The Cartoon Guide to Löb's Theorem

by Eliezer Yudkowsky1 min read17th Aug 2008103 comments

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Logic & Mathematics Art
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Lo!  A cartoon proof of Löb's Theorem!

Löb's Theorem shows that a mathematical system cannot assert its own soundness without becoming inconsistent.  Marcello and I wanted to be able to see the truth of Löb's Theorem at a glance, so we doodled it out in the form of a cartoon.  (An inability to trust assertions made by a proof system isomorphic to yourself, may be an issue for self-modifying AIs.)

It was while learning mathematical logic that I first learned to rigorously distinguish between X, the truth of X, the quotation of X, a proof of X, and a proof that X's quotation was provable.

The cartoon guide follows as an embedded Scribd document after the jump, or you can download as a PDF file.  Afterward I offer a medium-hard puzzle to test your skill at drawing logical distinctions.

Cartoon Guide to Löb's ... by on Scribd

Cartoon Guide to Löb's Theorem - Upload a Document to Scribd

Read this document on Scribd: Cartoon Guide to Löb's Theorem

And now for your medium-hard puzzle:

The Deduction Theorem (look it up) states that whenever assuming a hypothesis H enables us to prove a formula F in classical logic, then (H->F) is a theorem in classical logic.

Let ◻Z stand for the proposition "Z is provable".  Löb's Theorem shows that, whenever we have ((◻C)->C), we can prove C.

Applying the Deduction Theorem to Löb's Theorem gives us, for all C:

((◻C)->C)->C

However, those familiar with the logic of material implication will realize that:

(X->Y)->Y
  implies
(not X)->Y

Applied to the above, this yields (not ◻C)->C.

That is, all statements which lack proofs are true.

I cannot prove that 2 = 1.

Therefore 2 = 1.

Can you exactly pinpoint the flaw?

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"""(X->Y)->Y implies (not X)->Y"""

The arrow means "implies", right?

So,

(Smoke implies fire, therefore fire) implies (no smoke means fire)?

I don't get it.

1Ender8yI think that this is what the theorem means; If (X->Y) -> Y, then ~X -> Y (If it's true that "If it's true that 'if X is true, then Y is true,' then Y must be true," then Y must be true, even if X is not true). This makes sense because the first line, "(X->Y) -> Y," can be true whether or not X is actually true. The fact that ~X -> Y if this is true is an overly specific example of that "The first line being true (regardless of the truth of X)" -> Y. It's actually worded kind of weirdly, unless "imply" means something different in Logicianese than it does in colloquial English; ~X isn't really "implying" Y, it's just irrelevant. This doesn't mean that "(X -> Y) -> Y" is always true. I actually can't think of any intuitive situations where this could be true. It's not true that the fact that "if Jesus really had come back to life, Christians would be Less Wrong about stuff" implies that Christians would be Less Wrong about stuff even if Jesus really hadn't come back to life. Also, To anyone who wants to tell me I'm wrong about this; If I'm wrong about this, and you know because you've learned about this in a class, whereas I just worked this out for myself, I'd appreciate it if you told me and mentioned that you've learned about this somewhere and know more than I do. If logic is another one of those fields where people who know a lot about it HATE it when people who don't know much about it try to work stuff out for themselves (like Physics and AI), I'd definitely like to know so that I don't throw out wrong answers in the future. Thanks.
1lightlike8y"The fact that ~X -> Y if this is true is an overly specific example of that "The first line being true (regardless of the truth of X)" -> Y." This is basically correct; if ~X then X -> Y is always true because X never has the opportunity to be true, in a sense.
1VHanson7yI did study stuff like this a LONG time ago, so I respect your trying to work this out from 'common sense'. The way I see it, the key to the puzzle is the truth-value of Y, not 'whether or not X is actually true'. By working out the truth-tables for the various implications, the statement "((◻C)->C)->C" has a False truth-value when both (◻C) and C are False, i.e. if C is both unprovable and false the statement is false. Even though the 'material implication' "(X->Y)->Y implies (not X)->Y" is a tautology (because when the 1st part is false the 2nd part is true & vice-versa) that does not guarantee the truth of the 2nd part alone. In fact, it is intuitively unsurprising that if C is false, the premise that 'C is provable' is also false (of course such intuition is logically unreliable, but it feels nice to me when it happens to be confirmed in a truth table). What may seem counterintuitive is that this is the only case for which the premise is True, but the encompassing implication is False. That's because a 'logical implication' is equivalent to stating that 'either the premise (1st part) is False or the conclusion (2nd part) is True (or both)'. So, for the entire statement "((◻C)->C)->C" to be True when C is False, means "((◻C)->C)" must be False. "((◻C)->C)" is only False when "(◻C)" is True and "C" is False. Here's where the anti-intuitive feature of material implication causes brain-freeze - that with a false premise any implication is assigned a True truth-value. But that does NOT mean that such an implication somehow forces the conclusion to be true! It only affirms that for any implication to be true, it must be the case that "IF the premise is true then the conclusion is true." If the premise is False, the conclusion may be True or False. If the premise is True and the conclusion is False, then the implication itself is False! The translation of " (not ◻C)->C" as "all statements which lack proofs are true" is a ringer. I'd say the implication "If it is not t
0Ender7yJeepers. I haven't thought about this problem for a long time. Thanks. The answer that occurs to me for the original puzzle is that Yudkowsky never proved (◻(2 = 1) -> (2 = 1)). I don't know it that is actually the answer, but I really need to go do other work and stop thinking about this problem.

excellent south park reference, bro.

