Wednesday, September 29, 2010

Some Further Points About The Stone Paradox

In the "Analytic Philosophy" class I'm teaching in Korea this semester, this week we're covering the two chapters of Bertrand Russell's book My Philosophical Development concerning his and Whitehead's work writing Principia Mathematica. I always find the following fact fairly striking:

Following the discovery of Russell's Paradox, Russell got around it by means of his somewhat awkward and complicated theory of types. ZFC and other orthodox set theories got around it by the "hierarchical conception of sets" and similar means. In more recent decades, a few logicians on the radical fringe have argued for rehabilitating naive set theory at the expense of the Law of Non-Contradiction. Practically no one, though, seems to have thought of responding to the paradox by simply amending Frege's Basic Law V to something like Basic Law 5.1:

"Wherever it is logically possible for there to be a set of all and only the objects matching some description, there actually is such a set."

On a slightly different but closely related note, here's a true story about a friend of mine, D.:

One day in Hebrew school when D. was 12 or 13, the Rabbi was talking about how God can do anything. D. responded by pointing to the door of the classroom, which was always sticking and never quite closed. D. (who grew up to be a lawyer, and fondly refers to this as his "first cross-examination") pointed at the door and asked the Rabbi, "could God make that door closed all the way?" The Rabbi said "of course." Then D. asked, "could God make that door closed so that no one would open it?" Finally, D. asked, "could God make that door closed so that even God couldn't open it?" The Rabbi hemmed and hawed and never gave D. a good answer, and D. promptly gave up on belief in God.*

If, however, D. had gone into a Philosophy PhD program instead of law school, he might have discovered, through the writings of folks like Richard Swinburne, the standard theistic response to his point from Hebrew School, and that--again, the standard theistic response, indeed, almost the only response one ever encounters from philosophically trained theists to the worry--is, transparently, just a theological version of Basic Law 5.1 above. The previously mentioned Professor Swinburne tries to argue, in The Coherence of Theism, that God can perform any action, but that things like "create a stone an omnipotent being could not cause to rise" don't count as "actions." Other rather more intellectually honest theists just clarify that by "all-powerful" they mean that God can do anything that's logically possible given His omnipotence, not that God can do anything at all.

So, what, on earth, is the difference between the two cases?

Well, in terms of concern about ad-hocness, Basic Law 5.1 actually fares rather better than its theological counterpart, as it would represent a reversion to Cantor's original definition of "set," at the very dawn of naive set theory**, whereas the level of intellectual sophistication necessary for subtle caveats about "anything that's logically possible" comes rather late in the overall history of the Judeo-Christian religious tradition.

Other than that, I really have no idea. The uncharitable suggestion that rather forces itself on one after thinking about it for too long is that, by and large, set theorists are simply a bit more scrupulous than theistic philosophers when it comes to paying attention to epistemic standards like "general principles that need to have 'except when this produces a contradiction' caveats at the end of them are a lot less likely to be true than those that organically fail to produce contradictions." If one wanted to follow this thought to its natural conclusion, the sociological and psychological factors involved in filling out this story (even when it comes to very bright and otherwise epistemically scrupulous theists) aren't terribly hard to come up with.....bluntly, no one is, as a small child, indoctrinated by all the adult authority figures in their life to believe in naive set theory, so that, as an adult, they have an enormous antecedent emotional need to preserve their belief in Basic Law V.

Alternately, here's a (much more charitable) explanation:

The fact that the move from Basic Law V to Basic Law 5.1 is a bit ad hoc, that general principles are more plausible when they lack such epicycles, gives us some reason to think that naive set theory so amended gets things wrong. Similarly for the move from "naive omnipotence" to "God can do anything as long as no contradiction would follow from an omnipotent being performing the action." Still, the fact that the need for the epicycle at the end of that last sentence gives us some reason to believe that the theory in question is false isn't decisive if we have some tremendous independent reason to believe that the theory is true. Put crudely, you can get a point deducted for ad-hocness but still get more points overall than the alternative views.

If a theist had this view--they believed they had some excellent argument for the existence of God in their back pocket*** and that this justified belief in omnipotence-with-an-epicycle--that would be fair enough. Of course, that would involve acknowledging that the Stone Paradox gives us some reason to disbelieve in the existence of God, even if not a decisive one, and, at least anecdotally, that kind of attitude seems surprisingly rare.

In any case, though, it might be objected (indeed, commenters on previous posts where I've brought this sort of thing up have objected) that this whole discussion only applies to theists who take God to be omnipotent in all possible worlds, whereas a viable alternate version (we can call it Omnipotence 3.0) holds that God is omnipotent-without-epicycles in the actual case, and that He could indeed create an unliftable stone. If he chose to do so, He would, at that point, no longer be omnipotent--having just limited his power by creating a stone he couldn't lift--but that this counterfactual point doesn't bear on His actual omnipotence.

For a while, I thought this was a pretty good defense--certainly a lot more promising than the near-universal Standard Theistic Response--but, on second thought, I don't actually find it very plausible.

Here's why:

Talk of "abilities" or "powers" is counterfactual through and through.

Even if a man is part of a distant rainforest tribe that has never heard of baseballs, it can still be true of them that they have a powerful enough throwing arm to toss a baseball across a large field. This is an *actual* attribute of his, despite the fact that it doesn't come up in the circumstances of their life. If the tribesman in question was flown to America, shown a baseball, and he did indeed toss is across a large field, this would decisively confirm that he had the ability, but his having the ability is unaffected by the absence of the test.

