Thursday, December 31, 2009

Eastern APA (Part II)

On Monday night at the smoker, I finally did get a chance to get into it with Colin a bit about dialetheism, and, coming out of that discussion, a half-formed thought about Curry & contraction should be up on the blog next week.

On Tuesday, I went to see an "Author Meets Critics" session on Paul Redding's book "Analytic Philosophy and the Return of Hegelian Thought." Although I have read a good bit of Hegel in the distant past, the session mostly convinced me that I didn't remember as much about all that as I thought I did. I have flipped through the book a bit, though, and it does contain some interesting material on Hegel's attitude toward contradictions, picking up nuances rolled over both by some latter-day Hegel defenders who don't want to associate him with advocating inconsistencies on the one hand and by Graham Priest, who has made a big deal in a few places of claiming Hegel as one of his predecessors in affirming true contradictions on the other hand.

On Wednesday, I went to Brian Leiter's very interesting and entertaining talk on the foundations of religious liberty. Nussbaum argues in Liberty of Conscience that mere toleration is an insufficient moral basis on which to justify the legal tradition of singling out religion for particular constitutional protection in pretty much every western society. She thinks that we need something more, which is respect. Leiter argues against this on all the obvious grounds, given the epistemic and moral failings that religion tends to be associated with, and as far as that goes, it's hard to find fault with his argument. Where Leiter's argument went wrong in my view came in the last part of the talk where he (a) granted that even toleration is morally culpable given some utilitarian considerations, and sufficient harm caused by the thing being tolerated makes the toleration unhelpful, but (b) there's insufficient evidence that religion does more harm than good, so we don't really need to worry about (a).

Now, I am a card-carrying member of the ACLU and have been for a long time. (The EFF, too.) I'm emphatically in favor of religious freedom and other basic civil liberties. I am, however, pretty skeptical about (b). The problem, after all, about serious religious belief, is that when prophets and mystics think they're receiving direct communication about the nature and wishes of transcendent entities, the very nature of the process, insulated as it is from certain sorts of rational reflection (one doesn't second-guess the will of God), is such that it's inherently unpredictable. When you go up to the top of the mountain and fast and pray until you get a vision, by the time you come back down, you might end up telling your followers that all living things have a profound sense of connection and that we should all be peaceful and loving and compassionate, or you might end up telling them to go forth and kill all the unbelievers. Even if as a contingent fact of the matter, more of the former happens than the latter, the utilitarian calculations involved in continuing to play that particular game of societal russian roulette are deeply unclear to me.

None of which, from my perspective, means that we shouldn't tolerate religion, any more than the increased risk of street crime from respecting the 4th Amendment means that we should set up cameras on every street corner and frisk people coming onto and off of the city bus every day. It does, however, make me think that we should re-examine (a). Utilitarian considerations are probably insufficient to ground religious liberty. We need something a bit more robust and deontological.

Here endeth the sermon.

Next week, expect to see posts on Curry and on the classical re-capture.

Meanwhile, on the subject of religion, you should probably watch this video, which contains important public safety information about the dangers involved in mocking prophets.

Monday, December 28, 2009

Eastern APA (Part I)

Last night, I went to the event for New Waves In Philosophy Of Mathematics, where I got to hear Roy Cook talk about neo-logicism, Augustin Rayo talk about mathematical trivialism, Oystein Linnebo talk about the individuation of natural numbers and Otavio Bueno talk about mathematical fictionalism. I was at the original phil of math conference at the University of Miami the spring before last that the volume came out of (in fact, I had to get up early both days to go to Panera Bread to grab coffee and whatnot for the conference participants), and I indexed the book this last summer, and, yeah, Otavio is my dissertation advisor, so most of what I heard wasn't particularly new to me, but much of was interesting.

Note, in particular, that "trivialism" means something in this context totally unrelated to the sense usually used in my particular subarea--it doesn't mean that everything is true, it means that mathematical statements are just trivially true. Rayo argued (not quite convincingly to my way of thinking, but forcefully and entertainingly) that there are two kinds of mathematical Platonism: the kind that people argue for and the kind that people attack, and that the latter has very little to do with the former. They come apart because the former is non-trivial. In particular, Benaceraff-style worries about our epistemic access to mathematical objects are irrelevant to the former kind, because mathematical objects aren't an extra ingredient of reality above and beyond the more mundane sorts of objects. By way of illustrating this, he said that from the perspective of his sort of trivialist Platonism, to assert that there are no numbers is to commit oneself to a contradiction. After all, if you say that "there are no numbers," that means the same thing as "the number of numbers is zero," which means that at least one number--zero!--does exist, and you've just contradicted yourself.

Now, as an argument against someone who denies the existence of numbers, this seems transparently question-begging--why, after all, should they agree that "there are no Xs" and "the number of Xs is zero" are equivalent?--but my sense from the discussion in the Q & A was that Rayo understands this and wasn't using it in that way, but just giving an illustration of the trivialist Platonist's stance. Compare to an argument about whether Batman is Bruce Wayne or Harvey Dent. If someone thinks Dent is Batman, they will deny that Wayne is Batman, but one would be begging the question against them in a fairly silly way if one argued that to deny that Wayne is Batman is to deny that Batman is Batman and commit themselves to a logical absurdity. (After all, both sides could pull the same trick.) Like I said, Rayo's position on phil of mathematics is pretty distant from mine, but it was an interesting presentation of his view, and he concluded with some good remarks on the notions of "bloated ontology" and the like.

This morning, I went to see the "Author Meets Critics" session on Hartry Field's book "Saving Truth From Paradox." Some interesting points came out of the exchange, even if as a matter of personal taste I would have liked a bit more back-and-forth on bigger-picture philosophical issues about the book--Is Field's basic approach to the paradoxes on the right track? Is denying instances of the Law of the Excluded Middle sufficiently less counter-intuitive than accepting counter-examples to the Law of Non-Contradiction to make the considerable deficit in simplicity between dialetheism and Field's paracomplete approach worthwhile? Does his "algebraic" picture really capture the intuitive notion of truth?, etc., etc.--rather than quite so much focus on the intimate technical details. That said, some philosophical interesting points came through, particularly in the Q&A, and I finally got a chance to meet the man, who's the external reader for my dissertation committee but who I hadn't actually met in person before today.

....and, speaking of meeting people, I finally got a chance to meet Colin Caret--the guy who does the Inconsistent Thoughts blog and a regular commenter here. I guess I really should have seized the opportunity to argue about dialetheism. Instead, I ended up just exchanging a few pleasantries about the conference and whatnot then going off to (in theory) grab coffee with him and J.C. Beall and a few others, but actually just let Colin buy me a cup of coffee while I boorishly isolated myself at a corner of the table to continue a discussion with Otavio about Philip K. Dick and quantum logic.

Then came lunch with Jody Azzouni and Mark Colyvan, who I got to bore at length with exactly why I don't think Graham Priest's "classical re-capture" works. Then I ducked bag to my hotel in Koreatown to print some stuff out, put some stuff away, and blog.*

More on Wednesday.




*Just to be a complete jackass, I was sorely tempted to do a couple of Tristam Shandy-style stabs at descriptive completeness there, like "Then I wrote that sentence. Then that one...." But I just wouldn't feel right, gentle reader, wasting your time like that.

Thursday, December 24, 2009

This Is Not A Post

Not really, anyway. It's just an apology for the third consecutive week of not being great with the schedule and mostly posting things that are a bit slim on actual content when I do update, and a promise that the regular schedule will resume next week. I'll be at the Eastern APA, so I probably won't post anything too long and involved, but there should be at least somewhat substantive posts on both Monday and Wednesday.

Meanwhile, Merry Christmas, a (belated) Happy Hannukah, and, of course, a Joyous Smugmas*!



*Smugmas is the winter holiday I celebrate. It's all about sitting around the fireplace with your grad school friends, sipping a glass of good single malt and reminding each other of all the reasons why all the traditional defenses against the problem of evil fail, occasionally pausing to exchange high-fives or fist bumps.** Also, of course, occasionally gathering around the piano to sing old Smugmas favorites like "The Twelve Days Of Demonstrating The Incoherence Of Theism," "Rudolf The Red-Nosed Carnap," and "We're Smarter Than Everyone Else, Give Us Presents."

**The occasional grinch will ruin the general atmosphere of holiday cheer for everyone else by pointing out that the zombie theodicy actually holds up pretty well.

Monday, December 21, 2009

He'll Lure You In...

This is just awesome.

I mean, I disagree about analyticity, obviously, but I still love it.

The song is from The (21st Century) Mondads. Here's a bunch of their other stuff. They also provide the complete lyrics to the Quine song:

He’ll play upon
Your naturalistic intuitions
He’ll lure you in
With the promise of positions that you love
Like realism about numbers and sets
And the view that philosophy’s continuous with science.

And then you’re caught in the web
No support from the a priori
All aboard Neurath’s ship
It will all be fine
But you’re tied to the mast
It’s all a posteriori
Here on in; have a blast
With Willard Van Orman Quine.

He looks so good
Next to Carnap, Ayer and Chisholm
And there’s nothing wrong
With confirmational holism;
It’s quite nice as an account of how theories face the tribunal of
experience.
Mmm yeah.

But step away when he starts
Talking about the analytic.
Meaning’s not that strange in being tricky to define.
And it comes in small bits;
Not all holism’s terrific.
Keep your head; keep your wits
Round Willard Van Orman Quine.

Thursday, December 10, 2009

Language and Time (and Space)

Sorry about the delay. I've been pretty good about maintaining the every-Monday-and-Wednesday schedule since early October, but what with the end of the semester, grading, packing for an extended trip back to Michigan before the Eastern APA, etc., etc., etc., these things happen. Better late than never, though, so here's a post....

