Showing posts with label _G. Show all posts
Showing posts with label _G. Show all posts
Friday, December 26, 2008
Golden's paradox
Golden's paradox: Trey Golden's paradox is a paradox comprised of the question "What question has no answer?". Since all questions have answers, the answer to this particular question would be "This question." While the question still contains an answer, the answer to the question states that it does not exist.
Grelling-Nelson paradox
The Grelling-Nelson paradox is a semantic self-referential paradox formulated in 1908 by Kurt Grelling and Leonard Nelson and sometimes mistakenly attributed to the German philosopher and mathematician Hermann Weyl. It is thus occasionally called Weyl's paradox as well as Grelling's paradox. It is closely analogous to several other well-known paradoxes, in particular the Barber paradox and Russell's paradox.
Contents:
1. The Paradox
2. Similarities with Russell's paradox
3. See also
4. External links
1. The Paradox
Suppose one interprets the adjectives "autological" and "heterological" as follows:
1. An adjective is autological (sometimes homological) if and only if it describes itself. For example "short" is autological, since the word "short" is short. "English," "unhyphenated" and "pentasyllabic" are also autological.
2. An adjective is heterological if and only if it does not describe itself. Hence "long" is a heterological word, as are "abbreviated" and "monosyllabic."
All adjectives, it would seem, must be either autological or heterological, for each adjective either describes itself, or it doesn't. The Grelling-Nelson paradox arises when we consider the adjective "heterological". To test if the (imaginary) word "'foo" is autological one can ask: Is "foo" a foo word? If the answer is 'yes', "foo" is autological. If the answer is 'no', "foo" is heterological.
By comparison, one can ask: Is "heterological" a heterological word? If the answer is 'yes', "heterological" is autological (leading to a contradiction). If the answer is 'no', "heterological" is heterological (again leading to a contradiction).
But, then, we cannot say what is logical is not logical, and so on.
The paradox can be eliminated, without changing the meaning of "heterological" where it was previously well-defined, by modifying the definition of "heterological" slightly to hold of all nonautological words except "heterological." But "nonautological" is subject to the same paradox, for which this evasion is not applicable because the rules of English uniquely determine its meaning from that of "autological." A similar slight modification to the definition of "autological" (such as declaring it false of "nonautological" and its synonyms) might seem to fix that, but the paradox still obtains for synonyms of "autological" and "heterological" such as "selfdescriptive" and "nonselfdescriptive," whose meanings also would need adjusting, and the consequences of those adjustments would then need to be pursued, and so on. Freeing English of the Grelling-Nelson paradox entails considerably more modification to the language than mere refinements of the definitions of "autological" and "heterological," which need not even be in the language for the paradox to arise. The scope of these obstacles for English is comparable to that of Russell's paradox for mathematics founded on sets, argued as follows.
1. 1. Is "Autological" autological?
One may also ask if "autological" is autological. It can be chosen consistently to be either:
* if we say that "autological" is autological, and then ask if it applies to itself, then yes, it does, and thus is autological;
* if we say that "autological" is not autological, and then ask if it applies to itself, then no, it does not, and thus is not autological.
This is the opposite of the situation for heterological: while "heterological" logically cannot be autological or heterological, "autological" can be either. (It cannot be both, as the category of autological and heterological cannot overlap.)
In logical terms, the situation for "autological" is:
"autological" is autological if and only if "autological" is autological
A if and only if A, a tautology
while the situation for "heterological" is:
"heterological" is autological if and only if "heterological" is heterological
A if and only if not A, a contradiction.
2. Similarities with Russell's paradox
The Grelling-Nelson paradox can be translated into Bertrand Russell's famous paradox in the following way. First one must identify each adjective with the set of objects to which that adjective applies. So, for example, the adjective "red" is equated with the set of all red objects. In this way, the adjective "pronounceable" is equated with the set of all pronounceable things, one of which is the word "pronounceable" itself. Thus, an autological word is understood as a set, one of whose elements is the set itself. The question of whether the word "heterological" is heterological becomes the question of whether the set of all sets not containing themselves contains itself as an element.
3. See also
* List of autological words
* Metamagical Themas
4. External links
* Autological words
Contents:
1. The Paradox
2. Similarities with Russell's paradox
3. See also
4. External links
1. The Paradox
Suppose one interprets the adjectives "autological" and "heterological" as follows:
1. An adjective is autological (sometimes homological) if and only if it describes itself. For example "short" is autological, since the word "short" is short. "English," "unhyphenated" and "pentasyllabic" are also autological.
2. An adjective is heterological if and only if it does not describe itself. Hence "long" is a heterological word, as are "abbreviated" and "monosyllabic."
All adjectives, it would seem, must be either autological or heterological, for each adjective either describes itself, or it doesn't. The Grelling-Nelson paradox arises when we consider the adjective "heterological". To test if the (imaginary) word "'foo" is autological one can ask: Is "foo" a foo word? If the answer is 'yes', "foo" is autological. If the answer is 'no', "foo" is heterological.
By comparison, one can ask: Is "heterological" a heterological word? If the answer is 'yes', "heterological" is autological (leading to a contradiction). If the answer is 'no', "heterological" is heterological (again leading to a contradiction).
