What the Tortoise Said to Achilles
Encyclopedia
"What the Tortoise Said to Achilles", written by Lewis Carroll
in 1895 for the philosophical journal Mind
, is a brief dialogue which problematises the foundations of logic
. The title alludes
to one of Zeno's paradoxes of motion
, 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.
The Tortoise asks Achilles whether the conclusion logically follows from the premises, and Achilles grants that it obviously does. The Tortoise then asks Achilles whether there might be a reader of Euclid
who grants that the argument is logically valid, as a sequence, while denying that A and B are true. Achilles accepts that such a reader might exist, and that he would hold that if A and B are true, then Z must be true, while not yet accepting that A and B are true. (A reader who denies the premises.)
The Tortoise then asks Achilles whether a second kind of reader might exist, who accepts that A and B are true, but who does not yet accept the principle that if A and B are both true, then Z must be true. Achilles grants the Tortoise that this second kind of reader might also exist. The Tortoise, then, asks Achilles to treat him as a reader of this second kind, and then to logically compel him to accept that Z must be true. (The tortoise is a reader who denies the argument itself, the syllogism's conclusion, structure or validity.)
After writing down A, B and Z in his notebook, Achilles asks the Tortoise to accept the hypothetical:
The Tortoise agrees to accept C, if Achilles will write down what he has to accept in his note-book, making the new argument:
But now that the Tortoise accepts premise C, he still refuses to accept the expanded argument. When Achilles demands that "If you accept A and B and C, you must accept Z," the Tortoise remarks that that's another hypothetical proposition, and suggests even if he accepts C, he could still fail to conclude Z if he did not see the truth of:
The Tortoise continues to accept each hypothetical premise once Achilles writes it down, but denies that the conclusion necessarily follows, since each time he denies the hypothetical that if all the premises written down so far are true, Z must be true:
Thus, the list of premises continues to grow without end, leaving the argument always in the form:
At each step, the Tortoise argues that even though he accepts all the premises that have been written down, there is some further premise (that if all of (1)–(n) are true, then (Z) must be true) that he still needs to accept before he is compelled to accept that (Z) is true.
deductions.
P --> Q P
The regress problem arises, because, in order to explain the logical principle, we have to then propose a prior principle. And, once we explain 'that' principle, then we have to introduce another principle to explain that principle. Thus, if the causal chain is to continue, we are to fall into infinite regress. However, if we introduce a formal system where Modus Ponens is simply an axiom/rule, then we are to abide by it simply, because it is so. For example, in a chess game there are particular rules, and the rules simply go without question. As players of the chess game, we are to simply follow the rules. Likewise, if we are engaging in a formal system of logic, then we are to simply follow the rules without question. Hence, introducing the formal system of logic stops the infinite regression--that is, because the regress would stop at the axioms or rules, per se, of the given game, system, etc. Though, it does also state that there's problems with this as well, because, within the system, no proposition or variable carries with it any semantic content. So, the moment you add to any proposition or variable semantic content, the problem arises again, because the propositions and variables WITH semantic content run 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 conditional connective (the syntactic sign "
"), and the implication relation (the formal object denoted by the double arrow symbol "
"). These logicians use the phrase not p or q for the conditional connective and the term implies for the implication relation. Some explain the difference by saying that the conditional is the contemplated relation while the implication is the asserted relation. In most fields of mathematics, it is treated as a variation in the usage of the single sign "
", not requiring two separate signs. Not all of those who use the sign "
" for the conditional connective regard it as a sign that denotes any kind of object, but treat it as a so-called syncategorematic sign, that is, a sign with a purely syntactic function. For the sake of clarity and simplicity in the present introduction, it is convenient to use the two-sign notation, but allow the sign "
" to denote the boolean function that is associated with the truth table
of the material conditional.
These considerations result in the following scheme of notation.
The paradox ceases to exist the moment we replace informal logic with propositional logic. The Turtle and Achilles don't agree on any definition of logical implication. In propositional logic the logical implication is defined as follows:
P
Q if and only if the proposition P 
Q is a tautology
hence de modus ponens [P
(P 
Q)] 
Q, is a valid logical implication according to the definition of logical implication just stated. There is no need to recurse since the logical implication can be translated into symbols, and propositional operators such as 
. Demonstrating the logical implication simply translates into verifying that the compound truth table is producing a tautology.
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 is equivalent to, or dependent on, agreeing to the hypothetical "If A and B are true, then Z is true."
The Wittgensteinian
philosopher Peter Winch
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. 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. 53).