In Quentin Smith's book Language and Time, he argues for a moderate version of the A-Theory. Some background:

The B-Theory of Time holds that "now" is an indexical like "here," serving only to locate the speaker in time. The A-Theory is all about "the reality of temporal becoming," which basically means that there's a fact of the matter about which moment uniquely counts as "the present." The best argument for the B-Theory is that it's hard to square the admittedly compelling, common-sensical intuitions about time encoded in the A-Theory with the empirical deliverances of Einstein's Special Theory of Relativity (STR). The best argument for the A-Theory (beyond whatever immediate intuitive pull it might have) is that ordinary language sentences are tensed--that is to say, they seem to more or less constantly refer to properties of pastness, presentness and futurity--and that if we don't believe in the real objective existence of those properties, we have to either say that all tensed sentences of English and other languages are false (including, for example, the sentences physicists use to express the STR) or we have to come up with a way to paraphrase the sentences (or at least the ones we take to be true) in such a way that they no longer refer to pastness, presentness and futurity.

As it turns out, the project of coming up with such tenseless paraphrases is pretty hard. Smith is an A-Theorist, and he has a lot of fun pointing out the holes in various extant proposals to do this sort of thing. For example, the tenseless date theory says that we can translate "the meeting starts now," tokened at three o'clock as "the start of the meeting (is) at three o'clock." (The parentheses indicate that the (is) is tenseless.) So far, so good. But what, he asks, about "it is now three o'clock"? Apply the tenseless date theory, and we get "it (is) three o'clock at three o'clock." No good. For one thing, "it (is) three o'clock at three o'clock" is a tautology, and "it is now three o'clock" really really shouldn't come out that way. You need to check your cell phone* to see if "it is now three o'clock" is true, but you can be justifiably confident about "it (is) three o'clock at three o'clock" even if you just woke up in a doorless, windowless room with no memory of how you got there and your phone is out of power.

The tenseless token-reflexive theory does better with this example. It says that we should interpret "the meeting starts now" as "the start of the meeting (is) simultaneous with this utterance," which works just as well for sentences like "it is now three o'clock."

Smith, however, thinks that his counter-examples to this are just as good. For example, consider "it is true that it was true that the era devoid of linguistic utterances is present." Isn't this true? I mean, surely, there is language now, and there wasn't in the past. It isn't true now that there isn't any language, but it used to be. Right?

Well....one issue is that part of what's traditionally at stake between A-Theorists and B-Theorists is that the former take propositions to change truth-values over time (this is the point of developing tense logics) whereas the latter take them to have eternal unchanging truth-values, so Smith is at the very least dancing on the edge of begging the question here.**

Another issue comes to the fore when we start think about what the "it" is that was supposed to be once true and now false. It's certainly not a linguistic utterance like a sentence, since if so it--i.e. a sentence declaring the absence of sentences from the world--would never have been true, since such a sentence could never have been true, for obvious reasons. In fact, it looks like "it" has to be a proposition, and in fact a proposition that existed despite the non-existence of sentences. That is to say, Smith's move here strongly suggests a fairly extreme version of the proposition theory of truth-bearers, whereby we believe not only in the propositions expressed by sentences, but a sort of Platonic realm of un-instantiated propositions as well.

Now, there may be a lot to be said in favor of proposition theory--see the discussion with Emil on this blog a while back--but there's surely a lot to be said against it as well (e.g. if propositions are abstract objects we can't causally interact with, how would we ever find out that they existed?), and if the price of the best philosophical account of how to make sense of the picture of reality suggested by taking our best current science seriously is that we have to abandon belief in propositions, that strikes me as a price worth paying. But someone who starts from a posture more sympathetic to propositions, presentism or both than mine might not be convinced by this. As such, I'll take a stab at pointing out the problem that originally sparked my own slow move away from the A-Theory when I was an MA student at Western Michigan (where I took several classes from Smith, and spent a lot of time thinking about these issues):

On p. 129 of "Language and Time," Smith says that it would be an "interesting task" to provide an account of indexical terms like "I" and "here" in light of his account of "now," but that "this task falls beyond the purview of the present treatise."

Even when I was an A-Theorist***, this passage always struck me as extremely unsatisfying. The more I thought about it, the more dissatisfied I was. Note, for example, that Smith's criticisms of the date theory all go through just as well for what we can call the "place theory" of spatial indexicals. Imagine the following fairly banal conversation:

P1: "I just got here?"
P2: "Where's 'here'?"
P1: "Oh, sorry, 'here' is Miami."

We might be tempted to translate "I just got here," tokened in Miami, to "I just got to Miami." That's pretty clearly not going to work for reasons that exactly parallel the problem with temporal indexicals. "Miami is here" is only true when tokened in Miami but "Miami is in Miami" is true even if tokened in Hong Kong. The best way to go here looks very much like a token-reflexive view where "Miami is here" means "Miami is co-located with the person saying this" or something roughly along those lines. I feel absolutely no inclination to beieve that claims change their truth-value from location to location. Moreover, I feel even less inclination to think that there's some sort of property of "hereness" independently objectively existing in places where no speaker is located. As such, consideration of the sentence "it is true on Mars that 'Mars is here'" gives me no reason whatsoever to believe in propositions that have a special truth-value on Mars that they don't have on Earth, or to give up on the token-reflexive view for the sake of such strange Martian propositions.

Why, exactly, should "nowness" be any different in this respect? If it feels different, is there anything motivating that feeling other than residual folk belief in the claim that there is an ontologically privileged present moment, which is, after all, precisely the bone of contention here? If so, I'm having troubling seeing what.







*Ten years ago, you would have checked whatever those things were that people used to wear on their wrists. I forget.
**I'm putting things in terms of propositions here because that's the traditional way to frame it, not because I necessarily believe in propositions.
***We all get to have a few youthful indiscretions. Hell, just between you and me, there was even a time in my life when I was a libertarian about free will. Don't judge me. I was just a kid, and everyone else was doing it.

Monday, December 7, 2009

Quote of the Day

So, re-reading one of my favorite novels, I found this surprising foray into technical philosophy buried in a discussion of Wagner and Christianity....

"Parsifal is one of those corkscrew artifacts of culture in which you get the subjective sense that you've learned something from it, something valuable or even priceless; but on closer inspection you suddenly begin to scratch your head and say, 'Wait a minute. This makes no sense.' I can see Richard Wagner standing at the gates of heaven. 'You have to let me in,' he says, 'I wrote Parsifal. It has to do with the Grail, Christ, suffering, pity and healing. Right?' And they answer, 'Well, we read it and it makes no sense.' SLAM. Wagner is right and so are they. It's another Chinese finger trap.

"Or perhaps I'm missing the point. What we have here is a Zen paradox. That which makes no sense makes the most sense. I am being caught in a sin of the highest magnitude: using Aristotelian two-value logic. 'A thing is either A or not-A.' (The Law of the Excluded Middle.) Everybody knows that Aristotelian two-value logic is fucked."

--VALIS by Philip K. Dick, pp. 132-133

....to which I say....

"Huh."

Wednesday, December 2, 2009

The problem is U trying to fit set theory into ur mathematical box.

Naive set theory says that for every property, there's a set of things that have that property. Bertrand Russell asked, "what about the property of being a set that's not a member of itself?"

This is, traditionally, seen as a bit of a problem for naive set theory.

Classical theism says that an entity exists who can perform any action. Many people have asked, "what about the action of creating a stone so heavy that he himself can't lift it?"

This sort of thing used to be a bit of a problem for classical theism.

That was before Rick Warren solved all of these problems forever.

Check it out.

Monday, November 30, 2009

Why Truth-Functionality and the (Non-Epistemic) Analytic/Synthetic Distinction Make Strange Bedfellows

Discussions of "the" analytic/synthetic distinction tend to confuse things by using that definite article, but several distinct distinctions have been proposed using those terms. As far as I can tell, most of them are, in one form or another, epistemic distinctions--for example, we can be absolutely certain about analytic truths, whereas even the best-established synthetic truths are still known only in a fallible, probabilistic way (this is the sort of thing that seems to be built into standard Bayesian epistemology), or synthetic truths have to be discovered empirically while analytic truths are ones that we have epistemic access to merely by virtue of knowing the meanings of all of the terms involved (this seems to be about what Boghossian is getting at in his defense of the distinction), or analytic and synthetic statements are (rationally) revised away in different ways, such that synthetic claims can be empirically refuted, but belief-change about analytic matters has to be a change-of-meaning issue (this is the version of the distinction that Grice and Strawsson stamp their feet and insist on in what I can't help but think of as their very aptly-titled article "In Defense Of A Dogma"). Some of these epistemic formulations of the distinction are such that I think even the hard-core Quinean has relatively little quarrel with them, and others are, I think, deeply misguided and can lead to a dogmatic and undeserved sense of certainty about matters logical and mathematical. None of that, however, is anything that I'm going to get into right now.

What I'm interested on touching on instead is the more robust, more-than-just-epistemic sense of analyticity held by those who take the distinction to be about truth-making.* This is the strongest (but most famous) sense of the distinction, where synthetic statements are "made true by virtue of the way the world is," whereas analytic statements are "made true by virtue of the meanings of the terms involved." How to understand the second option is a complicated and confusing issue, and if my personal feeling is that the most sophisticated-sounding explications of it tend to devolve into hand-waving and places where "here a miracle occurs" is written on the conceptual chalkboard, I can happily accept for my present purposes that people who talk that way are talking sense, even if it is a deep and subtle sort of sense that I have troubling grasping.** At the moment, at any rate, I won't be arguing against the details of any such proposal for understanding the claimed truth-making distinction. My target here is much broader than that: I want to suggest a reason to be suspicious of the suggestion that there's any sort of truth-making distinction between "analytic" and "synthetic" truths, at least given a standard story about the truth-functional nature of logical connectives. If I'm not sure that it quite adds up to a full-fledged objection yet, I do think it at least strongly points the way to one.

Here it is.

Let P be a true statement--for the sake of convenience, let's use the old stock example "snow is white." In classical logic, (P v Q) follows from it, regardless of the content of Q. As long as we know that Q is some regular, meaningful, declarative sentence, the kind of thing that can be legitimately taken as the interpretation of a propositional variable, we know that the truth of P guarantees the truth of (P v Q), even though we don't know the specific content of Q, or even whether it is true or false.

Now, fiddle that picture just a little bit, to give us one new piece of information. We still know that P is "snow is white," and that that's true, and we're still in the dark about the specific content of Q, but now we also know that Q is false. At this point, we know not just that (P v Q) is true, but that it's true exclusively because P is true. Remember, whether or not there are any "analytic" statements, made true in a slightly mysterious secondary way, P is a completely banal "synthetic" statement, made true by the actual whiteness of the fluffy white stuff on the ground.

