## Monday, September 24, 2018

### The last word (I hope!) on Fitch's paradox

I was really hoping to leave Fitch's paradox in the rear view mirror, but like a moth to the flame (or perhaps a better metaphor would be like an alcoholic to the bottle) I find myself compelled to make one more observation.

First a quick review for those of you who haven't been following along: Fitch's paradox is a formal proof that starts with some mostly innocuous-seeming assumptions and concludes that all truths are known.  Since this conclusion is plainly false, the game (and it really is just a game) is to find the flaw in the reasoning.

There is some low-lying fruit: one of the assumptions that goes into the proof is that it is possible to know any truth.  That is plainly false because Godel, finite universe, yada yada yada.  You can try to do an end-run around this by restricting the domain of the logic to "tractable truths".  The problem with this is that tractability inherently involves time, but Fitch's logic does not model time.  So in some sense Fitch's conclusion in this case is actually true: if something is not known in the static situation described by the logic, then it cannot be known in that static situation.  Hence, all "tractable" truths (to the extent that it is possible to give that word a coherent meaning in a world without time) are in fact known.

An advocate of the tractability approach might try to rescue it by reconstructing Fitch's proof in a logic that did model time.  I suspect that this is not possible, and I even suspect that it's possible to prove that it is not possible, but I don't care anywhere near enough to actually try to prove it myself.

What I do want to point out here is that there is actually a much deeper problem: Fitch assumes that it is possible to assign coherent meanings to the words "possible" and "know".  In fact, not only does he assume it's possible, he assumes it's *trivial* because he doesn't even *try* to actually define these words.  He just tacitly assumes that they have meanings, that these meanings are common knowledge, and that they coincide with the semantics of his modal logic.

In fact, both "know" and "possible" are highly problematic.  What does it mean to know something?  Siri can tell you the temperature in Buffalo.  Does that mean that she "knows" the temperature in Buffalo?  Planets move according to Newton's laws, does that mean that they "know" how to solve differential equations?

Even among humans it is far from clear what it means to know something.  The subjective sensation of being absolutely convinced of a false proposition is generally indistinguishable from being convinced of a true one.  So can it be said that flat-earthers "know" that the earth is flat?  Did Ptolemy "know" that the sun revolves around the earth?  This matters because Fitch's proof depends on the assumption that anything that is known must be true (KP->P).

But even being true is not necessarily enough.  In 1653, Christian Huygens calculated the distance from the earth to the sun and got very nearly the correct answer, but he got it right by pure luck.  In fact his calculation was completely bogus, relying on numerology and mysticism to guess that Venus was the same size as the earth.  That just happens to be true, but can it be said that Huygens knew it?

The "state of the art" in defining knowledge is to add the condition that a belief must not only be true but properly justified.  But that just begs the question: what does it mean to be "properly justified"?

"Possible" is no less fraught.  Consider this simple situation: you are about to flip a coin.  Can it be said that "it is possible that the coin will land heads-up, and it is possible that the coin will land tails-up"?  Most people would say yes.  But now consider the situation after you have flipped the coin but before it lands, or after it lands but before you have looked at it.  Are both heads and tails still possible?  What about after you look?  Is it possible that you see "heads" but in fact the coin is "tails" and you are suffering from a hallucination?  Is it possible that the coin is neither heads nor tails, but has disappeared or turned into two coins?  Even before you flip, are both outcomes really possible, or is your perception that they are both possible merely a product of your inability to predict the outcome?

If you believe that both outcomes are possible before the flip but not after, at what point did the situation change?  At the instant the coin landed?  Why not a microsecond before, or when it left your hand, or when your brain sent the nerve impulse to your hand to start it spinning?

Possibility can only ever be assessed relative to either ignorance or willingness to suspend disbelief and consider counterfactuals.  As I write this, I am wearing a black T-shirt.  So relative to my knowledge state, it is not possible that I am wearing a red T-shirt, but relative to your knowledge state it is possible (because I could be lying about wearing a black T-shirt).  We can also imagine (counterfactually) some alternate reality that is identical to actual reality except for the color of my T-shirt.  These are very different senses of possibility.  On the possibility-from-ignorance view, it is not possible that Hillary Clinton won the 2016 election or to solve the halting problem, but on the possibility-from-counterfactuals view, both are possible.

There is an interesting interplay between knowledge and both kinds of possibility.  The relationship with possibility-from-ignorance is obvious.  If you know P, then relative to your knowledge state, ~P is not possible.  On the other hand, your willingness to entertain counterfactuals can also be constrained by your knowledge.  Is it possible that the earth is flat?  That the square root of two is a rational number?  That Santa is real?

Since this whole mess started with a formal proof, let me offer up one of my own.  A while back I opined that free will cannot exist in a universe where there is an omniscient, infallible deity.  It turns out that you can render the argument as a formal proof.  Let KP mean God knows P, and let LP mean P is possible.  Let P be an arbitrary universally quantified proposition, and S be the particular proposition "I will choose to sin."  Then:

1.  KP ∨ K~P (For every proposition, God knows whether or not it is true, definition of omniscience)

2.  KP -> ~L~P (If God knows P, then it is not possible that P is false, definition in infallibility)

3.  LS ∧ L~S (It is possible that I will choose to sin, and it is possible that I will not choose to sin, definition of free will)

From these three premises we can conclude:

4. KS ∨ K~S (from premise 1)

5. Assume KS (set up for conditional proof)

6. ~L~S (2, 5 modus ponens)

7. KS -> ~L~S (conditional proof, discharge assumption 5)

8. K~S -> ~LS (by analogous conditional proof starting with the assumption K~S, and the logical tautology ~~P -> P)

9. ~L~S ∨ ~LS (From 4, 7 and 8 by the Constructive Dilemma)

10. L~S (from 3, by conjunction simplification)

11. ~LS (from 9 and 10 by disjunctive syllogism)

12. LS (from 3, by conjunction elimination)

13. LS ∧ ~LS (from 11 and 12) -- contradiction

Therefore, premises 1-3 cannot all be simultaneously true, QED.

