READING: LEWIS CARROLL (1895): What The Tortoise Said To Achilles
I think the solution to this is Douglas Hofstadters’s: There are rules, and there are premises. There must be a bottom layer where the system simply acts on a rule, mechanically, rather than contemplating it as another proposition to assent to. You jump out of the loop, and the step from premises to conclusion is something you do, not something you believe:
The Regress That Ate Achilles: Why Logic Can’t Justify Its Own Last Step
What the Tortoise Knew: Lewis Carroll’s Proof That Reasoning Isn’t a Rule
A thousand and one premises: Carroll’s infinite note-book and the floor beneath logic; or, the regress and logic’s unsuccessful attempts to justify its own final operative step:
READING: LEWIS CARROLL (1895): What The Tortoise Said To Achilles
<https://math.dartmouth.edu/~matc/Readers/HowManyAngels/Tortoise.html>
ACHILLES had overtaken the Tortoise, and had seated himself comfortably on its back.
“So you’ve got to the end of our race-course?” said the Tortoise. “Even though it does consist of an infinite series of distances? I thought some wiseacre or other had proved that the thing couldn’t be done?”
“It can be done,” said Achilles. “It has been done! Solvitur ambulando. You see the distances were constantly diminishing; and so –”
“But if they had been constantly increasing?” the Tortoise interrupted. “How then?”
“Then I shouldn’t be here,“ Achilles modestly replied; “and you would have got several times round the world, by this time!”
“You flatter me – flatten, I mean,” said the Tortoise; “for you are a heavy weight, and no mistake! Well now, would you like to hear of a race-course, that most people fancy they can get to the end of in two or three steps, while it really consists of an infinite number of distances, each one longer than the previous one?”
“Very much indeed!” said the Grecian warrior, as he drew from his helmet (few Grecian warriors possessed pockets in those days) an enormous note-book and a pencil. “Proceed! And speak slowly, please! Shorthand isn’t invented yet!”
“That beautiful First Proposition of Euclid!” the Tortoise murmured dreamily. “You admire Euclid?”
“Passionately! So far, at least, as one can admire a treatise that wo’n’t be published for some centuries to come!”
“Well, now, let’s take a little bit of the argument in that First Proposition – just two steps, and the conclusion drawn from them. Kindly enter them in your note-book. And in order to refer to them conveniently, let’s call them A, B, and Z: –
(A) Things that are equal to the same are equal to each other.
(B) The two sides of this Triangle are things that are equal to the same.
(Z) The two sides of this Triangle are equal to each other.
Readers of Euclid will grant, I suppose, that Z follows logically from A and B, so that any one who accepts A and B as true, must accept Z as true?”
“Undoubtedly! The youngest child in a High School – as soon as High Schools are invented, which will not be till some two thousand years later – will grant that.“
“And if some reader had not yet accepted A and B as true, he might still accept the sequence as a valid one, I suppose?”
“No doubt such a reader might exist. He might say I accept as true the Hypothetical Proposition that, <em>if A</em> and <em>B</em> be true, <em>Z</em> must be true; but, I <em>don’t</em> accept <em>A</em> and <em>B</em> as true.’ Such a reader would do wisely in abandoning Euclid, and taking to football.”</p><p>“And might there not <em>also</em> be some reader who would sayI accept A and B as true, but I don’t accept the Hypothetical’?”
“Certainly there might. He, also, had better take to football.”
“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.”
“A tortoise playing football would be – “ Achilles was beginning
“ – an anomaly, of course,” the Tortoise hastily interrupted. “Don’t wander from the point. Let’s have Z first, and football afterwards!”
“I’m to force you to accept Z, am I?” Achilles said musingly. “And your present position is that you accept A and B, but you don’t accept the Hypothetical –”
“Let’s call it C,“ said the Tortoise.
“– but you don’t accept
(C) If A and B are true, Z must be true.”
“That is my present position,” said the Tortoise.
“Then I must ask you to accept C.“
“I’ll do so,” said the Tortoise, “as soon as you’ve entered it in that note-book of yours. What else have you got in it?”
“Only a few memoranda,” said Achilles, nervously fluttering the leaves: “a few memoranda of – of the battles in which I have distinguished myself!”
“Plenty of blank leaves, I see!” the Tortoise cheerily remarked. “We shall need them all!” (Achilles shuddered.) “Now write as I dictate:-
(A) Things that are equal to the same are equal to each other.
