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Eliezer_Yudkowsky comments on Mixed Reference: The Great Reductionist Project - Less Wrong
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Comments (353)
After it's been right the last 300 times or so, we should assess a substantial probability that it will be wrong before the 1,000th occasion, but believe much more strongly that it will be correct on the next occasion.
That doesn't seem to answer dspeyer's questions.
Okay. I'll bet with somewhere around 50% probability that the Great Reductionist Project as I've described it works, with reduction to a single thing counting as success, and requiring magical reality-fluid counting as failure. I'll bet with 95% probability that it's right on the next occasion for anthropics and magical reality-fluid, and with 99+ probability that it's right on the next occasion for things that confuse me less; except that when it comes to e.g. free will, I don't know who I'd accept as a judge that didn't think the issue already settled.
Can you expand on what you mean by this?
Either the Great Reductionist Thesis ("everything meaningful can be expressed by [physics+logic] eventually") is itself expressible with physics+logic (eventually) or it isn't. If it is, then it might be true.
If it isn't, then the great reductionist thesis is not true, because the proposition it expresses is not meaningful. I'm worried about this possibility because the phrase 'everything meaningful' strikes me as dangerously self-referential.
This is a reply to the long conversation below between Esar and RobbBB.
Let me first say that I am grateful to Esar and RobbBB for having this discussion, and double-grateful to RobbBB for steelmanning my arguments in a very proper and reasonable fashion, especially considering that I was in fact careless in talking about "meaningful propositions" when I should've remembered that a proposition, as a term of art in philosophy, is held to be a meaning-bearer by definition.
I'm also sorry about that "is meaningless is false" phrase, which I'm certain was a typo (and a very UNFORTUNATE typo) - I'm not quite sure what I meant by it originally, but I'm guessing it was supposed to be "is meaningless or false", though in the context of the larger debate now that I've read it, I would just say "colorless green ideas sleep furiously" is "meaningless" rather than false. In a strict sense, meaningless utterances aren't propositions so they can't be false. In a looser sense, an utterance like "Maybe we're living in an inconsistent set of axioms!" might be impossible to render coherent under strict standards of meaning, while also being colloquially called 'false' meaning 'not actually true' or 'mistaken'.
I'm coming at this from a rather different angle than a lot of existing philosophy, so let me do my best to clarify. First, I would like to distinguish the questions:
R1) What sort of things can be real?
R2) What thoughts do we want an AI to be able to represent, given that we're not certain about R1?
A (subjectively uncertain probabilistic) answer to R1 may be something like, "I'm guessing that only causal universes can be real, but they can be continuous rather than discrete, and in that sense aren't limited to mathematical models containing a finite number of elements, like finite Life boards."
The answer to R2 may be something like, "However, since I'm not sure about R1, I would also like my AI to be able to represent the possibility of a universe with Time-Turners, even though, in this case, the AI would have to use some generalization of causal reference to refer to the things around it, since it wouldn't live in a universe that runs on Pearl-style causal links."
In the standard sense of philosophy, question R2 is probably the one about 'meaning' or which assertions can be 'meaningful', although actually the amount of philosophy done around this is so voluminous I'm not sure there is a standard sense of 'meaning'. Philosophers sometimes try to get mileage out of claiming things are 'conceivable', e.g., the philosophical catastrophe of the supposed conceivability of P-zombies, and I would emphasize even at this level that we're not trying to get R1-mileage out of things being in R2. For example, there's no rule following from anything we've said so far that an R2-meaningful statement must be R1-possible, and to be particular and specific, wanting to conservatively build an AI that can represent Conway's Game of Life + Time-Turners, still allows us to say things like, "But really, a universe like that might be impossible in some basic sense, wihch is why we don't live there - to speak of our possibly living there may even have some deeply buried incoherence relative to the real rules for how things really have to work - but since I don't know this to be true, as a matter of my own mere mental state, I want my AI to be able to represent the possibility of time-travel." We might also imagine that a non-logically-omniscient AI needs to have an R2 which can contain inconsistent axiom sets the AI doesn't know to be inconsistent.
For things to be in R2, we want to show how a self-modifying AI could carry out its functions while having such a representation, which includes, in particular, being able to build an offspring with similar representations, while being able to keep track of the correspondence between those offspring's quoted representations and reality. For example, in the traditional version of P-zombies, there's a problem with 'if that was true, how could you possibly know it?' or 'How can you believe your offspring's representation is conjugate to that part of reality, when there's no way for it to maintain a correspondence using causal references?' This is the problem of a SNEEZE_VAR in the Matrix where we can't talk about whether its value is 0 or 1 because we have no way to make "0" or "1" refer to one binary state rather than the other.