Now, you'd think that at this point in the argument, we know everything there is to know about how (P v Q) becomes true. Given that we know that it's false, we know that Q does no work in the process, and we know that P is "made true by the world" in the boringly normal fashion.

But wait. Let's consider two scenarios. For the sake of simplicity, in both options, Q is a (false) statement about snow.***

(1) Q is "snow is green."

(2) Q is "snow is not white." (Q=~P)

Now, in scenario (1), (P v Q) is a "synthetic" statement, made true by the world through the truth of its first disjunct. In scenario (2), however, (P v Q) is an "analytic" statement, made true in some other way.

So my almost-objection is this: there's something deeply counter-intuitive about the suggestion that, despite the fact that we knew that (P v Q) was true before we knew whether Q was even a true statement or a false one, despite the fact that we knew that Q wasn't going to be doing any work in making (P v Q) true before we even knew what it's specific content was, and despite the fact that P itself isn't made true in different ways in scenarios (1) and (2), (P v Q) arrives at truth in fundamentally different ways depending on the specific content of Q.

So the challenge I'd throw to defenders of a (more-than-just-epistemic) analytic/synthetic distinction is approximately this:

If one takes confirmation to work on a case-by-case, statement-by-statement basis, no one would argue with the claim that the story we should tell about how to confirm P is very different from the story we should tell about how to confirm (P v ~P). Whether or not we should take confirmation to work that way is a question for another time. Your claim about truth-making, however, seems to put you in a very strange and awkward position, and it looks like a position you have to good reason to put yourself in.

You admit that P is made true by the way snow is, and that ~P is made false by the same thing. Why not extend that analysis in a straightforward way, given the way that the truth or falsity of disjunctions and conjunctions works as a function of the truth or falsity of their components, to cover the way that (P v ~P) is made true and (P & ~P) is made false? Why, in other words, take anything but the way that snow is to do any work in the story we tell about how complex statements entirely about snow are, or are not, made true?










*I take it that some (but not all) formulations of the epistemic distinction are at least strongly suggested by the truth-making distinction, but again, that's a separate issue. And, of course, the epistemic distinctions can be quite independently motivated, and it's not uncommon to read defenses of (epistemic senses of) the analytic/synthetic distinction by theorists who profess to be unsure about how seriously to take talk of things being made true by virtue of anything but the objects referred to in the sentences being the way the sentence says they are.

**By way of at least gesturing at an example of the sort of thing I'm talking about, an expressivist version of the truth-making distinction holds that the way that analytic statements become true has something to do with the way they express the speaker's commitment to certain vaguely-defined linguistic norms or "rules of use" that have somehow made it into our language. I think that the evidence for the existence of such "rules" is non-existent and that demands for precise explanation of what these "rules" would or could even consist of tend to be met with vague and rather unhelpful analogies. To say that two people who set up a chess board and then move around the pieces in disallowed ways are "violating the rules of chess" is simply to say that their game is inconsistent with a bunch of actual written rules explicitly stipulated and agreed on by a bunch of human beings, and that as such what they're doing doesn't count as an instance of chess. To say that asserting a contradiction amounts to violating "rules of use" or whatever, on the other hand, is to say nothing that even could be remotely similar, both because languages arise not from explicit stipulation and agreement but in a sprawling, unplanned way, and because--assuming a logically orthodox picture of things--contradictions definitely count as instances of language. The classical assumption that contradictions are always false entails that they are meaningful, understandable bits of whatever language they are asserted in.

***We're talking about very, very closely related falsehoods at that, considering that the second statement is actually entailed by the first one, which should underline the strangeness lurking in all of this.

Wednesday, November 25, 2009

No Philosophy Tonight

...on account of how it's been an extremely busy week, between the talk yesterday, job stuff, the holiday weekend, and so on, and I'm feeling pretty drained. I promise that there'll be a substantive post on Monday to make up for this almost content-free week.

So I'll just say happy Thanksgiving and leave it at that.*




*I think you can either get it from the website or just go to iTunes and search for "Diet Soap." It's the most recent episode (#33). If you don't feel like downloading the podcast, or you prefer reading fiction to listening to it, the story was originally published in Diet Soap's on-line edition last year, and you can still read it there.

Monday, November 23, 2009

Talk Tomorrow

Philosophy Colloquium
Ben Burgis
University of Miami

“Semantic Dialetheism & Philosophical Methodology”
Tuesday, November 24, 2009
University Center, Room 245
3:30 pm - 5:30 pm

Dialetheism is the position in the philosophy of logic according to which there are true statements of the form (P & ~P). Graham Priest has argued for it on the basis of the Liar and related semantic paradoxes, the antinomies (and intuitive appeal) of naïve set theory, the paradoxes of motion and change, and so on. Priest often argues that all consistent solutions to these problems necessarily purchase consistency at the expense of expressive power. This view, according to which any complete and accurate description of reality must involves contradictions, has been called "metaphysical dialetheism." By contrast, Edwin Mares has argued for "semantic dialetheism," the view that, even if there's nothing about extra-linguistic reality such that a complete and accurate description of it must be inconsistent, it is a contingent fact about natural languages that some predicates are "overdefined," meaning that the conditions for a a predicate mapping onto some bit of reality and the conditions for its negation mapping onto it can sometimes overlap. This view is a natural extension of the claim made by figures like Scott Soames and Jamie Tappenden that some predicates are "partially defined," and in many ways it may seem less counter-intuitive than metaphysical dialetheism. However, given the consequences of this view for the normal process of philosophical argumentation, we will see that there are principled reasons to reject it than don't beg the question against the semantic dialetheism by assuming the very principles in dispute.

Wednesday, November 18, 2009

Kirkegaard On The Sentence Theory Of Truth-Bearers

[UPDATE: Emil responds here. I'll just note that by "noneism" I mean the position usually called "Meinongianism," which is simply the view that some objects have the property of non-existence. The term was coined, I think, by Richard Routley in "Exploring Meinong's Jungle," and also used by Graham Priest to describe his version of the thesis. I guess I probably just should have said "Meingongianism." The only reason I didn't was that I wanted to make clear that I was talking about the general claim, not Meingong's specific version of it.]

Emil Kirkegaard has been posting about truth-bearer theory over at Clear Language, Clear Mind. In this post, he mentions dialetheism.

He's been arguing for a theory according to which propositions are the primary bearers of truth. In the post in quest, he objects to sentence theory on the basis of ambiguity, suggesting (if I'm understanding his argument correctly) that since sentences can be ambiguous, if they are the primary bearers of truth, truth itself must be ambiguous and we have to give up on classical logic.

Here's his example:

*****************

Consider the sentence:

S. It is logically possible that I exist and that I do not exist.

Is (S) true or false? I can’t tell because it is ambiguous. If you don’t see how it is ambiguous try deciding whether the predicate “It is logically possible” applies to only “I exist” or to both “I exist” and to “I do not exist”. Which is it? Logic helps us see the difference. We may formalize the two interpretations like this:

1. ◊Ei∧◊¬Ei
2. ◊(Ei∧¬Ei)

(Where “Ex” means x exists, “i” means I.)

We can translate these into english-ish:

1*. It is logically possible that I exist and it is logically possible that I do not exist.

2*. It is logically possible that (I exist and that I do not exist).

*****************

His line of thought seems to be that if this is just an ambiguity about which proposition the sentence expresses, and propositions are the things that are ultimately true or false, then what we could call the "Classical Principle" (Bivalence + Monaletheism) still holds--"every proposition is either true or false, but not both." If, on the other hand, the sentence itself is what's true or false, the "Classical Principle" formulated for sentences--"every sentence is either true or false, but not both" doesn't seem to hold. He speculates that perhaps we could say that S is both true and false, or that it's neither true nor false. Either way, the Classical Principle (and, given some basic assumptions about the relationship between falsehood and negation, etc.) classical logic itself, seems to be out the window.

I've got some further thoughts about this, but this post is going to be a bit hurried, so I'll just stick with making three quick points about this example. None of these really touch on the central issue--are sentences the bearers of truth?--but I'd be happy to get in to that in the comments if anyone's interested. Meanwhile, here's what I've got:

(1) The indexical phrasing might make things a bit confusing in this specific case. On one level, it's surely contingent that Ben Burgis exists, but one might argue that it's logically impossible that any instance of "I exist" tokened by anyone could ever be false. What one thinks about what to ultimately make of this might depend on what one thinks about the widely alleged essentialness of indexical claims--if "I exist" really *means* Ben Burgis exists, that's one thing, but given that I could forget that I'm Ben Burgis but still be quite sure that I exist, there are tricky issues at play here.

(2) Another complicating factor about the example is that existence is being treated as a predicate, which seems to assume "noneism," the view that there are objects that have some properties (like being referred to) but which don't exist. Anyone who agrees with Quine's claim in "On What There Is?" that the answer to the question of ontology ("what exists?") is "everything" would, while agreeing that it's possible for there to be no object that Ben-Burgisizes, strong object to ◊¬Ei.

(3) Where "I exist" is interpreted as "Ben Burgis exists," and we're assuming noneism, the "both" option seems very strange. After all, the "false" part comes from the alleged logical impossibility of my both existing and not existing, and anyone who thought that "both" is a conceptual possibility (i.e. who was a dialetheist) would pretty much by definition think that it was at least *logically* possible for me to both exist and not exist, even if as a matter of fact this never happens. Or maybe not...any dialetheists out there with specific reasons for thinking that contradictions about *existence* are logically impossible?

Monday, November 16, 2009

How Priest Begs The Question Against The Trivialist

[The following are excerpts from the portion of the rough draft of my dissertation where I discuss the Curry Paradox. Note that by "Liar-based dialetheists," I just mean any dialethest who takes Liar sentences to be both true and false. The numbering system is retained from the original.]

