Talk:What the Tortoise Said to Achilles

Modus ponens

Concerning the entry for "What the tortoise said to Achilles":

Section: Summary of the Dialogue Some errors of detail:

1. To describe the property of being Euclidean as a weakened form of transitivity is vague, and also inaccurate. Neither property standing alone implies the other; but given reflexivity, the Euclidean property is equivalent to the conjunction of transitivity and symmetry. In this latter sense, one could say, loosely, that being Euclidean is a strengthened (not weakened) form of transitivity.

2. “The transitive property” is poor English; one should say “the transitivity property”.

3. The argument form mentioned is not a syllogism in the technical sense of that term. It is an application of Modus Ponens, accompanied by implicit universal quantifier elimination and implicit conjunction introduction.

Section: Explanation Many weakness, some minor and some very serious:

1. The phrase “…that arises from modus ponens deductions” is sloppy. It should be “that arises when we attempt to justify modus ponens as a form of argument”.

2.There are several inaccuracies in the passage: “The regress problem arises because a prior principle is required to explain logical principles, here modus ponens, and once that principle is explained, another principle is required to explain that principle. Thus, if the causal chain is to continue,…”.

Specifically: (a) The word “because” is inappropriate; it should be “if”. (b) The word “explain” is vague and inappropriate, it should be “justify”. (c) The word “causal” is totally misguided – it is not a causal chain but a chain of justifications. One could say a chain of proximate grounds or, in the jargon, a doxastic chain, broadly speaking an epistemic chain, and also a “converse inferential” chain. But nothing to do with causality!

3. There are many errors in the paragraph: “Hence de modus ponens, [P ∧ (P → Q)] ⇒ Q, is a valid logical conclusion according to the definition of logical implication just stated. Demonstrating the logical implication simply translates into verifying that the compound truth table produces a tautology. But the tortoise does not accept on faith the rules of propositional logic that this explanation is founded upon. He asks that these rules, too, be subject to logical proof. The Tortoise and Achilles don't agree on any definition of logical implication”.

Specifically: (a) The sign ⇒ is inappropriate; it should be the standard turnstile sign for logical consequence or else the standard arrow sign → for material implication. (b) Under neither reading does the expression [P ∧ (P → Q)] (turnstile sign) Q resp. [P ∧ (P → Q)] → Q stand for “a valid logical conclusion”. In the case of the turnstile, it stands for an instance of the relation of logical consequence and, in the case of the arrow it stands for a certain tautology. (c) The alleged demonstration offered itself implicitly uses modus ponens (It proceeds: The expression is a tautolology; if it is a tautology then it is acceptable; so it is acceptable) and so is useless for the justificatory purpose in question. (d) The matter of whether the tortoise and Achilles have agreed on a specific definition of validity is irrelevant to the issue – the issue arises equally for any other logic, such as intuitionistic logic – that admits modus ponens.

4. The following paragraph of the entry (“In addition…)” is quite irrelevant to the issue.

5. The last paragraph is inaccurate. Specifically: The distinction mentioned is standard among logicians, not just something suggested by “some logicians” – and the two persons mentioned are not well-known logicians.

6. The last paragraph is too vague to be useful, and its term “more sober” is misplaced – Lewis Carroll’s piece is perfectly sober. “Less whimsical” would be a more accurate term here. David makinson (talk) 19:45, 6 March 2016 (UTC)

A,B,C,...,Z

Paullusmagnus: Although the original dialog does indicate the premises as you do, with A,B,C,...,Z, I think this is faulty, because it implies, on some level, that there couldn't be more than 26 stages to the argument. That is why I thought 1,2,3,...,Z would be better; in that case no confusion can arise. You might object to having Z be the name for the final conclusion, but couldn't you propose another name, rather than getting rid of the #s altogether? --Ryguasu 20:45, 20 Sep 2003 (UTC)