Since the problems of R2 are the AI-conjugates of problems of reference, designation, maintainance of a coherent correspondence, etcetera, they fall within the realm of problems that I think traditional philosophy considers to be problems of meaning.
I would say that in human philosophy there should be a third issue R3 which arises from our dual desire to:
In other words, we want to avoid the twin errors of (1) preemptively shooting down somebody who is making an honest effort to talk to us by claiming that all their words are meaningless noises, and (2) trying to extract info about reality just by virtue of having an utterance admitted into a debate, turning a given inch into a taken mile.
This leads me to think that human philosophers should also have a third category R3:
R3) What sort of utterances can we argue about in English?
which would roughly represent what sort of things 'feel meaningful' to a flawed human brain, including things like P-zombies or "I say that God can make a rock so heavy He can't lift it, and then He can lift it!" - admitting something into R3 doesn't mean it's logically possible, coherent, or 'conceivable' in some rigorous sense that you could then extract mileage from, it just means that we can go on having a conversation about it for a while longer.
When somebody comes to us with the P-zombie story, and claims that it's "conceivable" and they know this on account of their brain feeling able to conceive it, we want to reply, "That's what I would call 'arguable' (R3) and if you try to treat your intuitions about arguability as data, they're only directly data about which English sentences human brains can affirm. If you want to establish any stronger sense of coherence that you could get mileage from, such as coherence or logical possibility or reference-ability, you'll have to argue that separately from your brain's direct access to the mere affirmability of a mere English utterance."
At the same time, you're not shoving them away from the table like you would "colorless green ideas sleep up without clam any"; you're actually going to have a conversation about P-zombies, even though you think that in stricter senses of meaning like R2, the conversation is not just false but meaningless. After all, you could've been wrong about that nonmembership-in-R2 part, and they might be about to explain that to you.
The Great Reductionist Thesis is about R1 - the question of what is actually real - but it's difficult to have something that lies in a reductionist's concept of a strict R2, turn out to be real, such that the Great Reductionist Thesis is falsified. For example, if we think R1 is about causal universes, and then it turns out we're in Timetravel Life, the Great Reductionist Thesis has been confirmed, because Timetravel Life still has a formal logical description. Just about anything I can imagine making a Turing-computable AI refer to will, if real, confirm the Great Reductionist Thesis.
So is GRT philosophically vacuous from being philosophically unfalsifiable? No: to take an extreme case, suppose we have an uncomputable and non-logically-axiomatizable sensus divinatus enabling us to directly know God's existence, and by baptizing an AI we could give it this sensus divinatus in some way integrated into the rest of its mind, meaning that R2, R1, and our own universe all include things referrable-to only by a sensus divinatus. Then arguable utterances along the lines of, "Some things are inherently mysterious", would have turned out, not just to be in R2, but to actually be true; and the Great Reductionist Thesis would be false - contrary to my current belief that such utterances are not only colloquially false, but even meaningless for strict senses of meaning. But one is not licensed to conclude anything from my having allowed a sensus divinatus to be a brief topic of conversation, for by that I am not committing to admitting that it was strictly meaningful under strong criteria such as might be proposed for R2, but only that it stayed in R3 long enough for a human brain to say some informal English sentences about it.
Does this mean that GRT itself is merely arguable - that it talks about an argument which is only in R3? But tautologies can be meaningful in GRT, since logic is within "physics + logic". It looks to me like a completed theory of R2 should be something like a logical description of a class of universes and a class of representations corresponding to them, which would itself be in R2 as pure math; and the theory-of-R1 "Reality falls within this class of universes" could then be physically true. However, many informal 'negations' of R2 like "What about a sensus divinatus?" will only be 'arguable' in a human R3, rather than themselves being in R2 (as one would expect!).
R3) "What sort of utterances can we argue about in English?" is (perhaps deliberately) vague. We can argue about colorless green ideas, if nothing else at the linguistic level. Perhaps R3 is not about meaning, but about debate etiquette: What are the minimum standards for an assertion to be taken seriously as an assertion (i.e., not as a question, interjection, imperative, glossolalia, etc.). In that case, we may want to break R3 down into a number of sub-questions, since in different contexts there will be different standards for the admissibility of an argument.
