Grelling-Nelson paradox

The Grelling-Nelson paradox is a semantic paradox formulated in 1908 by Kurt Grelling and Leonard Nelson and sometimes mistakenly attributed to German philosopher and mathematician Hermann Weyl. It is thus occasionally called Weyl's paradox, as well as Grelling's paradox. Justice of attribution has increasingly encouraged the present name, however. It is closely analogous to several other well known paradoxes, in particular the Barber paradox and Russell's paradox.

Analysis

The Grelling-Nelson paradox is resolvable by simply admitting that the words "autological" and "heterological" do not in fact form two well-defined categories into which all adjectives fall. Thus the paradox can be taken as a demonstration that natural language does not necessarily partition all objects of thought into well-defined categories that will stand up to arbitrary logical scrutiny.

Related Topics:
Natural language - Logic

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

And a word is not itself the same as what it refers to.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The Grelling-Nelson paradox can be translated into Bertrand Russell's famous paradox in the following way: 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.

Related Topics:
Bertrand Russell - Famous paradox - Set

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The standard mathematical response to this paradox is to work within some axiomatic set theory which contains a rule, such as the axiom of regularity, stating or implying that sets may not be elements of themselves.

Related Topics:
Axiomatic set theory - Axiom of regularity

~ ~ ~ ~ ~ ~ ~ ~ ~ ~