...one could argue that it’s impossible to beg a question against a trivialist, because the trivialist accepts everything, necessarily including the wrongness of trivialism. This would, however, be entirely too hasty. The epistemic reasons that begging the question is barred from rational debate don’t cease to apply in the strange case of our opponents cheerfully granting whatever is throw at them. Priest himself has some very sensible things to say about precisely this subject in Doubt Truth To Be A Liar, and when looking at how he fares on these grounds, it will be sufficient to compare his actual performance with the standards he himself lays out in that book. What he asks us to do in that book is to imagine, when arguing against the trivialist, that we are not trying to convince the trivialist (who, after all, is already convinced of everything) but that we are trying to convince a neutral arbitrator who has cautiously not yet made up her mind about the issue of trivialism, and who will accept or reject arguments and evidence brought up one way or the other, according to normal rules applicable to any inquiry about anything. Convincing the trivialist that trivialism is wrong is a trivial matter. Convincing the imaginary arbitrator to rule in our favor is not. It is the latter, harder standard to which we should set ourselves.

"Convincing the arbitrator is…not as easy as it might first appear… For example, an obvious argument to use is that the trivialist’s position is inconsistent, and so ought not to be believed. (Naturally, a dialetheist is not going to make this objection; but a classical logician might.) It is clear that this argument fails, though, since it begs the question. The trivialist affirms…all contradictions. Simply to claim that these cannot be true is to take for granted part of what is at issue. "

Things are not so hopeless as they may appear, however. In fact, there is excellent reason to be confident that a nontrivialist with normal beliefs about the world could make a lot of good, convincing anti-trivialist arguments of the kind that the hypothetical neutral arbitrator would allow.

"The trivialist believes many strange things. They believe, for example, that you are a scrambled egg. This is objectionable since there is just no evidence to justify it. The rational person should not believe anything for which there is no reason. The arbitrator must agree. And if the trivialist argues that there is a reason, namely that it follows from trivialism, then it is they who now beg the question. The trivialist will also claim that there is independent evidence. (They will claim everything.) But claiming does not of course make it so. The evidence needs to be produced for the arbitrator…."

This should be enough to get across Priest’s general idea about how to argue against trivialism, and it is an idea that seems absolutely correct to me. Moreover, the anti-trivialist argument just quoted seems to be a good one, and many similarly good arguments can be made. Before concluding that we have a good, principled reason to reject trivialism, however, we must not just consider the case against trivialism but the case for it as well.

Just as the best argument for dialetheism is the Liar Paradox, the best argument for trivialism is the Curry Paradox. Liars say of themselves that they are false, or untrue, or (just) untrue, or something else of the same general kind. Curry sentences, by contrast, say of themselves that if they are true, some α is true as well, where α can be any arbitrarily chosen sentence that you like. For example, take sentence (45).

(45) If the sentence marked as (45) is true, then Graham Priest is a scrambled egg.

Symbolize (45) as α. The relevant instance of the biconditional truth schema for (45) is “‘If the sentence marked as (45) is true, then Graham Priest is a scrambled egg’ iff if the sentence marked as (45) is true, then Graham Priest is a scrambled egg.” If we name the sentence α, and the claim that Graham Priest is a scrambled egg β, then we can symbolize this as:

1. Tr<α> ↔ (Tr<α> →β)
Plugging in the definition of ↔, we get:
2. [Tr<α> → (Tr<α> → β)] & [(Tr<α> → β) → Tr<α>]
By Conjunction-Elimination, we get:
3. Tr<α> → (Tr<α> → β)
Now, from here, an easy conditional proof, using only Modus Ponens (MP), will get us to Tr<α> → β.
4. Tr<α> A for CP
5. Tr<α> → β 3, 4, MP
6. β 4, 5, MP
7. Tr<α> → β 4-6, CP
Now, applying Conjunction-Elimination once again to 2, we get:
8. (Tr<α> → β) → Tr<α>
From 7, 8 and Modus Ponens, we get:
9. Tr<α>
Now, from 7, 9 and Modus Ponens, we get:
10. β

Graham Priest is indeed a scrambled egg. In fact, since Curry sentences can be constructed with any arbitrary consequent that you care to pick, we can use this argument to prove all of the trivialist’s claims without exception, or just cut to the chase by having the consequent be “trivialism is true.”

This seems to be as good an argument for trivialism as the argument from the Liar Paradox is for dialetheism. In the latter case, in Chapter Three, we considered Gilbert Harman’s attempt to block the derivation of inconsistency by a rejection of the biconditional truth schema and rejected it as question-begging. Priest’s hypothetical neutral arbitrator would surely rule out a parallel rejection of the relevant schema instance in order to block the derivation of triviality for precisely the same reason. Similarly with any otherwise unmotivated tinkering with the formal machinery we used to derive triviality from Curry.

[....In the skipped section, I basically note that any consistent solution to the Liar worth its salt can be re-purposed as a non-trivial solution to Curry. Both Liar and Curry sentences, after all, purport to discuss their own truth-values. The two paradoxes are of a type.]

By contrast, the Liar-based dialetheist is in a tough spot here. They are clearly committed to the claim that self-referential truth talk is meaningful, and, in fact, truth-evaluable. Curry sentences must, then, be meaningful declarative sentences, as would the relevant instances of the biconditional truth schema, and no one who admits this can have a good, principled, nonadhoc and non-question-begging reason to avoid granting that these schema instances give the correct truth conditions for Curry sentences. Once that point has been granted, all we need is Modus Ponens and conditionalization (the rule that we can use conditional proofs to prove the truth of conditionals). Rejecting Modus Ponens is extremely implausible. In fact, no one is more adamant about this point than Priest, who says that,

"Any conditional worth its salt, →, should satisfy the modus ponens principle…This is, indeed, analytically part of what implication is."

Now, whatever one thinks of the analytic/synthetic distinction, or the complicated issue of the relationship between the meaning of logical constants and the rules governing their behavior that we looked at in Chapter One, Priest’s basic point in this passage is a good one, which we can re-phrase in weaker terms that are neutral on those other debates by saying that if someone denies that α, α→β ┠ β is a truth-preserving inference, this counts as pretty good evidence that they don’t mean the same thing by → that the rest of us do. We can extend (this charitably weakened version of) Priest’s point by saying that if someone denied conditionalization, that would be equally good evidence that they didn’t mean the same thing by → as the rest of us. The two rules are conceptually parallel to each other, a fact recognized by introductory logic texts that introduce conditionalization as “→-introduction” and Modus Ponens as “→-elimination.” In one case, the fact that α implies β is enough to guarantee that it entails it, and in the other case, the fact that α entails β is enough to guarantee that it implies it. This mutual relationship between entailment and implication seems obvious enough that no one thinks much of anything of talking about the “rule form” and the “conditional form” of various logical rules, and it’s not at all clear to me why a denial of one half of this relationship should count as better evidence than the denial of the other for the claim that the denier doesn’t mean what the rest of us mean by →.

Denying either of these rules in order to escape the inference from Curry sentences seems like a desperate, adhoc maneuver. This is, however, precisely the route that Priest takes. In In Contradiction, he rejects ‘absorption’ (also called ‘contraction’)—the inference from [α → (α → β)] to (α → β)—on the grounds that “entailment must not fall foul of Curry paradoxes.” Absorption, however, just is conditionalization plus Modus Ponens, so, given Priest’s endorsement of Modus Ponens, his rejection of absorption boils down to a rejection of conditionalization.

He just barely gets to keep Modus Ponens itself (in its rule form), but his anti-Curry reasoning has the further unattractive consequence that he must reject the conditional form of Modus Ponens--[α ∧ (α → β)] → β], sometimes called ‘assertion’--which, like absorption, is the product of accepting the validity of both Modus Ponens and conditionalization. Priest lays out a view about implication that avoid all of this, but the important point for the present discussion is that he justifies this theory by the need to bring about these counter-intuitive consequences, and that he justifies that with the need to get around Curry.

"It must be admitted that the assertion principle looks acceptable enough, but…a criterion of adequacy for a solution to the problem of formulating an account of implication is that it must not validate assertion (or absorption)."

Priest, in other words, is tinkering with otherwise intuitively plausible rules of inference in order to avoid being committed to triviality. If this is the best he can do when the trivialist brings her argument from the Curry Paradox to the neutral arbitrator, then the arbitrator will have no choice but to rule out this objection on the grounds of flagrant question-begging and, ultimately, rule in favor of the trivialist against the nontrivialist dialetheist.

Wednesday, November 11, 2009

A Simple Argument From Theism And Truth-Functionality To Bivalence

Classical logicians hold that statements can relate to truth in exactly two ways--'T' and 'F.' On the other hand, various deviant logics embody the assumption that the range of possible truth-values is wider than this. Perhaps "neither" or "both" is an option, or perhaps statements sometimes have a third truth-value that isn't best thought of us either the joint absence or the joint presence of the two traditional options, but as something else entirely. Maybe, when we figure all of this out, there are really five truth-values, or twelve, or....

Actually, no. Don't worry, gentle reader, I've come up with a devastatingly convincing deductive proof that this is not the case, that there are in fact two and only two real truth-values. In terms of logical machinery, this most excellent of all proofs relies on Modus Ponens alone, which is part of the overlapping consensus between all sorts of different logics, so no questions are begged. (Indeed, I often read claims in the literature that one of the conditions for a connective → in some system "counting" as actually being "a conditional" is that it satisfies Modus Ponens. This is terribly convenient for my purposes, so let's put aside any nit-picking issues about whether this is entirely reasonable given some solutions to various paradoxes involving implication and so on, and just assume that this claim is exactly right as stated.) As far as substantive assumptions, I'll assume truth-functionality--the truth-value of complex formulas is a function of the truth-values of their atomic components--which seems plausible enough, regardless of what you take the alethic options to be. Even less controversially, I'll assume that God exists and is just. Which is, like, obvious.

I mean, seriously, haven't you ever seen a beautiful sunset? Or a cute puppy? You have? Well, how do you explain those things without postulating an omnipotent and omnibenevolent being that created all things? Yeah, that's what I thought. You can't.

What? Some of you still aren't convinced? Really? For God's sake, haven't you people read any Plantinga? This could not be more straightforward. If I believe that God exists on the basis of the sunsets-and-puppies-argument, then I have every reason to be confident that my divinely-designed cognitive faculties got the job done and I'm not missing any possible objections. On the other hand, if you believe that God does't exist, and as such that your cognitive faculties arose naturalistically without any sort of benevolent entity supernaturally intervening to make sure they were reliable, then you can prattle on all you like about your "evidence" and your "arguments" against theism, and those of us who know we were created by God can just point and laugh. "Look at the evolved thing trying to come up with arguments with its puny little monkey brain! Haha!" And so on.