The natural replacement for "Z" in today's notation would be "ω" (lowercase Omega), although if Achilles were more clever, he could come up with an ωth axiom. (And indeed he does so, in one of Hofstadter's dialogues.) Of course, the use of "ω" by mathematicians today (originally chosen by Cantor, I believe, and probably at about the same time as Carroll wrote his dialogue) is inspired by the same idea as Carroll's use of "Z" -- it is the last letter. -- Toby Bartels 09:45, 30 Aug 2004 (UTC) An ωth axiom does not exist.Turtleguy1134 (talk) 22:46, 21 January 2012 (UTC)

Modus ponens

I don't get it. Basically, this is my interpretation:

A tortoise doesn't accept modus ponens arguments as being self-evident. And, in order to make the tortoise accept them, Achilles tries to use another modus ponens argument (knowing full well the tortoise doesn't accept them). And he fails, obviously. When it's all over, we come to the conclusion that Achilles can't prove the original argument to the tortoise.

So, Lewis Caroll basically proved that if someone doesn't accept your type of argument, using the same type of argument to prove the first argument won't work.

Three questions: Why, god forbid, does Achilles try proving a modus ponens argument with another one, even though the tortoise doesn't accept them? Why is this a paradox, and why would philosophers try to "solve" it? And isn't Achilles' failure self-evident, without the need for this story?

If I've interpreted this incorrectly, please tell me.

-Its not really about modus ponens, its about logical consequence

all premises from c onward are basically the previous premises are true. what it shows that you need an axiom for all proofs. or atleast thats my take on it. 24.237.198.91 05:32, 30 September 2006 (UTC)

Personally can't accept any of those statements as there are THREE sides to a triangle :-P UnseemlyWeasel 02:54, 9 October 2006 (UTC)

It seems that all of you bring up some fine points. I cannot grasp why this should be qualified as a genuine paradox either, and there doesn't seem to be a definitive answer that philosophers can agree on. But this mysterious short paper did draw the attention of countless influential philosphers and they took them seriously so there probably is some deep stuff going on here.

A few points I'd like to make after reading the comments written above:

1. First of all, you are right that this is not about modus ponens in particular. It is about inference rules in general.

2. I don't think it's the case that the premises from c onward say that the previous premises are true. They say that IF the previous premises are true THEN the conclusion is true. They are conditional statements and they say nothing about the truth values of those premises.

3. "If A and B, then Z" is not an argument and therfore it is neither valid nor invalid. It is a tautological statement.

Basically I think the most fundamental lesson we can draw from this is that we should not confuse inference (A, B therefore Z) with implication ((A.B)-->Z). I guess this is essentially what Bertrand Russell said in Principles of Mathematics. I have to write a paper on this dialogue and my professor suggested that I use the deduction theorem as a central idea in resolving this "paradox."

Sex

As of Nov 20, there is some material on the article about Hofstadter's change of mind regarding the tortoise's sex. Suggest deleting as not germane. Also, the article refers to tortoise as "he" (which Hofstadter would disapprove of), suggest changing to "it". elpincha 19:29, 20 November 2006 (UTC) Are boy turtles or girl turtles smarter?Turtleguy1134 (talk) 22:46, 21 January 2012 (UTC)

Football reference

Hey, Do we really need Carroll's in-joke about the hypothetical readers being better off if they were to play football instead of reading Euclid? It might read well in the original but here it looks very strange and is confusing. The text here should anyway not be a verbatim copy from Carroll's text and clearly the paradox doesn't suffer from having these references removed. Thoughts? Andeggs 11:33, 20 December 2006 (UTC)

Well, the paradox wouldn't suffer from having the entire context removed, we could state it entirely in the abstract. But what a sad thing that would be! If it confusing and doesn't read well, my preference would be to rewrite it to read better. But since I'm not volunteering to do that now, I won't stop you from removing it, if you must. 192.75.48.150 16:14, 20 December 2006 (UTC)

Yes sorry to be a party-pooper but I did have to read it several times to understand what the football reference was all about. Andeggs 12:44, 21 December 2006 (UTC)

Kurt Godel

Since Hofstadter has been cited here I think it makes sense to mention Godel's famous "Incompleteness" Proof that is given in his landmark paper "On Formally Undecidable Proposisions of Principia Mathematica". If no one objects, I'll add a few lines to the discussion section.