Another example:

Take an obese chain-smoking alcoholic named John. Despite his many health problems, he has the ability to climb a few flights of stairs without having a heart attack and dying. It seems fair, though, to say that John's stair-climbing abilities are limited. He could not, for example, climb a hundred flights of stairs without having a heart attack. Whether or not either of these situations will ever actually come up--e.g. whether John ever climbs stairs or he exclusively frequents buildings with elevators and escalators, whether John lives close enough to a city with a hundred-story building in it that he could attempt this feat if he were unwise enough to try it, etc.--seems quite irrelevant to our talk about John's powers. If John is a North Korean whose government will never allow him to travel to a place with hundred-story buildings, that doesn't seem to impact the truth of our statement about the limitations on his stair-climbing abilities. Nor would it, indeed, matter if, as a matter of contingent fact, the tallest building in the world happened to be ninety-eight floors tall.

To brings things closer to the God case, imagine a possible world where John--still an obese chain-smoking alcoholic--is the undisputed absolute ruler of the planet. Nothing can get built without his say-so, and he refuses to allow any building on earth to be constructed higher than four stories.

One day, two of his subjects--Jim and Jerry--are having a quiet conversation, perhaps in a quiet stairwell in one of the many four-story buildings where, as far as they know, John's secret police hasn't bothered to install any CCTV cameras or listening devices. They like to go there sometimes to hold the kind of private conversations that Winston Smith and Julia enjoyed in the early parts of Nineteen Eighty-Four.

At one point, Jim boldly speculates that, based on how pudgy and red-faced and out-of-breath Emperor John looks in the newsreels, the reason why he never allows buildings to be built over four feet high is that he doesn't have the ability to climb more flights of stairs than that and he wants to avoid the embarrassment. Jerry responds that, well, he could imagine John climbing as many as five or six flights of stairs without having heart attack, but there's no way he's healthy enough to climb, say, a hundred flights of stairs without collapsing.

At this point, of course, just like the capture scene in Nineteen Eighty-Four, Jim and Jerry find out that the secret police was listening all along, and both are tortured with rats in Room 101 until they admit that two and three make six if the Party says they do, and that Emperor John has the power to climb thousands of flights of stairs without physical setback.

Now, how would we evaluate Jerry's original claim about the limit's on John's stair-climbing abilities?

Given the innately counterfactual nature of all ability/power/powerfulness talk, the fact that John hasn't happened to create any such stairs, and has thus deprived himself of the opportunity to expose this particular limitation on his stair-climbing powers, seems quite irrelevant to the truth of Jerry's claim. Just so for God and unliftable stones.

*With D.'s permission, I used this incident in my short story Dark Coffee, Bright Light and the Paradoxes of Omnipotence, which is going to be reprinted this winter in the Prime Books' anthology People of the Book. (I'll admit to being pretty excited about that, since I get to share a Table of Contents with the likes of Neil Gaiman and Michael Chabon.) At the time, I asked him, "is it OK with you if I steal some of your life story, and portray it as part of the life story of a character who's (a) gay and (b) a terrorist?" His response was, "geez, Ben, how do you know I'm not either of those things? Also, yeah, sure."

**Cantor distinguished between "sets" whose members could be consistently jointly thought of as one thing, and "inconsistent multiplicities" that could not. I suspect that part of the reason that people didn't respond to Russell's Paradox with, "oh well, I guess Frege's wrong, but Cantor's right," is that, while he was certainly a brilliant mathematician who contributed many still-interesting proofs, Cantor's foundational ideas about the nature of set theory were never clearly and systematically laid out the axiomatic way that Frege's were, and a lot of his extant writings that touch on it (essays, letters to Dadekind, etc.) are full of unclear assumptions, weird religious baggage, quasi-mystical beliefs about "true infinity" and so on. For anyone interested in plunging into those waters, though, a good place to start is Michael Hallett's book Cantorian set theory and limitation of size.

***Given the large flaws in the standard contenders (cosmological, teleological and so on), I'm extremely skeptical that such an argument exists, but that, of course, is quite outside the subject matter of this post. At any rate, my (largely hypothetical) problem-acknowledging theist merely has to sincerely take themselves to have such a good argument, not to actually have one.

Monday, September 27, 2010

Quantum Logic, Part IV of IV

In Miami last year, I sat in on a few weeks of a seminar on philosophical issues about quantum mechanics, before I got too swamped with last-minute dissertation edits and whatnot to make the time. (Later, the professor did join me and some friends for an evening of drinking single malt and watching and making fun of "What The Bleep Do We Know?") On the first day, the professor went through a long and funny list of nonsense topics that might come to mind when one hears the phrase "philosophical issues about quantum mechanics" and announced that we wouldn't be talking about any of those. He then proceeded to list off a few "actually serious topics" we also wouldn't be going over, purely because of time constraints. One of the topics he listed off there was that of whether quantum results create a problem for classical logic.

A few minutes later, he moved on to describe the two path experiment we'd read about in a slightly whimsical way in the first chapter of David Albert's excellent book Quantum Mechanics and Experience. (The assignment was sent around by e-mail before the first day of class.) The punch line of the experiment, which establishes "superposition" (a term that often feels, at least from my non-physicist's perspective, more like a label slapped on the weirdness than anything particularly illuminating), goes like this:

Given the experimental evidence, it seems like we can absolutely rule out the possibility that the electron is passing through path A. Or path B.