But, then, we cannot say what is logical is not logical, and so on.
The paradox can be eliminated, without changing the meaning of "heterological" where it was previously well-defined, by modifying the definition of "heterological" slightly to hold of all nonautological words except "heterological." But "nonautological" is subject to the same paradox, for which this evasion is not applicable because the rules of English uniquely determine its meaning from that of "autological." A similar slight modification to the definition of "autological" (such as declaring it false of "nonautological" and its synonyms) might seem to fix that, but the paradox still obtains for synonyms of "autological" and "heterological" such as "selfdescriptive" and "nonselfdescriptive," whose meanings also would need adjusting, and the consequences of those adjustments would then need to be pursued, and so on. Freeing English of the Grelling-Nelson paradox entails considerably more modification to the language than mere refinements of the definitions of "autological" and "heterological," which need not even be in the language for the paradox to arise. The scope of these obstacles for English is comparable to that of Russell's paradox for mathematics founded on sets, argued as follows.
1. 1. Is "Autological" autological?
One may also ask if "autological" is autological. It can be chosen consistently to be either:
* if we say that "autological" is autological, and then ask if it applies to itself, then yes, it does, and thus is autological;
* if we say that "autological" is not autological, and then ask if it applies to itself, then no, it does not, and thus is not autological.
This is the opposite of the situation for heterological: while "heterological" logically cannot be autological or heterological, "autological" can be either. (It cannot be both, as the category of autological and heterological cannot overlap.)
In logical terms, the situation for "autological" is:
"autological" is autological if and only if "autological" is autological
A if and only if A, a tautology
while the situation for "heterological" is:
"heterological" is autological if and only if "heterological" is heterological
A if and only if not A, a contradiction.
2. Similarities with Russell's paradox
The Grelling-Nelson paradox can be translated into Bertrand Russell's famous paradox in the following way. First one must identify each adjective with the set of objects to which that adjective applies. So, for example, the adjective "red" is equated with the set of all red objects. In this way, the adjective "pronounceable" is equated with the set of all pronounceable things, one of which is the word "pronounceable" itself. Thus, an autological word is understood as a set, one of whose elements is the set itself. The question of whether the word "heterological" is heterological becomes the question of whether the set of all sets not containing themselves contains itself as an element.
3. See also
* List of autological words
* Metamagical Themas
4. External links
* Autological words
Saturday, August 30, 2008
Guilt by Association
Guilt by association is a version of the ad hominem fallacy in which a person is said to be guilty of error because of the group he or she associates with.
Example:
Secretary of State Dean Acheson is soft on communism as you can see by the fuzzy-headed liberals who come to his White House cocktail parties and the bleeding hearts of his Democratic Party who call for "moderation and constraint" against Soviet terror.Has any evidence been presented here that Acheson's actions are inappropriate in regards to communism? This sort of reasoning is an example of McCarthyism, the technique of smearing liberal Democrats that was so effectively used by the late Senator Joe McCarthy in the early 1950s. In fact, Acheson was strongly anti-communist and the architect of President Truman's firm policy of containing Soviet power.
Group Think
A reasoner commits the group think fallacy if he or she substitutes pride of membership in the group for reasons to support the group's policy. If that's what our group thinks, then that's good enough for me. It's what I think, too. "Blind" patriotism is a rather nasty version of the fallacy.
Example:
We K-Mart employees know that K-Mart brand items are better than Wall-Mart brand items because, well, they are from K-Mart, aren't they?
Genetic
A critic commits the genetic fallacy if the critic attempts to discredit or support a claim or an argument because of its origin (genesis) when such an appeal to origins is irrelevant.
Example:
Whatever your reasons are for buying that DVD they've got to be ridiculous. You said yourself that you got the idea for buying it from last night's fortune cookie. Cookies can't think!Fortune cookies are not reliable sources of information about what DVD to buy, but the reasons the person is willing to give are likely to be quite relevant and should be listened to. The speaker is committing the genetic fallacy by paying too much attention to the genesis of the idea rather than to the reasons offered for it. An ad hominem fallacy is one kind of genetic fallacy, but the genetic fallacy in our passage isn't an ad hominem. If I learn that your plan for building the shopping center next to the Johnson estate originated with Johnson himself, who is likely to profit from the deal, then my pointing out to the planning commission the origin of the deal would be relevant in their assessing your plan. Because not all appeals to origins are irrelevant, it sometimes can be difficult to decide if the fallacy has been committed. For example, if Sigmund Freud shows that the genesis of a person's belief in God is their desire for a strong father figure, then does it follow that their belief in God is misplaced, or does this reasoning commit the genetic fallacy?
Gambler's
This fallacy occurs when the gambler falsely assumes that the history of outcomes will affect future outcomes.
Example:
I know this is a fair coin, but it has come up heads five times in a row now, so tails is due on the next toss.The fallacious move was to conclude that the probability of the next toss coming up tails must be more than a half. The assumption that it's a fair coin is important because, if the coin comes up heads five times in a row, one would otherwise become suspicious that it's not a fair coin and therefore properly conclude that the probably is high that heads is more likely on the next toss.
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