**Followup to**: Logical Pinpointing, Causal Reference

Take the universe and grind it down to the finest powder and sieve it through the finest sieve and then *show* me one atom of justice, one molecule of mercy.

- Death, in *Hogfather* by Terry Pratchett

Meditation: So far we've talked about two kinds of meaningfulness and two ways that sentences can refer; a way of comparing to physical things found by following pinned-down causal links, and logical validity by comparison to models pinned-down by axioms. Is there anything else that can be meaningfully talked about? Where would you find justice, or mercy?

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Suppose that I pointed at a couple of piles of apples on a table, a pile of two apples and a pile of three apples.

And lo, I said: "If we took the number of apples in each pile, and multiplied those numbers together, we'd get six."

Nowhere in the physical universe is that 'six' written - there's nowhere in the laws of physics where you'll find a floating six. Even on the table itself there's only five apples, and apples aren't fundamental. Or to put it another way:

Take the apples and grind them down to the finest powder and sieve them through the finest sieve and then *show* me one atom of sixness, one molecule of multiplication.

Nor can the statement be true as a matter of pure math, comparing to some Platonic six within a mathematical model, because we could physically take one apple off the table and *make* the statement false, and you can't do that with math.

This question doesn't feel like it should be very hard. And indeed the answer is not very difficult, but it is worth spelling out; because cases like "justice" or "mercy" will turn out to proceed in a similar fashion.

Navigating to the six requires a *mixture *of physical and logical reference. This case begins with a physical reference, when we navigate to the physical apples on the table by talking about *the cause of* our apple-seeing experiences:

Next we have to call the stuff on the table 'apples'. But how, oh how can we do this, when grinding the universe and running it through a sieve will reveal not a single particle of appleness?

This part was covered at some length in the Reductionism sequence. Standard physics uses the same *fundamental *theory to describe the flight of a Boeing 747 airplane, and collisions in the Relativistic Heavy Ion Collider. Nuclei and airplanes alike, according to our understanding, are obeying special relativity, quantum mechanics, and chromodynamics.

We also use entirely different *models *to understand the aerodynamics of a 747 and a collision between gold nuclei in the RHIC. A computer modeling the aerodynamics of a 747 may not contain a single token, a single bit of RAM, that represents a quark. (Or a quantum field, really; but you get the idea.)

So is the 747 made of something other than quarks? And is the statement "this 747 has wings" meaningless or false? No, we're just *modeling *the 747 with *representational elements* that do not have a one-to-one correspondence with *individual *quarks.

Similarly with apples. To compare a mental image of high-level apple-objects to physical reality, for it to be true under a correspondence theory of truth, doesn't require that apples be *fundamental in physical law.* A single discrete element of fundamental physics is not the only thing that a statement can ever be compared-to. We just need truth conditions that categorize the low-level states of the universe, so that different low-level physical states are inside or outside the mental image of "some apples on the table" or alternatively "a kitten on the table".

Now we can draw a correspondence from our image of discrete high-level apple objects, to reality.

Next we need to count the apple-objects in each pile, using some procedure along the lines of going from apple to apple, marking those already counted and not counting them a second time, and continuing until all the apples in each heap have been counted. And then, having counted two numbers, we'll multiply them together. You can imagine this as taking the physical state of the universe (or a high-level representation of it) and running it through a series of functions leading to a final output:

And of course operations like "counting" and "multiplication" are pinned down by the number-axioms of Peano Arithmetic:

And we shouldn't forget that the image of the table, is being calculated from eyes which are in causal contact with the real table-made-of-particles out there in physical reality:

And then there's also the point that the Peano axioms themselves are being quoted inside your brain in order to pin down the *ideal *multiplicative result - after all, you can get multiplications *wrong* - but I'm not going to draw the image for that one. (We tried, and it came out too crowded.)

So long as the math *is* pinned down, any table of two apple piles should yield a single output when we run the math over it. Constraining this output constrains the possible states of the original, physical input universe:

And thus "The product of the apple numbers is six" is meaningful, constraining the possible worlds. It has a truth-condition, fulfilled by a mixture of physical reality and logical validity; and the correspondence is nailed down by a mixture of causal reference and axiomatic pinpointing.

I usually simplify this to the idea of "running a logical function over the physical universe", but of course the small picture doesn't work unless the big picture works.