#### 33 comments:

Luke said...

> You can try to do an end-run around this by restricting the domain of the logic to "tractable truths". The problem with this is that tractability inherently involves time, but Fitch's logic does not model time.

Why can't you just say the domain for p is "tractable truths" (candidate definition) and interact with the single place where one actually cares about the concept? In particular, for this point to really matter, I'm pretty sure it has to be the case that:

p   is tractable
p & ¬Kp   is intractable

Can you think of any instances where this is the case?

> What I do want to point out here is that there is actually a much deeper problem: Fitch assumes that it is possible to assign coherent meanings to the words "possible" and "know".

I was pretty much interpreting Kp as "have a proof that p" or "assume that p". Then, LKp would ostensibly be some much weaker proof that "maybe p". It would exist somewhere between (1) and (3):

(1) I have no idea whether p but it seems tractable so maybe I'll know in the future
(2) I have much more of a bead on the truth value of p
(3) I know whether p is true or false

(I already presented this (1)–(3) but with an empty (2); your response to that version was "I have no clue what you could possibly mean by that." If you don't like the above (2), I can work to see if I can come up with a more formal version. But the argument from Fitch is that it may not be possible to come up with a fully formal version of (2).)

The mechanics of L seem fairly straightforward:

(C) pLKp
(D) from ¬p, deduce ¬Lp

Do you think that possibility ought to be rendered differently? The trick here is that Fitch's paradox will obtain if you add any (E), (F), … So you have to question the very basic aspects offered, or offer some other way to render possibility that is anywhere close to the intuition which is [however questionably] modeled by the modal logic Fitch uses. If you cannot do either of these, then you are forced to admit that you just don't know if "possibly know" and "know" can be distinguished more strongly than Fitch's paradox permits.

It may be the case that I'm looking for the intermediate path from the energy state s1 to p1, where quantum mechanics prohibits us from giving anything but probabilities. This path can be a topological deformation; I don't need to assume the electron is in one place. Maybe the movement from "have no idea whether p" to "I know whether p" is just a quantum leap. Or maybe the only way to talk about intermediate states is something very different from the possibility Fitch used. I do suspect that the kind of thing being discussed here is very different from Bayesian inference.

Luke said...

> Since this whole mess started with a formal proof, let me offer up one of my own. A while back I opined that free will cannot exist in a universe where there is an omniscient, infallible deity.

That depends on how you define 'omniscient'. First let's talk about 'omnipotence': is it possible for God to withhold from using his full powers to create space for humans to act truly of their own accord? Parents can do this with their children; can God do it with his creation? If the answer is "no" let's explore. If the answer is "yes", then why cannot God prescind from knowing everything, to create the appropriate space for true free action (such that the moral responsibility is the human's instead of God's)?

I think that weak measurement and interaction-free measurement are two ways that God could gain some knowledge without fully determining the time-evolution of a system. This of course presupposes that he gains knowledge of reality in any way similar to how we do, but the point is that we really can think of means of interacting which are "partial" in the sense of determination. God could know enough and act enough to ensure that every promise made comes true, while simultaneously not needing to be in continuous control of every molecule in reality.

Phrased in terms of your KP ∨ K~P, not all P are yet known. This is trivially possible if we live in a growing block universe—if the universe is an open system instead of a closed system.

Ron said...

> Why can't you just say the domain for p is "tractable truths" (candidate definition)

I specifically said that you can do this: "You can try to do an end-run around this by restricting the domain of the logic to "tractable truths". Please go back and carefully re-read that sentence, its containing paragraph, and the paragraph that follows. Pay particular attention to the last sentence.

BTW, this is far from the first time (nor, sadly, the last -- see below) that one of your comments has made it obvious that you did not read the OP. My tolerance for that very near its end.

> That depends on how you define 'omniscient'.

See premise #1 and note that it is labelled "definition of omniscience". (See my point above about how your comments make it clear that you have not read the OP. I don't know whether it is your intent, but your behavior is becoming indistinguishable from a troll.)

> not all P are yet known

Then God is not omniscient according to my definition (and, I will wager, that of most Christians).

> is it possible for God to withhold from using his full powers to create space for humans to act truly of their own accord?

No. My proof does not assume God's omnipotence. Omniscience is sufficient to logically eliminate free will.

Luke said...

@Ron:

> > > You can try to do an end-run around this by restricting the domain of the logic to "tractable truths". The problem with this is that tractability inherently involves time, but Fitch's logic does not model time.

> > Why can't you just say the domain for p is "tractable truths" (candidate definition)

> I specifically said that you can do this: "You can try to do an end-run around this by restricting the domain of the logic to "tractable truths". Please go back and carefully re-read that sentence, its containing paragraph, and the paragraph that follows. Pay particular attention to the last sentence.

Erm, the bold makes clear that I quoted precisely that sentence. What I thought I was obviously objecting to was the next sentence: that to do so, you'd have to add *time* to Fitch's modal logic. I want to know why this is absolutely required for the proof to go through. The way I think of it is this way: as long as the p which shows up in step 1. is tractable, the only way intractable truths could be required for the proof to go through is to suppose that:

p   is tractable
p & ¬Kp   is intractable

If in fact there is no good argument to suppose this, then I don't see why one can't a priori restrict the domain to tractable truths, and then work inside a logic that does not include time. Maybe it's just that I'm not schooled in formal logic; if so, maybe you'll just have to throw up your hands and say that I'm "uncoachable". My guess is that without further explanation from me, not every rational person will be convinced by your mere utterance from Authority on High. But hey, maybe I'm deluded. I believe in God, after all!