Pretty sure I see it, not sure if I'm supposed to give it away in the comments.

(It did not seem all that medium hard to me, so maybe I didn't get it after all, and in that case I had better not post it and embarass myself.)

Löb proved the following: for any C, Provable(Provable(C)->C)->Provable(C).

So, we may derive from PA soundness, that for any C, Provable(Provable(C)->C)->C.

Nobody proved, as you stated, that for any C (Provable(C)->C)->C.

Sasha, why doesn't it follow from the Deduction Theorem?

1Andrew Jacob Sauer1yBecause assuming Provable(C)->C as a hypothesis doesn't allow you to prove C. Rather, the fact that a proof exists of Provable(C)->C allows you to construct a proof of C.

The confusing part is this:

(C is provable -> C) -> C

I can grok the part inside the parenthases - that's what we'd all like to believe; that this system can assure us that everything it proves is true. The weird step is that this statement itself implies that C is true - I guess we either take this on faith or read the Deduction Theorem.

What I find most strange, though, is the desire to have the proof system itself distinguish between provability and truth. Isn't that impossible? A formal system only provides proofs given axioms; to it, the only me... (read more)

Tiiba: "(Smoke implies fire, therefore fire) implies (no smoke means fire)?" "therefore fire" should be left out. i) (Smoke implies fire) implies fire. ii) No smoke implies fire.

You're confused because i) is false.

Medium puzzle:

"Applying the Deduction Theorem to Löb's Theorem gives us, for all C:

((◻C)->C)->C"

Wouldn't it be: ((?C) -> C) -> ?C If the provability of C implies C's truth, then C is provable.

In other words, where you write "(X->Y)->Y", it should be "(X -> Y) -> X".

I think the error is that you didn't prove it was unprovable -- all provably unprovable statements are also provable, but unprovable statements aren't necessarily true.

In other words, I think what you get from the Deduction Theorem (which I've never seen before, so I may have it wrong) is Provable((Provable(C) -> C) -> C). I think if you want to "reach inside" that outer Provable and negate the provability of C, you have to introduce Provable(not Provable(C)).

The hypothesis was that PA proves that "if PA proves C, then C" This enabled it to be proved that "PA proves C" So I think what we actually get applying the deduction theorem is ?((◻C)->C)->?C

So, I was thinking about the misapplication of the deduction theorem and suddenly had an insight into friendly AI. (Stop me if this has been discussed =D )

The problem is you give the AI a set of 'axioms' or goals and it might go off and tile the universe. Obviously we don't know the full consequences of the logical system we're initiating otherwise we wouldn't need the AI in the first place.

The problem already exists in humans. Evolution gave us 'axioms' or desires. Some people DO go off and tile their universe: drugs, sex or work addictions, etc, etc. Thu... (read more)

Peter: I THOUGHT that I'm supposed to assume that there's smoke. (DNRTFA, too hard for my little brain)

The flaw is the instant you used the deduction theorem. You used it to go from

PA |- "◻C -> C" -> PA |- "C"

to

PA |- "(◻C->C) -> C"

What the induction theorem really says is

T + X |- Y implies T |- "X -> Y"

so what you really would have needed to apply the deduction theorm would have been

PA + "◻C -> C" |- C

do I win?

oops that should be "what the deduction theorem really says"

Boiling it down to essentials, it looks to me like the key move is this:

  • If we can prove X, then we can prove Y.
  • Therefore, if X is true, then we can prove Y.

But this doesn't follow - X could be true but not provable.

Is that right? It's ages since I did logic, and never to a deep level, so excuse me if this is way off.

Hint: Pinpointing the exact flaw in the proof that 2=1, requires you to mention one of the steps 1-10 of the (valid) proof of Lob's Theorem.

Any supposed analysis that does not tell you to mention a particular step, is inexact at best, and more probably mistaken.

I'm having some technical issues reading this.

What character is ◻? On my browser (Firefox 3), it looks like a box with four symbols in it, like this:


|25|

|FB|

How do I get it to display properly?

2dlthomas9yhttp://www.fileformat.info/info/unicode/char/25fb/index.htm [http://www.fileformat.info/info/unicode/char/25fb/index.htm]

Eliezer: "Any supposed analysis that does not tell you to mention a particular step, is inexact at best, and more probably mistaken."

