Saw this (WP:"What the Tortoise said to Achilles") on the internet.

A summary is as follows. The common argument is:

A: If p then q B: p C: Therefore q.

This raises the following question: what if one were to object to this, i.e concede that A and B are true but object to C.

My question is this: could this objection be valid for use? How would you refute this objection?

So far I have been thinking that the only way you can refute it is by claiming that the person arguing it is ignorant.

UPDATE: A more detailed summary:

For most arguments in science, one uses the 'modus ponens' argument. Consider the following A: If it is night time, it will be dark. B: It is night time C: Therefore it is dark.

What if someone were to concede A and B, but object to C? In this case, you might consider adding the following argument.

A: If it is night time, it will be dark. B: It is night time C: Therefore it is dark. D: If A and B are true, C must be true.

Once again, what if someone accepts the first 3 arguments but objects to D.

Then, you might be tempted to add argument E.. and so on.

  • 1
    I am too lazy to read the article. But from what you are describing here, it seems that someone objecting simply does not agree with reasoning the way we accept it to be valid. He may do so, but he could never argue for his rejection because we wouldn't accept it to be valid. The way you are summarizing it, doesn't really pose a question. Could you expand on it and describe what your question is? Other people might just be as lazy as me!
    – Einer
    Sep 12, 2014 at 16:57
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    But you promised us a summary if we're too lazy to read the article. What's the argument here?
    – user4894
    Sep 12, 2014 at 20:31
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    It took me quite a while to figure out what the puzzle was about. It literally made me sweat at the coldest night, and hungry as soon as I filled up my stomach. Sep 13, 2014 at 22:02
  • I thought Carroll was poking fun at philosophers with their seemingly infinite regress of theories, meta-theories, meta-meta-theories, meta-meta-meta-theories... Sep 17, 2014 at 4:46

5 Answers 5


Lewis Carroll's puzzle first appeared in the April 1895 issue of Mind. It directly influenced the formulation of the first primitive proposition of Whitehead & Russell's Principia Mathematica.

This puzzle exposes the difference between implication and inference: an implication only tells you what follows your premise, but does not tell you whether your premise is true; an inference tells you what you can infer from a true proposition.

The reader of the second kind needs a hypothetical to enable her or him to infer (not just imply) Z from A and B. The hypothetical is Modus Ponens. Regression happens because Modus Ponens is presented in a wrong form:

IF P and (IF P Then Q) Then Q

It uses If-Then in the place of "therefore," thus both P and (If P Then Q) are presented in a hypothetical form. Regardless P or (IF P Then Q) is true or not, the implication (the outer IF-THEN statement) is always true, e.g. "if pigs fly then I am pope" is always true. Since an implication asserts neither the premise nor the conclusion, an endless regression will never reach Q's assertion.

The correct form of Modus Ponens should be like this:

If P Then Q
Therefore Q

Notice when "THEREFORE" replaces the outer IF-THEN, Q will not be asserted unless both P and (IF P Then Q) are really true. "THEREFORE" should be used only for inferences made from true proposition to true proposition.

In Whitehead & Russell's Principia Mathematica, Modus Ponens is presented in the form of ✳1.1 and ✳ 1.11. Both are primitive propositions, i.e. undemonstrated propositions.

✳1.1. Anything implied by true elementary proposition is true. Pp.

✳1.11. When ϕx can be asserted, where x is a real variable, and ϕ(x) implies Ψ(x) can be asserted, where x is a real variable, then Ψ(x) can be asserted, where x is a real variable.

Notice the absence of hypothesis.

  • Note: At the end of § 38 of The Principles of Mathematics, Russell wrote, "This seems to be the first step in answering Lewis Carroll's puzzle." Russell back then was young and giddy and flippant. He said this in jest to mock the endless steps in the puzzle. But the truth is he completely solved the puzzle once and for all in § 38. Jan 6, 2015 at 6:30

There is in a sense no rebuttal. If someone refuses to accept a basic law of logic (modus ponens), wilfully and without argument; then one can hardly use the laws of logic.

Rather then in engagingly in a pointless and long-winded argument that is leading nowhere one can exercise discretion and walk away.

Achilles being at the mercy of the Author (and the tortoise) obviously can't.

  • so if someone were to argue in this way, you would just have to say that they are ignorant? Sep 13, 2014 at 9:47
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    If some-one was arguing like this I would suspect their motivations; not that they're ignorant... Sep 13, 2014 at 9:49
  • You wouldn't have to say they are ignorant. They might be ignorant, trying to upset you, pretending to be ignorant, trying to win an argument by shouting louder or having a bigger bladder capacity than you.
    – gnasher729
    Sep 13, 2014 at 13:21
  • @gnasher729: sure - thats an interesting way of putting it; its a form of force as observed by Simone Weil; and common enough in politics of the big and micro kind. Sep 13, 2014 at 18:43

Could one not object on the grounds that they are using modus ponens to argue against its own validity?

The argument seems to be :

If (one objects to MP) then (one may reject C).

I object to MP.

Therefore I may reject C.

  • Haha I like that point! Sep 16, 2014 at 2:06

If someone rejects modus ponens, I would ask him what he would want to replace it with, because after all, we need some rule of inference that allows us to prove new statements from old ones, because otherwise there's no way to apply logical reasoning at all.

  • true, but that won't deal with their point - when criticising a point it is not conventionally necessary to provide an alternative. That being said, in practice we would definitely need a new rule. Sep 14, 2014 at 6:36
  • Well, if you want to argue their point, you need a rule of argument that they accept. Normally the common ground is modus ponens. But if they don't accept that, they have to establish another ground to be able to argue at all. So if they don't offer one, you cannot argue with them. Criticising modus ponens is not conventional in that way.
    – celtschk
    Sep 14, 2014 at 7:17
  • Yeah fair enough Sep 14, 2014 at 8:23

I am fascinated by this article of Lewis Carroll, and I think my ideas on it converge with the distinction George Chen has provided between knowing an implication (of hypothetical form: if this then that) and performing an inference (of, let's say, categorical form: this therefore that).

Here's the reasoning by which I've converged with George Chen: I thought Carroll's paradox shows us that we do different things with the knowledge of logical laws than with the knowledge of many other things. Consider that when we recognize the truth of "P", we often employ the fact that P as evidence for the rest of our beliefs. We infer from "P" to other things. For instance, we may infer from the statement that today is sunny to the statement that we will get sunburn by dwelling outdoors for too long.

When it comes to the recognition the truth of logical laws, however, we often do not employ such truths as the basis for inference. (Though of course we may.) The recognition of the logical law allows us to rightly and rightfully perform an inference. While knowing a logical law is an instance of propositional knowledge like any other, I think, — we know that if P and P → Q, then Q, for any P and Q, — its employment is a different kind of action than the employment of other sentences.

As I saw it, and still do, Achilles would be epistemically allowed to go from the (known) premises to the conclusion not because he has a further (known) premise stating such inference is truth-preserving, but rather because he's logically competent (i.e. has great epistemic access to logical facts) and this allows him to correctly do the inference.

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