The Paradox of the Court is a very old problem in logic stemming from ancient Greece. It is said that the famous sophist Protagoras took on a pupil, Euathlus, on the understanding that the student pay Protagoras for his instruction after he had won his first case. Some accounts claim that Protagoras demanded his money as soon as Euathlus completed his education, others say that Protagoras waited until it was obvious that Euathlus was making no effort to take on clients[1] and still others [2] assert that Euathlus made a genuine attempt but that no clients ever came. In any case, Protagoras decided to sue Euathlus for the amount owed.
Protagoras argued that if he won the case he would be paid his money. If Euathlus won the case, Protagoras would still be paid according to the original contract, because Euathlus would have won his first case.
Euathlus, however, claimed that if he won then by the court's decision he would not have to pay Protagoras. If on the other hand Protagoras won then Euathlus would still not have won a case and therefore not be obliged to pay.
The question is: which of the two men is in the right?
1. Analysis
From a moral standpoint it may be that either party was right, or that both were, or weren't, due to the ambiguous nature of the scenario. However, as a matter of law, if the Court were to rule in favor of Protagoras, the conditions of the original contract between him and his pupil would be invalid and Euathlus would have to pay Protagoras. If, on the other hand, Euathlus were to win, the Court could also void Euathlus's obligation of payment.
How, from an objective standpoint, the way the Court could make its ruling is not necessarily a paradox either. The Court would either rule that Euathlus (as the defendant) had violated the terms of the contract, or had not. The subsequent conundrum would have no legal bearing on the court's decision.
In some civil cases the respondent, if he receives the favor of the court, is also shielded from payments associated with the act of going to court. The Court could indeed rule that Protagoras, as the unsuccessful plaintiff, pay Euathlus the amount which it cost to win. In this case, Euathlus would pay Protagoras only to have the money returned by order of the court. The original contract would have been fulfilled, and Euathlus would bear no further obligation to pay Protagoras for his instruction. The net outcome for Protagoras would be to lose his case, receive payment per the original contract, and then have to pay for the defendant's losses due to his failed suit (Which would be equal to, or exceeding, the cost of Euathlus's education.)
Additionally, Euathlus could hire a lawyer to take on the case, thus invalidating this case as a standard for payment.
1. 1. Another theory
Another means of viewing this case is as follows:
Euathlus would win his case because Protagoras sued him BEFORE Euathlus won his first case. Protagoras would lose that particular case because Euathlus has not yet won a case, and therefore Protagoras's cause of action had not yet manifested itself.
The new victory of Euathlus would qualify as new evidence for Protagoras, thus constituting grounds for a new trial.
2. Notes
1. ^ Peter Suber, Department of Philosophy, Earlham College, Richmond, Indiana, 47374, U.S.A.
2. ^ Eugene P. Northrop, "Riddles in Mathematics", Penguin Books
Showing posts with label _C. Show all posts
Showing posts with label _C. Show all posts
Friday, December 26, 2008
Curry's paradox
Curry's paradox is a paradox that occurs in naive set theory or naive logics, and allows the derivation of an arbitrary sentence from a self-referring sentence and some apparently innocuous logical deduction rules. It is named after the logician Haskell Curry.
It has also been called Löb's paradox after Martin Hugo Löb. [1]
Contents:
1. In natural language
2. In formal language
3. In naive set theory
4. Discussion
5. See also
6. References
7. External links
1. In natural language
Claims of the form "if A, then B" are called conditional claims. It is not necessary to believe the conclusion (B) to accept the conditional claim (if A, then B) as true. For instance, consider the following sentence:
If a man with flying reindeer has delivered presents to all the good children in the world in one night, then Santa Claus exists.
Imagine that a man with flying reindeer has, in fact, done this. Does Santa Claus exist, in that case? It would seem so. Therefore, without believing that Santa Claus exists, or that this scenario is even possible, it seems that we should agree that if a man with flying reindeer has delivered presents to all the good children in the world in one night, then Santa Claus exists, and so the above sentence is true.
Now consider this other sentence:
If this sentence is true, then Santa Claus exists.
