In Lewis Carroll's story "What the Tortoise Said to Achilles," the swiftfooted
warrior has caught up with the plodding tortoise, defying Zeno's
paradox in which any head start given to the tortoise should rilake him
uncatchable. (In the time it would take for Achilles to close the gap, the
tortoise would have progressed a small amount; in the time it took to
close that gap, the tortoise would have moved a bit farther, ad infinitum.)
The tortoise offers Achilles a similar paradox from logic. Achilles pulls an
enormous notebook and a pencil from his helmet, and the tortoise dictates
Euclid's First Proposition:
(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.
The tortoise gets Achilles to agree that anyone who accepts A and B and
"If A and B then Z" must also accept Z. But now the tortoise disagrees
with Achilles' logic.
He says he is entitled to reject conclusion Z, because
no one ever wrote down the if-then rule on the list of premises he must
accept. He challenges Achilles to force him to conclude Z. Achilles
replies by adding C to the list in his notebook:
(C) If A and B are true, Z must be true.
The tortoise replies that he fails to see why he should assume that just because
A and B and C are true, Z is true. Achilles adds one more statement—
(D) If A and B and C are true, Z must be true.
—and declares that "Logic [must] take you by the throat, and force you"
to accept Z. The tortoise replies,
Thinking Machines 99
"Whatever Logic is good enough to tell me is worth writing down. 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."
"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.
I don't get it A, B does imply Z. Why need the third rule? C that A and B implies Z, and after that D that, A, B, and C implies Z and zo on?
I got this from pinker Stephen's How the Mind's work.
Best Answer
Of course $A$ and $B$ implies $Z$! That's not in question. But how do we get from the premisses $A$ and $B$ to the conclusion $Z$?
To avoid distracting clutter for a moment, let's change the example for a bit and consider
and
where $\to$ is some conditional. These evidently imply
But again, how and why? One thing to say is: because we can invoke a principle of inference, a permissive rule that says
That inference rule is of course the Modus Ponens rule. And the point of Lewis Carroll's 'What the Tortoise Said to Achilles' is to show us vividly that we can't here replace the rule by a proposition such as
to serve as a third premiss. For if we just accept this as a new premiss, we'll just have a list of three premisses, and will still need a permissive rule to allow us to get anywhere from them, e.g. the rule
Can we avoid appeal to that rule by instead accepting the proposition
as a new premiss. Of course not. To get to $q$ from $A', B', C', D'$ we'd need to invoke another rule! So we really, really, don't want to start down this regress!
In sum: we can't replace the modus ponens rule by a proposition such as $(C')$. Of course, $(C')$ is true, and the rule and the truth are intimately connected: that's why we might get confused here. But at some point, to get anywhere in a deduction, we need rules of inference like (MP), not just more premisses.
Likewise for Carroll's original example: how do we infer the original $Z$ from $A$ and $B$? We could add further propositional assumptions if we want, but at some point we have to appeal to a rule of inference. That's the moral that is being driven home.
(Of course, the rule/proposition distinction Carroll is driving at is built into every system of baby logic that beginners encounter, so -- looked at one way -- it can seem now that he is fussing about nothing. But looked at another way, this point explains why that fundamental distinction is compulsory.)