> > That depends on how you define 'omniscient'.

> See premise #1 and note that it is labelled "definition of omniscience". (See my point above about how your comments make it clear that you have not read the OP. I don't know whether it is your intent, but your behavior is becoming indistinguishable from a troll.)

So when I object to the soundness of an argument I'm trolling, but when you object to the soundness of an argument it's serious?

> > not all P are yet known

> Then God is not omniscient according to my definition (and, I will wager, that of most Christians).

Empirical claim requires empirical testing. If Christians realize that Ron-omniscience is not required for God to fulfill all of his promises in the Bible, maybe they won't see it as all that important. Maybe they'll see it as an approximation good for most purposes, but not when you really dig into the free will matter. (Well, at least non-Calvinists.)

> > is it possible for God to withhold from using his full powers to create space for humans to act truly of their own accord?

> No. My proof does not assume God's omnipotence. Omniscience is sufficient to logically eliminate free will.

I talked about omnipotence because I think it's easier for one to think about withholding power to do things than withholding ability to know everything. But I don't see why either is in principle impossible.

Ron said...

> I want to know why this [adding time] is absolutely required for the proof to go through.

It's not required for the proof to go through (obviously), it's required for the logic to model reality.

Maybe this is the part you missed:

"Since this conclusion is plainly false, the game (and it really is just a game) is to find the flaw in the reasoning."

> So when I object to the soundness of an argument I'm trolling

Yes, though I suspect that you meant something different from what the words you wrote actually mean.

To "object to the soundness of an argument" means that you concede that the argument is sound, and you object to the fact that it is sound. That is almost the *definition* of trolling.

What I suspect you meant to say was something like, "If I argue that an argument is not sound..." But that is not what you were doing. I doubt you even know what "soundness" means in the context of a formal proof. Under no circumstances can you legitimately address the soundness of a formal proof by talking about *definitions*.

Hint: Fitch's proof is sound.

> I talked about omnipotence

Yes, you did, and by so doing demonstrated that you have, once again, completely missed the point.

Luke said...

@Ron:

> [OP]: You can try to do an end-run around this by restricting the domain of the logic to "tractable truths". The problem with this is that tractability inherently involves time, but Fitch's logic does not model time.

> Ron: It's not required for the proof to go through (obviously), it's required for the logic to model reality.

"the proof"—which version? Here they are:

1. (C) applied to all truths, tractable and intractable
2. (C′) restricted to tractable truths

Actually, (C′) does not make explicit reference to tractable truths, just possibly knowable truths:

(C′) ∀(p & LKp)((p & ¬Kp) → LK(p & ¬Kp))

It's just convenient to speak of (p & LKp) as "tractable truths". But it's not at all clear to me that you accept that the (C′) version of the proof is valid. (I mean 'valid', not 'sound'.)

As to requiring *time* to model reality, one does not need it to model every aspect of reality in every circumstance. There are plenty of formal systems which leave out this or that bit because those bits are not required for the topic under discussion. As I made quite clear in the previous thread, unless you can demonstrate that *time* is required in order to distinguish between possible and actual knowledge, it is not known that it must be included in the model.

> To "object to the soundness of an argument" means that you concede that the argument is sound, and you object to the fact that it is sound. That is almost the *definition* of trolling.

My apologies, I meant to say that I object to the implied claim that the argument is sound. It saddens me that you think "Luke is a troll" has appreciable probability by this point in our interactions with each other, but I shall accept full responsibility for that. All I can say is that truly convincing people takes a lot more effort, in my experience, than you seem to expect. (I know, from all the effort it took to convince me away from being a creationist. But it worked, via pure online discussion. I am a counterexample to all those claims that internet discussion never convinces anyone about important issues.)

> Yes, you did, and by so doing demonstrated that you have, once again, completely missed the point.

On this logic, I can say that QFT and/or GR is unsound, because they provide conflicting predictions near the event horizons of black holes. But that's obviously a silly statement because they're really good models elsewhere. You don't seem to be admitting this kind of situation when it comes to the modal logic used in Fitch's paradox.

Ron said...

> I meant to say that I object to the implied claim that the argument is sound.

Fine. Then advance an argument that demonstrates its unsoundness. (You won't be able to because the proof is in fact sound.)

> On this logic, I can say that QFT and/or GR is unsound, because they provide conflicting predictions near the event horizons of black holes.

That's right. You can. And you would be correct.

> But that's obviously a silly statement because they're really good models elsewhere.

No, it's not a silly statement. It is a crucially important observation. The fact that these models are unsound *and* are nonetheless exceptionally good models of vast swaths of reality is a deep mystery. Resolving this mystery will be one of the most important advances in the history of physics, if not the single most important.

> You don't seem to be admitting this kind of situation when it comes to the modal logic used in Fitch's paradox.

The situation w.r.t. Fitch is exactly the opposite: Fitch's argument is sound, but it is obviously *not* a good model of reality. This too is a mystery, but a much less important one, which is why no one really cares about it. It's a fun little puzzle, but it's no more than that.

BTW:

> I think it's easier for one to think about withholding power to do things than withholding ability to know everything. But I don't see why either is in principle impossible.

I've decided to respond to this but I want to be absolutely clear: this is a tangent. It has absolutely nothing to do with Fitch or my proof. This is a new topic.

Suppose I claimed that *I* was omniscient and omnipotent, and that I have decided to deploy this power to withhold parts of it from myself. In fact, the parts that I have decided to withhold from myself are precisely those that make me appear to be an ordinary human, even though I could at any time change my mind about that and restore myself to full omniscience and omnipotence and smite you like a bug (but I'm not going to because I don't want to). Would you believe me? If not, why? How would you go about showing my claim to be false?

(Note that my claim is *actually true* in at least one world.)

Luke said...

