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.] 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?


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

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

(α ∧ β) ∧ ∂

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 ? O
T ? O ? T
T ? O ? O
T ? O ? F
T F F ? O
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 ? O
F ? O ? T
F ? O ? O
F ? O ? F
F F F ? O

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?


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.


Monday, November 9, 2009


(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.


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 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: 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.