We do have an article on Gödel's incompleteness theorems, and I don't see the point of adding them here. How would it be relevant? 192.75.48.150 15:00, 17 January 2007 (UTC)

Title Alluding

Not only the title alludes to the paradox stated. Within the first few lines of the text, the narrator says that Achilles has caught up to the turtle, and then they jabber about how the turtle thought the paradox couldn't be solved. Might want to include this in there, rather than just "the title alludes". 130.76.96.19 18:51, 13 March 2007 (UTC)

---

Of course, the problem with "ω" (from Cantor) and Godel's infamous theorem is that both are based on paradox themselves, so it is not entirely clear how they can be used to help resolve Carroll's Paradox.

Rosa Lichtenstein (talk) 13:38, 20 August 2008 (UTC)

Just to tell you, saying that Achilles runs at 10 meters per second and gives the 10 times slower tortoise a 10 meter head start, he will catch up in 11.11111111....... seconds.Turtleguy1134 (talk) 22:34, 21 January 2012 (UTC)

I'm sorry, if that is in response to me, it's still as clear as mud.

Rosa Lichtenstein (talk) 18:22, 12 October 2012 (UTC)

Confusing

I also don't seem to understand what the tortoise is saying. He accepts A as valid and true. He accepts B as valid and true.

He accepts that if he accepts A and B as valid and true, he must accept that Z is true. He accepts that A and B are valid and true, but still refuses to accept Z, even though he already stated that he must.

The only way that this would make any sense would be if he decides that C can be valid and true and false at the same time. For example, imagine that you accept A and B are true. I then attempt to prove C is incorrect with this logic:

If A and B are true, Z must be true.
A and B are true.
Therefore, Z must not true.

Or this:

If Z is false, either A or B must be false.
Neither A nor B are false.
Therefore, Z is false.

Or even this:

If A and B are true, Z must be true.
A and B are true.
If A and B are true, than it cannot be determined if A and B are true.
A and B are true.
It cannot be determined if A and B are true.
Therefore, Z cannot be proven.

Does this make any sense to anyone?

---- Dromioofephesus (talk) 12:54, 31 October 2008 (UTC)

Further, the only reason why it is true that if A, B, C, and D are true, then Z is true is because A and B and C are true.
If C, then Z.
C.
Therefore, Z.

I don't believe it is necessary to say that
If C is true, then Z is true.
C is true.
If C, D, E... are true, then Z is true.
D, E... cannot be proven, therefore Z cannot not be proven.
If Z is false, then C must be false.
C is true.
Therefore, Z cannot be proven.

As I see it, anything past C is unnecessary for proving Z, since Z has already been accepted as true a necessary consequence of C, which is a necessary consequence of A and B. C does not state "If A and B are true, and this is true, then Z is true." Why must we use logic that is based on the results of actions that follow the action in question?
If A is true, then B is true.
If A and C are true, then B is true.
If A is true and C is not true, then B is still true.
If A is true and this sentence is false, then A is true, then B is true.
If A is true and this sentence is true, then A is true, then B is true.
The above sentences are either true or false.
Therefore, C.

---

Dromioofephesus (talk) 13:31, 31 October 2008 (UTC)

Yep. He accepts that (C:) "If A and B are true then Z is true". But he does not see why this C together with A and B forces him to accept Z. It is of course obvious for us, but how would you explain it? You can say "Hey tortoise, accept it, it's so obvious" but it's not reasoning. You could say that about anything. If you want to explain it, you have to write down what you want to tell him. So this goes on infinitely! There is no end, you just keep going and going and nothing changes. The dialogue repeats itself while the list just gets longer and longer and nothing changes.