The Great Reductionist Project can be seen as figuring out how to express meaningful sentences in terms of a combination of *physical references* (statements whose truth-value is determined by a truth-condition directly correspnding to the real universe we're embedded in) and *logical references* (valid implications of premises, or elements of models pinned down by axioms); where both physical references and logical references are to be described 'effectively' or 'formally', in computable or logical form. (I haven't had time to go into this last part but it's an already-popular idea in philosophy of computation.)

And the Great Reductionist Thesis can be seen as the proposition that *everything* meaningful can be expressed this way *eventually.*

But it sometimes takes *a whole bunch of work.*

And to notice when somebody has *subtly* violated the Great Reductionist Thesis - to see when a current solution is *not* decomposable to physical and logical reference - requires a fair amount of self-sensitization before the transgressions become obvious.

**Example: Counterfactuals.**

Consider the following pair of sentences, widely used to introduce the idea of "counterfactual conditioning":

- (A) If Lee Harvey Oswald didn't shoot John F. Kennedy, someone else did.
- (B) If Lee Harvey Oswald hadn't shot John F. Kennedy, someone else would've.

The first sentence seems agreeable - John F. Kennedy definitely was shot, historically speaking, so if it wasn't Lee Harvey Oswald it was *someone*. On the other hand, unless you believe the Illuminati planned it all, it doesn't seem particularly likely that if Lee Harvey Oswald had been removed from the equation, somebody else would've shot Kennedy instead.

Which is to say that sentence (A) appears *true*, and sentence (B) appears *false*.

One of the historical questions about the *meaning* of causal models - in fact, of causal assertions in general - is, "How does this so-called 'causal' model of yours, differ from asserting a bunch of statistical relations? Okay, sure, these statistical dependencies have a nice neighborhood-structure, but why not just call them correlations with a nice neighborhood-structure; why use fancy terms like 'cause and effect'?"

And one of the most widely endorsed *answers,* including nowadays, is that causal models carry an extra meaning because they tell us about *counterfactual outcomes,* which ordinary statistical models don't. For example, suppose this is our causal model of how John F. Kennedy got shot:

Roughly this is intended to convey the idea that there are no Illuminati: Kennedy causes Oswald to shoot him, does not cause anybody else to shoot him, and causes the Moon landing; but once you know that Kennedy was elected, there's no correlation between his probability of causing Oswald to shoot him and his probability of causing anyone else to shoot him. In particular, there's no Illuminati who monitor Oswald and send another shooter if Oswald fails.

In any case, this diagram also implies that if Oswald hadn't shot Kennedy, nobody else would've, which is modified by a *counterfactual surgery *a.k.a. the do(.) operator, in which a node is severed from its former parents, set to a particular value, and its descendants then recomputed:

And so it was claimed that the *meaning* of the first diagram is embodied in its implicit claim (as made explicit in the second diagram) that "if Oswald *hadn't* shot Kennedy, nobody else would've". This statement is *true*, and if all the other implicit counterfactual statements are also true, the first causal model as a whole is a true causal model.

What's wrong with this picture?

Well... if you're *strict* about that whole combination-of-physics-and-logic business... the problem is that *there are no counterfactual universes* for a counterfactual statement to correspond-to. "There's apples on the table" can be true when the particles in the universe are arranged into a configuration where there's some clumps of organic molecules on the table. What arrangement of the particles in this universe could directly make true the statement "If Oswald hadn't shot Kennedy, nobody else would've"? In this universe, Oswald *did* shoot Kennedy and Kennedy *did* end up shot.

But it's a subtle sort of thing, to *notice* when you're trying to establish the truth-condition of a sentence by comparison to counterfactual universes that are not measurable, are never observed, and do not in fact actually exist.

Because our own brains carry out the same sort of 'counterfactual surgery' automatically and natively - so natively that it's embedded in the syntax of language. We don't say, "What if we perform counterfactual surgery on our models to set 'Oswald shoots Kennedy' to false?" We say, "What if Oswald *hadn't *shot Kennedy?" So there's this counterfactual-supposition operation which our brain does very quickly and invisibly to *imagine* a hypothetical non-existent universe where Oswald doesn't shoot Kennedy, and our brain very rapidly returns the supposition that Kennedy doesn't get shot, and this seems to be a fact like any other fact; and so why couldn't you just compare the causal model to this fact like any other fact?