@Ron:

I have several clarifying questions. First: are you presupposing that there is a meaningful distinction between 'actual knowledge' and 'possible knowledge'? Fitch can be seen as basically collapsing any real distinction. If you are presupposing that such a meaningful distinction exists, do you think there is any obligation for me to grant that presupposition in our discussion?

> Then advance an argument that demonstrates its unsoundness. (You won't be able to because the proof is in fact sound.)

Are you making a sharp distinction between soundness and validity?

> > On this logic, I can say that QFT and/or GR is unsound, because they provide conflicting predictions near the event horizons of black holes.

> That's right. You can. And you would be correct.

Do you recognize a difference between the binary judgment of sound/​unsound vs. the "analog" judgment of "how good of a model is X given these conditions and those purposes"? Something can be unsound and yet a very good model, in some situation. Since you seem to like arguing with 100% precision, you claiming that X is unsound will probably be rather uninformative.

> Suppose I claimed that *I* was omniscient and omnipotent, and that I have decided to deploy this power to withhold parts of it from myself. In fact, the parts that I have decided to withhold from myself are precisely those that make me appear to be an ordinary human, even though I could at any time change my mind about that and restore myself to full omniscience and omnipotence and smite you like a bug (but I'm not going to because I don't want to). Would you believe me? If not, why? How would you go about showing my claim to be false?

Did you get the negations (or lack thereof) correct? What I'm seeing here is a godlike being who is appearing to be an ordinary human but has intentionally forgotten that he/​she/​it merely appears to be an ordinary human. I'm confused.

Ron said...

> Are you making a sharp distinction between soundness and validity?

In the case of a reductio proof (which my proof is) these are synonyms.

In the case of Fitch, I should have said that his argument is valid rather than sound.

> Something can be unsound and yet a very good model

Yes, that is exactly what I said:

"The fact that these models are unsound *and* are nonetheless exceptionally good models..."

I've had it. Since you are obviously not bothering to read what I write I see no point in engaging with you further. Life is too short for this sort of nonsense.

Luke said...

@Ron:

> > Something can be unsound and yet a very good model

> … Since you are obviously not bothering to read what I write

Says the person who wrote:

> Ron: Please go back and carefully re-read that sentence, its containing paragraph, and the paragraph that follows. Pay particular attention to the last sentence.

In my case, it's just one paragraph and its final sentence which you chose to ignore:

> Luke: Do you recognize a difference between the binary judgment of sound/​unsound vs. the "analog" judgment of "how good of a model is X given these conditions and those purposes"? Something can be unsound and yet a very good model, in some situation. Since you seem to like arguing with 100% precision, you claiming that X is unsound will probably be rather uninformative.

All I was trying to do was to stop using the term 'unsound' when it is uninformative. But apparently, this is a High Crime to you, even though you conflated soundness and validity yourself. Good grief.

Peter Donis said...

@Ron has bowed out but I'll give a try at a couple of points that strike me:

are you presupposing that there is a meaningful distinction between 'actual knowledge' and 'possible knowledge'? Fitch can be seen as basically collapsing any real distinction.

The fact that Fitch's paradox basically collapses this distinction was, I thought, a key thing that came out of previous threads on this topic. And since it seems obvious--at least to me, and I'm guessing to Ron based on those previous threads--that in the real world, there is a meaningful distinction between actual knowledge and possible knowledge, any argument that seems to be showing that there is no meaningful distinction must be wrong.

If you honestly believe that there is no meaningful distinction between actual knowledge and possible knowledge, then I don't see how we're going to resolve that by further discussion. We'll just have to agree to disagree and drop the whole thing.

Do you recognize a difference between the binary judgment of sound/​unsound vs. the "analog" judgment of "how good of a model is X given these conditions and those purposes"?

Sure, there's a difference, but it's irrelevant to the topic of discussion here, since Fitch's paradox and the argument supporting it use binary logic, not "analog" goodness of fit.

If your point in the sentence quoted above is that physical theories like QFT and GR are not judged by binary "true/false" logic but "analog" goodness of fit, I agree, but again, that's irrelevant to the topic here, and so is the point you raise about QFT and GR making different predictions in certain domains. They do (actually there are a lot of technicalities lurking here, but I don't see the point of going into them in this discussion), but that has nothing to do with Fitch's paradox that I can see.

Luke said...

@Peter Donis:

> The fact that Fitch's paradox basically collapses this distinction was, I thought, a key thing that came out of previous threads on this topic.

I am not at all convinced that Ron agrees. He seems to jump back and forth between formal/​abstract-land and empirical-land at the drop of a hat. It makes it very hard for me to see what his position is. But perhaps I'm just retarded.

> And since it seems obvious--at least to me, and I'm guessing to Ron based on those previous threads--that in the real world, there is a meaningful distinction between actual knowledge and possible knowledge, any argument that seems to be showing that there is no meaningful distinction must be wrong.

I too am inclined to think that there is a meaningful distinction between actual and possible knowledge. But if one cannot be constructed with formal logic, then I don't think one should assume or presuppose that in fact there is such a distinction. And in this conversation, it's just not clear to me how adding *time* actually helps one get such a distinction. Maybe that is because my ability to imagine in this realm is limited, but it's all I have and I've seen no other decent attempts be put forward. In light of that, I don't think I'm under any intellectual obligation to just grant that there does exist such a formally describable distinction.

> If you honestly believe that there is no meaningful distinction between actual knowledge and possible knowledge, then I don't see how we're going to resolve that by further discussion. We'll just have to agree to disagree and drop the whole thing.

I said in the previous thread that I reject (B) K(p & q) → (Kp & Kq). However, it seems to me that Ron does not want to reject that premise and I find that rather fascinating.

> Sure, there's a difference, but it's irrelevant to the topic of discussion here, since Fitch's paradox and the argument supporting it use binary logic, not "analog" goodness of fit.

