Grelling-Nelson paradox
Encyclopedia
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
.
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". One can ask: Is "heterological" a heterological word?
If the answer is 'no', "heterological" is autological. This leads to a contradiction! In this case, "heterological" does not describe itself: it must be a heterological word.
If the answer is 'yes', "heterological" is heterological. This again leads to a contradiction, because if the word "heterological" describes itself, it is autological.
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.
It can be chosen consistently to be either:
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:
while the situation for "heterological" is:
'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.
Paradox
Similar 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...
formulated in 1908 by Kurt Grelling
Kurt Grelling
Kurt Grelling was a logician, philosopher and member of the Berlin Circle.- Life and work :Shortly after his arrival in 1905 at University of Göttingen, Grelling began a collaboration with philosopher Leonard Nelson, with whom he tried to solve Russell's paradox, which had shaken the foundations...
and Leonard Nelson
Leonard Nelson
Leonard Nelson was a German mathematician and philosopher. He was part of the Neo-Friesian School and a friend of the mathematician David Hilbert, and devised the Grelling–Nelson paradox with Kurt Grelling...
and sometimes mistakenly attributed to the German philosopher and mathematician Hermann Weyl
Hermann Weyl
Hermann Klaus Hugo Weyl was a German mathematician and theoretical physicist. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His...
. 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
Barber paradox
The Barber paradox is a puzzle derived from Russell's paradox. It was used by Bertrand Russell himself as an illustration of the paradox, though he attributes it to an unnamed person who suggested it to him. It shows that an apparently plausible scenario is logically impossible.- The Paradox...
and Russell's paradox
Russell's paradox
In the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction...
.
The paradox
Suppose one interprets the adjectives "autological" and "heterological" as follows:- An adjectiveAdjectiveIn grammar, an adjective is a 'describing' word; the main syntactic role of which is to qualify a noun or noun phrase, giving more information about the object signified....
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. - An adjective is heterological 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". One can ask: Is "heterological" a heterological word?
If the answer is 'no', "heterological" is autological. This leads to a contradiction! In this case, "heterological" does not describe itself: it must be a heterological word.
If the answer is 'yes', "heterological" is heterological. This again leads to a contradiction, because if the word "heterological" describes itself, it is autological.
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
Russell's paradox
In the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction...
for mathematics founded on sets, argued as follows.
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 tautologyTautology (logic)In logic, a tautology is a formula which is true in every possible interpretation. Philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921; it had been used earlier to refer to rhetorical tautologies, and continues to be used in that alternate sense...
while the situation for "heterological" is:
- "heterological" is heterological if and only if "heterological" is autological
- A if and only if not A, a contradictionContradictionIn classical logic, a contradiction consists of a logical incompatibility between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which form the logical, usually opposite inversions of each other...
.
Similarities with Russell's paradox
The Grelling–Nelson paradox can be translated into 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...
's famous paradox
Russell's paradox
In the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction...
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.