For us it is easy to see the similarity and connection of the sentences, we recognise patterns, we see categories, and that is what moves mathematics; maths does not stand all alone, you have to use "common sense". Maths is simply a tool to use common sense practically and usefully. It is harder to solve problems in mind directly; harder to jump long so we create middle "stations" on which we can walk. These stations are mathematics, but it's still you who walks among them, using the original common sense. Mathematics is useless without thinking! Qorilla (talk) 15:54, 8 December 2008 (UTC)

Let's assume that the Tortoise does not accept Z from merely A and B. We include C, and get this:

• (A) Things that are equal to the same are equal to each other.

• (B) The two sides of the Triangle are things that are equal to the same.

• (C) If A and B are true, Z must be true.

• (Z) The two sides of this Triangle are equal to each other.

Now the Tortoise does not accept Z. We realise the following: A is true. B is true. C states that if A and B are true, which they are, then Z must be true. Therefore Z must be true. The Tortoise is still claiming not to accept Z, which can only be true if the Tortoise has not accepted C. Accepting C, by definition, includes accepting Z. Yet the Tortoise accepts C but not Z. There isn't really a paradox - just a Tortoise who fails at logic. MeTheGameMakingGuy (talk) 11:59, 25 July 2009 (UTC) Yep, just a Tortoise who fails at logic.

Solution by using the liar paradox

I removed the following paragraph from the article:

"A possible solution to this paradox can be reached, but only by the use of another paradox (The liar paradox [1]) Consider the following statements.

A) This and some and all following statement are false. B) Things that are equal to the same are equal to each other. C) The two sides of this triangle are thing that are equal to the same. Therefor Z) The two side of the triangle are equal to each other.

Statment A) in this version of the liar paradox. If statement A) is false, all part of the statement are therefor false making the meaning of the statement become. This and some and all following statements are true. If statement A) is true, the fact it calls itself false means all part of the statment are false again again making the statements meaning. This and some and all following statements are true. This statment inself means that all following statement must be true and no addition statment like the Tortoise used are needed to prove the truth in the other statements.

Staement A) is worded the way it is, instead of 'This and all following statements are false' This is because when the stsatement if false it is possible to say that some of the following statement are false instead. That is why the statement is worded in this particualar way.

The main problem of this solution is that some may argue against it because while it does in essence solve the paradox another paradox has been introduced instead."

This is clearly original research, prohibited here [[1]].

It may also be worth pointing out that this argument is flawed. The claim that "If statement A) is false, all part of the statement are therefore false" does not follow. -(A&B) is equivalent to -A OR -B. (To simply illustrate this point: "It is not the case that Earth is round and Mars is square" doesn't imply that "Earth is not round and Mars is not square". Only one of the conjuncts need be false.)

Additionally even if the logic in the attempted solution was correct the turtle could presumably ask you to write "This statment inself means that all following statement must be true" as a proposition itself, thus continuing the cycle.--Rxtreme (talk) 06:55, 27 March 2009 (UTC)

A possible solution?

The Tortoise's hangup seems to be on "Whatever Logic is good enough to tell me is worth writing down." Therefore, step C could be written as, "If everything that is written before Z is true, then Z must be true." When the Tortoise asks Achilles to write down the step that leads from that to Z, Achilles can then point to what he has already written down at C. --BaruMonkey (talk) 15:40, 8 September 2009 (UTC)

Shut Up, Hannibal!

"Do I?" said the Tortoise innocently. "Let's make that quite clear. I accept A and B and C and D. Suppose I still refused to accept Z?" "Then Logic would take you by the throat, and force you to do it!" Achilles triumphantly replied. 74.128.56.194 (talk) 00:23, 16 June 2011 (UTC)

My interpretation

OK, I think I got this. C:If A and B are true then Z is true, the turtle does not think that is right. But then D:if A and B and C are true then Z is true. Since the inputs to the and logic gate are all accepted by the tortoise, he should accept the output, Z. But he does not.Turtleguy1134 (talk) 22:40, 21 January 2012 (UTC)

You're neglecting that in this context, the statement "if … then" is a 'logic gate' in itself. Now, the tortoise accepts that it gives a 'true' output. But if we try to say that this means Z is true, then aren't we just trying to use another "if … then" statement to justify the first one? — NuclearDuckie (talk) 14:49, 15 June 2014 (UTC)