That's not the point, the point is whether the modal logic used in Fitch is a good enough model for discussing possible vs. actual knowledge. Ron has claimed "no", but he hasn't actually provided a better way to construct a distinction. His claim that you need *time* is rather hand-wavy. For example, it still makes intuitive sense to me that one should need to do substantially less work to show possible knowledge than actual knowledge. But Ron and I cannot even agree on a way to construe "substantially less work". He's fighting me at every single turn, faulting me for failing to be utterly and perfectly logically rigorous while not adhering to the same standards himself—from my perspective which is apparently worthless, of course. Because I never listen, never pay attention, quote-mine, etc. (Sigh.)

> but that has nothing to do with Fitch's paradox that I can see.

Compare:

(I) Fitch's logic is unsound.
(II) Fitch's logic is not a good fit in the actual/​possible knowledge domain.

These are different statements. The first is as true as "(QFT ∧ GR) is unsound". And as [ir]relevant.

Peter Donis said...

@Luke:
I am not at all convinced that Ron agrees.

I thought his posts in previous threads were pretty clear. But in any case, I agree.

I too am inclined to think that there is a meaningful distinction between actual and possible knowledge. But if one cannot be constructed with formal logic, then I don't think one should assume or presuppose that in fact there is such a distinction.

Then, as I said, I don't think further discussion is going to be helpful. To me it's obvious that there is such a distinction. I don't think everything can be captured with formal logic.

I said in the previous thread that I reject (B) K(p & q) → (Kp & Kq).

I don't see how this relates to the distinction between actual and possible knowledge; in the logical language of the paradox, "actual knowledge" is the K operator and "possible knowledge" is the L operator. As I noted in previous discussions, when you look at the actual semantics of these operators, they amount to the same thing. But that has nothing to do with whether premise (B) is accepted or not, since that premise only relates to the K operator.

the point is whether the modal logic used in Fitch is a good enough model for discussing possible vs. actual knowledge. Ron has claimed "no"

Yes, and I agree with him.

he hasn't actually provided a better way to construct a distinction

So what? That just means none of us has *any* good model for discussing possible vs. actual knowledge, at least not in the context of formal logic. It doesn't mean we should use the obviously not good enough model.

Compare:

(I) Fitch's logic is unsound.
(II) Fitch's logic is not a good fit in the actual/​possible knowledge domain.

These are different statements.

I agree.

The first is as true as "(QFT ∧ GR) is unsound".

I agree in the sense that they make contradictory predictions in some domains. But I don't agree with the implied claim you are making that (QFT ∧ GR) is a formal logic model in the same sense as Fitch's modal logic is a formal logic model. See below.

And as [ir]relevant.

I disagree. Neither QFT nor GR are formal logic models; they are physical theories. Physical theories never claim to give exact, binary true/false answers. They always make predictions that have confidence intervals; how useful the predictions are depends on how much accuracy you need and how narrow the confidence intervals are. There is no real concept of "sound" vs. "unsound" at all; the domain is simply not formal logic and can't be treated as if it is.

None of that is true of Fitch's argument; it is explicitly framed in a formal logic model and should be treated as such. So saying that his logic is unsound (because at least one premise is false in the domain of interest) is a valid criticism of his model, even though it is not for a physical theory.

Luke said...

@Peter Donis:

> I thought his posts in previous threads were pretty clear.

Care to pick one out which you think unambiguously shows that Ron believes "Fitch's paradox basically collapses this distinction"? There's some subtlety here, because the less of a good match there is between K and L and what we intuitively think of as the distinction, the less we can say it even talks about the distinction—which is a big point of this very blog post.

> To me it's obvious that there is such a distinction. I don't think everything can be captured with formal logic.

I don't think it's helpful to make the reference domain "everything". Why shouldn't we expect a distinction between actual and possible knowledge to be capturable by formal logic?

> "possible knowledge" is the L operator

Actually, "possible knowledge" is LK, not L. Lp reads "possibly p".

> As I noted in previous discussions, when you look at the actual semantics of these operators, they amount to the same thing.

I see this comment, but I don't see why LKpp unless you accept all of (A)–(D).

> I agree in the sense that they make contradictory predictions in some domains.

Contradiction is a validity issue; I was talking of soundness—of the formalism matching reality. The contradiction with GR and QFT just means they cannot possibly both be sound near the event horizons of black holes. Sometimes talking to Ron is frustrating in this sense: I will talk about a model/​formalism which seems a good fit for the present conversation, and then he'll pick an example I judge to be way over there and say, "Therefore the model is unsound and we can just throw it away." Without justifying that the example actually isn't "way over there", this is an invalid form of arguing for many situations. How frequently do we require models to be utterly sound in all situations? Almost never, as far as I can tell.

> Physical theories never claim to give exact, binary true/false answers.

I don't see why that's relevant; the question at hand is whether the modal logic used in Fitch's paradox is a good enough model of the relevant bits of reality for the relevant purposes.

Peter Donis said...

The contradiction with GR and QFT just means they cannot possibly both be sound near the event horizons of black holes.

As a side note: that's actually not the case for all QFTs. It's only the case for certain hypothetical QFTs that are supposed to "solve" the black hole information paradox by making quantum effects non-negligible near the horizon. But nobody has ever experimentally tested any such QFT. All of the QFTs that have been experimentally tested--which means the Standard Model of particle physics, which includes all of them--predict that quantum field effects are negligible near the horizons of black holes with masses much greater than the Planck mass.

The place where GR has an issue no matter what QFT you look at is near the singularity of a black hole.

Peter Donis said...

Why shouldn't we expect a distinction between actual and possible knowledge to be capturable by formal logic?