And in one sense, "If Oswald hadn't shot Kennedy, nobody else would've" *is* a fact; it's a *mixed reference* that starts with the causal model of the *actual* universe where there are *actually* no Illuminati, and proceeds from there to the *logical* operation of counterfactual surgery to yield an answer which, like 'six' for the product of apples on the table, is not actually present anywhere in the universe. But you can't say that the causal model is true *because* the counterfactuals are true. The truth of the counterfactuals has to be calculated from the truth of the causal model, followed by the implications of the counterfactual-surgery axioms. If the causal model couldn't be 'true' or 'false' on its own, by direct comparison to the actual real universe, there'd be no way for the counterfactuals to be true or false either, since no actual counterfactual universes exist.

So that business of counterfactuals may sound like a relatively obscure example (though it's going to play a large role in decision theory later on, and I expect to revisit it then) but it sets up some even larger points.

For example, the Born probabilities in quantum mechanics seem to talk about a 'degree of realness' that different parts of the configuration space have (proportional to the integral over squared modulus of that 'world').

Could the Born probabilities be *basic* - could there just be a *basic law of physics* which just says directly that to find out how likely you are to be in any quantum world, the integral over squared modulus gives you the answer? And the same law could've just as easily have said that you're likely to find yourself in a world that goes over the integral of modulus to the power 1.99999?

But then we would have 'mixed references' that mixed together *three kinds of stuff* - the Schrodinger Equation, a deterministic causal equation relating complex amplitudes inside a configuration space; logical validities and models; *and *a law which assigned fundamental-degree-of-realness a.k.a. magical-reality-fluid. Meaningful statements would talk about some mixture of physical laws over particle fields in our own universe, logical validities, and *degree-of-realness.*

This is just the same sort of problem if you say that causal models are meaningful and true relative to a mixture of three kinds of stuff, actual worlds, logical validities, and counterfactuals, and logical validities. You're only supposed to have *two* kinds of stuff.

People who think qualia are fundamental are also trying to build references out of at least three different kinds of stuff: physical laws, logic, and experiences.

Anthropic problems similarly revolve around a mysterious degree-of-realness, since presumably when you make more copies of people, you make their experiences more anticipate-able somehow. But this doesn't say that anthropic questions are meaningless or incoherent. It says that since we can only *talk about* anthropic problems using three kinds of stuff, we haven't finished Doing Reductionism to it yet. (I have not yet encountered a claim to have finished Reducing anthropics which (a) ends up with only two kinds of stuff and (b) does not seem to imply that I should expect my experiences to dissolve into Boltzmann-brain chaos in the next instant, given that if all this talk of 'degree of realness' is nonsense, there is no way to say that physically-lawful copies of me are more common than Boltzmann brain copies of me.)

Or to take it down a notch, naive theories of free will can be seen as *obviously not-completed Reductions* when you consider that they now contain physics, logic, and this third sort of thingy called 'choices'.

And - alas - modern philosophy is *full* of 'new sorts of stuff'; we have modal realism that makes *possibility* a real sort of thing, and then other philosophers appeal to the truth of statements about *conceivability* without any attempt to reduce conceivability into some mixture of the actually-physically-real-in-our-universe and logical axioms; and so on, and so on.

But lest you be tempted to think that the correct course is always to just envision a simpler universe without the extra stuff, consider that we do *not* live in the 'naive un-free universe' in which all our choices are constrained by the malevolent outside hand of physics, leaving us as slaves - *reducing* choices to physics is not the same as taking a naive model with three kinds of stuff, and deleting all the 'choices' from it. This is confusing the project of getting the gnomes out of the haunted mine, with trying to unmake the rainbow. Counterfactual surgery was eventually given a formal and logical definition, but it was a lot of work to get that far - causal models had to be invented first, and before then, people could only wave their hands frantically in the air when asked what it meant for something to be a 'cause'. The overall moral I'm trying convey is that the Great Reductionist Project is *difficult;* it's not a matter of just proclaiming that there's no gnomes in the mine, or that rainbows couldn't possibly be 'supernatural'. There are all sorts of statement that were not originally, or are presently not *obviously* decomposable into physical law plus logic; but that doesn't mean you just give up immediately. The Great Reductionist Thesis is that reduction is always *possible eventually.* It is nowhere written that it is easy, or that your prior efforts were enough to find a solution if one existed.

Continued next time with justice and mercy (or rather, fairness and goodness). Because clearly, if we end up with meaningful moral statements, they're *not *going to correspond to a combination of physics and logic *plus *morality.

Part of the sequence *Highly Advanced Epistemology 101 for Beginners*

Next post: "By Which It May Be Judged"

Previous post: "Causal Universes"

Not to be obnoxious, but...

