By Lewis Caroll in 1895. [1]
The Tortoise assumes a proposition \(p\) and a material conditional \(p \implies q\).
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: \(p \land (p \implies q) \implies q\).
Achilles says that, now, you really have to accept \(q\), given that you’re committed to:
\(p\)
\(p \implies q\)
\(p \land (p \implies q) \implies q\).
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:
\(p \land (p \implies q) \land (p \land (p \implies q)) \implies q\)
This can go ad infinitum; the Tortoise wins.
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 his Sellars lectures)
This seems analogous to descriptivism vs expressivism.
It illustrates what we lose when we reduce all discourse to descriptive discourse. When we choose our definitions such that natural laws are facts in the world, just like any other ordinary empirical fact, we lose both:
Their role in reasoning
A story for our knowledge/justification of them.
The story shows how treating a rule as a fact strips it of its normative force. It is the problem of conflating description in the narrow sense with description in the wider sense: see here.
The last line of this commentary makes me think this can also be used to counter some forms of radical skepticism, i.e. to recover an air of dignity to making working assumptions.