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Disclaimer: I have absolutely no background in philosophy or logic.

I've just randomly read the short story "What the Tortoise said to Achilles" by Lewis Carrol. As far as I have understood this, Achilles tries to force the tortoise to accept a simple conclusion, but fails because the tortoise leads him into an infinite regression.

But I couldn't stop thinking about the following: Couldn't Achilles force the turtle to accept the statement with the help of self reference? Like, by writing (C) If A and B and C are true, Z must be true.

Something in my gut tells me this is not possible, but I can't figure out why. Why would this solution not work?

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    The gist of the paradox is that there must be a difference in "status" between logic "assertions" like e.g. axioms and logical laws and inference rules, and the latter cannot be "deduced" from logical laws, because this produces an infinite regress. Jul 21, 2022 at 12:29
  • Thank you for bringing this up; I had never heard of this little story before. Jul 21, 2022 at 17:37

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The problem is that the turtle essentially rejects the concept of implication itself. That is basically the section where Achilles gives away the game:

"And neither of these readers," the Tortoise continued, "is as yet under any logical necessity to accept Z as true?"

"Quite so," Achilles assented.

"Well, now, I want you to consider me as a reader of the second kind, and to force me, logically, to accept Z as true."

First he agrees that logic doesn't force anybody to accept Z and then he accepts to try to force Z via logic.

And in terms of your example:

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

Well the tortoise could again force him into an infinite regression by checking the premises, A is accepted to be true, B is accepted to be true, C ... well let me check the premises, A is accepted to be true, B is accepted to be true, C ... well let me check the premises...

But the tortoise likely accepts that to be true and asks you to call that D because she still doesn't believe that a true implication forces her to accept Z.

Also a funny tidbit because the turtle and the Greek warrior change the names at the end of the story, the story itself becomes some sort of infinite regression where Achilles is carrying the turtle.

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Solution for the “What the Tortoise Said to Achilles” paradox

I see the resolution as follows. The original argument is AAA in the first figure, which is valid. If someone does not accept the rules of inference, then no, the argument is not valid; but neither is any other argument. All human communication is rendered impossible.

The original argument is:

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

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

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

Here, “The two sides of this Triangle” is the Subject. ”are equal to each other” is the Predicate. And the Middle Term connecting the two is “Things that are equal to the same “.

Perhaps what Carroll was getting at was the need for a defense of the rules of inference.

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