Or neither.

Or both.

Which, um....

....would seem to me to kind of create a problem for classical logic.


That's such a good note to end on that I was tempted to it there, but I should probably say some more.

As I've indicated before, the Hartry-Field-style "paracomplete" approach to the Liar Paradox--whereby we set up an elaborate formal apparatus and use it to reject that Liar sentences are true, reject that they're false, reject that they're neither, and so on--isn't particularly attractive to me, both because I think that better options are on the table and because of the difficulties its proponents face in saying anything particularly intuitively plausible about sentences like this one:

"This sentence would not be accepted by a being who accepted all true sentences."

...and, of course, my standard complaint about non-classical solutions to the Liar, which is that the Liar and Curry are obviously instances of the same phenomenon, and a solution to the Liar that doesn't apply to Curry is no better than a solution to the Simple Liar that doesn't apply to the Strengthened Liar.

All of that said, quantum superposition state weirdness does strike me as a much more promising application of parcompleteness. I think that Priest & Routley have an old paper suggesting a dialetheic approach to quantum mechanics, but that seems to get the intuitive situation exactly wrong. It's not that all the possibilities can be jointly ruled *in*, it's that they can all be jointly ruled *out.*

Moreover, the quantum analogy to my view of the Liar Paradox would fall completely flat. Claiming that any of the statements involved commit category mistakes is (a) incompatible with the claim that they can be empirically ruled out, and (b) hard to square with the way we talk about electrons that *aren't* in superposition states. It's meaningless to say that some ideas are yellow, or to deny it, since color talk just doesn't apply to ideas, but position talk clearly *does* apply to particles.

Now, I'm certainly not endorsing a paracomplete approach to quantum weirdness--I'm not ready to give up on classical logic just yet, and the empirical and conceptual issues involved in arguing about this one way or the other get pretty murky pretty fast--but I do think there's an obvious prima facie case for some such logical revision. (Even if I hold out hope for its defeat.) Notice, though, that, as I've been arguing in the last few posts, none of this remotely threatens Distribution.

Wednesday, September 22, 2010

Quantum Logic, Part III of IV

Quantum logicians argue that, when it comes to statements about the behavior of subatomic particles, Distribution--the inference from [P & (Q v R)] to [(P & Q) v (P & R)]--fails.

In Part II, I suggested that it's very hard to make sense of this claim if you take validity to be a matter of universal truth-preservation. If one extends the classical truth tables for conjunction and disjunction with a third truth value, it's hard to see which one would or could "get the job done." I quickly surveyed various obvious candidates for a third truth-value--gaps, gluts, undecidedness, on-the-border-between-true-and-falsiness and so on. In none of these cases does it seem plausible that any of these additions to the classical truth tables would make Distribution invalid.

Of course, the quantum logician could revise the classical truth tables instead of extending them--they could, for example, argue that it's sometimes possible for (P v Q) to be true even if neither P nor Q is true--but this makes them vulnerable to "change of meaning" charges in a way that, for example, as we saw in Part II, even as extreme a heterodox logician as the dialetheist is not vulnerable.

(In his argument against Putnam's early quantum logic stuff, Dummett claims that the whole notion of truth tables and truth-functionality subtly relies on Distribution and that, as such, the quantum logician is not entitled to it. This charge seems to me to rely on an illegitimate leap from the notion that Distribution fails in some contexts to the notion that it fails in all contexts. It is, in other words, like anti-dialetheist arguments that rely on equating dialetheism with trivialism.)

Now, it's certainly possible to concoct truth tables that fit the bill--retaining the classical lines while invalidating Distribution and keeping most of the rest of the classical laws we care about. I wrote a paper last year suggesting one scheme for that--treated as a formal exercise, because I've never been convinced by the quantum case against Distribution--and in the comments on Part II Brandon suggests another, while making clear that it's just an example, and raising the concern about his example that the "true if..." truth-values might be a bit too crudely rigged to get the desired result but speculating that there were probably more sophisticated ways to go about it. (In so far as I might have come off too strongly last time, sounding like there was literally no way to set up truth tables to get the desired results, Brandon's point was well-taken.) For the sake of elegance at least, my favorite way of invalidatng Distribution like this:

Assume there are three truth-values, 1, 0 and .5. For disjunctions to have value 1, the value of their conjuncts must add up to at least 1, for them to have value 0, the sum up of the conjuncts must be 0, and otherwise, they have value .5. Conjunctions have value 1 if the sum of their conjuncts is 2 and 0 otherwise. The value of the negation of any claim is the absolute difference between 1 and the value of the claim. Validity is preservation of value 1.