I'm pretty sure my answer was completely correct. Care to point out the inexactness/mistakes?

The problem is that while Lob's theorem says that whenever we have ((◻C)->C), C is provable, it does not mean that there is a proof of C from ((◻C)->C). This means that you're use of the deduction theorem is incorrect. You can only say (((◻C)->C)->(◻C)). It is clear that (X->Y)->X does not imply (not X)->X, so your little trick doesn't work.

-1dlthomas9y(X->Y)->X and (X->Y)->Y have the same truth table.
2TobyBartels9yNot when X is false and Y is true.
3dlthomas9yMy bad :-\

If L:ob's theorem is true, then surely we can use it without any reference to the details of its proof? Isn't the point of a proof to obtain a black box, so that we do not have to worry about all the various intermediate steps? And in this case, wouldn't it defeat the purpose of having a proof if, to explain how the proof was misapplied, we had to delve deep into the guts of the proof, rather than just pointing to the relevant features of its interface?

Opps, I made an error when I said: "It is clear that (X->Y)->X does not imply (not X)->X, so your little trick doesn't work." This is false. This implication does exist, so we get:

((not ◻C)->(◻C))

This sentence is true just in case C is in fact provable, so we don't have a problem with your 2=1 cases.

Robert: "You can only say (((◻C)->C)->(◻C))"

No that's not right. The theorem says that if PA proves "◻C -> C" then PA proves C. so that's ◻(◻C -> C) -> ◻C.

The flaw is that the deduction theorem does not cross meta levels. Eliezer says "Löb's Theorem shows that, whenever we have ((◻C)->C), we can prove C." and goes on to claim that (◻C->C)->C. But he's intentionally failed to use quotes and mixed up the meta levels here. Lob's Theorem does not give us a proof in first order logic from ((◻C)->C) to C. It gives us a proof that if there is a proof of ((◻C)->C) then there is a proof of C. Which is an entirely diffirent thing altogether.

so what you really would have needed to apply the deduction theorm would have been PA + "◻C -> C" |- C do I win?

Why doesn't the given proof of Lob's Theorem, the steps 1-10, show that PA + "◻C -> C" |- C?

Eliezer: "Why doesn't the given proof of Lob's Theorem, the steps 1-10, show that PA + "◻C -> C" |- C?"

That's just not what it says. The hypothesis in step 2 isn't "◻C -> C" it's "◻(◻C -> C)". I suppose if the Hypothesis were "◻C -> C" we could try to find where it breaks. Step 7 is the only step that depends on 2 so it has to be there. Translating the amusing cartoons, we have

  1. ◻(◻C -> C)
  2. ◻(◻L -> ◻C)
  3. ◻(◻L -> C)

Lets say that instead we have

2'. ◻C -> C

Well what are we supposed to do with it? We're stuck. Just because ◻C -> C doesn't mean that PA can prove it.

Eliezer's password is [correct answer deleted, congratulations Douglas --EY].

Larry, interpret the smiley face as saying:

PA + (◻C -> C) |-

Then the steps 1-10 of Lob's proof would seem to take us from:

PA + (◻C -> C) |- (◻C -> C)

to

PA + (◻C -> C) |- C

If this were the case, the deduction theorem would in fact come into play and give us:

PA |- (◻C -> C) -> C

So there must be at least one step of the reasoning 1-10 that does not apply. Which one and why?

Eliezer: "Larry, interpret the smiley face as saying:

PA + (◻C -> C) |-"

OK ignoring the fact that this is exactly what it doesn't say, I suppose we could notice that every step is surrounded at the top by a happy-face-says, so, if we may be inclined to think we can take those off and get a proof inside PA, starting with "◻C -> C" as a hypothesis. Lets see what happens.

  1. ◻L <-> ◻(◻L -> C)
  2. ◻C -> C

ok these are our hypothesis.

  1. ◻(◻L->C) -> (◻◻L -> ◻C)

Modus Ponens works in PA, fine

  1. ◻L -> (◻◻L -> ◻C)

or... (read more)

1[anonymous]9yI tried summarizing it independently and came up with fewer steps and more narrative. Hope you don't me replying here or stealing your lovely boxes. PA lets us distribute boxes across arrows and add boxes to most things. Lob's hypothesis says we can remove boxes (or at least do around statement C in the following). 1. L is defined (in a difficult, confusing, self-referential way that this margin is to narrow to contain) such that ◻L is equivalent to ◻(◻L -> C) 2. Distribute the box over the second form: ◻L -> (◻◻L -> ◻C) 3. ◻L also means ◻◻L since we can add boxes, so by the inference "A->(B->C) + (A->B) -> (A->C)" we know ◻L -> ◻C 4. All of that was standard PA. Here it becomes funny; Lob's hypothesis tells us that ◻C can become C, so ◻L -> C 5. Adding a box to that gives ◻(◻L->C), which in 1. we said was the same as ◻L 6. And in 4. we found out that ◻L gives C, so finally C

Doug S.:

What character is ◻?
That's u+25FB ('WHITE MEDIUM SQUARE').

