Editing What the Tortoise Said to Achilles

{{Short description|1895 allegorical dialogue by Lewis Carroll}}
{{EngvarB|date=September 2013}}
{{Use dmy dates|date=December 2023}}

"'''What the Tortoise Said to Achilles'''",<ref name=":0">{{Cite web |last=Carroll |first=Lewis |author-link=Lewis Carroll |date=1895 |title=What the Tortoise Said to Achilles |url=https://math.dartmouth.edu/~matc/Readers/HowManyAngels/Tortoise.html |access-date=2024-03-25 |website=math.dartmouth.edu}}</ref> written by [[Lewis Carroll]] in 1895 for the philosophical journal ''[[Mind (journal)|Mind]]'',<ref name=":0" /> is a brief allegorical dialogue on the foundations of [[logic]].<ref name=":0" /> The title [[allusion|alludes]] to one of [[Zeno's paradoxes#Paradoxes of motion|Zeno's paradoxes of motion]],<ref>{{Cite book |last=Tsilipakos |first=Leonidas |title=Clarity and confusion in social theory: taking concepts seriously |date=2021 |publisher=Routledge |isbn=978-1-032-09883-8 |series=Philosophy and method in the social sciences |location=Abingdon New York (N.Y.) |pages=48}}</ref> in which [[Achilles]] could never overtake the [[tortoise]] 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 an [[infinite regression]].<ref name=":0" />

==Summary of the dialogue==
The discussion begins by considering the following logical argument:<ref name=":0" /><ref name=":1">{{Cite book |last=Gratton |first=Claude |title=Infinite regress arguments |date=2010 |publisher=Springer |isbn=978-90-481-3340-6 |series=Argumentation library |location=Dordrecht |pages=38-44}}</ref>
* ''A'': "Things that are equal to the same are equal to each other" (a [[Euclidean relation]])
* ''B'': "The two sides of this triangle are things that are equal to the same"
* Therefore, ''Z'': "The two sides of this triangle are equal to each other"

The tortoise accepts premises ''A'' and ''B'' as true but not the hypothetical:
* ''C'': "If ''A'' and ''B'' are true, ''Z'' must be true"<ref name=":0" /><ref name=":1" />

The Tortoise claims that it is not "under any logical necessity to accept ''Z'' as true". The tortoise then challenges Achilles to force it logically to accept ''Z'' as true. Instead of searching the tortoise’s reasons for not accepting ''C'', Achilles asks it to accept ''C'', which it does. After which, Achilles says:
* ''D'': "If ''A'' and ''B'' and ''C'' are true, ''Z'' must be true"<ref name=":0" /><ref name=":1" />

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?"<ref name=":0" /><ref name=":1" />

Again, instead of requesting reasons for not accepting ''D'', he asks the tortoise to accept ''D''. And again, it is "quite willing to grant it",<ref name=":0" /><ref name=":1" /> but it still refuses to accept Z. It then tells Achilles to write into his book,

* ''E:'' If A and B and C and D are true, Z must be true.

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".<ref name=":0" /> With a touch of sadness, Achilles sees the point.<ref name=":0" /><ref name=":1" />

The story ends by suggesting that the list of premises continues to grow without end, but without explaining the point of the regress.<ref name=":0" /><ref name=":1" />
==Explanation==
{{unreferenced section |date=April 2023}}
Lewis Carroll was showing that there is a regressive problem that arises from ''[[modus ponens]]'' deductions. 
:<math>\frac{P \to Q,\; P}{\therefore Q}</math>
Or, in words: proposition ''P'' (is true) implies ''Q'' (is true), and given ''P'', therefore ''Q''.

The regress problem arises because a prior principle is required to explain logical principles, here ''modus ponens'', and once ''that'' principle is explained, ''another'' principle is required to explain ''that'' principle. Thus, if the argumentative chain is to continue, the argument falls into infinite regress. However, if a formal system is introduced whereby ''modus ponens'' is simply a [[rule of inference]] defined within the system, then it can be abided by 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, the [[Constructivist logic|constructivist]]'s rejection of the [[law of the excluded middle]] and the [[Dialetheism|dialetheist]]'s rejection of the [[law 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 of ''modus ponens'' is eschewed without the system.

In propositional logic, the logical implication is defined as follows:

P implies Q  if and only if the proposition ''not P or Q'' is a [[Tautology (logic)|tautology]].

