"What the Tortoise Said to Achilles",[1] written by Lewis Carroll in 1895 for the philosophical journal Mind,[1] is a brief allegorical dialogue on the foundations of logic.
In Carroll's dialogue, the tortoise challenges Achilles to use the force of logic to make him accept the conclusion of a simple deductive argument.
[1][3] Lewis Carroll was showing that there is a regressive problem that arises from modus ponens deductions.
Demonstrating the logical implication simply translates into verifying that the compound truth table produces a tautology.
Some logicians (Kenneth Ross, Charles Wright) draw a firm distinction between the conditional connective and the implication relation.
These logicians use the phrase not p or q for the conditional connective and the term implies for an asserted implication relation.
Bertrand Russell discussed the paradox briefly in § 38 of The Principles of Mathematics (1903), distinguishing between implication (associated with the form "if p, then q"), which he held to be a relation between unasserted propositions, and inference (associated with the form "p, therefore q"), which he held to be a relation between asserted propositions; having made this distinction, Russell could deny that the Tortoise's attempt to treat inferring Z from A and B as equivalent to, or dependent on, agreeing to the hypothetical "If A and B are true, then Z is true."
Winch goes on to suggest that the moral of the dialogue is a particular case of a general lesson, to the effect that the proper application of rules governing a form of human activity cannot itself be summed up with a set of further rules, and so that "a form of human activity can never be summed up in a set of explicit precepts" (p. 53).
Carroll's dialogue is apparently the first description of an obstacle to conventionalism about logical truth,[4] later reworked in more sober philosophical terms by W.V.O.