I'm sorry I had to slap you down like that, atheist readers, but I get so impatient with your failure to recognize the obvious truth of theism. So, that's enough of that, eh? We can assume the existence of a gloriously powerful and benevolent creator deity and go from there?

Good.

Now, assume there are only two truth-values. That means that for any formula with two propositional variables in it, the truth table will be four lines long. For example, take a simple truth table for conjunction.

α ∧ β
T T T
T F F
F F T
F F F

Nice, simple, clean little truth table there. Now, add another propositional variable.

(α ∧ β) ∧ ∂
T T T T T
T T T F F
T F F F T
T F F F F
F F T F T
F F T F F
F F F F T
F F F F F

Still only eight lines long. Takes about twenty seconds to write it all out long-hand. Pretty straightforward.

Now, assume that there's even one more possible truth-value. Since this argument should apply to any and all many-valued proposals, it doesn't matter exactly how we understand this third option--gappy, glutty, undecided, partially true and partially false, something else entirely--so we'll just write it as "O" for "other." Since "O" could be all sorts of things, and different ways of filling it in could have different consequences for how it impacts the truth-values of larger formulas O-valued statements enter into, and we want to be absolutely general here, we'll err on the side of extreme caution and just write down a ? for every formula with an O-valued component. So, here's a 3-valued version of the same truth table we just did.

(α ∧ β) ∧ ∂
T T T T T
T T T ? O
T T T F F
T ? O ? T
T ? O ? O
T ? O ? F
T F F F T
T F F ? O
T F F F F
O ? T ? T
O ? T ? O
O ? T ? F
O ? O ? T
O ? O ? O
O ? O ? F
O ? F ? T
O ? F ? O
O ? F ? F
F F T F T
F F T ? O
F F T F F
F ? O ? T
F ? O ? O
F ? O ? F
F F F F T
F F F ? O
F F F F F

Now, you want to know what the difference is between this truth table and previous one? They were for the same formula, but that second one was a huge pain in the ass to write up. And that's even typing it up, when you can copy and paste chunks and then go back to change bits. Imagine being a student in an introductory symbolic logic class in a world where some three-valued logic had replaced classical logic as the orthodox, establishment choice taught to beginning students with Hurley-type textbooks. You're given a quiz, and for something as simple as a fucking three-way conjunction, you have to write out that whole thing, long hand, and keep track of it all?

Yikes.

So, I submit the following obvious truth, which we can call the Principle of Divine Justice:

PDJ: A just God would arrange the universe in such a way that people could accurately represent its logical structure without going through the hassle of writing twenty-seven line truth tables for simple three-variable statements.

Now, since we know that God is just, it follows that God has arranged reality in such a way that the only ways that statements can be are "true" and "false." There are no other options.

QED.

Monday, November 9, 2009

Links

(1) My post on Probability Theory and Logical Fallibilism made it into the Carnival of Mathematics. I have to admit that I'd never actually heard of it before this month's host e-mailed me to ask if he include my post, but I guess it's the mathematical equivalent of the Philosopher's Carnival. Anyway, I'm tickled as hell to be included.

"The Law of the Excluded Middle states that if a mathematical proposition is not true, it’s false. Mathematicians have experimented with dropping this requirement; Ben Burgis considers the ramifications for probability."

Of course, strictly speaking, my post was mostly about the ramifications for probability of being a fallibilist about the LEM and similar laws, not dropping them entirely (although I did briefly touch on the latter), but that extremely trivial nit-pick aside, I'm happy to do my bit for cross-disciplinary communication about these issues.

(2) When I'm not here, I also blog (along with co-contributors Dave, Matt, Bryan and Andrew) at Awkward Haiku, where we boldly explore the frontiers of the post-ironic world we live in. My most recent post there is in large part about philosophy (it includes an extended discussion of Judith Jarvis Thompson, a link to Chaospet, and so on), so I guess it's worth linking to it here.

Abort A Fetus For Jesus! (Why Abortion Should Be Safe, Legal and Extremely Common)

Wednesday, November 4, 2009

What The Explosion Proof Isn't

The following is an extract from the first chapter of the rough draft of my dissertation.

#

In classical logic, anything and everything follows from any contradiction. To make things concrete, take the following example. If we start by assuming that the Russell Set (R={x : x ∉ x}) is both a member of itself and not a member of itself, we can, in a few easy steps, derive the result that John McCain is a 400-year-old vampire. After all, if the Russell Set is both a member of itself and not a member of itself, then by conjunction-elimination we derive the consequence that the Russell Set is a member of itself. From there, by Disjunction-Addition, we conclude that either the Russell Set is a member of itself or John McCain is a 400-year-old vampire. From a second application of conjunction-elimination to the first premise, we conclude that the Russell Set is not a member of itself. From the previous two results and Disjunctive Syllogism, we get the startling result that John McCain is indeed a 400-year-old vampire.

Formally,

1. α ∧ ¬α A
2. α 1, Conjunction-Elimination
3. α ∨ β 2, Disjunction-Addition
4. ¬α 1, Conjunction-Elimination
5. β 3,4, Disjunctive Sylloigsm

Now, this proof is a familiar one that, in one form or another, most introductory logic students are exposed to in their first course, but it is odd in light of that that it is terribly unclear what sort of proof it is supposed to be. It is, obviously, not supposed to be the sort of proof that functions to convince anyone of the conclusion, since the typical logic instructor showing it to her students does not herself believe that any statement of the form (α ∧ ¬α) is true, and she believes still less that any and every possible β is true. It is often referred to as a reductio proof, but strictly speaking, it is nothing of the sort. A reductio proof that the first premise is not true would be spectacularly short and uninteresting:

1. α ∧ ¬α A for RAA
2. ¬(α ∧ ¬α) 1, RAA

Perhaps it is a conditional proof, to get us the result that (α ∧ ¬α)→β. If so, again, it is a bit like using a sledgehammer to crack a nut. The falsehood of (α ∧ ¬α) is alone sufficient to guarantee the truth of that conditional in classical logic.

We are running out of formal options. Turning to the informal ones, we could speculate that it is supposed to show that there are not and could not be any true statements of the form (α ∧ ¬α) because, if any such statement were true, everything would be true, and at least some things pretty clearly fail to be true. Or, more weakly than this counterpossible claim, it could be seen as showing that anyone who believed that some such statement was true would be rationally compelled to believe that absolutely everything is true. The problem is that if it is supposed to be an argument for either of these things, it is a terribly unconvincing one. As an argument against the possibility of true contradictions, it begs the question by employing a rule (Disjunctive Syllogism) that is only universally truth-preserving if there are no true contradictions. After all, if for some α and some β, α is both true and false, whereas β is just false, then (α ∨ β) will be true, ¬α will be true and β will nevertheless fail to be true. For the same reason, someone who believed some statement (α ∧ ¬α) was true would not be rationally compelled to conclude any and every random β, because they would have no rational reason to believe that Disjunctive Syllogism was a valid rule of inference.

Reflection on this point shows that, even from a classical monist perspective, ‘from a contradiction, anything follows’ is no more interesting a logical principle than ‘from Hitler winning World War II, anything follows,’ which could be proven as follows. ‘If Hitler won World War II, then β’ is true for any and every β, because all conditionals with false antecedents come out as true in classical logic. Now, assume that Hitler won World War II. By Modus Ponens, we are able to conclude any and every random β.

The obvious objection is that we have switched positions midway through the argument on the subject of whether Hitler won World War II. Quite so. However, this is not a bit different in principle from the way we have switched positions on the possibility of true contradictions when we write down (α ∨ ¬α) as our first premise in the explosion proof, and then proceeded to use Disjunctive Syllogism as if it were a valid rule of inference, which it would not be if the first premise was true.

As such, while the chain of inferences connecting the Russell Set’s membership conditions to John McCain’s age and supernatural status is perfectly valid as far as it goes (if classical logic is right), the answer to what the proof tells us one way or the other about any remotely philosophically interesting subject whatsoever is, I think, ‘not a damned thing.’ It’s valid if monaletheism is correct, and by helping ourselves to explosion in order to argue against dialetheism, we would be putting the cart before the horse and egregiously begging the question. Dialetheism cannot be batted away with the blunt instrument of triviality-avoidance. It needs to be engaged on the arguments.

Monday, November 2, 2009

Nietzsche, Truth and Meaning



I just listened to this extremely interesting interview with Brian Leiter over at Philosophy Bites about "Nietzsche Myths." The "Nietzsche wasn't actually an anti-Semite whose philosophy was all about the notion of the Overman" part was pretty much stuff that I knew, although it was explained nicely, in a concise, entertaining and well-reasoned way. The interesting part to me was the last bit, about the way that folks like Derrida and Foucault tried to claim Nietzsche as a proto-post-modern skeptic about truth and meaning. Leiter argues that this doesn't make a hell of a lot of sense. The one essay in which Nietzsche says post-modern-sounding things about truth was one that he never gave anyone permission to publish, whose themes he never returned to in his later work, and which flatly contradicts and makes nonsense of a great number of the views that he implicitly assumed or even explicitly endorsed in the work that he did think enough of to publish.

Thinking about this point, it kinda occurred to me that...uh....

...come to think of it....

....if one of the main themes of post-modern skepticism about truth and meaning is that texts don't really have any kind of fixed meaning having to do with anything as banal as context or consistency or authorial intent, but that they can be legitimately interpreted in a more or less infinite number of incompatible ways...and the argument between advocates of this view who see Nietzsche as an early-co-thinker and boring "analytic" Nietzsche scholars like Leiter is about what the textually-best-supported reading of Nietzsche's views is...it shouldn't exactly be shocking that they come to different conclusions. In fact, it's amazing that the differences aren't a lot greater.

You'd sort of expect to have a lot of discussions between Post-modern Nietzsche Enthusiasts (PNEs) and Normal Nietzsche Scholars (NNSs) along the lines of:

PNE: In Thus Spoke Zarathustra, Nietzsche makes the bold assertion that President Obama is actually a lizard-like alien creature, a la the classic science fiction television series V.

