"What the Tortoise Said to Achilles" is a brief dialog by
Lewis Carroll which playfully problematizes the foundations of
logic. The conversation takes place between the tortoise and Achilles, the two runners in
Zeno's racecourse paradox. The tortoise challenges Achilles to use the force of logic to convince him that a particular deductive inference of the
modus ponens variety is a
valid argument. Ultimately, Achilles fails, because the clever tortoise leads him into an infinite regression.
The discussion concerns three statements:
- (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"
which could be formalized with mathematical symbols as:
- (A') ∀x∀y:equalsame(x,y) ⇒ x=y
- (B') equalsame(a,b)
- (Z') a=b
Achilles' task is to make the Tortoise agree that, if he accepts A and B, he must necessarily accept Z. The Tortoise sees no particularly good reason to agree to this, but takes it up as an additional premise C in order to humor Achilles:
- (C) "If A and B are true, Z must be true"
Achilles argues that because the Tortoise now accepts A, B, and C, he must now necessarily accept Z. The Tortoise sees no particularly good reason to agree to this, but takes it up as an additional premise D in order to humor Achilles:
- (D) "If A and B and C are true, Z must be true"
Ultimately, the Tortoise grows the list of premises ad infinitum, and never accepts the conclusion Z.
The tortoise does not acknowledge modus ponens as a generally valid argument form. Carroll demonstrates the great difficulty of deducing modus ponens from premises that do not include modus ponens. This suggests that, more generally, valid logical argument forms cannot be derived "from nothing". ...
Several philosophers have tried to resolve the Carroll paradox. Isashiki Takahiro (1999) summarizes past attempts and concludes they all fail before beginning yet another.
...
- Carroll, Lewis. "What the Tortoise Said to Achilles". Mind, n.s., 4 (1895), pp. 278-80.
- Hofstadter, Douglas. Gödel, Escher, Bach: an Eternal Golden Braid. See the second dialog, entitled "Two-Part Invention".
- any number of websites, including [1] (http://www.lewiscarroll.org/achilles.html), [2] (http://www.ditext.com/carroll/tortoise.html), and [3] (http://home.earthlink.net/~lfdean/carroll/essays/achilles.html)
- Isashiki Takahiro (1999). "What Can We Learn from Lewis Carroll's Paradox?". In Memoirs of the Faculty of Education, Miyazaki University: Humanities, no. 86, pp. 79-98. The paper is in Japanese, although an extremely condensed summary by the author is availible from [4] (http://www.miyazaki-u.ac.jp/~e02702u/papers/eng_carroll.html). Another author provides a more extended summary at [5] (http://homepage2.nifty.com/Workshop-Alice/click/m-t.html)