Hence ''modus 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 the [[material conditional|conditional connective]] and the [[Logical consequence|implication relation]]. These logicians use the phrase ''not p or q'' for the conditional connective and the term ''implies'' for an asserted implication relation.

==Discussion==
Several philosophers have tried to resolve Carroll's paradox. [[Bertrand Russell]] discussed the paradox briefly in [http://fair-use.org/bertrand-russell/the-principles-of-mathematics/s.38 § 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."

[[Peter Winch]], a [[Ludwig Wittgenstein|Wittgensteinian]] philosopher, discussed the paradox in ''The 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 learning ''to do'' something" (p.&nbsp;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 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.&nbsp;53).

Carroll's dialogue is apparently the first description of an obstacle to [[conventionalism]] about logical truth,<ref name="Maddy">{{cite journal | title=The Philosophy of Logic | author=Maddy, P. | author-link=Penelope Maddy | journal=Bulletin of Symbolic Logic |date=December 2012  | volume=18 | issue=4 | pages=481–504 | doi=10.2178/bsl.1804010 | jstor=23316289| s2cid=28202258 }}</ref> later reworked in more sober philosophical terms by [[Willard Van Orman Quine|W.V.O. Quine]].<ref name="Quine">{{cite book | title=The Ways of Paradox, and Other Essays | publisher=Harvard University Press | author=Quine, W.V.O. | year=1976 | location=Cambridge, MA | isbn=9780674948358 | oclc=185411480 | url-access=registration | url=https://archive.org/details/waysofparadox00quin }}</ref>

==See also==
*[[Deduction theorem]]
*[[Homunculus argument]]
*[[Münchhausen trilemma]]
*[[Paradox]]
*[[Regress argument]]
*[[Rule of inference]]
*''[[Gödel, Escher, Bach]]'', a book by [[Douglas Hofstadter]] which includes this story (though retitled as "Two-Part Invention"), and original stories with the Tortoise and Achilles characters

==Sources==
{{cite journal|author1=Lewis Carroll|title=What the Tortoise Said to Achilles|journal=[[Mind (journal)|Mind]]|date=April 1895|volume=IV|issue=14|pages=278–280|doi=10.1093/mind/IV.14.278}}

Reprinted:
* in the New Series of the same journal, a century later: {{cite journal|author1=Lewis Carroll|title=What the Tortoise Said to Achilles|journal=[[Mind (journal)|Mind]]|date=1 October 1995|volume=104|issue=416|pages=691–693|jstor=2254477|doi=10.1093/mind/104.416.691}}
* in {{cite book|title=The Penguin Complete Lewis Carroll|publisher=Harmondsworth, Penguin|year=1982|pages=1104–1108}}
* on a number of websites, including [http://www.ditext.com/carroll/tortoise.html "What the Tortoise Said to Achilles"] at [http://www.ditext.com/ Digital Text International], and [http://fair-use.org/mind/1895/04/what-the-tortoise-said-to-achilles "What the Tortoise Said to Achilles"] at [http://fair-use.org Fair Use Repository].
* on [[Wikisource]]: {{wikisource-inline|What the Tortoise Said to Achilles|single=true}}.
* in {{cite book|author1=[[Douglas Hofstadter]]|title=[[Gödel, Escher, Bach|Gödel, Escher, Bach: an Eternal Golden Braid]]|publisher=Basic Books|year=1979|isbn=978-0-465-02656-2}} Reprinted as the second dialogue, entitled "Two-Part Invention – or What the Tortoise Said to Achilles", between chapters 1 and 2. Hofstadter appropriated the characters of Achilles and the Tortoise for other, original, dialogues in the book which alternate contrapuntally with prose chapters.

As audio:
* {{librivox book|author=Lewis Carroll|title=What the Tortoise Said to Achilles}}

==References==
{{Reflist}}

==Further reading==
* Moktefi, Amirouche & Abeles, Francine F. (eds.). “‘What the Tortoise Said to Achilles’: Lewis Carroll's Paradox of Inference.” ''The Carrollian: The Lewis Carroll Journal'', No. 28, November 2016. [Special issue.] {{ISSN|1462-6519}} {{ISBN|978-0-904117-39-4}}

{{Lewis Carroll}}

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