NNS: What? How is that possible? Nietzche died in 1900, and Barack Obama wasn't even born until 1961.

PNE: Here we see the typical obtuseness of the analytic philosopher who cannot think outside of the arbitrary and culturally-determined narratives of "temporal causation" and...

NNS: Yeah, OK. Whatever. But where are you even getting this?

PNE: From the passage where Zarathustra talks about his admiration for tight-rope-walkers.

NNS: Wait. The passage about tight-rope-walkers was a metaphor for Obama being an alien lizard creature?

PNE: Why privilege that interpretation of what I just said? My statement "From the passage where Zarathustra talks about his admiration for tight-rope-walkers" could just as easily be read as a statement about about the war in Afghanistan, in which the signifiers "Zarathustra" and "tight-rope-walkers" were used to dialogue with Vice President Biden's support for a strategy of relying on Predator drone attacks rather than more troops on the ground.

NNS: OK...uh...is that what you meant? Because it seemed more like we were talking about....

PNE: Are you honestly assuming some restrictive narrative of authorial intent whereby my reading of my statement at the time I said it is privileged over any other?

NNS: Never mind.

Wednesday, October 28, 2009

Standards For Claims Of Retroactive Implicit Paraconsistency

In Mark Colyvan's article "Who's Afraid Of Inconsistent Mathematics?", he starts things off with a snarky "five line proof" of Fermat's Last Theorem.

1. The Russell Set is both a member of itself and not a member of itself.
2. (From 1, and Conjunction-Elimination): The Russell Set is a member of itself.
3. (From 2, and Disjunction-Addition): Either the Russell Set is a member of itself or Fermat's Last Theorem is true.
4. (From 1, and Conjunction-Elimination): The Russell Set is not a member of itself.
5. (From 3, 4 and Disjunctive Syllogism): Fermat's Last Theorem is true.

Now, he argues, the first premise is easily demonstratable given naive set theory. Why is it, then, that no mathematician ever tried to prove FLT in this way, and that FTL was considered un-proved until Andrew Wiley came along with a proof running over a hundred pages and employing all kinds of sophsiticated, recently developed mathematical machinery?

Part of the explanation is that the first premise relies on naive set theory, and mathematical orthodoxy has, in no small part *because* of Russell's Paradox, abandoned the naive conception of sets in favor of the hierchical conception of ZFC set theory. Fair enough, but what about the three decades between the discovery of Russell's Paradox and the consistent re-formulation of set theory in terms of ZFC's cumulative hierarchy? Why didn't anyone try to prove FTL this way during the lag period?

The lesson Colyvan draws is more or less standard for historical examples used in apologias for paraconsistency:

Mathematicians during the lag period implicitly treated the contradiction exposed by Russell's Paradox as entailing some conclusions but not others. In other words, the standards of reasoning in play in the mathematical community of the time are best captured by paraconsistent logic rather than classical logic. Moreover, so goes the story, they were not being irrational in implicitly employing these standards of reasoning. Therefore, paraconsistent logic represents the appropriate logic for at least some domains of inquiry, at least sometimes.

I'm picking on Colyvan's paper because it's an extraordinarily clear, clean, chemically pure example of its type, but this sort of thing is a very common manuever in the literature on paraconsistency. We can call these Retroactive Implicit Paraconsistency (RIP) claims. To make an RIP claim is to say that, in the past, some investigator or community of investigators about some topic reasoned in a way best codified by paraconsistent logic. (Not, of course, that they actually reasoned using formally explicit deductive arguments of any kind, or that they were aware that their practices were, in some, sense in conflict with the norms of classical reasoning--generally speaking, RIP claims are made about people who predate the explicit formulation of paraconsistent logics--but that the implicit standards of reasoning their practices seemed to conform to were ones that fit better with a paraconsistent logical consequence relationship than with one in which anything can be derived from any contradiction.) In most cases, a normative element is at least implicit--do we really want to say that these people *should* have concluded everything from the contradicitons inherent in their theories?--but for the sake of simplicity, let's put that aside and just deal with RIP claimss on a purely descriptive level.

I should have some more to say about this soon, but for now, I just want to note that the standards we use to evaluate RIP claims should be a bit more rigorous than the standards that tend to be appealed to in standard presentations of them.

Theorists often take it to be sufficient for an RIP claim about Person X that:

(a) Person X accepted some overall package of theories involving inconsistencies between its elements, or from which some sort of contradiction could be derived, or even is just used some sort of theoretical tools that they know *could* be used to generate contradictions, but:
(b) They didn't use the relevant contradiction to prove random arbitrary conclusions.

I think this is grossly insufficient. As a first stab as something a bit more substantive, I'd argue that we also need:

(c) That they were aware of the entailment of inconsistency (note that this is a standard that the Colyvan example meets, but which other historical examples used in this literature may not) and, crucially, that
(d) That they explicitly believed the contradiction, and also crucially, that
(e) They used both halves of the contradiction (conjunctively, or even one half at a time) as a premise in at least some of their reasoning about the subject.

Also, on an even more basic level, and as an absoltuely minimal standard, I'd suggest:

(f) That, in general or at *least* when dealing with the relevant subject matter (e.g. in other set-theoretic reasoning during the lag time Colyvan talks about), they did *not* implicitly reason according to classical rules of derivation that are paraconsistently invalid, like Disjunctive Syllogism and Reudctio Ad Absurdum.

Thoughts?

Sunday, October 25, 2009

Absolutely Unrestricted Quantification, the Classical Principle, and Non-Existence Solutions to the Liar Paradox

Graham Priest has argued that the structure of paradoxes like the Liar (and similar paradoxes, like Russell’s Paradox for naïve set theory) is best understood in terms of his ‘Inclosure Schema.’ The three elements of the Schema are:

(1) Existence
(2) Closure
(3) Transcendence

(1) is the condition that the relevant object (like the Liar Sentence, or the Russell Set) exists. (2) is the condition that it be a member of some category (being true, being a member of itself), and (3) is the condition that it also ‘transcend’ (i.e. fail to be part of) that category (being untrue, being a non-member of itself). Whatever one thinks of the Inclosure Schema as an analysis of what’s going on with Liar-type paradoxes, and whatever one thinks of Priest’s ‘Principle of Unified Solution’ whereby all paradoxes modeled by the Inclosure Schema should be solved in the same way, it’s at least a useful way of classifying various solutions.

A type-(2) solution to the Liar Paradox would be standard truth-value gap proposals, that say that the key place where the argument from the Liar to inconsistency goes wrong is the assumption that it is either true or false. We can keep the assumption that it’s true iff it’s false, but deny that it’s either one of those things. Presto. Done!

Of course, famously, while this works for the Simple Liar:

"This sentence is false."

….it runs into problems for the Strengthened Liar:

"This sentence is not true."

A type-(3) solution to the Liar Paradox would be a denial of the relevance instance of the Biconditional Truth Schema. (For example, Gil Harman takes this line in Change Of View.) That way, we get to keep the classical assumption that it’s either true or false, but ditch the assumption that it’s true iff it’s false, and thus avoid the contradiction. A problem with this is that it seems terribly ad hoc, and, when we approach the inference from Liar to inconsistency as an argument for dialetheism, it begs the question. If all you can say in response to an argument for a position that you don’t like is that some extremely basic, obviously true-sounding conceptual principle must have exceptions, because if it applied in this case, we’d get that conclusion, you’re not going to win any arguments.

Type-(1) solutions to the Liar Paradox are a bit trickier to formulate. To see why, consider the contrast to Russell’s Paradox. The standard, orthodox mathematical and philosophical response to it is simply to abandon the assumption that sets like the Russell Set exist. And, after all, that’s sort of plausible. If the Russell Set did exist, how would we ever know? If the best reason to think that any sets exist is (a la Quine) because they’re indispensible to our best scientific theories, then the restricted universe of ZFC hierarchical set theory should be quite sufficient. Given that, all else being equal, it seems un-parsimonious to also postulate a multitude of other sets, outside of the ZFC hierarchy.

Whatever one thinks of this argument, my point here is not to defend it but to underline the contrast to the Liar Paradox. There’s no denying that sentence tokens stringing together the words “this," "sentence," "is," and "false” in the right order exist, or even that, in some cases, the intended referent of the “this” is that sentence itself. (By contrast, of course, you can point at a sentence about some empirical subject written on a chalkboard and say “this sentence is false” without any sort of paradox being in the offing.) Non-existence claims, therefore, aren’t quite literally claims that the relevant sentences don’t exist. They’re claims that the relevant truth-bearers don’t exist. If you think that the bearers of truth are propositions, the non-existence claim is the claim that “this sentence is false” fails to express a proposition. If you think that sentences are the bearers of truth (or sentences-in-contexts, or precissified sentences, or any of the many variants of sentence theory out there), then the non-existence claim is that, for whatever reason, sentences like the Liar are actually meaningless. Exactly how to motivate this claim, respond to the obvious objections (don’t such sentences seem meaningful?, etc.), get off the hook of various revenge paradoxes and so on is a terribly difficult and complicated matter, and I don’t want to get into ay of that here.

I do, however, want to note that any remotely plausible non-existence story will apply just as well to the Truth-Teller:

"This sentence is true."

…as it does to the Liar. If a sentence can still be meaningful, or express a proposition, or whatever, when it isn’t trying to assert anything above and beyond its own truth, why not the same for falsity? Any non-existence solution that doesn’t treat the two cases similarly starts to look entirely too much like an ad hoc maneuver to save consistency. So a non-existence solution that even begins to be plausible must apply to all sentences that do nothing but talk about their own truth-status, or the truth-status of some other sentence (as in, say, the Yablo series), without ever "grounding out" in some other subject.

OK, but wait a second. Presumably, part of the motivation for any non-existence solution is likely a desire to defend the following principle about how truth and falsity relate to each other, which we can call the Classical Principle (CS), and, for the sake of simplicity, express in terms of “statements.” When reading “statement,” substitute in whatever your favorite candidate for the role of truth-bearers may be.

CS: "Every statement is either true or false (but not both)."