In terms of the most important rules about Disjunction and Conjunction, Disjunction-Addition and Conjunction-Elimination obviously both fall neatly out of that scheme. Disjunctive Syllogism is also safe. Here's the truth table for that:

( P v Q ) / ~ P // Q
1 1 1 0 1 1
1 1 0 0 1 0
1 1 .5 0 1 .5
.5 1 1 .5 .5 1
.5 1 .5 .5 .5 .5
.5 0 0 .5 .5 0
0 1 1 1 0 1
0 .5 .5 1 0 .5
0 0 0 1 0 0

Distribution, though, is invalid, the relevant line of the truth table being:

P & ( Q v R) // ( P & Q) v ( P & R )
1 1 .5 1 .5 1 0 .5 0 1 0 .5

A slightly less elegant way to do it is just to call the third value "O," not say much about it's relationship to T and F, and just choose a bunch of classical claims about truth-functionality while ignoring others--e.g. it will still be true that "conjunctions are only T if their conjuncts are both T", but it won't still be true that "conjunctions are only F if their conjuncts are both F," it'll still be true that "disjunctions are only F if their disjuncts are both F", but it won't still be true that "disjunctions are only T if their disjuncts are both T," etc.

So it's certainly possible to put together such truth tables, invalidating Distribution while retaining some plausible-sounding principles linking them to bivalent truth tables. The difficulty in each case is about putting conceptual meat on the bones, philosophically justifying the strange behavior of the new truth-value.

Intuitively, if .5 is read as "sort-of-true" or "half-true" or something, then, if P is completely true and Q is "sort-of-true" or "half-true" or something, then the joint statement of both should be "sort-of-true" or "half-true" or something as well. That won't do, though, in terms of the mathematicized truth tables given above, because if, when P is 1, Q is .5 and R is .5., (P & Q) and (P & R) both get value .5, their disjunction gets value 1 and Distribution goes through.

In general, however one sets things up, the sticky question is this:

What could we possibly say about the third truth-value that justifies its strange behavior here? How can we make sense of saying that the value is close enough to truth that two disjuncts that each have it add up to a disjunction which is fully true, but far enough from truth that one conjunct with the value, in combination with one fully true conjunct, add up to a conjunction that's not only untrue but fully false? Without this awkward combination, after all, Distribution goes through.

Again--if anyone has anything to offer in the comments by way of a plausible story about the nature of the proposed value, I'm all ears. I certainly don't take the limits of my creativity here to be the last word, and one of the joys of philosophy blogging is getting to brainstorm with people about this stuff, so seriously, if you've got something, I'd love to hear it. From my perspective, though, I don't see how it could work.

Of course, it could be that the crucial mistake happened at the beginning, when we assumed that validity was about truth-preservation, and so, for Distribution to be invalid, we had to find an instance of it where a true premise delivered a non-true conclusion. As widespread as this view of validity is--it certainly the official dogma of introductory logic textbooks, and it's a subject that Graham Priest and I agree about, which should indicate a certain kind of consensus--it's certainly not the only view out there. There is, for example, the truth-preservation-plus view often assumed by relevance logicians, according to which truth-preserving inferences can still fail to be valid if the premises and conclusion aren't sufficiently connected given some sort of relevance constraint (e.g. they don't share any non-logical terms in common). That clearly won't do, here, though, since instances of Distribution concerned with the behavior of quantum particles will pass those sorts of relevance tests with flying colors.

What, though, if we switched over to an important secondary tradition, the inferentialist one according to which certain inference rules are basic and analytically constitutive of the meaning of the relevant logical connectives?

One problem here will be a general one afflicting all change-of-logic proposals made by inferentialists--"given that you think that the inference rules are constitutive of the meaning of the connectives, and that you're changing your mind about the inference rules, aren't you just talking about new connectives, not making heterodox claims about the behavior of the old connectives?"

Now, I think this is a serious problem, and I don't want to dismiss it too quickly, but there is a way around it that's at least plausible enough to tentatively assume for the purposes of this discussion (as I assume for the sake of argument that--contrary to my views on the matter--inferentialism is correct):

We can bite half the bullet, in a way that makes room for arguing about logic, but saying that, for example, quantum-logical disjunction and classical disjunction really are two different connectives with two different meanings, but both represent analyses of our ordinary intuitive notion of (inclusive) disjunction. In general, we all start out with intuitive proto-versions of all the logical concepts, and the meaning-constituting inference rules of various formal systems all represent something like conceptual analyses of those proto-concepts.

(That was a bit rough, but it's probably good enough for our purposes here. Also note that a truth-preservationist quantum logician who was willing to simply revise the classical truth tables instead of expanding them could presumably make the same sort of move to justify themselves.)

Now, this sort of move may seem plausible when it comes to Van McGee's counter-example to Modus Ponens or relevance logicians' arguments against the classical notions of implication and entailment and so on, but it seems to me that the quantum context is the one in which this move falls the most spectacularly flat.

Surely, our ordinary-language, intuitive notions of "and" and "or" predate the discovery of subatomic particles, superposition and the rest by....

...well, by pretty much all of human history. These notions were developed to talk about the ordinary properties of ordinary "mid-sized dry goods."

Now, taking the "quantum leap" in our understanding of the physical world surely poses challenges to many of our ordinary intuitions about things like location, causation at a distance and so on. It's plausible that it turns out to upset all sorts of metaphysical applecarts. Some of these could even have consequences for logic, if logic is understood in truth-preservationist terms and odd possibilities, not accounted for in logical schemes developed in a previous era, turn out to sometimes be true if this strange domain of physical reality.

But. Surely the one thing that we won't discover from our experiments about the behavior of previously-unknown subatomic particles is that we were mistaken about what "and" and "or" meant in our ordinary pre-philosophical, pre-scientific discourse.

