What the Tortoise said to Achilles

Written by Lewis Caroll in 1895.

Summary §

• The Tortoise assumes a proposition $p$ and a material conditional $p⟹q$.

  • The exact $p$ and $q$ aren’t important to the moral of the story, though it’s something like “If $A=B$ and $B=C$ ($p$), then $A=C$ ($q$)”
  • The Tortoise is playing a game: I’ll do anything you tell me to do, so long as you make explicit the rule you’re asking me to follow.

• Achilles tries to convince the Tortoise to accept $q$.

  • He says that logic obliges you to acknowledge $q$ in this case.

• The Tortoise is willing to go along with this but demands that this rule be made explicit:

  • Achilles adds an extra axiom: $p\wedge (p⟹q)⟹q$.

• Achilles says that, now, you really have to accept $q$, given that you’re committed to:

  • $p$
  • $p⟹q$
  • $p\wedge (p⟹q)⟹q$.

• But the Tortoise notes that, if taking those three propositions and concluding $q$ is really something logic obliges one to do, then it bears writing down:

  • $p\wedge (p⟹q)\wedge (p\wedge (p⟹q))⟹q$

• This can go ad infinitum; the Tortoise wins.

Brandom commentary §

• The most influential pragmatist work in the philosophy of logic.

• The lesson:

  • in any particular case, you can substitute a rule (that tells you you can go from this to that) with an axiom.
  • But there have got to be some moves you can make without having to explicitly license them by a principle.
  • I.e. you’ve got to distinguish between 1.) premises from which to reason 2.) principles in accordance with which to reason.

• This teaches an un-get-over-able lesson about the necessity for an implicit practical background of making some moves that are just okay. Things that would be put in a logical system, not in the forms of axioms, but in the form of rules.

(This is from one of Brandom’s lectures on Sellars)