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An influential idea in the philosophy of language, first proposed by Ludwig Wittgenstein in his book Philosophical Investigations.

Wittgenstein discussed examples of terms which he argued would not admit of a full and complete definition. How, he asks, would one go about giving a definition of "game"? He argued that there is nothing that is common to all games, but rather that games held certain similarities and relations with each other. He admonished his reader not to think, but to look, at the vast range of things that we call games. Some games involve winning and losing, but not all; some are entertaining, but not all; some require skill or luck, but not all. ^[1]

Similarly, he argued that there is nothing that all "numbers" have in common; but furthermore that we regularly extend the notion of 'number'.^[2] So we might start by thinking only of the natural numbers, and later learn to extend this to rational numbers, integers, cardinal numbers; but then to irrational numbers, complex numbers, surcomplex numbers, surreal numbers and so on, the only limit being the capacity of mathematicians to innovate.

Nor will it suffice to define "number" as the disjunction of each of these types, as:

A number =_def (a natural number) ∨ (an irrational number) ∨ (an integer)...

Here he says '...you are only playing games with words. One might as well say: "Something runs through the whole thread - namely the continuous overlap of those fibres"'.^[3]

The third example he uses, and the one that provides him with the name, is a "family".

I can think of no better expression to characterise these similarities than "family resemblance"; for the various resemblances between members of a family: build, features, colour of eyes, gait, temperament, etc. overlap and criss-cross in the same way...^[4]

Prior to Philosophical Investigations the ideal way to give the meaning of something had been thought to be by specifying both genus and differentia. So a 'triangle' is defined as 'a plane figure (genus) bounded by three straight sides (differentia)'. Logically, this sort of definition can be seen as a series of conjunctions; A triangle is a plane figure and has three sides. More generally, "P" might be defined using a simple conjunction of "A" and "B":

P =_def A ∧ B

By examining closely the use of terms such as 'game', 'number' and 'family', Wittgenstein showed that for a large number of terms such a definition is not possible. Rather, in some cases a definition needs to be a disjunction of conjuncts,

P =_def (A ∧ B) ∨ (C ∧ D)

but furthermore the way we use such terms means that we can both extend and detract from the series by adding or removing some of the conjunctions.

P =_def (A ∧ B) ∨ (C ∧ D)...

Nor should we conclude that because we cannot give a definition of "game" or "number" that we do not know what they are: "But this is not ignorance. We do not know the boundaries because none have been drawn".^[5]

Family resemblances might be taken to have 'blurred edges'. Wittgenstein points out that in such cases the term nevertheless has a sense; for example one can quite sensibly say 'stand roughly there', indicating a spot by pointing. The lack of precision does not make the expression meaningless. Similarly, even though the definition of 'game' may be imprecise, it is still meaningful.^[6] Furthermore a sharp boundary can be chosen, to suit whatever purpose one has to hand. In such cases, it is the way in which the term is employed, and how it is learned, that are pivotal, rather than any precise meaning. ^[7]

Remarks in Part I of Investigations are preceded by the symbol "§". Remarks in Part II are referenced by their Roman numeral or their page number in the third edition.

1. ^ §66
2. ^ §67
3. ^ §67
4. ^ §67
5. ^ §69
6. ^ §71
7. ^ §76-77

Wittgenstein, Ludwig (1953/2001). Philosophical Investigations. Blackwell Publishing. ISBN 0-631-23127-7.