Stop. Wait. Is the statement of CS a “statement” (i.e. does the sentence express a proposition, or is it meaningful, or whatever)? It seems like it had better be, if we want to express our anti-gap, anti-glut commitments by asserting it! But if so, is it mixing use and mention in the context of talking about truth, and thus violating the requirement that truth-talk ultimately ground out in a subject other than truth? If so, it looks like, on the assumptions that got us to non-existence, it’s not eligible for truth-bearing status.

To get a grip on what’s going on here, it’s useful to start thinking about the general semantics of claims that involve quantification (some, all, etc.) For example, think about the following statements:

(1) All of the milk is gone!
(2) Everyone but John and Jack was eliminated in the first round of the game.
(3) All sentences are composed of words.
(4) Everything the Pope says is true.

In (1) and (2), interpretive charity should warn us away from seeing the domain over which the “all” and “everyone” are intended to quantify as absolutely unrestricted, including everything whatsoever, and context gives us good, solid clues about what the intended domains are. In the case of (1), if we take the domain being quantified over to be universal and unrestricted, then this is an obviously, bizarrely false claim, given the existence of full bottles and cups of milk in grocery stores and houses all around the world. Obviously, however, we take the intended domain to be something more like the objects that are inside the house of the people who are having the conversation. For parallel reasons, we shouldn’t take the “everyone” in (2) to quantify over all persons everywhere, but instead take it to quantify over the restricted domain of people who were playing the game with John and Jack on the particular occasion in question.

(3), on the other hand, seems to be a different sort of case. There’s no reason not to take the domain which the “all” is quantifying over—“for all x, if x is a sentence, x is composed of words”—to include itself. Use and mention seem to be innocently combined here.

How about (4)? Well, in the logical translation “for every x, if x is something the pope said, then x is true”, if we take the domain that “every x” is quantifying over to include questions the Pope asks, commands he gives, meaningless jumbles of word thats he murmurs in his sleep, etc, etc., then (4) is meaningless. Moreover, if he endorses Papal infallibility himself (as Popes are prone to do), then he might do so (oddly and pretentiously enough) in the third person, and thus himself utter (4). Whatever considerations led us to a non-existence solution to the Liar should rule out (4)-as-uttered-by-the-Pope-when-the-domain-over-which-he-is-quantifying-includes-his-assertion-of-(4). But it seems implausible that we should take any of these considerations to render standard statements of Papal infallibility not just false but meaningless. As such, it seems reasonable to charitably interpret the domain over which a non-Pope’s assertion of (4) quantifies as failing to include the Pope’s questions, commands, meaningless jumbles of words uttered during his holy sleep, and so on, and the domain over which a Pope’s assertion of it quantifies as also failing to include his own sentence asserting it. Which is not to say, of course, that the Pope doesn’t think his assertion of (4) is also true, but just that to express that bit of information, he’s going to need to construct a new sentence. And if he wants to express his belief in the truth of that sentence, he’s going to need to construct a new sentence to do it with, and so on.

None of this limits the Pope’s expressive power. He can express any bit of information that strikes his holy fancy. He can’t, however, do it all at once—even if he was infallible, he still wouldn't be capable of the miracle of expressing all of his beliefs with one short sentence.

The road from this to a plausible claim about the CS and non-existence solutions should be starting to get pretty clear. Before we get there, though, one reminder of the stakes.

When the antecedent and the consequent of a conditional are both true, the conditional itself is true. (Material conditionals work that way, anyway. The question of whether the material conditional is the right analysis of the conditional of ordinary language is actually going to be irrelevant to the problem I'm getting at…all I’ll need is the assumption that the conditional we end up with is at least false.) Now, say that someone has the false belief that McCain won the 2008 election. (Perhaps they just woke up from a coma they’ve been in since before the election, and the last polling data they saw showed McCain in the lead.) Given that they have that belief, and standard classical beliefs about the semantics of conditionals, and the plausible belief that “P” and “”P’ is true” are universally intersubstitutable, they (validly) conclude the following McCain Principle (MP):

MP: "For every statement, if that statement is true, then McCain won the election."

If you take the domain over which “every statement” quantifies to include the statement of the MP itself, then it asserts (among other things) that if it is true, then McCain won the election. As anyone who’s familiar with the Curry Paradox knows, that claim is one that has some very strange properties (if it's meaningful). Given the Biconditional Truth Schema and any logic strong enough to include Modus Ponens and Conditional Proof, the truth of its consequent can be derived from the simple statement of its truth conditions. If we want the MP to be false (and thus meaningful, truth-evaluable, etc.), and we don’t want to believe that it's also true*, then it might be a good idea to take the domain over which the MP quantifies as one that does not include itself. (Or, rather, to say that it's only meaningful if the domain over which it is supposed to be quantifying fails to include itself, and that when we say that it is false, we're reading it in this way.) Of course, this isn’t to say that the person who mistakenly believes McCain to have won the election doesn’t believe about the statement of the MP itself that if that statement is true, then McCain won the election, or that they can’t or shouldn't express this belief of theirs. They will, however, have to construct a new statement to express it with. It’s not always possible to say everything you want to say all at once.

Similarly for the CS. If, for example, the gap theorist expresses their view by saying that “all statements are either true, false or neither” and we (as non-existence theorists) take this as a meaningful, truth-bearing statement (since we think it’s false), we won’t take the domain over which the gap theorist’s statement is quantifying to include itself. Thus, our counter-statement that “every sentence is either true or false” will be presumed to quantify over exactly the same domain, and we’ve succeeded in disagreeing with each other. Moreover, when someone asks about the truth-status of the sentence we constructed to express our disagreement with the gap theorist, we can attribute truth to it, and when someone asks about the truth-status of the sentence attributing truth to that sentence, we can attribute truth to it, and so on. Again, expressive power is unlimited, but we can’t necessarily express all of our relevant beliefs all at once. Of course, as we've seen, we have excellent independent reasons to think that we'll often be unable to do so.

Now, note that I’m not actually asserting the non-existence solution here, or defending it against the multitude of other obvious objections that one could make to it. My point is a much narrower one, which is this:

There’s no special objection to the non-existence solution lurking in a non-existence advocate uttering the sentence “every sentence is true or false (but not both),” or in the fact that they are using that sentence to express their disagreement with gap and glut theorists despite the fact that it doesn't quantify over itself. Given that they wouldn’t be asserting it if they didn’t take it to be true, we have really excellent third person justification for believing that their attitude to the truth-status of that sentence isn’t an exception to the general pattern of their attitudes about the truth-status of every other sentence. However, this extra belief that we should so obviously attribute to the non-existence theorist who claims that "every sentence is either true or false (but not both)" isn't expressed by that very sentence. That’s OK. It doesn’t mean they can’t express that belief as well. It just means that they can’t say everything that they have to say all at once.






*....and, moreover, that every other false statement is also true, which could be shown with the production of exactly similar sentences for them.

Tuesday, October 20, 2009

Logical Pluralism, Monism and Logically Impossible Worlds

Logics are either:

(a) Overall theories of reality, relative to the high level of generality and abstractness at which formal languages operate, or
(b) Codifications of what we take to be good reasoning, or
(c) Something else

For the moment, let’s put aside (c) and focus on the debate between (a) and (b). On initial consideration, (b) may sound more natural and obvious. We do, after all, often use “logical” and “reasonable” as if they were synonymous, and the use of terms like “inductive logic” and “informal logic” certainly lends itself most easily to (b). On the other hand, normal talk of “logically possible worlds,” “logical impossibility” and so on most naturally lends itself to (a), and (a) has a long and glorious philosophical lineage, it or something along very much the same lines having been advocated (in one form or another) by figures as diverse in their other views as, for example, Frege, Quine, Penelope Maddy and Graham Priest.*

That list includes two people who believe that logical truths are “analytic” in some sense and two who reject the very notion of analyticity, two who accept the web-of-belief model of logical theory change and two who do not, two people who accept classical logic and two who reject it, one who believes in truth-value gaps and three who do not and one who believes in true contradictions and three who do not. Moreover, those categories and battle lines crisscross in dizzingly complex enough ways just among the overall views of those four thinkers to pretty much demand philosophical baseball cards to keep it all straight....the point is just that, far from being an eccentric consequence of any one perspective, it can fit with a variety of different overall packages of beliefs.

Further, note that (a) is even compatible with a wide variety of views about how logic and reasoning do relate to each other. That said, we might be able to say some general things about that relationship without bringing in very many extra assumptions. The theory of what it means to say that one thing follows from another that most naturally fits with (a) it the traditional view that validity is truth-preservation. (Roughly: Q follows from P iff P has never been and will never be true without Q being true as well. This gives us an obvious way to connect the claim that logic is all about giving us rules of entailment with the claim that logic is an overall theory about how the world is. For example, a philosopher who takes classical logic to give us the right overall theory of reality will claim that any time the world is configured in such a way as to make a disjunction true and one of its disjuncts false, it will be configured in a way that makes the other disjunct true. The dialetheist, who takes it that sometimes the world makes one disjunct both true and false and the other disjuct false only, a possibility the classical theorist rules out, will disagree with that, and the two sides proceed to engage in an argument about which way the world actually is.) This, in turn, generates at least one obvious point of connection between logic and good reasoning: logically valid arguments will never lead you from true premises to false conclusions, so if it’s reasonable to believe a bunch of premises, it’s also reasonable to (on the basis of the logical connection) believe the conclusions that can be validly inferred from those premises, hence Frege’s observation that although logic is not about how men think, it is about how they must think “if they are not to miss the truth.” So even if logically valid inferences don’t exhaust the category of reasonable inferences (as, on this view, it’s pretty clear that they don’t), they still constitute an important subcategory of it.

With the conceptual distinction between (a) and (b) firmly in mind, let’s move on to a related distinction. Logical pluralists believe that various logics are suitable for various purposes, but that there is no such thing as the One True Logic. By contrast, logical monists believe that there is some sense in which one logic is uniquely correct.

(a) seems to pretty much demand logical monism, unless of course one takes reality to be spectacularly inconsistent in a way that it’s hard to make sense of even dialetheically. (Note, for example, that Priest is a monist.) On the other hand, (b) fits very naturally with pluralism. Think about the notion of underlying logics, where “underlying logic” means, very roughly, something like the following.

*Logic L is the underlying logic of some context C iff L formally captures the standards of reasoning that are relevant to C.