I'm not sure what exactly a sensus divinatus is, or why it wouldn't be axiomatizable. Perhaps it would help flesh out the Great Reductionist Thesis if we evaluated which of these phenomena, if any, would violate it:
Objective fuzziness. I.e., there are entities that, at the ultimate level, possess properties vaguely; perhaps even some that exist vaguely, that fall in different points on a continuum from being to non-being.
Ineffable properties, i.e., ones that simply cannot be expressed in any language. The specific way redness feels to me, for instance, might be a candidate for logico-physical inexpressibility; I can perhaps ostend the state, but any description of that state will underdetermine the precise feeling.
Objective inconsistencies, i.e., dialetheism. Certain forms of perspectivism, which relativize all truths to an observer, might also yield inconsistencies of this sort. Note that it is a stronger claim to assert dialetheism (an R1-type claim) than to merely allow that reasoning non-explosively with apparent contradictions can be very useful (an R2-type claim, affirming paraconsistent logics).
Nihilism. There isn't anything.
Eliminativism about logic, intentionality, or computation. Our universe lacks logical structure; basic operators like 'and' and 'all' and 'not' do not carve at the joints. Alternatively, the possibility of reference is somehow denied; AIs cannot represent, period. This is perhaps a stronger version of 2, on which everything, in spite of its seeming orderliness, is in some fashion ineffable.
Are these compatible with GRT? What else that we can clearly articulate would be incompatible? What about a model that is completely expressible in classical logic, but that isn't ontologically 'made of logic,' or of physics? I intuit that a classically modelable universe that metaphysically consists entirely of mind-stuff (no physics-stuff) would be a rather severe break from the spirit of reductive physicalism. But perhaps you intended GRT to be a much more modest and accommodating claim than everyday scientific materialism.
I have no objection to your description of R3 - basically it's there so that (a) we don't think that something not immediately obviously being in R2 means we have to kick it off the table, and (b) so that when somebody claims their imagination is giving them veridical access to something, we can describe the thing accessed as membership in R3, which in turn is (and should be) too vague for anything else to be concluded thereby; you shouldn't be able to get info about reality merely by observing that you can affirm English utterances.
Insofar as your GRT violations all seem to me to be in R3 and not R2 (i.e., I cannot yet coherently imagine a state of affairs that would make them true), I'm mostly willing to agree that reality actually being that way would falsify GRT and my proposed R2. Unless you pick one of them and describe what you mean by it more exactly - what exactly it would be like for a universe to be like that, how we could tell if it were true - in which case it's entirely possible that this new version will end up in the logic-and-physics R2, and for similar reasons, wouldn't falsify GRT if true. E.g., a version of "nihilism" that is cashed out as "there is no ontologically fundamental reality-fluid", denial of "reference" in which there is no ontologically basic descriptiveness, eliminativism about "logic" which still corresponds to a computable causal process, "relativized" descriptions along the lines of Special Relativity, and so on.
This isn't meant to sneak reductionism in sideways into universes with genuinely ineffable magic composed of irreducible fundamental mental entities with no formal effective description in logic as we know it. Rather, it reflects the idea that even in an intuitive sense, sufficiently effable magic tends toward science, and since our own brains are in fact computable, attempts to cash out the ineffable in greater detail tend to turn it effable. The traditional First-Cause ontologically-basic R3 "God" falsifies reductionism; but if you redefine God as a Lord of the Matrix, let alone as 'natural selection', or 'the way things are', it doesn't. An irreducible soul falsifies GRT, until I interrogate you on exactly how that soul works and what it's made of and why there's still such a thing as brain damage, in which case my interrogation may cause you to adjust your claim and adjust it some more and finally end up in R2 (or even end up with a pattern theory of identity). It should also be noted that while the adjective "effable" is in R2, the adjective "ineffable" may quite possibly be in R3 only (can you exhibit an ineffable thing?)
What does it mean to consist entirely of mind-stuff when all the actual structure of your universe is logical? What is the way things could be that would make that true, and how could we tell? This utterance is not yet clearly in my R2, which doesn't have anything in it to describe "metaphysically consists of'". (Would you consider "The substance of the cracker becomes the flesh of Christ while its accidents remain the same" to be in your equivalent of R2, or only in your equivalent of R3?)