Eliezer Yudkowsky:

Larry, interpret the smiley face as saying:

PA + (◻C -> C) |- I'm still struggling to completely understand this. Are you also changing the meaning of ◻ from 'derivable from PA' to 'derivable from PA + (◻C -> C)'? If so, are you additionally changing L to use provability in PA + (◻C -> C) instead of provability in PA?

As near as I can make out, the flaw is the assumption that there isn't a proof of 2=1.

There's no proof of Peano arithmetic being consistent, right? (There's a proof of that lack of proof, of course. :))

Psy-Kosh: No that isn't the problem. If there is a proof that 1=2, then 1=2. If there isn't, then 1=2. Either way 1=2. The problem is the mixing of meta-levels / inappropriate use of the deduction theorem / tacit assumption that ◻a -> ◻b is the same as "a -> b".

Sebastian Hagen: no ◻X is PA |- "X". My best guess as to what Eliezer meant by "interpret the smiley face as saying.." is that he meant to interpret the cartoonproof as a proof from the assumption "(◻C -> C)" to the conclusion "C", rather than a proof from "◻(◻C -> C)" to "◻C".

Loeb's theorem shows that if PA proves "not P(C)->C", then PA proves C (using P(X) for "X is provable in PA").

It follows from that, using the Deduction Theorem, that PA proves "(not P(C)->C)->C"; it does not follow that "(not P(C)->C)->C" is a logical tautology.

From that, you obtain, similarly, that PA proves "(not P(C)) -> C". And you say "I cannot prove that 2 = 1". Sure, you cannot, but can PA? That's the real question.

There's nothing particularly surprising about PA proving &... (read more)

Larry, a simple 8 would have sufficed. :) (That's what Douglas wrote.)

I just tested and anecdotally confirmed a hypothesis made with very little data: I suspected that neither Douglas Knight nor Larry D'Anna, the two who pinpointed 8 as the critical step, would be among the objectors to my metaethics. (Either of them can torpedo this nascent theory by stating otherwise.)

"I just tested and anecdotally confirmed a hypothesis made with very little data: I suspected that neither Douglas Knight nor Larry D'Anna, the two who pinpointed 8 as the critical step, would be among the objectors to my metaethics. (Either of them can torpedo this nascent theory by stating otherwise.)"

I like your metaethics just fine.

Okay, I still don't see why we had to pinpoint the flaw in your proof by pointing to a step in someone else's valid proof.

Larry D'Anna identified the step you were looking for, but he did it by trying to transform the proof of Lob's Theorem into a different proof that said what you were pretending it said.

I think, properly speaking, the flaw is pinpointed by saying that you were misusing the theorems, not that the mean old theorem had a step that wouldn't convert into what you wanted it to be.

I've been looking more at the textual proof you linked (the cart... (read more)

How is the problem not going from 6 to 7? The false argument you gave uses PA + ((◻C)->C)|- C instead of PA |- ?((◻C)->C) when invoking the deduction theorem. In other words, the false argument is exactly the same except when it uses step 2 (which is Löb's Hypothesis). More specifically, making L doesn't depend on Löb's Hypothesis, and steps 1-6 don't depend on it either (as in they are still true in PA without the Hypothesis). Furthermore, assuming step 7 (ie PA + whatever you add |- step 7) step 8 must follow by A1.

Larry: I'm a bit confused, could you clarify what you mean? Thanks. (Incidentally, I'm pretty sure what Brain said is basically a much more detailed way of saying what I said. That is, I think Brian's right, and, further, that what I was saying was similar (though, obviously, far less detailed)

So what am I misunderstanding?

Again, thanks.

Fascinating that you could present Lob's theorem as a cartoon, Eliezer.

One tiny nitpick: The support for statement 9 seems to be wrong. It reads (1, MP) but that doesn't follow. Perhaps you mean (8, 1, MP)

1Cyan8yI was just working through the cartoon guide pursuant to this recent discussion post [http://lesswrong.com/r/discussion/lw/h9k/probabilistic_l%C3%B6b_theorem/], and I found this minor mistake too.

Psy-Kosh: There are two points of view of where the flaw is.

My point view is that the flaw is here:

"Löb's Theorem shows that, whenever we have ((◻C)->C), we can prove C. Applying the Deduction Theorem to Löb's Theorem gives us, for all C: ((◻C)->C)->C"

Because, in fact Lob's Theorem is: (PA |- "◻C -> C") => (PA |- "C") and the Deduction theorem says (PA + X |- Y) => (PA |- "X->Y"). We don't have PA + X proving anything for any X. The deduction theorem just doesn't apply. The flaw is that the ... (read more)

Larry D'Anna: Thanks, I think I understand the Deduction Theorem now.

