Carroll’s Paradox
In the journal Mind in 1895 the Oxford logician and author of Alice in Wonderland Lewis Carroll published a playful dialogue between Achilles and the Tortoise that brought to light a central problem in logic as it was understood at the time. Specifically, he showed that merely having axioms—even the best and most perfect axioms—is not sufficient for determining truth in a system of logic, for one also must be very careful about one’s choice of rules of inference. In other words, one’s assumptions must be explicitly augmented by the exact mechanisms by which one is to deduce consequences from those assumptions.
In his dialogue, which is fully reproduced below, Carroll tackles the single most important rule of first-order logic, modus ponens, which says that if a statement P is assumed, and if the conditional statement “P implies Q” is also assumed (or previously proved), then the statement Q itself is a logical consequence and may therefore be considered proved. What Achilles learns, to his lasting regret, is that modus ponens must be first granted as a rule of inference, for otherwise no conclusion can ever be reached.
Readers may wish to note that Carroll is drawing on what has become a tradition, starting with Zeno of Elea and continuing with the modern author Douglas Hofstadter, of using a dialogue between the Homeric hero Achilles and the comical figure of the Tortoise to make an important philosophical point. Compare with Zeno’s Paradox of the Tortoise and Achilles.
What the Tortoise Said to Achilles
by Lewis Carroll
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 pencil. “Proceed! And speak SLOWLY, please! SHORTHAND isn’t invented yet!”
“That beautiful First Proposition by Euclid!” the Tortoise murmured dreamily. “You admire Euclid?”
“Passionately! So far, at least, as one CAN admire a treatise that won’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, if A and B be true, Z must be true; but I DON’T accept A and B as true.’ Such a reader would do wisely in abandoning Euclid, and taking to football.”
“And might there not ALSO be some reader who would say ‘I 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 notebook 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 can’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 can’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 notebook, 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 renamed 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!”
* * *
Carroll’s humor in this piece disguises a point that is essential to understanding modern logic. Unlike in the classical, Aristotelian conception, modern mathematics relies ultimately on pure formalism in its use of logic. This avoids the infinite regress in which the Tortoise traps Achilles.
This trap is impossible to avoid if logic is not formalized, because, as Douglas Hofstadter points out in Gödel, Escher, Bach (p. 170), in order to know how to use a rule (such as a rule of inference) you need a rule telling you how to apply the rule. And then a rule telling you how to apply that rule, and so on. By contrast, in formal logic, rules of inference are reduced to rules of symbol manipulation. Since the symbols themselves are uninterpreted (which is what we really mean by “formal”), we have a system as austere and elegant as chess, where it is understood that the game arises from—and entirely consists in—the rules for moving the pieces on the board.
The formalist solution, while effective, has its own philosophical drawbacks. Not the least of these is that, by reducing logic to uninterpreted symbols, all semantic content is removed from the conclusions of formal logic. In other words, what we would ordinarily consider meaning is lost. How to restore meaning to systems of inference while still avoiding difficulties such as Carroll’s Paradox remains a thorny question for philosophers of mathematics.