PS: I misspelled it, it's http://en.wikipedia.org/wiki/Sensus_divinitatis
Here are three different doctrines:
Expressibility. Everything (or anything) that is the case can in principle be fully expressed or otherwise represented. In other words, an AI is constructible-in-principle that could model every fact, everything that is so. Computational power and access-to-the-data could limit such an AI's knowledge of reality, but basic effability could not.
Classical Expressibility. Everything (or anything) that is the case can in principle be fully expressed in classical logic. In addition to objective ineffability, we also rule out objective fuzziness, inconsistency, or 'gaps' in the World. (Perhaps we rule them out empirically; we may not be able to imagine a world where there is objective indeterminacy, but we at least intuit that our world doesn't look like whatever such a world would look like.)
Logical Physicalism. The representational content of every true sentence can in principle be exhaustively expressed in terms very similar to contemporary physics and classical logic.
Originally I thought that your Great Reductionist Thesis was a conjunction of 1 and 3, or of 2 and 3. But your recent answers suggest to me that for you GRT may simply be Expressibility (1). Irreducibly unclassical truths are ruled out, not by GRT, but by the fact that we don't seem to need to give up principles like Non-Contradiction and Tertium Non Datur in order to Speak Every Truth. And mentalistic or supernatural truths are excluded only insofar as they violate Expressibility or just appear empirically unnecessary.
If so, then we should be very careful to distinguish your confidence in Expressibility from your confidence in physicalism. Neither, as I formulated them above, implies the other. And there may be good reason to endorse both views, provided we can give more precise content to 'terms very similar to contemporary physics and classical logic.' Perhaps the easiest way to give some meat to physicalism would be to do so negatively: List all the clusters that do seem to violate the spirit of physicalism. For instance:
A list like this would give us some warning signs that a view, even if logically specifiable, may be deviating sharply from the scientific project. If you precisely stipulated in logical terms how Magic works, for instance, but its mechanism was extremely anthropocentric (e.g., requiring that Latin-language phonemes 'carve at the joints' of fundamental reality), that would seem to violate something very important about reductive physicalism, even if it doesn't violate Expressibility (i.e., we could program an AI to model magical laws of this sort).
I'm not sure what you mean by 'actual structure.' I would distinguish the Tegmark-style thesis 'the universe is metaphysically made of logic-stuff' from the more modest thesis 'the universe is exhaustively describable using purely logical terms.' If we learned that all the properties of billiard balls and natural numbers are equally specifiable in set-theoretic terms, I think we would still have at least a little more reason to think that numbers are sets than to think that billiard balls are sets.
So suppose we found a way to axiomatize 'x being from the perspective of y,' i.e., a thought and its thinker. If we (somehow) learned that all facts are ultimately and irreducibly perspectival (i.e., they all need an observer-term to be saturated), that might not contradict the expressibility thesis, but I think it would violate the spirit of physicalism.
I'm not sure. I doubt our universe has 'substance-accident' structure, but there might be some negative way to R2ify transubstantiation, even if (like epiphenomenalism or events-outside-the-observable-universe) it falls short of verifiability. Could we coherently model our universe as a byproduct of a cellular automaton, while lacking a way to test this model? If so, then perhaps we could model 'substance-properties' as unobservables that are similarly Behind The Scenes, but are otherwise structurally the same as accidents (i.e., observables).
So... in my world, transubstantiation isn't in R2, because I can't coherently conceive of what a substance is, apart from accidents. For a similar reason, I don't yet have R2-language for talking about a universe being metaphysically made of anything. I mean, I can say in R3 that perhaps physics is made of cheese, just like I can say that the natural numbers are made of cheese, but I can't R2-imagine a coherent state of affairs like that. A similar objection applies to a logical universe which is allegedly made out of mental stuff. I don't know how to imagine a logically structured universe being made of anything.
Having Latin-language phonemes carve at the joints of fundamental reality seems very hard, because in my world Latin-language phonemes are already reduced - there's already sequential sound-patterns making them up, and the obvious way to have a logic describing the physics of such a world is to have complex specifications of the phonemes which are 'carving at the joints'. It's not totally clear to me how to make this complex thing a fundamental instead, though perhaps it could be managed via a logic containing enough special symbols - but to actually figure out how to write out that logic, you would have to use your own neuron-composed brain in which phonemes are not fundamental.
