Speech Communities and Natural Languages

Let's suppose, with Chomsky, that the basic explanatory entities of scientific linguistics are individual languages spoken by individual speakers. These are idiolects, or micro-idiolects, or Chomskyan I-languages, and are somehow "cognized" or implemented by the cognitive systems of individuals.

Then one wonders how to make sense of the collective notions of:

• "speech community"
• "natural language". 

It seems that these notions are intimately connected: for a natural language, such as Punjabi, French, Middle English, Euskadi, etc., is usually identified in terms of some speech community of its speakers.

Here are two attempted explications:

A set $C$ of agents is a speech community if and only if, for any pair $x,y \in C$, the speech behaviour of $x$ and $y$ is mutually interpretable.

$L$ is a natural language if and only if there is a speech community $C$ each member of which uses/cognizes $L$.

The first uses the notion of "speech behaviour being pairwise mutually interpretable", and this has genuine empirical content, in terms of the observable overall ease of social co-operation between agents. The second uses the notion of "speech community" and the hard to define, but crucial, notion of "using/cognizing a language".

Both notions are vague. Mutual interpretability is a rather vague and context-dependent matter; consequently, what sets count as speech communities will inherit this vagueness. And even when there is a speech community, it is usually the case that it is somewhat heterogeneous, and therefore for no idiolect pair $L_1,L_2$ do we have $L_1 = L_2$, strict dictu. So, only at some idealized level is there some single "external" language $L$ that all members of $C$ speak. Rather, this $L$ is an idealization that somehow approximates the varying idiolects $L_1, L_2$, etc., spoken within the speech community.

There are some refinements that might be introduced to the above rough ideas (see the comments below). The most obvious would be treat the "mutual interpretability" relation as a matter of degree. Instead of being modelled by a graph (nodes representing speakers; edges representing mutual interpretability), in which a speech community is a maximal clique, a collection of speakers might be modelled by a weighted graph; and speech community is then like a kind of weighted clique.

Finally, probably one ought to be sceptical about attempts to make precise the notions of "natural language" or "speech community". A similar view can be found in Chomsky's work. For example, in his "Knowledge of Language" (1975):

The notion of ‘language’ as a common property of a speech community is a construct, perfectly legitimate, but a higher-order construct. In the real world, there are no homogeneous speech communities, and no doubt every speaker controls several grammars, in the strict sense in which a grammar is a formal system meeting certain fixed conditions.

[UPDATE (29th March): I've added two paragraphs at the end.]