As before, imagine that the antecedent is true - in this case, "this sentence is true". Does Santa Claus exist, in that case? Well, if the sentence is true, then what it says is true: namely that if the sentence is true, then Santa Claus exists. Therefore, without necessarily believing that Santa Claus exists, or that the sentence is true, it seems we should agree that if the sentence is true, then Santa Claus exists.
But then this means the sentence is true. So Santa Claus does exist. Furthermore we could substitute any claim at all for "Santa Claus exists". This is Curry's paradox.
2. In formal language
In formal languages, we sometimes interpret "If X then Y" as a material conditional. On this reading, it simply means "Y, or else not X". Here we would read the sentence as "Santa Claus exists, or this sentence is false". On this reading, Curry's paradox is simply a variant on the liar paradox. However, in natural language this is not usually what we mean by "If X then Y". For instance, "if 6*7=42, then the moon exists" is true as a material implication, but is generally not considered true in natural language, because the moon's existence does not seem to be related to this fact of arithmetic.
Nevertheless we arrived at paradox in natural language. In fact, not only did we arrive at a contradiction, but we were actually able to prove anything at all, without relying on such principles as the principle of explosion which are generally held to be false in accounts of natural language. Thus Curry's paradox poses an additional problem.
To arrive at this formally, let us denote by Y the proposition to prove, in this case "Santa Claus exists". Then, let X denote the statement that asserts that Y follows from the truth of X. Mathematically, this can be written as X = (X → Y), and we see that X is defined in terms of itself. The proof proceeds:
1. X → X
rule of assumption, also called restatement of premise or of hypothesis
2. X → (X → Y)
substitute right side of 1, since X = X → Y
3. X → Y
from 2 by contraction
4. X
substitute 3, since X = X → Y
5. Y
from 4 and 3 by modus ponens
3. In naive set theory
Even if the underlying mathematical logic does not admit any self-referential sentence, in set theories which allow unrestricted comprehension, we can nevertheless prove any logical statement Y from the set
The proof proceeds:
This can be seen as a variant on Russell's paradox, but is in an important way more general. Some proposals for set theory have attempted to deal with Russell's paradox not by restricting the rule of comprehension, but by restricting the rules of logic so that it tolerates the contradictory nature of the set of all sets that are not members of themselves. This reasoning shows that such a task is not so simple, because again, we have not only a contradiction, but we have in fact proved any statement whatsoever, without recourse to the full apparatus of the propositional calculus.
4. Discussion
Curry's paradox can be formulated in any language meeting certain conditions:
1. The language must contain an apparatus which lets it refer to, and talk about, its own sentences (such as quotation marks, names, or expressions like "this sentence");
2. The language must contain its own truth-predicate: that is, the language, call it "L", must contain a predicate meaning "true-in-L", and the ability to ascribe this predicate to any sentences;
3. The language must admit the rule of contraction, which roughly speaking means that a relevant hypothesis may be reused as many times as necessary; and
4. The language must of course admit the rules of identity (if A, then A) and modus ponens (from A, and if A then B, conclude B).
Various other sets of conditions are also possible. Natural languages nearly always contain all these features. Mathematical logic, on the other hand, generally does not countenance explicit reference to its own sentences, although the heart of Gödel's incompleteness theorems is the observation that usually this can be done anyway; see Gödel number. The truth-predicate is generally not available, but in naive set theory, this is arrived at through the unrestricted rule of comprehension. The rule of contraction is generally accepted, although linear logic (more precisely, linear logic without the exponential operators) does not admit the reasoning required for this paradox.
Note that unlike the liar paradox or Russell's paradox, this paradox does not depend on what model of negation is used, as it is completely negation-free. Thus paraconsistent logics can still be vulnerable to this, even if they are immune to the liar paradox.
The resolution of Curry's paradox is a contentious issue because resolutions (apart from trivial ones such as disallowing X directly) are difficult and not intuitive. Logicians are undecided whether such sentences are somehow impermissible (and if so, how to banish them), or meaningless, or whether they are correct and reveal problems with the concept of truth itself (and if so, whether we should reject the concept of truth, or change it), or whether they can be rendered benign by a suitable account of their meanings.
Linear logic disallows contraction and so does not admit this paradox directly, but one must remove its exponential operators, or else the paradox reappears in a modal form.