Lewis Carroll
Charles Lutwidge Dodgson , better known by the pseudonym Lewis Carroll , was an English author, mathematician, logician, Anglican deacon and photographer. His most famous writings are Alice's Adventures in Wonderland and its sequel Through the Looking-Glass, as well as the poems "The Hunting of the...
in 1895 for the philosophical journal Mind
Mind (journal)
Mind is a British journal, currently published by Oxford University Press on behalf of the Mind Association, which deals with philosophy in the analytic tradition...
, is a brief dialogue which problematises the foundations of logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
. The title alludes
Allusion
An allusion is a figure of speech that makes a reference to, or representation of, people, places, events, literary work, myths, or works of art, either directly or by implication. M. H...
to one of Zeno's paradoxes of motion
Zeno's paradoxes
Zeno's paradoxes are a set of problems generally thought to have been devised by Greek philosopher Zeno of Elea to support Parmenides's doctrine that "all is one" and that, contrary to the evidence of our senses, the belief in plurality and change is mistaken, and in particular that motion is...
, 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.
Summary of the dialogue
The discussion begins by considering the following logical argument:- A: "Things that are equal to the same are equal to each other" (Euclidean relationEuclidean relationIn mathematics, Euclidean relations are a class of binary relations that satisfy a weakened form of transitivity that formalizes Euclid's "Common Notion 1" in The Elements: things which equal the same thing also equal one another.-Definition:...
, a weakened form of the transitive property) - 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 asks Achilles whether the conclusion logically follows from the premises, and Achilles grants that it obviously does. The Tortoise then asks Achilles whether there might be a reader of Euclid
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
who grants that the argument is logically valid, as a sequence, while denying that A and B are true. Achilles accepts that such a reader might exist, and that he would hold that if A and B are true, then Z must be true, while not yet accepting that A and B are true. (A reader who denies the premises.)
The Tortoise then asks Achilles whether a second kind of reader might exist, who accepts that A and B are true, but who does not yet accept the principle that if A and B are both true, then Z must be true. Achilles grants the Tortoise that this second kind of reader might also exist. The Tortoise, then, asks Achilles to treat him as a reader of this second kind, and then to logically compel him to accept that Z must be true. (The tortoise is a reader who denies the argument itself, the syllogism's conclusion, structure or validity.)
After writing down A, B and Z in his notebook, Achilles asks the Tortoise to accept the hypothetical:
- C: "If A and B are true, Z must be true"
The Tortoise agrees to accept C, if Achilles will write down what he has to accept in his note-book, making the new argument:
- A: "Things that are equal to the same are equal to each other"
- B: "The two sides of this triangle are things that are equal to the same"
- C: "If A and B are true, Z must be true"
- Therefore Z: "The two sides of this triangle are equal to each other"
But now that the Tortoise accepts premise C, he still refuses to accept the expanded argument. When Achilles demands that "If you accept A and B and C, you must accept Z," the Tortoise remarks that that's another hypothetical proposition, and suggests even if he accepts C, he could still fail to conclude Z if he did not see the truth of:
- D: "If A and B and C are true, Z must be true"
The Tortoise continues to accept each hypothetical premise once Achilles writes it down, but denies that the conclusion necessarily follows, since each time he denies the hypothetical that if all the premises written down so far are true, Z must be true:
- "And at last we've got to the end of this ideal race-course! Now that you accept A and B and C and D, of course you accept Z."
- "Do I?" said the Tortoise innocently. "Let's make that quite clear. I accept A and B and C and D. Suppose I still refused to accept Z?"
- "Then Logic would take you by the throat, and force you to do it!" Achilles triumphantly replied. "Logic would tell you, 'You can't help yourself. Now that you've accepted A and B and C and D, you must accept Z!' So you've no choice, you see."
- "Whatever Logic is good enough to tell me is worth writing down," said the Tortoise. "So enter it in your note-book, please. We will call it
- (E) If A and B and C and D are true, Z must be true.
- Until I've granted that, of course I needn't grant Z. So it's quite a necessary step, you see?"
- "I see," said Achilles; and there was a touch of sadness in his tone.
Thus, the list of premises continues to grow without end, leaving the argument always in the form:
- (1): "Things that are equal to the same are equal to each other"
- (2): "The two sides of this triangle are things that are equal to the same"
- (3): (1) and (2) ⇒ (Z)
- (4): (1) and (2) and (3) ⇒ (Z)
- …
- (n): (1) and (2) and (3) and (4) and ... and (n − 1) ⇒ (Z)
- Therefore (Z): "The two sides of this triangle are equal to each other"
At each step, the Tortoise argues that even though he accepts all the premises that have been written down, there is some further premise (that if all of (1)–(n) are true, then (Z) must be true) that he still needs to accept before he is compelled to accept that (Z) is true.
Explanation
Lewis Carroll was showing that there's a regress problem that arises from modus ponensModus ponens
In classical logic, modus ponendo ponens or implication elimination is a valid, simple argument form. It is related to another valid form of argument, modus tollens. Both Modus Ponens and Modus Tollens can be mistakenly used when proving arguments...
deductions.