If revisionary moves about logic are justified as a result of the deliverances of quantum physics, it looks like we're going to have to justify them in terms of the truth-preservationist conception of validity. And the prospects for making sense of that are looking pretty dim as well.

So, if Distribution is safe after all, does that mean that classical logic has been comfortably and securely squared with quantum weirdness, that there are no remaining problems or challenges there?


Stay tuned for next week!

Monday, September 20, 2010

In The Absence Of A Real Monday Post...

....I'll just note that barking mad seems like a severe understatement when it comes to describing this.

To put things a bit more bluntly:

If you think that the proposals in question fall short of being near-psychotically racist and evil, you are a fucking idiot who should be shunned in all important contexts. Obviously.

We'll be back to quantum logic next time.

Wednesday, September 15, 2010

Quantum Logic, Part II of IV

Assume (until we get to Part III) that validity is a matter of truth-preservation.

Now, for any proposal that we "change logics" (i.e. revise our current ideas about what the true laws of logic are) to succeed in doing more than just changing the subject, we need to make sure the terms we're using mean the same things after the revision that they did before. For example, if I suggest that "P & Q" is true iff P is true, Q is true, or both, whereas "P v Q" is true iff P and Q are jointly true, my proposal amounts to reversing the usual notation for conjunction and disjunction, not changing our ideas about them. By contrast, if someone denies that the inference from any and every premise to any instance of the disjunction "P v ~P" is truth-preserving, and justifies it with some philosophical story about the unevenness with which vague predicates map onto the world, they really are rejecting classical logic.

Quine, in a famous passage in Philosophy of Logic, seems to suggest that all revisionary proposals about logic fall into the first category, that "the deviant logician" always succeeds merely in changing the subject. He argues, in particular, that when the dialetheist says that some contradictions are true and not everything follows from them, they have stopped using "not" and "~" in the usual ways.

For a variety of reasons, I don't this is one of Quine's better or more plausible moments. For my tastes, Quine is at his best in Two Dogmas of Empiricism, where he suggests that perhaps quantum physics will one day force us to abandon classic logic, and provides a holistic framework for thinking about questions of confirmation and belief revision within which that makes sense.

Without getting into a longer critique of Quine here, it's worth noting, as at least one major test of sameness of meaning, that the dialetheist, who takes some contradictions to be true and denies that anything and everything follows from any given contradiction--because she denies that Disjunctive Syllogism is universally truth-preserving--actually affirms all of the relevant classical truth tables. She just adds some extra lines. Where the classical logician says that ~P is true iff P is false and ~P is false iff P is true, the dialetheist agrees, and, given that they think that P can be both true or false, take the step they are required to take by the assumptions they share with the classical logician and say that ~P can be both true and false as well. Where the classical logician says that a disjunction is true iff at least one of the disjuncts is true--indeed, given that the disjunction symbol refers to "inclusive disjunction" (P or Q or both), "P v Q" and "at least one of these things is true: P, Q" seem to mean precisely the same thing--the dialetheist grants this, assents to all the lines of the classical truth table (where "F" is read as "just false") and adds some extra ones, getting the result that "at least one of these things is true: P, Q" can be true even if P and Q are both false and Q fails to be true, if P is both true and false. Once all that is in place, it's a straightforward, principled consequence of the view--with the truth tables for the two classical truth values (just true and just false) intact and the meaning securely unchanged--that, given the assumption that truth and falsity can overlap, Disjunctive Syllogism is not universally truth-preserving.

A similar story could be told about the inferences denied by deviant logicians who do things like deny the Law of the Excluded Middle, but the example of the dialetheist makes the point:

Even very radical proposals for logic change--and, in some ways, the claim that there are true contradictions seems much more radical than the quantum logician's rejection of Distribution--can pass the "not just changing the subject" test with flying colors. Whatever one thinks of their claim to have discovered new logical possibilities ignored by orthodoxy, their views about how the classical truth-values interact are thoroughly orthodox. Crucially, they respect the fact that "P or Q" in the sense captured by "P v Q" seems to mean the same thing as "at least one of these things is true: P, Q."

The quantum logician, on the other hand, seems to fail the test.

For the premise of some instance of Distribution--i.e. [P & (Q v R)]--to be true, surely P must be true. (If a conjunction can be true even though one of its conjuncts is not, the "change of meaning" charge starts to sound pretty convincing.) For the same reason, "Q v R" must be true. How, then, can the conclusion--[(P & Q) v (P & R)] be false? Well, we've already said that P is true. And if Q and R were both false, then (Q v R) would be false, and hence the premise as a whole would be false. Moreover, if one of them were true, the conclusion would be true!

In order for the premise to be true and the conclusion to have some other status, then, both Q and R must have some other truth-value. What other truth-value, though, would do the trick? If Q and R were both both true and false, the conclusion would be true. If Q and R were both neither true nor false, then how could "at least one of these things is true: P, Q" possibly be true? If Q and R were both somehow undecided or unsettled between truth and falsity, or on the vague borderline between them or something, why wouldn't their disjunction similarly be undecided or unsettled between truth and falsity, or on the vague borderline between them or something?

If someone has a proposal for what truth-value Q and R could have that would make the premise of non-truth-preserving instances of Distribution true without the conclusion being true, I'd love to hear about it in the comments, but right now it looks like nothing fits the bill.