I do agree that - if it were possibly to rule out the Matrix, I mean, if spells not only work but the incantation is "Stupefy" then I know perfectly well someone's playing an S-day prank on me - that finding magic work would be a strong hint that the whole framework is wrong. If we actually find that prayers work, then pragmatically speaking, we've received a hint that maybe we should shut up and listen to what the most empirically powerful priests have to say about this whole "reductionism" business. (I mean, that's basically why we're listening to Science.) But that kind of meta-level "no, you were just wrong, shut up and listen to the spiritualist" is something you'd only execute in response to actually seeing magic, not in response to somebody hypothesizing magic. Our ability to hypothesize certain situations that would pragmatically speaking imply we were probably wrong about what was meaningful, doesn't mean we're probably wrong about what's meaningful. More along the lines of, "Somebody said something you thought was in R3(only), but they generated predictions from it and those predictions came true so better rethink your reasons for thinking it couldn't go in R2."
With all that said, it seems to me that R3-possibilities falsifying 1, 2, or (a generalization of 3 to other effectively or formally specified physics (e.g. Time-Turners)), and with the proviso that we're dealing in second-order logic rather than classical first-order logic, all seem to me to pretty much falsify the Great Reductionist Thesis. Some of your potential examples look to me like they're not in my R2 (e.g. mental facts that can't be expressed in non-mental terms) though I'm perfectly willing to discuss them colloquially in R3, and others seem relatively harmless (effects which aren't further causes of anything? I could write a computer program like that). I am hard-pressed to R2-meaningfully describe a state of affairs that falsifies R1, though I can talk about it in R3.
I have an overall agenda of trying to think like reality which says that I want my R1 to look as much like the universe as possible, and it's okay to contemplate restrictions which might narrow my R2 a lot relative to someone's R3, e.g. to say, "I can't seem to really conceive of a universe with fundamentally mental things anymore, and that's a triumph". So a lot of what looked to me years ago like meaningful non-reductionism, now seems more like meaningless non-reductionism relative to my new stricter conceptions of meaning - and that's okay because I'm trying to think less like a human and more like reality.
Many mathematicians, scientists, and philosophers believe in things they call 'sets.' They believe in sets partly because of the 'unreasonable effectiveness' of set theory, partly because they help simplify some of our theories, and partly because of set theory's sheer intuitiveness. But I have yet to hear anyone explain to me what it means for one non-spatiotemporal object to 'be an element of' another. Inasmuch as set theory is not gibberish, we understand it not through causal contact or experiential acquaintance with sets, but by exploring the theoretical role these undefined 'set' thingies overall play (assisted, perhaps, by some analogical reasoning).
'Substance' and 'accident' are antiquated names for a very commonly accepted distinction: Between objects and properties. (Warning: This is an oversimplification. See The Warp and Woof of Metaphysics for the historical account.) Just as the efficacy of mathematics tempts people into reifying the set-member distinction, the efficacy of propositional calculus (or, more generally, of human language!) tempts people into reifying the subject-predicate distinction. The objects (or 'substances') are whatever we're quantifying over, whatever individual(s) are in our domain of discourse, whatever it is that predicates are predicated of; the properties are whatever it is that's being predicated.
And we don't need to grant that it's possible for there to be an object with no properties (∃x(∀P(¬P(x)))), or a completely uninstantiated property (∃P(∀x(¬P(x)))). But once we introduce the distinction, Christians are free to try to exploit it to make sense of their doctrines. If set theory had existed in the Middle Ages, you can be sure that there would have been attempts to explicate the Trinity in set-theoretic terms; but the silliness of such efforts would not necessarily have bled over into delegitimizing set theory itself.
That said, I sympathize with your bafflement. I'm not committed to taking set-membership or property-bearing completely seriously. I just don't think 'I can't imagine what a substance would be like!' is an adequate argument all on its own. I'm not sure I have a clear grasp on what it means for a set to have an element, or what it means for a number line to be dense and uncountable, or what it means for my left foot to be a complexly-valued amplitude; but in all these cases we can gain at least a little understanding, even from initially undefined terms, based on the theoretical work they do. Since we rely so heavily on such theories, I'm much more hesitant to weigh in on their meaninglessness than on their evidential justification.
You sound like a structural realist. On this view, as I understand it, we don't have reason to think that our conceptions straightforwardly map reality, but we do have reason to think that a relatively simple and uniform transformation on our map would yield a pattern in the territory.