We don't have PA + X proving anything for any X.

It seems to me that we do have (PA + "?(◻C -> C)" |- "?C")

from which the deduction theorem gives: (PA |- "?(◻C -> C) -> ?C") which is Löb's Theorem itself.

simon: Let me explain some of the terminology here, because that may be where the confusion lies.

A scentence is a finite string symbols that satisfies a certain set of syntactic constraints.

A theory is a (possibly infinite) set of sentences. PA is a theory.

A proof from a theory T is a finite set of sentences, each of which is either an element of T, or follows from the ones before according to a fixed set of rules.

The notation PA + X, where X is a sentences is just the union of PA and {X}.

The notation PA |- Y means that a proof from PA that ends in Y ex... (read more)

Hmm. I was thinking that Löb's Theorem was a theorem in PA, in which case the step going from

PA + ?(?C -> C) |- ?(?L -> C)

to

PA + ?(?C -> C) |- ?(?(?L -> C))

seems legitimate given

PA |- (?X -> ?(?X))

which we ought to be able to use since PA is part of the theory before the |- symbol.

If we don't have PA on the left, can we use all the "ingredients" without adding additional assumptions?

In any case, if we do not use the deduction theorem to derive the implication in Löb's Theorem, what do we use?

simon:

I was thinking that Löb's Theorem was a theorem in PA

It is a theorem about PA, although everything the statement and the proof of it can be expressed in PA if you like, in which case it is a theorem inside of PA about PA. There's no contradiction there, as long as you have everything quoted properly.

in which case the step going from PA + ?(?C -> C) |- ?(?L -> C) to PA + ?(?C -> C) |- ?(?(?L -> C)) seems legitimate given PA |- (?X -> ?(?X))

That's a valid deduction, but I don't think it's a step that anyone has written down ... (read more)

I know nothing about math, and cannot even follow what's being argued (keep this in mind when programming the fAI), but it's really funny to see how one of the 2 guys that got the right answer is on this like white on rice. Is that part of the experiment?

It's not clear to me where you are going with it.

To argue that a proof is being made concluding ?C using the assumption ?(◻C -> C) given the theory PA, to which proof we can apply the deduction theorem to get (PA |- "?(◻C -> C) -> ?C") (i.e. my interpretation of Löb's Theorem)

We use 10 steps, 9 of which are proofs inside of PA

But the proof uses an additional assumption which is the antecedent of an implication, and comes to a conclusion which is the consequent of the implication. To get the implication, we must use the deduction theorem ... (read more)

simon:

To argue that a proof is being made concluding ?C using the assumption ?(◻C -> C) given the theory PA, to which proof we can apply the deduction theorem to get (PA |- "?(◻C -> C) -> ?C") (i.e. my interpretation of Löb's Theorem)

OK so the question marks are boxes right? In that case then yes, PA |- "?(?C -> C) -> ?C". This is OK. The contradiction comes if PA |- "(?C->C)->C". But morally this doesn't have anything to do with the deduction theorem. PA proves Lob because everything in the proo... (read more)

I went to the carnival and I met a fortune-teller. Everything she says comes true. Not only that, she told me that everything she says always comes true.

I said, "George Washington is my mother" and she said it wasn't true.

I said, "Well, if George Washington was my mother would you tell me so?" and she refused to say she would. She said she won't talk about what would be true if George Washingto was my mother, because George Washington is not my mother.

She says that everything she says comes true. She looked outside her little tent, and ... (read more)

2TobyBartels9yYou can tell that J_Thomas2's "if" (which is the counterfactual "if") is not the "if" of material implication (which is what appears in Loeb's theorem) from the grammar: "if George Washington was my mother" rather than "if George Washington is my mother".

Let me try to say that clearer.

Suppose that A is false.

How the hell are you going to show that if PA proves A true then A will be true, when A is actually false?

If you can't prove what would happen if PA proved A true when A is actually false, then if you can prove that if PA proves A is true then A has to be true, it must be that A is true in the first place.

If this reasoning is correct then there isn't much mystery involved here.

One more time. If PA proves you are a werewolf, then you're really-and-truly a werewolf. PA never proves anything that isn't ac... (read more)

J Thomas:

Once more through the mill. If PA proves that 6 is a prime number, then 6 is really a prime number. Can you prove that if PA proved 6 was a prime number then 6 would be a prime number? How would you do it?

If PA |- "forall x y . x y = 6 => |x|=1 || |y|=1" then N |= "forall x y . x y = 6 => |x|=1 || |y|=1" (N = the natural numbers equiped with + and ) so for all x and y in N, N |= ",x ,y = 6 => |,x|=1 || |,y|=1" (where ,x means a constant symbol for x) if xy = 6 then N |= ",x ,y = 6" so theref... (read more)

But Larry, PA does not actually say that 6 is prime, and 6 is not prime.

You could say that if PA proved that every theorem is false then every theorem would be false.