Of course, what it means to be “relevant to C” here is importantly non-obvious. Are we talking about the standards implicit in what most people treat as good reasoning about C, or if not most people, then some special class of people (e.g. subject matter experts)? Or are we, maybe, talking about something a bit more objective, like “the standards of reasoning that are the most likely to lead to the truth about C” or “the standards of reasoning that are most likely to lead to progress towards some goal (other than truth) that’s important in C” or something like that?

These are important distinctions, and it’s usually hard to tell, when reading bits of underlying-logic-talk, exactly which position is implicitly being taken on these issues. Fortunately, at the moment, we won't have to clarify them for the purposes this discussion, so let’s put all of those issues to one side, and consider one last distinction. Given that we decide to get into the business of using underlying-logic-talk, either we take it to be the case that:

(d) Different contexts of reasoning involve different underlying logics, or
(e) There’s just one logic which is the unique underlying logic of every context of reasoning, or perhaps just of all the ones that are worth engaging with.

If we make the background assumption that (b) is the correct view about what logics are, then (d) just is logical-pluralism-expressed-in-underlying-logic-talk and (e) is logical-monism-expressed-in-underlying-logic-talk. Now, again relative to the assumption of (b), it’s pretty easy to argue for (d) and against (e).

After all, any given context of actual human reasoning is going to be finite. (Note that this is true even in the total context of "all reasoning that has ever, or will ever occur in any particular context whatsoever.”) Only so many inferences will be made, and thus only so many inferences will be classified as reasonable or unreasonable relative to that context. Formal logics, on the other hand, are total systems whose rules classify every inference as valid or invalid. As such, a variety of logics—in fact, an infiite number of possible systems of formal logic—will license all of the inferences regarded as good reasoning in that context. Thus, far from (e) making sense, it doesn’t look like we can specify a singular underlying logic even within a single context.

If you start out with a strong committment to (b), the obvious move is to take this as a reductio against logical monism. On the other hand, if you don’t start by making that crucial background assumption, it might make more sense to take it as a reductio against the usefulness of underlying-logic-talk.

With all that kept in mind, let’s backtrack from the special difficulties faced by (b)-plus-(e)-monism (which, obviously, don't touch (a)-monism) to a more general issue.

One might plausibly think that counterpossible reasoning is a problem for any sort of logical monism. Here's why:

The process of debate between deeply entrenched philosophical views seems to necessarily involve reasoning about cases in which each side takes it not just that the other side’s views are wrong, but that they *couldn’t* be right. (Although this sort of thing goes on all the time in other areas of philosophy, it’s particularly unavoidable in phil of logic, for obvious reasons.) That said, if we’re going to actually reason with each other about which of the disputing views is correct, rather than simply retreating behind our respective barricades to glare at each other, than we’ve got to make the case for our view (in part) by drawing out unappealing consequences of the views we reject. “X is absurd. After all, if X were true, then Y would be true.”

Now, on the most orthodox theory of counterfactual conditionals, “if X were true, Y would be true” just means that the closest possible worlds at which X is true are also worlds at which Y is true. This has the unfortunate consequence that all counterpossible conditionals come out as (vacuously) true. This is both wildly counterintuitive and (for the reasons just discussed) would, if taken seriously, make philosophical debate about basic issues impossible. As such, an attractive move to make here is to tweak the Lewissian rules for evaluating counterfactuals, such that a statement of the form “if X were true, Y would be true” is true iff the closest worlds, possible or impossible, at which X is true are also worlds at which Y is true. (A natural assumption to make is that possible worlds are just definitionally always closer to the actual world than even the closest impossible worlds.) So, while arguing against quantum logician’s claim that the distributive law (the conjunction of “A” and “B or C” entails the disjunction of “A and B” or “A and C”) fails, a partisan of distribution can say that “if the distributive law failed, then counter-intuitive consequence C would hold,” and we take it that that conditional is true if and only if the closest (impossible) worlds where the conjunctions and disjunctions of the properties of subatomic particles really do fail to distribute in the way the quantum logician claims that they fail to in the actual world are also worlds at which C holds.

So far, so good.

But wait.

One could make the following objection here. “On the old Lewissian picture where only logically possible worlds are allowed, the logical monist can evaluate whether the closest worlds at which X is true are worlds at which Y is true by using whatever they regard as the actually correct logic to reason about whether Y follows from X. Now, the (b)-plus-(d)-pluralist, who changes his logic due to variations of contexts of reasoning as easily as the rest of us change our shirts to due to variations in weather conditions, can handle this in an easy, straightforward way. When reasoning about whether C holds in the closest quantum-logical worlds, he can just reason about these worlds using quantum logic. But the logical monist faces a dilemma. Either she uses her favored (non-quantum) logic to reason about what’s true at the logically impossible world where distribution fails, and, given this inconsistency, she comes to many strange and irrelevant conclusions, or else she uses quantum logic when reasoning about the world where quantum logic is correct, and thus in effect becomes a logical pluralist.”**

Now, there are two things wrong with all of this. The first is that it’s not entirely true, even on the old only-logically-possible-worlds-need-apply model of counterfactual reasoning, that anyone reasoned about whether the closest worlds at which X was true were worlds at which Y was true by “using” the logic they used to determine the conditions for counting as a “logically possible world,” or, for that matter, by using any logic at all. In fact, it looks to me like interesting and informative counterfactual conditionals are very rarely ones in which the consequent is logically derivable from the antecedent. (After all, if Y is logically derivable from X, then Y is true at all of the logically possible worlds at which X is true, not juts the closest ones.) For example “if Hitler had won World War II, Germany would be the dominant power in post-War Europe” certainly seems to be true, but not because there’s a logically valid inference connecting the antecedent to the consequent.

Now, *if* we’re OK with underlying logic talk, we can re-phrase the point to get around this issue by saying that at least, there’s no reason to think that the underlying logic of reasoning about the actual world needs to be different from the underlying logic of reasoning about other logically possible worlds, whereas reasoning about what's true at some logically impossible world seems to pretty much demand a different underlying logic. Given this new phrasing, the first problem with the objection disappears. The problem posed for underlying logic talk by the failure of strictly logically valid inferences to exhaust the whole category of intuitively rational inferences doesn’t look like more of a problem for ordinary counterfactual reasoning than it does for factual reasoning, and assuming that there’s some sort of good solution to that problem when it comes to factual reasoning, that solution should carry over just as well to counterfactual reasoning.

To get a handle on the second and more important thing that’s wrong with the objection, we need to first answer the following question: Which sort of logical monism is this directed against? If the answer is that it’s a problem for any logical monist who wants to engage in counterpossible reasoning, we should still disambiguate monisms and try it out separately as an objection against each kind.

Taken as an objection to (b)-plus-(e) monists who are interested in counterpossible reasoning, it looks to me like it goes through. On the other hand, it’s a bit redundant. After all, as we saw above, (b)-plus-(e) monism fails on much more basic grounds than this, and the more basic objection works just as well even against (b)-plus-(e) monists who don’t think that it’s possible or desirably to (non-trivially) reason about counterpossible scenarios.

OK, how about (a)-monists? Well, an (a)-monist who accepts the notion of “underlying logics” can cheerfully admit that, when they reason about what’s the case in logically impossible worlds, the underlying logic of their reasoning is something other than the logic they take to be the uniquely correct overall theory of our reality. This semi-pluralism doesn’t, however, threaten, or indeed have much of anything to do with, the sense of logical monism that they endorse. (I say 'semi-pluralism' because to get to logical pluralism from these considerations, you need to combine your endorsement of (d) with an endorsement of (b).) On the other hand, I’ve already suggested that, if we don’t start out by assuming (b), the problem about uniquely specifying underlying logics sketched out above might be more naturally taken as a reductio against the very idea of underlying logics. An (a)-monist who took that route would not only be able to resist the push towards pluralism on the basis of counterpossible reasoning, but even the push towards the semi-pluralism involved in combining (a) with (d).

One way or another, anyone who accepts (a) has no reason to accept the notion that all reasoning involves “using a logic,” but, crucially, for the evaluation of what would and would not be the case in certain counterpossible scenarios, an (a)-monist might find it useful to consult the rules of a logic other than the one they take to the One True Logic. For example, when trying to figure out what’s the case in the failure-of-distribution-worlds that are the closest to the actual world (despite their logically impossibility), an important subcategory of good inferences will be those that are valid in quantum logic. Assuming that classical logic gives us the correct theory of the actual case, quantum logic will play the same role in relationship to our reasoning about what’s true at failure-of-distribution worlds that classical logic plays in relationship to our reasoning about what’s actually true. What of it? Once we reject (b), nothing in that picture gives us any more reason to abandon classical monism about logic than the uses of the details of Lemarckian biological theory for determining what’s true at the closest worlds at which acquired traits are genetically inherited gives us the slightest reason to abandon Darwinian monism about biology.







*This is clearest in the cases of Maddy, who explicitly claims in her book Second Philosophy that the laws of logic are like the laws of physics at a different level of generality, and Quine, who famously argues in Two Dogmas Of Empiricism that even logic is (rationally) revisable in the light of empirical evidence, and explicitly points to the possibility that classical logic will be falsified by quantum anomolies. Priest and Frege are more difficult cases, given that both men subscribe to some notion of "analytic truth," but at least in Priest's case, given his endorsement of the web-of-belief model in Two Dogmas Of Quineanism, and the explicit arguments he has made in various contexts for rejecting classical logic on empirical grounds, seems to more or less comfortably fit in the Quine/Maddy camp on this issue. As far as Frege goes, the evidence is weaker, but a very natural reading of claims like "logic is the science of the general laws of truth" and that logic gives us not the laws of "how men think, but of how they must think if they are not to miss the truth" is to see him as a type-(a) theorist, albeit a very un-Quine-like type-(a) theorist who takes logical laws to describe reality at a "third realm," non-empirical level. At the very least, without getting too deeply into thorny questions of Frege-interpretation, we can point out that the quoted statements are ones that a theorist who was solidly in the Quine/Maddy/Priest camp would enthusiastically endorse.

**This is certainly an objection that could be levelled against Graham Priest's overall package of views, given his explicit endorsement of logical monism in Doubt Truth To Be A Liar and of logically impossible worlds (at which, for example, there are no true contradictions) in Towards Non-Being.