(B) The two sides of this 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.”
“You should call it D, not Z,“ said Achilles. “It comes next to the other three. If you accept A and B and C, you must accept Z.“
“And why must I?”
“Because it follows logically from them. If A and B and C are true, Z must be true. You don’t dispute that, I imagine?”
“If A and B and C are true, Z must be true,” the Tortoise thoughtfully repeated. “That’s another Hypothetical, isn’t it? And, if I failed to see its truth, I might accept A and B and C, and still not accept Z, mightn’t I?”
“You might,” the candid hero admitted; “though such obtuseness would certainly be phenomenal. Still, the event is possible. So I must ask you to grant one more Hypothetical.”
“Very good. I’m quite willing to grant it, as soon as you’ve written it down. We will call it
(D) If A and B and C are true, Z must be true.
Have you entered that in your note-book?”
“I have!” Achilles joyfully exclaimed, as he ran the pencil into its sheath. “And at last we’ve got to the end of this ideal race-course! Now that you accept A and B and C and D, of course you accept Z.“
“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. “Logic would tell you `You ca’n’t help yourself. Now that you’ve accepted A and B and C and D, you must accept Z!’ So you’ve no choice, you see.”
“Whatever Logic is good enough to tell me is worth writing down,“ said the Tortoise. “So enter it in your book, please. We will call it
(E) If A and B and C and D are true, Z must be true. Until I’ve granted that, of course I needn’t grant Z. So it’s quite a necessary step, you see?”
“I see,” said Achilles; and there was a touch of sadness in his tone.
Here the narrator, having pressing business at the Bank, was obliged to leave the happy pair, and did not again pass the spot until some months afterwards. When he did so, Achilles was still seated on the back of the much-enduring Tortoise, and was writing in his note-book, which appeared to be nearly full. The Tortoise was saying “Have you got that last step written down? Unless I’ve lost count, that makes a thousand and one. There are several millions more to come. And would you mind, as a personal favour, considering what a lot of instruction this colloquy of ours will provide for the Logicians of the Nineteenth Century – would you mind adopting a pun that my cousin the Mock-Turtle will then make, and allowing yourself to be re-named Taught-Us?”
“As you please!” replied the weary warrior, in the hollow tones of despair, as he buried his face in his hands. “Provided that you, for your part, will adopt a pun the Mock-Turtle never made, and allow yourself to be re-named A Kill-Ease!”
<https://math.dartmouth.edu/~matc/Readers/HowManyAngels/Tortoise.html>
Wikipedia <https://en.wikipedia.org/wiki/What_the_Tortoise_Said_to_Achilles> tells me:
Wikipedia: ‘Several philosophers have tried to resolve Carroll’s paradox. Bertrand Russell discussed the paradox briefly in § 38 of The Principles of Mathematics (1903), distinguishing between implication (associated with the form “if p, then q“), which he held to be a relation between unasserted propositions, and inference (associated with the form “p, therefore q“), which he held to be a relation between asserted propositions; having made this distinction, Russell could deny that the Tortoise’s attempt to treat inferring Z from A and B as equivalent to, or dependent on, agreeing to the hypothetical “If A and B are true, then Z is true.”
Peter Winch, a Wittgensteinian philosopher, discussed the paradox in The Idea of a Social Science and its Relation to Philosophy (1958), where he argued that the paradox showed that “the actual process of drawing an inference, which is after all at the heart of logic, is something which cannot be represented as a logical formula … Learning to infer is not just a matter of being taught about explicit logical relations between propositions; it is learning to do something” (p. 57). Winch goes on to suggest that the moral of the dialogue is a particular case of a general lesson, to the effect that the proper application of rules governing a form of human activity cannot itself be summed up with a set of further rules, and so that “a form of human activity can never be summed up in a set of explicit precepts” (p. 53).
Carroll’s dialogue is apparently the first description of an obstacle to conventionalism about logical truth,[4] later reworked in more sober philosophical terms by W. V. O. Quine.[5]
Perhap the best thumbnail is: Achilles has all the premises and the conclusion staring him in the face—yet the Tortoise will not budge, and every reason he offers only becomes one more premise to grant. Carroll’s 1895 joke turns out to be a load-bearing wall of modern logic. Every attempt to make the Tortoise reason produces another proposition he can politely refuse to grant. The way out isn’t more logic; it’s noticing that drawing a conclusion was never a belief in the first place.