Or what would it mean if PA proved that Lob's theorem was false?

It's customary to say that any conclusion from a false premise is true. If 6 is prime then God's in his heaven, everything's right with the world and we are all muppets. Also God's in hell, everything's wrong with the world, and we are all mutant ninja turtles. It doesn't really matter what conclusions you draw from a false premis... (read more)

"But Larry, PA does not actually say that 6 is prime, and 6 is not prime."

Well of course 6 isn't prime. But if PA said it was, then it would be. There's nothing invalid about proving that A -> B if you know ~A. It's just not very useful. But for a somewhat less vacuous example, let RH be the riemann hypothesis. Then if PA |- RH then RH is true and if PA |- ~RH then RH is false. At least one of these implications has a false hypothesis, but they are both perfectly valid.

Larry, one of them is counterfactual.

If you draw implications on a false asumption then the result is useful only to show that an assumption is false.

So if PA -> 1=2 then PA -> 1<>2. How is that useful?

If PA -> 6 is prime then PA also -> 6 is not prime.

Once you assume that PA implies something that PA actually implies is false, you get a logical contradiction. Either PA is inconsistent or PA does not imply the false thing.

How can it be useful to reason about what we could prove from false premises? What good is it to pretend that PA is inconsistent?

J Thomas: "How is that useful?"

I'm just answering your question

"Can you prove that if PA proved 6 was a prime number then 6 would be a prime number? How would you do it?"

I'm not stating that proving implications with false antecedent is particularly useful, just that it is valid. Also aside from 6 being prime it is true that for any sentence phi, ZF |- "if PA |- phi then phi", but that ZF cannot even say, yet alone prove that "forall phi. if PA |- phi then phi". But it can prove "forall phi. if PA |- phi then N |= phi".

Larry, you have not proven that 6 would be a prime number if PA proved 6 was a prime number, because PA does not prove that 6 is a prime number.

The theorem is only true for the phi that it's true for.

The claim that phi must be true because if it's true then it's true, and if it's false then "if PA |- phi then phi" has an officially true conclusion whenever PA does not imply phi, is bogus.

It's simply and obviously bogus, and I don't understand why there was any difficulty about seeing it.

I thought of a simpler way to say it.

If Hillary Clinton was a man, she wouldn't be Bill Clinton's wife. She'd be his husband.

Similarly, if PA proved that 6 was prime, it wouldn't be PA. It would be Bill Clinton's husband. And so ZF would not imply that 6 is actually prime.

J Thomas
Larry, you have not proven that 6 would be a prime number if PA proved 6 was a prime number, because PA does not prove that 6 is a prime number.

No I'm afraid not. You clearly do not understand the ordinary meaning of implications in mathematics. "if a then b" is equivalent (in boolean logic) to ((not a) or b). They mean the exact same thing.

The claim that phi must be true because if it's true then it's true

I said no such thing. If you think I did then you do not know what the symbols I used mean.

It's simply and obviously bogus, and I... (read more)

From the medium hard question, we have "Löb's Theorem shows that, whenever we have ((◻C)->C), we can prove C." From the cartoon, the hypothesis of Lob's theorem is, " If PA proves 'PA proves X, then X' then PA proves X", so to apply the theorem to the situation outlined in the question we would be required to write instead of "((◻C)->C)" in the first sentence the following "PA proves '((◻C)->C)' ". Then by the deduction theorem, we'd have the following conclusion: '◻('(◻C)->C')->◻C' and so the trouble with material implication wouldn't arise because the first '->' is in a quoted sentence.

Trying the puzzle, before I read the other comments:

Löb's Theorem shows that, whenever we have ((◻C)->C), we can prove C.

I think this is the flaw. Löb's Theorem shows that whenever we can prove ((◻C)->C) in PA, then we can prove C in PA. So instead of ((◻C)->C)->C, it would actually be ◻((◻C)->C)->◻C.

At first I looked at ◻((◻C)->C)->◻C and thought it had a similar problem — if we let C be the statement 2 = 1, then clearly C = false, so ◻((◻C)->false)->◻C; so ◻(true)->◻C; and presumably you can prove tautologies in PA, s... (read more)

(not ◻C)->C

I was confused, because that statement is correct. Let F be a canonical false statement. Then F->C, and hence (not ◻C)->(not ◻F). But (not ◻F) is the consistency statement of PA. So (not ◻C) implies consistency of PA, which implies anything, including C.