C: If A, B, and C are true, then Z is true

Why doesn't this work? — Preceding unsigned comment added by 200.252.231.131 (talk) 17:03, 1 February 2012 (UTC)

Because the infinite regress problem still occurs; now it's just bundled into one statement. Expanded out, it would read something like:

• 'If A, B, and the statement "If A, B, and the statement 'If A, B, … are true, then Z is true' are true, then Z is true" are true, then Z is true'

It's problematic for self-recurrence to be allowed in logic, as many contradictory claims (notably the liar paradox) can be made. — NuclearDuckie (talk) 04:20, 10 May 2014 (UTC)

Paradox or Insanity?

It is clear to me that the Tortoise violates argument C by lying about his belief or through mental insanity. Achilles provides perfectly sound reasoning which is true regardless of whether or not the Tortoise accepts it. In fact, the tortoise's belief is completely irrelevant to the soundness of the reasoning, if anyone can point out how the Tortoise could be correct in his assertions, please do so. Additionally, this is the second paradox in a row by Lewis Carroll that is logically unsound due to fallacies rather than contradictions, I have yet to see a paradox from this man that is a logically sound self-contradiction.--Kodiak42 (talk) 01:22, 21 February 2013 (UTC)

What is the nature of a logical operator? In [ P => Q ], for example, what does it mean for P to imply Q? What is implication? — Preceding unsigned comment added by 184.179.86.135 (talk) 22:23, 29 June 2018 (UTC)

The thing is, why must we accept logical rules? This is the point: habitually, they are just taken on faith, they're not provably sound. Sure, this is why they are called "axioms", but this does not bring relief... - 92.100.183.188 (talk) 14:38, 30 October 2013 (UTC)

Because they are sound? since "sound" here also means logical. so yeah. the point is that dodgson made a logical/unsound error here83.209.66.117 (talk) 04:38, 3 February 2014 (UTC)

Original research

Statements and arguments that propositional logic resolves the paradox are original research. Floorsheim (talk) 19:41, 28 February 2013 (UTC)

Removing a link

I'm going to make "Carroll's paradox" in the first sentence of Discussion not a link. The article the link points to is about something else entirely, which happens to be called Carroll's paradox but is named for a different Carroll. Cognita (talk) 05:05, 20 December 2013 (UTC)

Thoughts

Some thoughts: What Achilles refers to when he says "Then logic will take you by the throat and force you to accept it!" is my answer to the problem. Consider statement D: If I accept A, B, and C, then I accept Z. Expanding: If I accept A, B, and (If A and B then Z), then I accept Z. I replace (A and B) with (AB) to get: If I accept AB and (If AB then Z) then I accept Z. This is getting close to: If I accept x, and I accept that if x, then y, then I accept y. This is the definition of the word "if": it is that accepting the if's condition means accepting it's output. In other words, statement D is the definition of the word "if". If the Tortoise does not accept D, then it is admitting it cannot use "if" in a logical way.

Achilles should flip the turtle over and get on his way. --Turtleguy1134 (talk) 19:43, 12 April 2015 (UTC)

Propositional logic resolution

"The paradox ceases to exist the moment informal logic is replaced with propositional logic."

This isn't really accurate because the paradox can be reintroduced by having the tortoise accept a particular propositional logic but refusing to verify that the proposition is tautological.

The heart of the paradox is that the tortoise has a notebook of logically sound propositions but he can't be forced to write any derivation of those propositions in the notebook. Any form of logic is implicitly dependent on the cooperation of a reader since it's possible to construct a reader that transitions betweens states logically but does not flesh out the current state by augmenting it with conclusions of accepted propositions.

108.93.144.184 (talk) 13:31, 12 January 2016 (UTC)

I'm not sure if I'm observing the same thing as you, but as for the sentence:

"However, if a formal system is introduced where modus ponens is simply a rule of inference defined within the system, then it can be abided by simply by reasoning within the system."