First consider what "possible" means for a simple proposition that doesn't involve knowledge: for example, "Donald Trump is President of the United States". Call this proposition TP. In possible world semantics, TP is "possible" if it is true in at least one possible world. (It is "necessarily true" if it is true in all possible worlds; but I think we can assume for this discussion that TP is not necessarily true.)

Now consider knowledge: for example, "it is known that Donald Trump is President of the United States". To say that TP is "possibly known" is to say that it is known in at least one possible world. But to say that TP is "actually known" is...what? To say that it is true in all possible worlds? First, that would be "necessarily true", not just "true"; and second, KTP can't be true in all possible worlds because TP itself is not true in all possible worlds. (This assumes that only a true proposition can be known, but I don't think that premise is in dispute in this discussion.) But the only other possibility is that TP is "actually known" if it is known in at least one possible world--and that is the same as "possibly known".

(Note that the same problem appears for any proposition if we try to draw a distinction between it being "actually true" and "possibly true". The only real distinction that can be drawn in modal logic is between "possibly true" and "necessarily true", and that's not sufficient to capture "actual" vs. "possible".)

"possible knowledge" is LK, not L

Yes, you're right, I should have written LK.

Peter Donis said...

@me:
To say that it is true in all possible worlds? First, that would be "necessarily true", not just "true"; and second, KTP can't be true in all possible worlds because TP itself is not true in all possible worlds.

To clarify, by "KTP" I mean the proposition "TP is known". Or, to rephrase the sentences quoted above: "To say that TP is known in all possible worlds? First, that would be necessarily known, not just known; and second, TP can't be known in all possible worlds because it is not true in all possible worlds."

Luke said...

@Peter Donis:

> As a side note: that's actually not the case for all QFTs. It's only the case for certain hypothetical QFTs that are supposed to "solve" the black hole information paradox by making quantum effects non-negligible near the horizon. But nobody has ever experimentally tested any such QFT. All of the QFTs that have been experimentally tested--which means the Standard Model of particle physics, which includes all of them--predict that quantum field effects are negligible near the horizons of black holes with masses much greater than the Planck mass.

Ok, but if the experimentally-tested QFTs fail to resolve the black hole information paradox, then they appear to be poor models around black hole event horizons—or am I in error?

> In possible world semantics, TP is "possible" if it is true in at least one possible world.

I'm not sure why possible world semantics are required. If one discards (B), it seems one can obtain a meaningful distinction between possible and actual knowledge. Furthermore, there seem to me to be very good reasons to discard (B); I can present some from Robert B. Laughlin's A Different Universe: Reinventing Physics from the Bottom Down, if you'd like.

Notice that your shift to possible world semantics doesn't actually solve the problem; it doesn't introduce any meaningful gradations between possible and actual knowledge. It's still as binary as ever: some people know, the rest are completely in the dark.

> (Note that the same problem appears for any proposition if we try to draw a distinction between it being "actually true" and "possibly true". The only real distinction that can be drawn in modal logic is between "possibly true" and "necessarily true", and that's not sufficient to capture "actual" vs. "possible".)

I just don't see why I should accept this. The Gettier problem seems to provide a pretty good example of possible knowledge that isn't guaranteed to be actual.

Peter Donis said...

if the experimentally-tested QFTs fail to resolve the black hole information paradox, then they appear to be poor models around black hole event horizons—or am I in error?

We don't know. It's possible that a resolution to the black hole information paradox will be found that doesn't change anything at the event horizon, only near the singularity. If that's the case, then our current experimentally tested QFTs will work just fine at the horizon.

Many physicists have been making claims in pop science venues that amount to saying that the black hole information paradox has been resolved and we know the resolution requires changing our predictions about black hole horizons. But none of these "resolutions" have been experimentally tested, and none of them have any prospect of being experimentally tested any time soon. Physicists are notorious for trying to make it seem to lay people that their pet theories that they *hope* will turn out correct when they are experimentally tested, are on the same footing as theories that already *have* been experimentally tested. But such claims are not valid, though they are distressingly common.

your shift to possible world semantics doesn't actually solve the problem

I wasn't trying to present a solution to the problem; I was trying to describe the problem. If you don't like possible world semantics, translate it into any semantic model you like that satisfies the premises. The same problem will be there.

The Gettier problem seems to provide a pretty good example of possible knowledge that isn't guaranteed to be actual.

I don't see how this has anything to do with possible knowledge vs. actual. From the Wikipedia article, at least, it seems like quibbling over the meaning of "justified" with respect to beliefs. (Also, the examples given do not satisfy the Fitch premise that any known proposition must be true; they contain propositions that are considered "known"--the "justified beliefs"--but turn out to be false. To me, this means the beliefs weren't justified in the first place--how can you have a justified belief in a false proposition?)

Luke said...

@Peter Donis:

> We don't know. It's possible that a resolution to the black hole information paradox will be found that doesn't change anything at the event horizon, only near the singularity. If that's the case, then our current experimentally tested QFTs will work just fine at the horizon.

My real point is that the GR and QFT we know and love and use in all our GPS devices today won't both remain unscathed in resolving the black hole information paradox. Is that correct? I would prefer to have a way to phrase things which shows up as "differing predictions", but perhaps that simplicity is not available if I am to be correct according to the highest standards.

> If you don't like possible world semantics, translate it into any semantic model you like that satisfies the premises. The same problem will be there.

Why do I need to translate it to any semantic model? Why can't I talk about the amount of logical work done so far and compare the amount of logical work required to prove LKp vs. Kp? This permits a kind of "analog" measurement which doesn't exist if some people Kp while the rest do not and the jump from not knowing to knowing is instantaneous (= has no intermediate states which can be described).

> I don't see how this has anything to do with possible knowledge vs. actual.

If you see a barn while on the highway, you don't know whether it's a faÃ§ade or a real barn. So, reject (B) K(p & q) → (Kp & Kq). Separate out appearances from … substance. (I'm not sure I like the term 'substance' and there are hints of Kant's Ding an sich, but appearances really can deceive so we need to be able to talk about what is underneath.)