5. See also
* Richard's paradox
* Kleene-Rosser paradox
6. References
1. Barwise, Jon and John Etchemendy 1987: The Liar, p. 23. Oxford University Press.
7. External links
* The Stanford Encyclopedia of Philosophy: "Curry's Paradox" -- by J. C. Beall.
* Grossman, Jason, Australian National University: A Proof that Penguins Rule the Universe. A brief and entertaining discussion of Curry's paradox.
"[1]" "[2]" Relevant First-Order Logic LP# and Curry's Paradox http://front.math.ucdavis.edu/0804.4818
It has also been called Löb's paradox after Martin Hugo Löb. [1]
Contents:
1. In natural language
2. In formal language
3. In naive set theory
4. Discussion
5. See also
6. References
7. External links
1. In natural language
Claims of the form "if A, then B" are called conditional claims. It is not necessary to believe the conclusion (B) to accept the conditional claim (if A, then B) as true. For instance, consider the following sentence:
If a man with flying reindeer has delivered presents to all the good children in the world in one night, then Santa Claus exists.
Imagine that a man with flying reindeer has, in fact, done this. Does Santa Claus exist, in that case? It would seem so. Therefore, without believing that Santa Claus exists, or that this scenario is even possible, it seems that we should agree that if a man with flying reindeer has delivered presents to all the good children in the world in one night, then Santa Claus exists, and so the above sentence is true.
Now consider this other sentence:
If this sentence is true, then Santa Claus exists.
As before, imagine that the antecedent is true - in this case, "this sentence is true". Does Santa Claus exist, in that case? Well, if the sentence is true, then what it says is true: namely that if the sentence is true, then Santa Claus exists. Therefore, without necessarily believing that Santa Claus exists, or that the sentence is true, it seems we should agree that if the sentence is true, then Santa Claus exists.
But then this means the sentence is true. So Santa Claus does exist. Furthermore we could substitute any claim at all for "Santa Claus exists". This is Curry's paradox.
2. In formal language
In formal languages, we sometimes interpret "If X then Y" as a material conditional. On this reading, it simply means "Y, or else not X". Here we would read the sentence as "Santa Claus exists, or this sentence is false". On this reading, Curry's paradox is simply a variant on the liar paradox. However, in natural language this is not usually what we mean by "If X then Y". For instance, "if 6*7=42, then the moon exists" is true as a material implication, but is generally not considered true in natural language, because the moon's existence does not seem to be related to this fact of arithmetic.
Nevertheless we arrived at paradox in natural language. In fact, not only did we arrive at a contradiction, but we were actually able to prove anything at all, without relying on such principles as the principle of explosion which are generally held to be false in accounts of natural language. Thus Curry's paradox poses an additional problem.
To arrive at this formally, let us denote by Y the proposition to prove, in this case "Santa Claus exists". Then, let X denote the statement that asserts that Y follows from the truth of X. Mathematically, this can be written as X = (X → Y), and we see that X is defined in terms of itself. The proof proceeds:
1. X → X
rule of assumption, also called restatement of premise or of hypothesis
2. X → (X → Y)
substitute right side of 1, since X = X → Y
3. X → Y
from 2 by contraction
4. X
substitute 3, since X = X → Y
5. Y
from 4 and 3 by modus ponens
3. In naive set theory
Even if the underlying mathematical logic does not admit any self-referential sentence, in set theories which allow unrestricted comprehension, we can nevertheless prove any logical statement Y from the set
The proof proceeds:
This can be seen as a variant on Russell's paradox, but is in an important way more general. Some proposals for set theory have attempted to deal with Russell's paradox not by restricting the rule of comprehension, but by restricting the rules of logic so that it tolerates the contradictory nature of the set of all sets that are not members of themselves. This reasoning shows that such a task is not so simple, because again, we have not only a contradiction, but we have in fact proved any statement whatsoever, without recourse to the full apparatus of the propositional calculus.
