What The Tortoise Said To Achilles - Summary of The Dialogue

Summary of The Dialogue

The discussion begins by considering the following logical argument:

• A: "Things that are equal to the same are equal to each other" (Euclidean relation, a weakened form of the transitive property)
• B: "The two sides of this triangle are things that are equal to the same"
• Therefore Z: "The two sides of this triangle are equal to each other"

The Tortoise asks Achilles whether the conclusion logically follows from the premises, and Achilles grants that it obviously does. The Tortoise then asks Achilles whether there might be a reader of Euclid who grants that the argument is logically valid, as a sequence, while denying that A and B are true. Achilles accepts that such a reader might exist, and that he would hold that if A and B are true, then Z must be true, while not yet accepting that A and B are true. (A reader who denies the premises.)

The Tortoise then asks Achilles whether a second kind of reader might exist, who accepts that A and B are true, but who does not yet accept the principle that if A and B are both true, then Z must be true. Achilles grants the Tortoise that this second kind of reader might also exist. The Tortoise, then, asks Achilles to treat the Tortoise as a reader of this second kind. Achilles must now logically compel the Tortoise to accept that Z must be true. (The tortoise is a reader who denies the argument itself, the syllogism's conclusion, structure or validity.)

After writing down A, B and Z in his notebook, Achilles asks the Tortoise to accept the hypothetical:

• C: "If A and B are true, Z must be true"

The Tortoise agrees to accept C, if Achilles will write down what it has to accept in his notebook, making the new argument:

• 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"
• Therefore Z: "The two sides of this triangle are equal to each other"

But now that the Tortoise accepts premise C, it still refuses to accept the expanded argument. When Achilles demands that "If you accept A and B and C, you must accept Z," the Tortoise remarks that that's another hypothetical proposition, and suggests even if it accepts C, it could still fail to conclude Z if it did not see the truth of:

• D: "If A and B and C are true, Z must be true"

The Tortoise continues to accept each hypothetical premise once Achilles writes it down, but denies that the conclusion necessarily follows, since each time it denies the hypothetical that if all the premises written down so far are true, Z must be true:

"And at last we've got to the end of this ideal racecourse! 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 notebook, 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.

Thus, the list of premises continues to grow without end, leaving the argument always in the form:

• (1): "Things that are equal to the same are equal to each other"
• (2): "The two sides of this triangle are things that are equal to the same"
• (3): (1) and (2) ⇒ (Z)
• (4): (1) and (2) and (3) ⇒ (Z)
• ...
• (n): (1) and (2) and (3) and (4) and ... and (n − 1) ⇒ (Z)
• Therefore (Z): "The two sides of this triangle are equal to each other"

At each step, the Tortoise argues that even though he accepts all the premises that have been written down, there is some further premise (that if all of (1)–(n) are true, then (Z) must be true) that it still needs to accept before it is compelled to accept that (Z) is true.