So is this a fair characterization of the Great Reductionist Thesis?: "Anything that is the case can in principle be exhaustively expressed in classical second-order predicate logic, relying only on predicates of conventional mathematics (identity, set membership) and of a modestly enriched version of contemporary physics."
We could then elaborate on what we mean by 'modest enrichment' if someone found a good way to add Thoroughly Spooky Doctrines (dualism, idealism, traditional theism, nihilism, trivialism, ineffable whatsits, etc.) into our language. Ideally, we would do this as un-ad-hocily as possible.
I think we both agree that 'meaning' won't ultimately carve at the joints. So it's OK if R2 and R3 look a bit ugly; we may be eliding some important distinctions when we speak simply of a 'meaningful vs. meaningless' binary. It's certainly my own experience that I can incompletely grasp a term's meaning, and that this is benign provided that the aspects I haven't grasped are irrelevant to what I'm reasoning about.
Can I run something by you? An argument occurred to me today that seems suspect, but I don't know what I'm getting wrong. The conclusion of the argument is that GRTt entails GRTm. For the purposes of this argument, GRTt is the statement that all true statements have a physico-logical expression (meaning physical, logical, or physical+logical expression). GRTm is the statement that all true and all false statements have a physico-logical expression.
P1) All true statements have a physico-logical expression. (GRTt)
P2) The negation of any false statement is true.
P3) If a statement has a physico-logical expression, its negation has a physico-logical expression.
P4) All false statements have a physico-logical expression.
C) All true and all false statements have a physical-logical expression. (GRTm)
So for example, suppose XYZ is false, and has no physico-logical expression. If XYZ is false, then ~XYZ is true. By GRTt, ~XYZ has a physico-logical expression. But if ~XYZ has a physico-logical expression, then ~(~XYZ), or XYZ, does. Throwing a negation in front of a statement can't change the nature of the statement qua reducible.
Therefore, GRTt entails GRTm. What do you think?
I think your argument works. But I can't accept GRTm; so I'll have to ditch GRTt. In its place, I'll give analyzing GRT another go; call this new formulation GRTd:
This is quite unlike (and no longer implies) GRTm, 'Every meaningful statement is expressible in purely physical and logical terms.'
The problem for GRTt was that statements like 'there are no gods' and 'there are no ghosts' seem to be true, but cast in non-physical terms; so either they are reducible to physical terms (in which case both GRTt and GRTm are true), or irreducible (in which case both GRTt and GRTm are false). For GRTd, it's OK if 'there are no ghosts' can't be analyzed into strictly physical terms, provided that 'there are no ghosts' is entailed by a statement of what 'ghost' means plus all the purely physical and logical truths.
For example, if part of what 'ghost' means is 'something non-physical,' then 'there are no ghosts' will be derivable from a complete physical description of the world provided that such a description includes a physical/logical totality fact. You list everything that exists, then add the totality fact 'nothing except the above entities exists;' since the semantic of 'ghost' ensures that 'ghost' is not identical to anything on the physicalism list, we can then derive that there are no ghosts.
Note that the semantic 'bridge laws' are themselves entailed by (and, in all likelihood, analyzable into) purely physical facts about the brains of English language speakers.
I don't see anything wrong with this kind of self-reference. We can only explain what generalizations are by asserting generalizations about generalization; but that doesn't undermine generalization itself. GRT would only be an immediate problem for itself if GRT didn't encompass itself.
Okay, so lets assume that the generalization side of things is not a problem, though I hope you'll grant me that if a generalization about x's is meaningful, propositions expressing x's individually are meaningful. That is, if 'every meaningful proposition can be expressed by physics+logic (eventually)', then 'the proposition "the cat is on the mat" is meaningful' is meaningful. It's this that I'm worried about, and the generalization only indirectly. So:
1) A proposition is meaningful if and only if it is expressible by physics+logic, or merely by logic.
2) If a proposition is expressible by physics+logic, it constrains the possible worlds.
3) If the proposition "the cat is on the mat" is meaningful, and it is expressible by physics+logic, then it constrains the possible worlds.
4) If the proposition "the cat is on the mat" constrains the possible worlds, then the proposition "the proposition 'the cat is on the mat' is meaningful" does not constrain the possible worlds. Namely, no proposition of the form '"XYZ" constrains the possible worlds' itself constrains the possible worlds.
So if 'XYZ' constrains the possible worlds, then for every possible world, XYZ is either true of that world or false of that world. But if the proposition '"XYZ" constrains the possible worlds' expresses simply that, namely that for every possible world XYZ is either true or false of that world, then there is no world of which '"XYZ" constrains the possible worlds' is false.