4. Discussion
Curry's paradox can be formulated in any language meeting certain conditions:
1. The language must contain an apparatus which lets it refer to, and talk about, its own sentences (such as quotation marks, names, or expressions like "this sentence");
2. The language must contain its own truth-predicate: that is, the language, call it "L", must contain a predicate meaning "true-in-L", and the ability to ascribe this predicate to any sentences;
3. The language must admit the rule of contraction, which roughly speaking means that a relevant hypothesis may be reused as many times as necessary; and
4. The language must of course admit the rules of identity (if A, then A) and modus ponens (from A, and if A then B, conclude B).
Various other sets of conditions are also possible. Natural languages nearly always contain all these features. Mathematical logic, on the other hand, generally does not countenance explicit reference to its own sentences, although the heart of Gödel's incompleteness theorems is the observation that usually this can be done anyway; see Gödel number. The truth-predicate is generally not available, but in naive set theory, this is arrived at through the unrestricted rule of comprehension. The rule of contraction is generally accepted, although linear logic (more precisely, linear logic without the exponential operators) does not admit the reasoning required for this paradox.
Note that unlike the liar paradox or Russell's paradox, this paradox does not depend on what model of negation is used, as it is completely negation-free. Thus paraconsistent logics can still be vulnerable to this, even if they are immune to the liar paradox.
The resolution of Curry's paradox is a contentious issue because resolutions (apart from trivial ones such as disallowing X directly) are difficult and not intuitive. Logicians are undecided whether such sentences are somehow impermissible (and if so, how to banish them), or meaningless, or whether they are correct and reveal problems with the concept of truth itself (and if so, whether we should reject the concept of truth, or change it), or whether they can be rendered benign by a suitable account of their meanings.
Linear logic disallows contraction and so does not admit this paradox directly, but one must remove its exponential operators, or else the paradox reappears in a modal form.
5. See also
* Richard's paradox
* Kleene-Rosser paradox
6. References
1. Barwise, Jon and John Etchemendy 1987: The Liar, p. 23. Oxford University Press.
7. External links
* The Stanford Encyclopedia of Philosophy: "Curry's Paradox" -- by J. C. Beall.
* Grossman, Jason, Australian National University: A Proof that Penguins Rule the Universe. A brief and entertaining discussion of Curry's paradox.
"[1]" "[2]" Relevant First-Order Logic LP# and Curry's Paradox http://front.math.ucdavis.edu/0804.4818
Tuesday, December 23, 2008
Carroll's Paradox
"What the Tortoise Said to Achilles" is a brief dialogue by Lewis Carroll which playfully 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.
Contents:
1. Summary of the dialogue
2. Discussion
3. See also
4. Where to find the article
5. References
1. 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" (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 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.
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.
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.
2. Discussion
Several philosophers have tried to resolve the Carroll paradox. 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".
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).
Isashiki Takahiro (1999) summarizes past attempts and concludes they all fail before beginning yet another.
3. See also
* Deduction theorem
* Münchhausen Trilemma
* Paradox
4. Where to find the article
* Carroll, Lewis. "What the Tortoise Said to Achilles". Mind, n.s., 4 (1895), pp. 278-80.
* Hofstadter, Douglas. Gödel, Escher, Bach: an Eternal Golden Braid. 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 [1], [2], and [3]
5. References
* Isashiki Takahiro (1999). What Can We Learn from Lewis Carroll's Paradox?. In Memoirs of the Faculty of Education, Miyazaki University: Humanities, no. 86, pp. 79-98. The paper is in Japanese only, except for the abstract. A slightly extended version of the English-language abstract is available from [4].
Another author provides a more extended summary at [5] (currently down, and unavailable from archive.org).
Contents:
1. Summary of the dialogue
2. Discussion
3. See also
4. Where to find the article
5. References
1. 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" (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 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.
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.
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.
2. Discussion
Several philosophers have tried to resolve the Carroll paradox. 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".
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).
Isashiki Takahiro (1999) summarizes past attempts and concludes they all fail before beginning yet another.
3. See also
* Deduction theorem
* Münchhausen Trilemma
* Paradox
4. Where to find the article
* Carroll, Lewis. "What the Tortoise Said to Achilles". Mind, n.s., 4 (1895), pp. 278-80.