Peter Donis said...

@Luke:
My real point is that the GR and QFT we know and love and use in all our GPS devices today won't both remain unscathed in resolving the black hole information paradox. Is that correct?

It's correct even more generally than that: the GR and QFT we use today are inconsistent with each other, so at least one of them can't be exactly right, and at least one of them will have to change, even if we didn't know about the black hole information paradox. The BH information paradox is just one illustration of the inconsistency.

Why do I need to translate it to any semantic model? Why can't I talk about the amount of logical work done so far

I don't know how you would quantify "logical work" without a semantic model.

If you see a barn while on the highway, you don't know whether it's a faÃ§ade or a real barn.

What does this have to do with possible knowledge vs. actual? You're not even talking about the difference between K(this is a barn) and LK(this is a barn). You're talking about the difference between K(this is a barn) and K(this is a facade).

Peter Donis said...

@Luke:
If you see a barn while on the highway, you don't know whether it's a faÃ§ade or a real barn. So, reject (B) K(p & q) → (Kp & Kq).

I don't see the connection here either. If you see an object and you don't know whether it's a facade or a real barn, the proposition that describes your state of knowledge (assuming those are the only two possibilities) is K(p V q), not K(p & q). And nobody is claiming that K(p V q) -> Kp V Kq.

Luke said...

@Peter Donis:

> It's correct even more generally than that: the GR and QFT we use today are inconsistent with each other, so at least one of them can't be exactly right, and at least one of them will have to change, even if we didn't know about the black hole information paradox.

But where do the inconsistencies become measurable—that is, rise above the noise floor?

> I don't know how you would quantify "logical work" without a semantic model.

How about # of steps required? I imagine one would run into the equivalent of the uncomputability of Kolmogorov complexity in figuring out just how to do this, but that doesn't stop people like Ron from thinking you really can use Kolmogorov complexity in discussions like this.

> What does this have to do with possible knowledge vs. actual?

That which leads me to possibly know p but not actually know p can only be a significant difference if maybe ¬p. Otherwise, you're back at LKpKp.

> If you see an object and you don't know whether it's a facade or a real barn, the proposition that describes your state of knowledge (assuming those are the only two possibilities) is K(p V q), not K(p & q).

That's not what I get from (B) K(p & q) → (Kp & Kq). Instead, that premise indicates that if I know something at the aggregate/​high-level, then necessarily I know the component pieces. It means I can judge from appearance to constitution. This doesn't match my everyday experience of reality. Appearances can very easily mislead.

Peter Donis said...

where do the inconsistencies become measurable—that is, rise above the noise floor?

We don't know. All we know is that this doesn't happen in any domain that's accessible to our current experiments; everywhere we can actually measure, there is no problem.

Peter Donis said...

How about # of steps required?

Which requires you to have a semantic model of the computer that's doing the steps. (And yes, that makes the number of steps model-dependent.)

Peter Donis said...

That's not what I get from (B) K(p & q) → (Kp & Kq).

See, this is why Ron gets so frustrated with you. You're completely ignoring my response to what you said in your last post, and responding to something I didn't even say. *You* are the one who brought up seeing an object that could be a real barn or could be a facade, not me. If that is irrelevant to K(p & q) → (Kp & Kq), then why did you bring it up? And if you think it is relevant, why have you not responded to my post explaining why it seems obviously irrelevant to me, because when I translate "that object I see could be a real barn or could be a facade" into symbolic logic, I don't get K(p & q) → (Kp & Kq), but something else?

Peter Donis said...

that premise indicates that if I know something at the aggregate/​high-level, then necessarily I know the component pieces. It means I can judge from appearance to constitution.

That's not what I get from K(p & q) → (Kp & Kq). Let me translate back into plain English to explain why:

K(p & q) → (Kp & Kq): I know that object I see out there is a red barn; therefore I know that the object I see out there is a barn, and I know that the object I see out there is red.

Knowing something at an "aggregate/high-level", but not the component pieces, would be something like, in plain English: (a) I know this program is a text editor for code, but (b) I don't know the specific libraries it uses to do syntax highlighting or filesystem operations. But I don't see any way to translate (a) into a form K(p & q), where the two parts of (b) translate to not Kp and not Kq. So I don't see how your description matches the symbolic logic of premise (B).

Ron said...

> that doesn't stop people like Ron from thinking you really can use Kolmogorov complexity in discussions like this.

That depends on what you want to use it *for*. If you want to use it as a theoretical construct to illustrate an abstract point, then sure. If you want to use it in a context that would require you to actually know its numerical value, then no.

For example:

> > I don't know how you would quantify "logical work" without a semantic model.

> How about # of steps required?

And how exactly are you going to determine how many steps are required?

And why does this matter anyway? I am willing to stipulate for the sake of argument that it is possible to define a set of "tractable truths" in a way that has some reasonable correspondence to someone's intuition about the meaning of the word "tractable". Call that set TT. As long as you accept that TT must be finite, then I accept that TT can be rigorously defined somehow, even if you are too lazy and/or incompetent to actually come up with an acceptable definition. So what? How does this advance the point you are trying to make? (What the fuck *is* the point you are trying to make???)

Luke said...

@Peter Donis:

> Which requires you to have a semantic model of the computer that's doing the steps. (And yes, that makes the number of steps model-dependent.)

Ok, so what about counting the number of steps required in the smallest currently known proof? There are 11 steps in WP: Fitch's paradox of knowability § Proof. Fitch shows that how many ever steps is required to possibly know something, it is one more step to actually know it. (Or if one wants to count all the step in Fitch's proof, than a constant number of additional steps.)