Why two?

ETA: I feel like I may have distracted from the thrust of the post. I think the main point was that there really really probably shouldn't be

morethen two stuffs, which is legit.Because Tegmark 4 isn't mainstream enough yet to get it down to one.

If there is a way to reduce it to zero or not is one discovery I'm much looking forward to, but there probably isn't. It certainly

seemstotally impossible, but that only really means "I can't think of a way to do it".How confident are we in the Great Reductionist Thesis? Short of the Great Reductionist Project's success, what would be evidence for or against it?

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.

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.

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

strictstandards of meaning, while also being... (read more)From the logic point of view, counterfactuals are unproblematic, in that I can prove consistency of my favorite counterfactual logic by exhibiting a model. Then as far as a logician is concerned, we are done: our counterfactual worlds live in the mathematical structure of the exhibited model.

From the computer science point of view a little more is required, but as luck would have it, we can implement counterfactuals in some causal models. If your causal model is an

actual circuit, then not only is it perfectly meaningful to ask "the output of the circuit is 1, what would be the output if I changed gate_0212 from OR to AND?" but it is possible to implement the counterfactual directly, and check. This is because we know enough about the causal model to ensure counterfactual invariance (e.g. other gates do not change). People use this kind of counterfactual reasoning to debug programs and circuits all the time! So from the "comp. sci" point of view, counterfactuals are unproblematic. The counterfactual universe "exists" in the operational sense of us having an effective procedure to get us there.The problem arises when you are trying to deal wit... (read more)

Further to my other comment, how would one define a counterfactual in the Game of Life? Surely we should be able to analyze this simple case first if we want to talk about counterfactuals in the "real world"?

I am finding the same problem with all articles in this sequence that I find with the explanation of Bayes' Theorem on Yudkowsky's main site. There are parts that seem so blindingly obvious they don't bear mentioning.

Yet soon thereafter, all of a sudden, I find myself completely lost. I can understand parts of the text separately, but can't link them together. I don't see where it comes from, where it's going, what problems it's addressing. I find it especially difficult to relate the illustrations to what's going on in the text.

I seldom have had this problem with the blog posts from the classical sequences (with some exceptions, such as his quantum physics sequence, which left me similarly confused).

Am I the only one who feels this way?

EDIT: upon reflection, this phenomenon, of feeling like there was a sudden, imperceptible jump from the boringly obvious to the utterly confusing, I've already experienced it before: in college, many lessons would follow this pattern, and it would take intensive study to figure out the steps the professor merrily jumped between what is, to them, two categories of the set of blindingly obvious things they already know and need to explain

again. Maybe there's some sort of pattern there?This is a problem known as "bad writing" which I continue to struggle with, even after many years. Can you list

the first partwhere you felt lost? Somewhere between there and the previous part, I must have skipped something.I do hope people appreciate that all the "blindingly obvious" parts are parts where (at least in my guesstimation, and often in my actual experience) somebody else would otherwise get lost. The "obvious" is not the same for all people.

Take the universe and grind it down to the finest powder and sieve it through the finest sieve and then show me one atom of

time, one molecule ofvelocity. Oh wait...Incidentally, I'd give a probability of about 0.1 to the statement "If Lee Harvey Oswald hadn't shot John F. Kennedy, someone else would have" - there have been many people who have tried to assassinate Presidents.

I was going to challenge you to a wager, but then I realized that (1) I agree with your estimate, and (2) I don't know how we'd settle a wager about a counterfactual.

This list contains duplicate elements.

I think it was Vladimir Nesov who said something like the following: Anticipation is just what it feels like when your brain has decided... (read more)

I realize this is a small thing, but this essay appears to use "fact" to mean "a statement sufficiently well-formed to be either true or false" rather than "a statement which is true" and that kept distracting me from its actual point. Can some other word be found?

I like how you frame this discussion. At this stage, I'd like to see more LessWrongers spending sleepless nights pondering how we want to renegotiate our correspondence theory to keep our theory and jargon as clean and useful as possible. Calling ordinary assertions 'true/false' and logical ones 'valid/invalid' isn't satisfactory. Not only does it problematize mixed-reference cases, but it also confusingly conflates a property of structured groups of assertions (arguments, proofs, etc.) with a property of individual assertions.