5) The proposition 'the proposition "the cat is on the mat" is meaningful' is not both meaningful and expressible by physics+logic. But it is meaningful, and therefore (as per premise 1) it is expressible by mere logic.
6) Every generalization about a purely logical claim is itself a purely logical claim (I'm not sure about this premise)
7) The GRT is a purely logical claim.
I'm thinking EY wants to get off the GRT boat here: I don't think he intends the GRT to be a logical axiom or derivable from logical axioms. Nevertheless, if he does want the GRT to be an axiom of logic, and in order for it to be a meaningful axiom of logic, it still has to pick out one logical model as opposed to another.
But here, the problem simply recurs. If 'The proposition 'GRT' is meaningful' is meaningful then it doesn't, in the relevant respect, pick out one logical model as opposed to another.
Does that make sense?
I don't think we need this rule. It would make logical truths / tautologies meaningless, inexpressible, or magical. (We shouldn't dive into Wittgensteinian mysticism that readily.)
That depends on what you mean by "proposition." The written sentence "the cat is on the mat" could have been ungrammatical or semantically null, like "colorless green ideas sleep furiously." After all, a different linguistic community could have existed in the role of the English language. So our semantic assertion could be ruling out worlds where "the cat is on the mat" is ill-formed.
On the other hand, if by "proposition" you mean "the specific meaning of a sentence," then your sentence is really saying "the meaning of 'the cat is on the mat' is a meaning," which is just a special case of the tautology "meanings are meanings." So if we aren't committed to deeming tautologies meaningless in the first place, we won't be committed to deeming this particular tautology meaningless.
This looks like a problem of self-reference, but it's really a problem of essence-selection. When we identify something as 'the same thing' across multiple models or possible worlds, we're stipulating an 'essence,' a set of properties providing identity-conditions for an object. Without such a stipulation, we couldn't (per Leibniz's law) identify objects as being 'the same' while they vary in temporal, spatial, or other properties. If we don't include the specific meaning of a sentence in its essence, then we can allow that the 'same' sentence could have had a different meaning, i.e., that there are models in which sentence P does not express the semantic content 'Q.' But if we instead treat the meaning of P as part of what makes a sentence in a given model P, then it is contradictory to allow the possibility that P would lack the meaning 'Q,' just as it would be contradictory to allow the possibility that P could have existed without P existing.
What's important to keep in mind is that which of these cases arises is a matter of our decision. It's not a deep metaphysical truth that some essences are 'right' and some are 'wrong;' our interests and computational constraints are all that force us to think in terms of essential and inessential properties at all.
Only because you've stipulated that meaningfulness is essential to GRT (and to propositions in general). This isn't a spooky problem; you could have generated the same problem by claiming that 'all cats are mammals' fails to constrain the possible worlds, on the grounds that cats are essentially mammals, i.e., in all worlds if x is a non-mammal then we immediately know it's a non-cat (among other things). Someone with a different definition of 'cat,' or of 'GRT,' would have arrived at a different conclusion. But we can't just say willy-nilly that all truths are essentially true; otherwise the only possible world will be the actual world, perhaps a plausible claim metaphysically but not at all a plausible claim epistemically. (And real possibility is epistemic, not metaphysical.)
Also, 'GRT' is not in any case logically true; certainly it is not an axiom, and there is no reason to treat it as one.
No, I didn't say that constraining possible worlds is a necessary condition on meaning. I said this:
This leaves open the possibility of meaningful, non-world-constraining propositions (e.g. tautologies, such as the claims of logic), only they are not physics+logic expressible, but only logic expressible.
That's not relevant to my point. I'd be happy to replace it with any proposition we can agree (for the sake of argument) to be meaningful. In fact, my argument will run with an unmeaningful proposition (if such a thing can be said to exist) as well.
No, this isn't what I mean. By 'proposition' I mean a sentence, considered independently of its particular manifestation in a language. For example, 'Schnee ist weiss' and 'Snow is white' express the same proposition. Saying and writing 'shnee ist weiss' express the same proposition.
I didn't understand this. Propositions (as opposed to things which express propositions) are not "in" worlds, and nothing of my argument involved identifying anything across multiple worlds. EY's OP stated that in order for an [empirical] claim to be meaningful, it has to constrain possible worlds, e.g. distinguish those worlds in which it is true from those in which it is false. Since a statement about the meaningfulness of propositions doesn't do this (i.e. it's a priori true or false of all possible worlds), it cannot be an empirical claim.