2Will_Sawin10yPeano's consistence doesn't imply anything, the provability of Peano's consistency implies anything.
0Stuart_Armstrong10yYes, but you can get from "there is no proof of proposition C" to "there is no proof of the canonical false proposition F". The second is often taken to be the definition of the provability of Peano's consistency, I believe.
2Will_Sawin10yIt think the provability of consistency would be: "There is a proof that there is no proof of the canonical false proposition F"
0Stuart_Armstrong10yI'm using Peter Smith's definition (see http://www.logicmatters.net/resources/pdfs/gwt/GWT.pdf [http://www.logicmatters.net/resources/pdfs/gwt/GWT.pdf] , Godel without too many tears). In definition 59 (on page 1 of part 10), he identifies consistency with "not a proof of the canonical false statement". This is a valid statement within Peano arithmetic. And it's logical consequences are... anything.
2Will_Sawin10yYou're confusing consistency with a proof of consistency. Theorem 56: Consistency implies no proof of consistency. Which is of course where you get: Proof of consistency implies inconsistency. Which gives you: Proof of consistency implies anything.
2Stuart_Armstrong10yI think you're right.... I was commiting the same mistake is above, using the first derivability condition and assuming that Peano arithmetic could treat it as a statement in Peano arithmetic - which it isn't.

If you really want to see the truth of the theorem at a glance, I found a much simpler proof on page 7 of this writeup by Aaronson.

Posting before reading other comments...

The flaw is in the very last step: that there is no proof of 2=1. If PA is inconsistent then contradictions are provable in PA and then anything else is too, including 2=1. To prove that there is no proof of 2=1 is to prove that PA is consistent, which is impossible by Godel's theorem.

[-][anonymous]9y 0
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You once wrote on the SL4 mailing list

It appears to me that for the GM to execute a change that switches to a new theorem-prover or proof-verifier, the Gödel Machine must prove that if the proof-verifier accepts P, then P.

The verifier has to translate a meta-language result into an object-language result.

Löb's Theorem establishes that for any proof system capable of encoding its own proof process, |- Provable('P') -> P if and only if |- P.

But if an AI does believe that (AI |- P) -> P, Löb's construction permits an automatic proof of P

You didn'... (read more)

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[-][anonymous]9y 1

Here's the solution for your AI reflection problem.

In this SL4 post you equivocate ⊢ ◻P and ⊢ P . We are are allowed to believe P from a proof of P, just not from a proof of provability of P. Of course, whenever we have ⊢P, we'll have ◻P, so it's an understandable attribute substitution. Modifying your minimal rewrite criterion with this in mind, we might require that, for interpretations supporting derivability, S1 and a subsystem of S2 be mutually interpretable. Makes sense?

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[-][anonymous]9y 0

The drawing convention used in the cartoon guide strongly suggests that you're making an incorrect substitution when you think about this.

When you draw "PA says (X)", that corresponds in the symbolic proof of Löb's theorem to "PA⊢X", and when you draw "PA says (PA says X)" that corresponds to PA⊢◻X. The notation makes it look like "PA⊢◻X" is just the second order version of "PA⊢X", i.e. a second application of "PA says" (PA⊢). But of course "PA⊢(PA⊢X)" is the same as "P⊢X", not PA⊢◻X.

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Is "I cannot prove that 2 = 1." Supposed to mean that a you can prove that it is impossible to prove 2=1?

0Manfred8yA proof that you cannot prove that 2=1 would be a proof of consistency, so certainly this is a problematic statement.

Sanity check: If I'm understanding Eleizer and D'Anna correctly, step 8 is valid in PA but not in classical (= first order?) logic. The deduction theorem only applies to proofs in classical logic, therefore this part:

"Applying the Deduction Theorem to Löb's Theorem gives us, for all C:

((◻C)->C)->C"

is incorrect; you can't use the deduction theorem on Lob's Theorem. Everything after that is invalidated, including 2=1.

Is that more or less the point here? I have a good head for math but I'm not formally trained, so I'm trying to be sure I ac... (read more)

0VAuroch7yPeano Arthimetic is a first-order logic, so I'm reasonably certain the deduction theorem applies.

A simple explanation of a flaw that makes no reference to Lob's Theorem, meta-anythings, or anything complicated. Of course, spoilers.

"Let ◻Z stand for the proposition "Z is provable". Löb's Theorem shows that, whenever we have ((◻C)->C), we can prove C."

This statement is the source of the problem. For ease of typing, I'm going to use C' = We can prove C. What you have here is (C'->C)->C'. Using Material Implication we replace all structures of the from A->B with ~A or B (~ = negation). This gives:

~(~C' or C) or C`

Usin... (read more)

0VAuroch7yYou are wrong, and I will explain why. If you "have ((◻C)->C)", that is an assertion/assumption that ◻((◻C)->C). By Loeb's theorem, It implies that ◻C. This is different from what you wrote, which claims that ((◻C)->C) implies ◻C.
0Zaq4yWow. I've never run into a text using "we have" as assuming something's provability, rather than assuming its truth. So the application of the deduction theorem is just plain wrong then? If what you actually get via Lob's theorem is ◻((◻C)->C) ->◻C, then the deduction theorem does not give the claimed ((◻C)->C)->C, but instead gives ◻((◻C)->C)->C, from which the next inference does not follow.
1VAuroch3yI don't think I've ever used a text that didn't. "We have" is "we have as a theorem/premise". In most cases this is an unimportant distinction to make, so you could be forgiven for not noticing, if no one ever mentioned why they were using a weird syntactic construction like that rather than plain English. And yes, rereading the argument that does seem to be where it falls down. Though tbh, you should probably have checked your own assumptions before assuming that the question was wrong as stated.