But wait! Maybe we went wrong in our initial assumption that validity is truth-preservation. Perhaps, once we switch over to one of the other theories of validity, the quantum logician's move will start making more sense.

To which all I can say is...

Stay tuned!

Monday, September 13, 2010

Quantum Logic, Part I of IV

So when I asked for reader requests last week, a couple of people asked for a post about quantum logic. The phrase "quantum logic" means a lot of different things these days, as indeed the word "logic" can mean a lot of different things. (See, for example, inductive "logic", computer-programming "logics" and so on.) So when many people talk about "quantum logic", they're talking about various formal or even mathematical constructions that model certain kinds of experimental results, the kind of thing for which no deeper philosophical justification is either offered nor required.

My interest in logic, however, veers towards what we can think of as 'logical metaphyics,' questions like, 'Do the inferences that we classically take to be universally truth-preserving really universally preserve truth? Are the claims we take to be logical truths really true?' (Hence my central research focus is in the semantic paradoxes, the question of whether there are any true contradictions, whether the Law of the Excluded Middle always holds and so on.) Because of that--and because I'm a Quineish confirmational holist--my interest in quantum logic is specifically on the question of whether the best explanation of the relevant physics might involve rejecting some of our current ideas about logical truth.

Historically, the most common proposal along these lines has been that, in response to quantum phenomena, we should reject Distribution, the principle that [P & (Q v R)] entails [(P & Q) v (P & R)]. Unless otherwise indicated, when I talk about "quantum logicians" or "the quantum logician", I'll be talking about the Distribution-rejecting quantum logician. In the new few posts, I'll argue that the prospects for their proposal are fairly bleak.

If we understand entailment in terms of truth-preservation (as I think we should), then, no matter how creative we get about adding in extra truth-values, there doesn't seem to be any plausible way to (a) get the result that the usual inferences about conjunction and disjunction that the quantum logician doesn't want to revise away remain valid, while (b) getting the result that Distribution is invalid. Without (a)--if, for example, it turns out that the truth of P isn't enough to guarantee the truth of (P v Q) in some truth-functional quantum logic--the 'change of meaning' charges often lobbed against quantum logic start seeming pretty hard to refute.

Of course, one could take all of this (in combination with whatever empirical case one thinks there is for quantum logic) as a good reason to reject truth-preservationism in favor of switching over to an inferentialist account of logical consequence, where primitive inference rules are taken to be "meaning-constituting" for logical connectives. Unfortunately, on closer inspection, things look even worse for the quantum logician here. It seems terribly implausible that experimental results about esoteric sub-atomic phenomena should show us that we were mistaken about the meaning of the terms "and" and "or."

At the level of description I'm giving here, it might seem like these are generic criticisms that would apply to *any* proposal to revise logic--"aren't classical logicians always accusing people with heterodox views of these sorts of things?"--but this isn't the case. Comparisons to other revisionary proposals will be instructive. After all, as we'll see, paracomplete theorists who reject instances of the Excluded Middle and Disjunctive Syllogism-rejecting dialetheists both pass the tests which (I argue) the Distribution-rejecting quantum logician fails.

Finally, though, I'll argue that even if the proposal that we reject Distribution to make sense of quantum phenomena isn't particularly plausible, that doesn't let classical orthodoxy 'off the hook.' Given the experiments that establish superposition and the rest, there are well-grounded worries that some sort of logically revisionary solution may be needed, even if rejecting Distribution doesn't fit the bill. I'll conclude with some tentative thoughts about that.

Meanwhile, though, I have to prep for teaching Philosophy of Art to some Koreans, so this post will have to remain nothing more than a preview for coming attractions. Stay tuned for Wednesday!

Wednesday, September 8, 2010

Ethical Quasi-Realism and Logical Truth

Moral realists think that there are mind-independent facts of the matter that ground the truth-values of claims like "killing small children for sport is wrong" and "playing Wii Golf is wrong." Whether one is the more austere kind of minimalist about truth or the most extreme, early-Wittgenstein-style correspondence theorist, or anything in between, if one is a moral realist, one will--at least on the most basic level of description--tell pretty much the same story about why "killing small children for sport is wrong" is true and "playing Wii Golf is wrong" is false. In each case, the type of event being referred to either is the way it is being described as being (in which case it's true) or it isn't (in which case it's false). For moral error theorists, again, regardless of their precise position on truth as long as it's within the usual range, the story about how to evaluate such statements is precisely the same as for the moral realist. The only difference is that, according to the error theorist, *both* of the statements just mentioned are false, since neither type of event has the property in question. According to the error theorist, after all, *no* type of event has that property, since there is no such property as (moral) wrongness.

(An important sidenote is that people sometimes speak as if the moral error theorist had to give up on *all* evaluative language, but that's absurd. The usual J.L. Mackie-style arguments against moral properties are obviously inapplicable to other sorts of evaluative properties--e.g. ones pertaining to epistemic matters--that can be reduced to ordinary, non-"queer" properties far more easily and less problematically than *morally* evaluative terms can. Anyone who thinks that the moral error theorist is saying something false-according-to-error-theory when she says "you shouldn't be a moral realist, since there's no evidence for the existence of moral properties" hasn't thought very hard about the variety of different things that the word "should" can do in different contexts. I'm not an error theorist, but I do take it a bit more seriously than all that.)