So I haven't said anything about essence, nor does any part of my argument require reference to essence.
Agreed, it is not a merely logical claim. Given that it is also not an empirical (i.e. a physics+logic claim), and given my premise (1), which I take EY to hold, then we can conclude that the GRT is meaningless.
My mistake. When you said "physics+logic," I thought you were talking about expressing propositions in general with physics and/or logic (as opposed to reducing everything to logic), rather than talking about mixed-reference assertions in particular (as opposed to 'pure' logic). I think you'll need to explain what you mean by "logic"; Eliezer's notion of mixed reference allows that some statements are just physics, without any logical constructs added.
What 'Schnee ist weiss' and 'Snow is white' have in common is their meaning, their sense. A proposition is the specific meaning of a declarative sentence, i.e., what it declares.
Then they don't exist. By 'the world' I simply mean 'everything that is,' and by 'possible world' I just mean 'how everything-that-is could have been.' The representational content of assertions (i.e., their propositions), even if they somehow exist outside the physical world, still have to be related in particular ways to our utterances, and those relations can vary across physical worlds even if propositions (construed non-physically) cannot. The utterance 'the cat is on the mat' in our world expresses the proposition <the cat is on the mat>. But in other worlds, 'the cat is on the mat' could have expressed a different proposition, or no proposition at all. Now let's revisit your (4):
A clearer way to put this is: If the proposition p, <the cat is on the mat>, varies in truth-value across possible worlds, then the distinct proposition q, <p is meaningful>, does not vary in truth-value across possible worlds. But what does it mean to say that a proposition is meaningful? Propositions just are the meaning of assertions. There is no such thing as a 'meaningless proposition.' So we can rephrase q as really saying: <p exists>. In other words, you are claiming that all propositions exist necessarily, that they exist at (or relative to) every possible world, though their truth-value may or may not vary from world to world. Once we analyze away the claim that propositions are 'meaningful' as really just the claim that certain propositions/meanings exist, do you still have any objections or concerns?
(Also, it should be obvious to anyone who thinks that 'possible worlds' are mere constructs that do not ultimately exist, that 'propositions' are also mere constructs in the same way. We can choose to interrelate these two constructs in various ways, but if we endorse physicalism we can also reason using one while holding constant the fact that the other doesn't exist.)
No, GRT is an empirical claim. You defined GRT as the proposition <everything meaningful can be expressed by [physics+logic] eventually>. But the actual Great Reductive Thesis says: <everything true can be expressed by [physics+logic] eventually>. Everything true is meaningful, so your formulation is part of GRT; but it isn't the whole thing. An equivalent way to formulate GRT is as the conjunction of the following two theses:
Both 1 and 2 are empirical claims; we could imagine worlds where either one is false, or where both are. But we may have good reason to suspect that we do not inhabit such a world, because there are no inexpressible truths and no irreducibly neither-physical-nor-logical truths. For example, we could have lived in a world in which qualia were real and inexpressible (which would violate Expressibility), and/or one in which they were real and irreducible (which would violate Logico-Physicalism). But the physicalistically inclined doubt that there are such qualia in our universe.
We have a couple of easy issues to get out of the way. The first is the use of the term 'proposition'. That term is famously ambiguous, and so I'm not attached to using it in one way or another, if I can make myself understood. I'm just trying to use this term (and all my terms) as EY is using them. In this case, I took my cue from this: http://lesswrong.com/lw/eqn/the_useful_idea_of_truth/
EY does not seem to intend 'proposition' here to be identical to 'meaning'. At any rate, I'm happy to use whatever term you like, though I wish to discuss the bearers of truth value, and not meanings.
I don't want to define the GRT at all. I'm using EY's definition, from the OP:
You might want to disagree with EY about this, but for the purposes of my argument I just want to talk about EY's conception of the GRT. Nevertheless, I think EY's conception, and therefore mine, follows from yours, so it may not matter much as long as you accept that everything false should also be expressible by physics+logic (as EY, I believe, wants to maintain).
I'd like to get these two issues out of the way before responding to the rest of your interesting post. Let me know what you think.
Only because you're cheating by reclassifying all cases where it was wrong as cases where we haven't figure out how to properly apply it yet.