* Hofstadter, Douglas. Gödel, Escher, Bach: an Eternal Golden Braid. 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 [1], [2], and [3]
5. References
* Isashiki Takahiro (1999). What Can We Learn from Lewis Carroll's Paradox?. In Memoirs of the Faculty of Education, Miyazaki University: Humanities, no. 86, pp. 79-98. The paper is in Japanese only, except for the abstract. A slightly extended version of the English-language abstract is available from [4].
Another author provides a more extended summary at [5] (currently down, and unavailable from archive.org).
Monday, August 25, 2008
Cum Hoc, Ergo Propter Hoc
Latin for "with this, therefore because of this." This is a false cause fallacy that doesn't depend on time order (as does the post hoc fallacy), but on any other chance correlation of the supposed cause being in the presence of the supposed effect.
Example:
Gypsies live near our low-yield cornfields. So, gypsies are causing the low yield.
Converse Accident
If we reason by paying too much attention to exceptions to the rule, and generalize on the exceptions, we commit this fallacy. This fallacy is the converse of the accident fallacy. It is a kind of Hasty Generalization.
Example:
I've heard that turtles live longer than tarantulas, but the one turtle I bought lived only two days. I bought it at Dowden's Pet Store. So, I think that turtles bought from pet stores do not live longer than tarantulas.The original generalization is "Turtles live longer than tarantulas." There are exceptions, such as the turtle bought from the pet store. Rather than seeing this for what it is, namely an exception, the reasoner places too much trust in this exception and generalizes on it to produce the faulty generalization that turtles bought from pet stores do not live longer than tarantulas.
Consensus Gentium
Fallacy of argumentum consensus gentium (argument from the consensus of the nations). See Traditional Wisdom.
Confirmation Bias
The tendency to look only for evidence in favor of one's controversial hypothesis and not to look for disconfirming evidence, or to pay insufficient attention to it. This is the most common kind of Fallacy of Selective Attention.
Example:
Committing the fallacy of confirmation bias is often a sign that one has adopted some belief dogmatically and isn't seriously setting about to confirm or disconfirm the belief.She loves me, and there are so many ways that she has shown it. When we signed the divorce papers in her lawyer's office, she wore my favorite color. When she slapped me at the bar and called me a "handsome pig," she used the word "handsome" when she didn't have to. When I called her and she said never to call her again, she first asked me how I was doing and whether my life had changed. When I suggested that we should have children in order to keep our marriage together, she laughed. If she can laugh with me, if she wants to know how I am doing and whether my life has changed, and if she calls me "handsome" and wears my favorite color on special occasions, then I know she really loves me.
Composition
The composition fallacy occurs when someone mistakenly assumes that a characteristic of some or all the individuals in a group is also a characteristic of the group itself, the group "composed" of those members. It is the converse of the division fallacy.
Example:
Each human cell is very lightweight, so a human being composed of cells is also very lightweight.
Complex Question
You commit this fallacy when you frame a question so that some controversial presupposition is made by the wording of the question.
Example:
[Reporter's question] Mr. President: Are you going to continue your policy of wasting taxpayer's money on missile defense?The question unfairly presumes the controversial claim that the policy really is a waste of money. The fallacy of complex question is a form of begging the question.
Common Cause
This fallacy occurs during causal reasoning when a causal connection between two kinds of events is claimed when evidence is available indicating that both are the effect of a common cause.
Example:
Noting that the auto accident rate rises and falls with the rate of use of windshield wipers, one concludes that the use of wipers is somehow causing auto accidents.However, it's the rain that's the common cause of both.
Circular Reasoning
Circular reasoning occurs when the reasoner begins with what he or she is trying to end up with. The most well known examples are cases of the fallacy of begging the question. However, if the circle is very much larger, including a wide variety of claims and a large set of related concepts, then the circular reasoning can be informative and so is not considered to be fallacious. For example, a dictionary contains a large circle of definitions that use words which are defined in terms of other words that are also defined in the dictionary. Because the dictionary is so informative, it is not considered as a whole to be fallacious. However, a small circle of definitions is considered to be fallacious.
Example:
Definition: A couch is a sofa.For additional difficulties in deciding whether an argument is deficient because it is circular, see Begging the Question.
Definition: A sofa is a davenport.
Definition: A davenport is a couch.
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