I don't see why 'reasoning within the system' avoids the problem. For example, in most proof systems I know, one creates a sequence of formulae, wherein each formula is either an axiom or follows from preceding formulas by a 'rule of inference'. One of those rules of inference might be, "If the sequence contains P -> Q and P, then we may append Q.". So I look at my sequence, and indeed, it contains P -> Q and P. What allows me to append Q? Of course, it is the human principle of Modus Ponens, applied to this rule of inference. And if we need the human principle of Modus Ponens then Carroll's argument still applies.

The point is that the pattern of Modus Ponens is not axiomatizable, if what we mean is that we wish to subsume the pattern of reasoning in an axiom. Indeed, Modus Ponens encapsulates what we mean by human logic: it is in a sense the definition of the words "if" and "then". When using these words, we are describing a pattern by which we can move from one idea to another, similar to how we move physically from one place to another by walking.

The same is true for rules of inference like, "if the sequence contains Ax P(x) then we may append P(c) for any c" or "if the sequence contains X /\ Y then we may append X". Of course it is useful to study inference systems but we should not pretend like these can circumvent the normal human meanings of these words. For example, the first rule allows us to append Pc "for any c". How do we appeal to this rule to append, say, P(3)? By using the human rule of instantiation. This is just the meaning of the words 'for any' or 'for all'.

I've always taken Carroll to be saying that axioms cannot replace meaning. Axioms abstract from meaning, make meaning precise, so that we can study meaning mathematically. 96.224.255.217 (talk) 03:08, 14 October 2022 (UTC)

P implies Q if and only if the proposition not P or Q is a tautology

This statement seems ambiguous to me. What is the tautology here? Given the proposition A below,

A: P -> Q <=> (¬P v Q),

is this statement saying that A is a tautology or that (¬P v Q) is a tautology? Rgiusti (talk) 19:01, 21 May 2016 (UTC)

Neither is a tautology, but the whole P -> Q <=> (¬P v Q) is a tautology. 86.132.221.253 (talk) 20:05, 14 April 2017 (UTC)

It's saying that the definition of "P implies Q" is taken to be that "P implies Q" holds precisely when (¬P v Q) is a tautology.2.24.113.223 (talk) 18:49, 12 August 2016 (UTC)

Not so. 86.132.221.253 (talk) 20:05, 14 April 2017 (UTC)

Source which is no source

Why is Hofstadter, Douglas. Gödel, Escher, Bach: an Eternal Golden Braid, listed in the sources? It has no relevance to the subject matter. — Preceding unsigned comment added by 86.132.221.253 (talk) 20:03, 14 April 2017 (UTC)

Sure it does; it reproduces the original piece (which I believe was little known until its publication there) and includes a series of dialogues inspired by it (featuring Achilles, the Tortoise, and other fanciful characters), each of which playfully illustrates the theme of a chapter or section. Overall the book can be viewed as an extended exploration of the ‘bootstrapping’ problem the Tortoise raises in Carroll’s dialogue, with respect to logic, intelligence, and creativity. It might be better described in a “Legacy” section, or listed under “Further reading” rather than “Sources“, but I think it‘s quite relevant regardless.—Odysseus1479 21:05, 14 April 2017 (UTC)

Reversion of annotated links

Deacon Vorbis, I see that you have reverted the annotated links in the see also section, without providing alternative annotation. Do you claim MoS guidance or other policy or guidance justifying this reversion? If so, please link me to it as I am unaware of such justification existing. Otherwise please explain your edit summary "annotated link shouldn't be used" which appears to be stated as if a fact. · · · Peter Southwood ^(talk): 18:35, 13 September 2020 (UTC)

The annotated link template is inappropriate for "see also" sections. If you want to add manual ones, please feel free, but the template shouldn't be used. –Deacon Vorbis (carbon • videos) 18:37, 13 September 2020 (UTC)

Deacon Vorbis, As far as I can determine that is a minority opinion currently not supported by policy or guidance. If I am wrong, please link to the relevant policy or guidance. Please ping with reply. Cheers, · · · Peter Southwood ^(talk): 10:03, 16 September 2020 (UTC)