosion is (vacuously) truth-preserving, since, while there are no true contradictions, it is true that works of fiction sometimes portray contradictions as being true. (I've advocated this approach here in the past.) Of course, as a couple of commenters then indicated, there are real questions about the feasibility of developing rigorous formal tools with which to reason about fictional worlds, but let's leave that aside for now, since right now, we're focussed on the narrower project of weighing the respective virtues of the two proposals on the table for how to do so.

Now, at least for anyone who agrees with Priest (and I) about the impossibility of truth-value gaps, I think the following consideration should tilt the balance of things in favor of explosion-fails-in-fiction-but-Sylvan's-box-tells-us-nothing-about-whether-it-really-fails approach:

Fictional worlds are chock-full of (non-glutty) truth-value gaps.

Let's take some obvious examples.

(1) "Sherlock Holmes' brother Mycroft secretly liked to dress up in women's clothing."
(2) "Watson's great-grandfather on his maternal grandmother's side had blue eyes."
(3) "Professor Moriarty once wrote a paper anticipating Russell's Paradox, but he never published it."

Now, since none of these people existed to do any of these things, (1)-(3) are all false, but are they true or false in the world of the stories? It seems to me that, far from our inability to answer these questions being a matter of ignorance, there is absolutely no way that there could be a fact of the matter one way or the other on these issues. The world of these stories is an incomplete one that, far from including information that Mycroft liked to dress in women's clothing *and* that he never did so, fails to include anything on the subject.**** While these issues give us no good reason to doubt [F(P) v ~F(P)] (in each case, I'd say that ~F(P) is true), [F(P) v F(~P)] simply fails to hold. As such, the logical structure of the world of the Holmes stories includes not only (non-trivial) gluts but also (non-glutty) gaps. None of that tells us much of anything about which inferences are valid in the actual case.

*(And yes, I know I promised a follow-up to the post about the metaphysics of change, and yes, I know that it's been a while since I did so, but the thought is fresh and I'm weirdly excited about the point, so this one comes first.)
**(The morals that Priest draws from the story are many and complicated, but for the moment let's leave it at that, since it's at least what many readers take the story to show.)
***(At the time, he asked me a question about Bohr's of the atom to which I never responded, since I forgot about it until now. My punctuality, in general, seems to be sucking this year. In any case, it's the kind of thing that deserves its own post, which I hope to have in the near future.)
****(Part of the reason that the case for the failure of explosion in the world of the Holmes stories is so compelling is that it seems implausible that Watson's inconsistently-placed war wound would entail, e.g. Mycroft's cross-dressing. This appealing feature would be quite lost if we stuck to our guns and started postulating that it was true--and false--in the world because of its incompleteness.)