Gur snpg gung lbh crefbanyyl pnaabg cebir 2=1 qbrf abg zrna gung vg vf hacebinoyr. Vs lbh rire cebir (va CN) gung gurer vf ab cebbs (va CN) gung 2=1, gura lbh unir vaqrrq cebirq CN vapbafvfgrag.

Gur qrqhpgvba gurberz qbrf abg fgngr gung G ∪ {◻⊥→⊥} ⊢ ⊥, orpnhfr ◻ vf n cebinovyvgl cerqvpngr sbe G. Naq, va snpg, CN ∪ {◻⊥→⊥} vf whfg CN+Pba(CN), juvpu vf bs pbhefr pbafvfgrag.

I may not removed the flaw entirely, but I have definitely changed it into a less-obviously bad flaw. And also used Loeb's Theorem to derive Goedel's, or a close analogue.

The summary statement of Loeb is wrong. It is not the case that (◻X->X) -> X, it is only the case that ◻(◻X->X) -> ◻X. That is to say, that if (a proof of X implies X) is provable, then X is provable.

Using Deduction here, we get only ◻(~◻X) -> ◻X, which in English is "If it is provable that X is unprovable, then X is provable", or in other words "If PA is proved complete, it is proved inconsistent."

The download link for the Cartoon PDF Guide is outdated, it should be: http://yudkowsky.net/assets/pdf/LobsTheorem.pdf

Yrg'f frr, V xabj sbe n snpg gung ¬(CN ⊢ (◻P → P) → P), gung vf, gung guvf fhccbfrq "nccyvpngvba" bs gur Qrqhpgvba Gurberz qb Yöo'f Gurberz vf abg n gurberz bs CN.

Yöo'f Gurberz vgfrys vf whfg gung -

Buuuu tbg vg.

Yöo'f Gurberz fnlf gung vs CN ⊢ ◻P → P, gura CN ⊢ P; be, fvzcyl, gung CN ⊢ ◻(◻P → P) → ◻P. Gur Qrqhpgvba Gurberz fnlf gung sbe "ernfbanoyr" (va n pregnva frafr) qrqhpgvir flfgrzf va svefg-beqre ybtvp naq nal frg bs nkvbzf Δ, vs vg'f gur pnfr gung, sbe fbzr N naq O, Δ ∪ {N} ⊢ O, gura Δ ⊢ N → O. Ubjrire, vg vf abg gur pnfr gung CN ... (read more)

Did EY post the correct answer anywhere? I've been thinking for hours and give up. The comments make no sense. The best I can tell from Larry_D'Anna's comment is that axiom 1 is wrong. But it doesn't seem wrong and I am very confused. Or maybe he's saying the way it's used in step 8 is wrong. But that seems a very straightforward inference from it. Either way, it must be correct, because Lob's theorem is an actual theorem. EY didn't just make it up to trick us.

So I went to wikipedia and it does say the same thing:

◻(◻C->C)->◻C

If you subtract the quota... (read more)

1AlexMennen5yStep 8 relies on the lemma that ◻X→◻◻X. If you try running through the proof with the ◻ removed from both sides, the corresponding lemma you would need to make step 8 work is X→◻X, but this is not true.
0[anonymous]5y[deleted]

I don't think there's a flaw in this argument. I'm pretty sure that is a theorem of PA for any sentence . However, even if we consider to be a sentence like , that does not mean that we can conclude that is a theorem of PA, since proving that there doesn't exist a proof for a given sentence in PA is equivalent to proving that PA is consistent, which is not possible if we assume that PA is consistent.

4SamEisenstat7mo*I* think that there's a flaw in the argument. I could elaborate, but maybe you want to think about this more, so for now I'll just address your remark about ¬□C→C, where C is refutable. If we assume that ¬□ C→C, then, since C is false, ¬□C must be false, so □C must be true. That is, you have proven that PA proves □C, that is, since C is contradictory, PA proves its own inconsistency. You're right that this is compatible with PA being consistent--PA may be consistent but prove its own inconsistency--but this should still be worrying.
1jpulgarin6moThanks Sam. I thought about this some more and realized where I went wrong - I was applying the deduction theorem incorrectly (as other comments in this thread have pointed out). By the way, when you say that PA proves its own inconsistency, do you mean that PA⊢□(¬Con(PA)) or that PA⊢¬Con(PA)? From your reasoning I agree that if we assume ¬□C→C then we can arrive at PA⊢□C and PA⊢□(¬C), from which we can conclude that PA⊢□(¬Con(PA)). If you meant that PA⊢¬Con(PA) though, could you explain how you arrived at that?