P --> Q P
- ---------------
- Therefore, Q.
The regress problem arises, because, in order to explain the logical principle, we have to then propose a prior principle. And, once we explain 'that' principle, then we have to introduce another principle to explain that principle. Thus, if the causal chain is to continue, we are to fall into infinite regress. However, if we introduce a formal system where Modus Ponens is simply an axiom/rule, then we are to abide by it simply, because it is so. For example, in a chess game there are particular rules, and the rules simply go without question. As players of the chess game, we are to simply follow the rules. Likewise, if we are engaging in a formal system of logic, then we are to simply follow the rules without question. Hence, introducing the formal system of logic stops the infinite regression--that is, because the regress would stop at the axioms or rules, per se, of the given game, system, etc. Though, it does also state that there's problems with this as well, because, within the system, no proposition or variable carries with it any semantic content. So, the moment you add to any proposition or variable semantic content, the problem arises again, because the propositions and variables WITH semantic content run 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 conditional connective (the syntactic sign "
"), and the implication relation (the formal object denoted by the double arrow symbol ""). These logicians use the phrase not p or q for the conditional connective and the term implies for the implication relation. Some explain the difference by saying that the conditional is the contemplated relation while the implication is the asserted relation. In most fields of mathematics, it is treated as a variation in the usage of the single sign "", not requiring two separate signs. Not all of those who use the sign "" for the conditional connective regard it as a sign that denotes any kind of object, but treat it as a so-called syncategorematic sign, that is, a sign with a purely syntactic function. For the sake of clarity and simplicity in the present introduction, it is convenient to use the two-sign notation, but allow the sign "" to denote the boolean function that is associated with the truth tableTruth table
A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—to compute the functional values of logical expressions on each of their functional arguments, that is, on each combination of values taken by their...
of the material conditional.
These considerations result in the following scheme of notation.
The paradox ceases to exist the moment we replace informal logic with propositional logic. The Turtle and Achilles don't agree on any definition of logical implication. In propositional logic the logical implication is defined as follows:
P
Q if and only if the proposition P Q is a tautologyhence de modus ponens [P
(P Q)] Q, is a valid logical implication according to the definition of logical implication just stated. There is no need to recurse since the logical implication can be translated into symbols, and propositional operators such as . Demonstrating the logical implication simply translates into verifying that the compound truth table is producing a tautology.Discussion
Several philosophers have tried to resolve the Carroll paradox. Bertrand RussellBertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
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 is equivalent to, or dependent on, agreeing to the hypothetical "If A and B are true, then Z is true."
The Wittgensteinian
Ludwig Wittgenstein
Ludwig Josef Johann Wittgenstein was an Austrian philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. He was professor in philosophy at the University of Cambridge from 1939 until 1947...
philosopher Peter Winch
Peter Winch
Peter Guy Winch was a British philosopher known for his contributions to the philosophy of social science, Wittgenstein scholarship, ethics, and the philosophy of religion...
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. 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. 53).
See also
- Deduction theoremDeduction theoremIn mathematical logic, the deduction theorem is a metatheorem of first-order logic. It is a formalization of the common proof technique in which an implication A → B is proved by assuming A and then proving B from this assumption. The deduction theorem explains why proofs of conditional...
- Münchhausen Trilemma
- ParadoxParadoxSimilar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...
- Homunculus argumentHomunculus argumentThe homunculus argument is a fallacy arising most commonly in the theory of vision. One may explain vision by noting that light from the outside world forms an image on the retinas in the eyes and something in the brain looks at these images as if they are images on a movie screen The homunculus...
Where to find the article
- Carroll, Lewis. "What the Tortoise Said to Achilles". Mind, n.s., 4 (1895), pp. 278–80.
- Hofstadter, DouglasDouglas HofstadterDouglas Richard Hofstadter is an American academic whose research focuses on consciousness, analogy-making, artistic creation, literary translation, and discovery in mathematics and physics...
. Gödel, Escher, Bach: an Eternal Golden BraidGödel, Escher, BachGödel, Escher, Bach: An Eternal Golden Braid is a book by Douglas Hofstadter, described by his publishing company as "a metaphorical fugue on minds and machines in the spirit of Lewis Carroll"....
. See the second dialogue, entitled "Two-Part Invention." Dr. Hofstadter appropriated the characters of Achilles and the Tortoise for other, original, dialogues in the book which alternate contrapuntally with prose chapters. - A number of websites, including "What the Tortoise Said to Achilles" at the Lewis Carroll Society of North America, "What the Tortoise Said to Achilles" at Digital Text International, and "What the Tortoise Said to Achilles" at Fair Use Repository.