Moral "quasi-realists" argue that this story about truth only works for one kind of truth--"descriptive truth," and that it fails for another kind of truth--"evaluative truth." (Everyone else, of course, thinks that the very idea of a "non-descriptive" form of truth is deeply confused.) When it comes to "evaluative truths", the statement is true iff it (depending on one's preferred flavor of quasi-realism) expresses the speaker's moral attitudes, or the moral attitudes the speaker would have under certain sorts of idealized circumstances, or something like that.

Now, generally speaking, quasi-realists are realistic enough about human psychology to grant that there's no particular reason to believe that, even under idealized circumstances, we'd all have precisely the same moral attitudes, and of course, no one claims that we all have the same moral attitudes right now. As such, any form of quasi-realism about morality automatically adds up to a sort of relativism--not relativism about moral properties, mind you, but relativism about truth. If you and I have different deep moral attitudes, still would under idealized circumstances, etc., it's that from my perspective, killing small children for sport is right-for-me and wrong-for-you, but that, from my perspective, my claim that it's right is true and your claim that it's wrong is false, and the opposite is true from your perspective. Although few would like to put it quite like this, the quasi-realist avoids right-for-you and wrong-for-me at the expense of embracing true-for-you and false-for-me.

Now, relativism about truth might seem like an intuitively unappealing enough consequence on its own, but relativism about logical truth would take the quasi-realist to a really awkward place. After all, a big part of any story about "idealized circumstances" is presumably that, under these circumstances, one has, for example, chosen between moral claims and their negations in every case, made sure that all of their moral claims are consistent with each other, and so on. Logical matters are a key part of how quasi-realists can still make sense of criticizing claims that (according to them) aren't in the business of trying to correctly describe anything. You might not know what someone's deep attitudes are, but surely you can know that they're saying two things that are implicitly inconsistent with each other.

Moreover, they seem to be on solid ground here. It might seem like almost nothing is universally agreed on in the philosophy of logic. (After all, it's arguably the area of philosophy where the claims in dispute are the most basic.) Are contradictions ever true? Are instances of Excluded Middle always true, or should we sometimes simultaneously reject claims are their negations? What does it mean to say that one thing follows from another? Is there One True Logic, or should we be pluralists?

In all of this chaos, however, one of the few things that nearly everyone agrees on is what we could call the Universality of Logical Truth Thesis (ULTT). Dialetheists believe that some contradictions are absolutely, perspective-independently true no less than their orthodox opponents believe that all contradictions are are absolutely, perspective-independently false. Paracomplete theorist reject the view that, say, the instances of Excluded Middle that are relevant to the Liar Paradox are true for *anyone.* Even logical pluralists--e.g. Greg Restall and JC Beall, currently the best-known advocates of logical pluralism--generally focus on questions of validity. They argue that there are a plurality of genuine, legitimate logical consequence relationships, that that each one really does delineate a class of "valid" inferences is universally true and that these claims do not contradict each other. It's simply that some logical consequence relationships are appropriate for regulating our reasoning about some areas and that others are appropriate for regulating our reasoning about other areas. The truth of these appropriateness claims themselves will, again, be universal.

Of course, most philosophers believe in what we could call the UATT--the Universality of All Truth Thesis--and wouldn't bother thinking of the ULTT as a separate matter. The ULTT is important, though, because even quasi-realists, having rejected the UATT, still have good reason to cling tight to the ULTT. In fact, the ULTT looks like it's going to be absolutely central to their project, for cashing out what "idealized circumstances" look like, for making sense of why we should criticize people for having internally inconsistent moral stances, etc.

But there's a problem.

What about people who literally don't have *any* moral attitudes, and who are psychologically constituted in such a non-standard way that they're just totally incapable of forming any attitude of that type--they're morally color-blind? Parsing the existing scientific literature on people who have deeply non-standard psychological make-ups in ways that are morally relevant--say, psychopaths--raises all kinds of complicated conceptual and empirical problems that we don't need to get into here, but surely it's at least *possible* in principle for such people to exist.

From their perspective, no matter of expressing attitudes that they do have or under any circumstances would have could possibly decide between the truth of a moral claim and the truth of its negation. It seems like, given the overall story, the most natural thing to do would be to say that, from the perspective of the morally disengaged observer listening to a moral debate, either the person who says "killing small children for sport is wrong" and the person who says "killing small children for fun isn't wrong" are both saying true things, or neither of them is saying a true thing, or perhaps there's simply no fact of the matter about whether any such statements are true. All of these options, of course, get you logically heterodox general results.

And that seems to show that quasi-realists really aren't entitled to the full ULTT. The universality of logical truth at least can break down.

And, given the ULTT's apparently centrality to the quasi-realist's project, that seems like a problem.

*Remember, on a quasi-realist story, "killing children for fun" isn't a descriptive statement--it's not even a descriptive statement about the speaker's attitudes--but an expression of those attitudes. As such, "killing children for fun is always wrong" is *also* an expression of an attitude, so (at the very least) it's not entirely clear that one can take a classical way out here and claim that all negations of positive moral claims are true (for someone totally incapable of forming attitudes of the relevant type) according to the quasi-realist, the way that all such statements are true (for everyone) according to the error theorist.

Monday, September 6, 2010

Reader Requests?

In recent weeks, I've done two four-part sets of posts, one about the Liar Paradox and one about set theory. Is there anything in particular that anyone would like to hear about in upcoming posts?