> See, this is why Ron gets so frustrated with you. You're completely ignoring my response to what you said in your last post, and responding to something I didn't even say. *You* are the one who brought up seeing an object that could be a real barn or could be a facade, not me. If that is irrelevant to K(p & q) → (Kp & Kq), then why did you bring it up?

Peter, you and Ron portray yourselves as [vastly?] smarter than I am. I'm going to make certain assumptions in my interactions when that impression is in play. One is that you can easily make logical steps which I see as rather small. In this case, it is that (B) K(p & q) → (Kp & Kq) denotes a kind of logical connectivity between aggregate-level logical propositions and their constituent parts which we humans don't always have. That which appears to be an X is sometimes an X but sometimes it is not. Suppose that to really be an X, (q & r) have to obtain. And yet, (q & p) obtaining can look like X until I do additional work to distinguish. Well, this is a way I can envision possibly knowing X but not actually knowing X. In order to test whether it's really X I can undertake further actions.

I realize a response here will be: "But if you find out it isn't an X, then it was never an X." This would be an insistence that we never apply the same label in a way which could possibly be ambiguous. But I just don't see how this matches everyday experience at all. It also seems predicated upon LKpp; if I can prove that possibly I know p, then it must be that p. However, this seems to eviscerate the very notion of possibility.

> Knowing something at an "aggregate/high-level", but not the component pieces, would be something like, in plain English: (a) I know this program is a text editor for code …

I disagree: you're assuming it always is an X, but maybe it's an X of this type rather than that type. I'm saying it can appear to be an X but actually not be one at all. Hence my use of "real barn" (not "red barn").

Luke said...

@Ron:

> And how exactly are you going to determine how many steps are required?

I already linked you the 1989 paper On the number of steps in proofs.

> And why does this matter anyway?

Because an intuitive understanding has it that it should take more work to actually know something than possibly know something, than permitted via Fitch's paradox.

> As long as you accept that TT must be finite, then I accept that TT can be rigorously defined somehow, even if you are too lazy and/or incompetent to actually come up with an acceptable definition.

Here's how I started my first comment on this blog post:

> > You can try to do an end-run around this by restricting the domain of the logic to "tractable truths". The problem with this is that tractability inherently involves time, but Fitch's logic does not model time.

> Luke: Why can't you just say the domain for p is "tractable truths" (candidate definition) and interact with the single place where one actually cares about the concept? In particular, for this point to really matter, I'm pretty sure it has to be the case that:
>
> p   is tractable
> p & ¬Kp   is intractable
>
> Can you think of any instances where this is the case?

Given that I offered a candidate definition, why exactly did you write "even if you are too lazy and/or incompetent to actually come up with an acceptable definition"? I don't recall you offering any concrete objection to that candidate definition.

> So what? How does this advance the point you are trying to make? (What the fuck *is* the point you are trying to make???)

I have tried to explain it in many different ways; my response to you is now your response to me: "you are obviously not bothering to read what I write". Peter seems to have more patience than you ("Life is too short for this sort of nonsense."), so why not let us hack at this for a while longer to see if the result is something that is sufficiently interesting to you?

Peter Donis said...

@Luke:
you and Ron portray yourselves as [vastly?] smarter than I am.

You need to stop worrying about who is smarter and start concentrating on the actual subject of discussion. Perhaps Ron and I seem smarter to you because we are better at doing that. I can't speak for Ron, but thoughts about which of us is smarter, you or me, have not even crossed my mind. I have no interest in such questions, and they're irrelevant to the discussion anyway.

In this case, it is that (B) K(p & q) → (Kp & Kq) denotes a kind of logical connectivity between aggregate-level logical propositions and their constituent parts which we humans don't always have.

And my point, which I have made at least twice now and you have not responded to, is that I don't see the connectivity here that you are claiming. I've already given an example in plain English of K(p & q) → (Kp & Kq). It looks nothing like anything you've described. What you originally described was not knowing whether the object you see is a real barn or a facade. Logically, as I said already, that translates to K(p V q), where p is "this object I see is a real barn" and q is "this object I see is a facade". Or, if you insist on including the appearance, we could add proposition b, "this object I see looks like a real barn", and we would have Kb & K(p V q). Which doesn't look like K(p & q) → (Kp & Kq) or its negation.

That which appears to be an X is sometimes an X but sometimes it is not.

And logically, the way we capture this is would be to write, for example:

p: "That appears to be an X"
q: "That is an X"

Then the state of knowledge described would be Kp & ~Kq. Which still doesn't look the same as K(p & q) → (Kp & Kq) or its negation.

The rest of your post is just as irrelevant to the question of whether K(p & q) → (Kp & Kq) is true as what I quoted above, so I don't see the point of going on with it.

Ron said...

> Given that I offered a candidate definition, why exactly did you write "even if you are too lazy and/or incompetent to actually come up with an acceptable definition"?

Because the "candidate definition" you offered was:

> can be computed within the next 200 years, barring civilization collapse

This makes reference to TIME. But Fitch's logic is not a temporal logic. The worlds modeled by Fitch's logic are static. They have no time. The set TT is fixed. It does not change.

Recall from the OP:

"You can try to do an end-run around this by restricting the domain of the logic to "tractable truths". The problem with this is that tractability inherently involves TIME, but Fitch's logic does not model TIME." {Emphasis added.]

Does that help to clarify my frustration with you? I have said this over and over and over again and yet you either do not understand this very elementary point or you choose to ignore it.

Another example:

> it should take more work to actually know something than possibly know something

Work is a concept that only makes sense in a world with TIME. Without time there can be no change and without change there can be no work.

Peter Donis said...

@Luke:
It also seems predicated upon LKp → p

LKp → p is not a premise of Fitch's Paradox. What is a premise of Fitch's Paradox is p -> LKp. I trust you are aware that the latter is not logically equivalent to the former. :-)