"What the Tortoise Said to Achilles",[1] written byLewis Carroll in 1895 for the philosophical journalMind,[1] is a brief allegorical dialogue on the foundations oflogic.[1] The titlealludes to one ofZeno's paradoxes of motion,[2] in whichAchilles could never overtake thetortoise in a race. 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. Ultimately, Achilles fails, because the clever tortoise leads him into aninfinite regression.[1]
The discussion begins by considering the following logical argument:[1][3]
The tortoise accepts premisesA andB as true but not the hypothetical:
The Tortoise claims that it is not "under any logical necessity to acceptZ as true". The tortoise then challenges Achilles to force it logically to acceptZ as true. Instead of searching the tortoise’s reasons for not acceptingC, Achilles asks it to acceptC, which it does. After which, Achilles says:
The tortoise responds, "That's another Hypothetical, isn't it? And, if I failed to see its truth, I might accept A and B and C, and still not accept Z, mightn't I?"[1][3]
Again, instead of requesting reasons for not acceptingD, he asks the tortoise to acceptD. And again, it is "quite willing to grant it",[1][3] but it still refuses to accept Z. It then tells Achilles to write into his book,
Following this, the Tortoise says: "until I’ve granted that [i.e.,E], of course I needn’t grant Z. So it's quite a necessary step".[1] With a touch of sadness, Achilles sees the point.[1][3]
The story ends by suggesting that the list of premises continues to grow without end, but without explaining the point of the regress.[1][3]
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Lewis Carroll was showing that there is a regressive problem that arises frommodus ponens deductions:
or, in words: propositionP (is true) impliesQ (is true), and givenP, thereforeQ.
The regress problem arises because a prior principle is required to explain logical principles, heremodus ponens, and oncethat principle is explained,another principle is required to explainthat principle. Thus, if the argumentative chain is to continue, the argument falls into infinite regress. However, if a formal system is introduced wherebymodus ponens is simply arule of inference defined within the system, then it can be abided simply by reasoning within the system. That is not to say that the user reasoning according to this formal system agrees with these rules (consider, for example, theconstructivist's rejection of thelaw of the excluded middle and thedialetheist's rejection of thelaw of noncontradiction). In this way, formalising logic as a system can be considered as a response to the problem of infinite regress:modus ponens is placed as a rule within the system, the validity ofmodus ponens is eschewed without the system.
In propositional logic, the logical implication is defined as follows:
P implies Q if and only if the propositionnot P or Q is atautology.
Hencemodus ponens, [P ∧ (P → Q)] ⇒ Q, is a valid logical conclusion according to the definition of logical implication just stated. Demonstrating the logical implication simply translates into verifying that the compound truth table produces a tautology. But the tortoise does not accept on faith the rules of propositional logic that this explanation is founded upon. He asks that these rules, too, be subject to logical proof. The tortoise and Achilles do not agree on any definition of logical implication.
In addition, the story hints at problems with the propositional solution. Within the system of propositional logic, no proposition or variable carries any semantic content. The moment any proposition or variable takes on semantic content, the problem arises again because semantic content runs outside the system. Thus, if the solution is to be said to work, then it is to be said to work solely within the given formal system, and not otherwise.
Some logicians (Kenneth Ross, Charles Wright) draw a firm distinction between theconditional connective and theimplication relation. These logicians use the phrasenot p or q for the conditional connective and the termimplies for an asserted implication relation.
Several philosophers have tried to resolve Carroll's paradox.Bertrand Russell discussed the paradox briefly in§ 38 ofThe Principles of Mathematics (1903), distinguishing betweenimplication (associated with the form "ifp, thenq"), which he held to be a relation betweenunasserted propositions, andinference (associated with the form "p, thereforeq"), which he held to be a relation betweenasserted propositions; having made this distinction, Russell could deny that the Tortoise's attempt to treatinferringZ fromA andB as equivalent to, or dependent on, agreeing to thehypothetical "IfA andB are true, thenZ is true."
Peter Winch, aWittgensteinian philosopher, discussed the paradox inThe Idea of a Social Science and its Relation to Philosophy (1958), where he argued that the paradox showed that "the actual process of drawing an inference, which is after all at the heart of logic, is something which cannot be represented as a logical formula ... Learning to infer is not just a matter of being taught about explicit logical relations between propositions; it is learningto do something" (p. 57). Winch goes on to suggest that the moral of the dialogue is a particular case of a general lesson, to the effect that the properapplication of rules governing a form of human activity cannot itself be summed up with a set offurther 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 toconventionalism about logical truth,[4] later reworked in more sober philosophical terms byW.V.O. Quine.[5]
Lewis Carroll (April 1895). "What the Tortoise Said to Achilles".Mind.IV (14):278–280.doi:10.1093/mind/IV.14.278.
Reprinted:
The full text ofWhat the Tortoise Said to Achilles at Wikisource.As audio:
What the Tortoise Said to Achilles public domain audiobook atLibriVox