Obviously, my likelihood of actually taking a suggestion is correlated with the degree to which I have much of anything to say about the topic....e.g. last time I asked for requests, someone wanted a post about theories of truth-makers, but sadly, all I have to say about truth-makers is "I'm a bit leery about propositions for the usual reasons that people are often a bit leery of propositions or other kinds of abstract objects, and I'm somewhat agnostic about precisely what the primary bearers of truth actually are." Which wouldn't have made a very interesting post. That said, I'm open to suggestions about anything from Yablo's Paradox to the new Of Montreal album, quantum logic to the mid-term elections, Wittgenstein to whiskey, so ask away.

Wednesday, September 1, 2010

Bertrand Russell: Portrait Of The Philosopher As A Young Man

One of the classes I'm teaching in Korea this semester is called "Analytic Philosophy", and one of the textbooks I'm assigning is Bertrand Russell's book "My Philosophical Development."

In Chapter 3, "First Efforts," Russell records his first youthful doubts about conventional ideas.

"I began thinking about philosophical questions at the age of fifteen. From then until I went to Cambridge, three years later, my thinking was solitary and completely amateurish, since I read no philosophical books, before I read Mill's Logic in the last months before going to Trinity... I minded my theological doubts, not only because I had found comfort in religion, but also because I felt that these doubts, if I revealed them, would cause pain and bring ridicule, and I therefore became isolated and solitary. Just before and just after my sixteenth birthday, I wrote down my beliefs and un-beliefs, using Greek letters and phonetic spelling for purposes of concealment."

To which I have to say, uh, really?

The Greek letters thing is a nice, vivid, picturesque image, but you have to wonder if someone as smart as Russell obviously was, even at the age of 16, would have thought that this method would actually fool anyone in his household.

To review some relevant facts:

Russell's grandfather had been the Prime Minister in the 1840s and again in the 1860s. The family had been raised to peerage with the rise of the Tudor dynasty....i.e. a few centuries before Young Master Russell turned 16. It's safe to say that every male in the Russell clan since time immemorial would have received a good classical education. It doesn't seem like much of a stretch to say that they would have all been sufficiently used to reading Greek that any of them would have been able to tell that they were reading English words transliterated into the Greek alphabet if they'd just glanced at the page for long enough to read a sentence while shuffling around papers looking for a misplaced cup of tea.

(And that's just the boys. I also wonder if, in a family as progressive as the Rusells--keep in mind that Russell's godfather was John Stuart Mill--the education of the girls might have been considerably better than average as well.)

All of which makes me wonder: was young Bertie really particularly concerned about concealment, or did he just enjoy the romantic gesture of making a big elaborate show of concealment?

Moving on to the actual contents of the journal--which Russell faithfully reproduces in full in My Philosophical Development, while making embarrassed noises about the confused, undeveloped nature of a lot of the ideas therein--we find a lot of skepticism about traditional Christian dogmas, but Russell doesn't go quite so far at this point as to doubt the existence of God per se. When it comes to morality, we see a lot of the the sharp polemical humor you get in his later writings. For example, in one passage, he talks about his Presbyterian grandmother's view that, instead of using reason to tell right from wrong, one should follow the 'inner voice' of conscience, then a few paragraphs down he casually refers to "this inner voice, this God-given conscience which made Bloody Mary burn the Protestants..."

Everywhere, he protests in a fairly hyperbolic way about his dedication to rationality, e.g. "April 29. In all things, I have made the vow to follow reason, not the instincts inherited partly from my ancestors and gained gradually by selection and partly due to my education. How absurd it would be to follow these in the questions of right and wrong."

Keep that passage in mind while we go back and take a closer look at the bit about his grandmother:

"My rule of life which I guide my conduct by, and a departure from which I consider as a sin, is to act in the manner which I believe to be most likely to produce the greatest happiness considering both the intensity of the happiness and the number of people made happy. I know that my grandmother considers this an impractical rule of life and says that, since you can never know the thing which will produce the greatest happiness, you do much better in following the inner voice. The conscience, however, can easily be seen to depend mostly upon education (as, for example, common Irishmen do not consider lying wrong) which fact alone seems to be quite sufficient to disprove the divine nature of conscience."

Some thoughts about this:

(1) He considers his ideas about this subject to be shocking enough to go in his secret journal of forbidden thoughts, but he had at least one argument about it with grandma?

(2) The racism here is pretty awesome. It seems like a safe guess that the young English aristocrat writing this journal had never actually met a 'common Irishman', nor quite likely had he ever met anyone who had ever met one, so you have to wonder where exactly he got his information about The Irish And Their Propensity To Lie.

(3) He claims to have read no philosophy books at this time, and maybe he hadn't, but somehow or another he seems to have absorbed the utilitarian ideas of Bentham and Mill in full, complete with the precise characteristic turns of phrase and careful qualifications--"the greatest happiness considering both the intensity of the happiness and the number of people made happy."

(4) However that may have come about, it's awfully interesting that Russell's steadfast dedication to his sacred vow to follow reason alone in determining the difference between right and wrong, sweeping aside all the mental clutter derived from his ancestors and his education, led him to replicate, in a meticulously exact fashion, the precise moral opinions of his godfather, John Stuart Mill, and Mill's godfather, Jeremy Bentham.