Our prototype for 'truth' is ... (read more)

Mainstream status:AFAIK, the proposition that "Logical and physical reference together comprise the meaning of any meaningful statement" is original-as-a-whole (with many component pieces precedented hither and yon). Likewise I haven't elsewhere seen the suggestion that the great reductionist project is to be seen in terms of analyzing everything into physics+logic.

An important related idea I haven't gone into here is the idea that the physical and logical references should be

effectiveorformal, which has been in the job description since, if I... (read more)This seems awfully similar to Hume's fork:

An Enquiry Concerning Human Understanding(1748)As Mardonius says, 20th century logical empiricism (also called logical positivism or neopositivism) is basically the same idea with "abstract reasoning" fleshed out as "tautologies in formal systems" and "experimental reasoning" fleshed out initially as " statements about sensory experiences". So the neopositivists' original plan was to analyz... (read more)

Not sure how good an example apple multiplication is, given that if you multiply 2 apples by 3 apples, you are supposed to get 6

square apples.Hence my careful specification that you're multiplying the numbers, not the piles.

This is an interesting post but, I have to say, kind of frustrating. I have tried to follow the discussions between Esar and RobbBB and your substantial elucidation as well as many other great comments, but I remain kind-of in the dark. Below are some questions which I had, as I read.

What question? What exactly is the problem you are purporting to solve, here? If it is, "What is the truth condition of 'If we took the number of apples in each pile, and multiplied those numbers together, we'd ge... (read more)

Can anyone say a bit more about why physical references would need to be described 'effectively'/computably? Is this based on the assumption that the physical universe must be computable?

I have had this question in my mind for ages. You say that these counterfactual universes don't actually exist. But, according to Many-Worlds, don't

alllawful Universes actually really really exist? I mean, isn't theresomeamplitude for Mr. Oswald to not have shot Kennedy, and then you get a blob where Kennedy didn't get murdered?I've been banging my head against a wall on this and still can't come to a conclusion. Are the decoherent blobs actually capable of creating multiple histories on the observable level, up here? It

looks, to me, that they should ... (read more)Discworld reference FTW. I would suspect that Pratchett's Death, being the secular humanist and life enthusiast that he is, would strongly approve of our efforts here to eventually render him irrelevant.

It may not be possible to draw a sharp line between things that exist from the things that do not exist. Surely there are problematic referents ("the smallest triple of numbers in lexicographic order such that a^3+b^3=c^3", "the historical jesus", "the smallest pair of numbers in lexicographic order such that a^3+24=c^2", "shakespeare's firstborn child") that need considerable working with before ascertaining that they exist or do not exist. Given that difficulty, it seems like we work with existence explicitly, as a... (read more)

I am not convinced -- by this article, at least -- that there could only be two kinds of stuff. It sounds like the answer to the question, "why two and not one or possibly three ?" is, "because I said so", and that's not very convincing.

I am also not entirely sure what the Great Reductionist Project is, or why it's important.

Note that I'm not arguing against reductionism, but solely against this post.

Great post as usual, Eliezer! I have to admit that I never thought of logical and causal references being mixed before, but truly that is often exactly how we use them.

I have one question, though: I read through the quantum physics sequence, and I just don't understand - why are the Born probabilities such a problem? Aren't there just blobs of amplitude decohering? Is the problem that all the decoherence is already predicted to happen, without implying the Born rule? If someone could clarify this for me, I'd greatly appreciate it.

I am somewhat confused about the nature of logical axioms. They are not reducible to physical laws, and physical laws are not reducible to logic. So then, it what sense are they (axioms) real? I don't think you are saying that they are "out there" in some Platonic sense, but it also seems like you are taking a realist or quasi-empirical approach to math/logic.

You have a typo there, I think. "Logical validities" appears twice. If it's not a type, the sentence is very unclear.

Tangential: I keep not understanding counterfactuals intuitively, not because of the usual reason, but simply because if I take my best model of the past and rerun it towardsthe future I do not arive at the present due to stochastic and chaos elevents.

Aka, trying to do the standard math: I throw a 100 sided dice, it comes out 73, "If 2+2 were equal to 4, the dice would with 99% certainty have come out 73".

Technically you'd get 6apples^2 sorry just making a joke.. :)

EDIT: Doubly sorry, someone else already made this comment! How can this be deleted? :)

If great minds think alike, then from this evidence we can conclude that puny minds joke alike :)

Well, you've certainly ground the 'hogfathers' argument into dust, but I've gotta point out that 2 apples times 3 apples isn't 6 apples; it's 6 SQUARE apples. Just for what it's worth.

Er... Why? :-)