Chomsky (2015:13) says that "It is intensional in the technical sense
that the I-language is a function specified in intension, not
I think he just means “internally” rather than “externally”. https://www.rep.routledge.com/articles/thematic/intensional-logics/v-1
It could probably use a bit of development, but the idea is, that “I-language”, or, the actual rule-governed system specifying how language works (according to Chomsky here), is “self-complete”. It is a system more or less of inherently logical character (not because the brain wrote down the rules of formal logic, but because formal logic is an attempt to write down the rules by which anything simply must be, if there are any universal assertions that can be made about how the world or anything must be - ie, logic as the observation of necessity.)
So, “language” sort of comes with its own batteries. There is a set of axioms and of elements and they define how the system actually works. The axioms are apparently some kind of composition rules, like functions on those elements.
I’m actually not sure it’s that interesting of an idea, it depends on the context. He is refuting the idea of an “E-language” being significant (an epiphenomenon). It could have elements of a “straw man argument”, in that it would only be a compelling claim if you assumed there was some false notion needing to be refuted.
But arguably, Chomsky could probably “formalize” his ideas even more, so that he could just define what something is, rather than assert something polemically.
To me, within this limited context, a shortcoming would appear to me to be, that it is not that clear what the opposite of an “I-language” is, what it would be saying to claim it is something ephemeral.
Here my understanding of function is a function of inputting computer
programming within I-language, and the output is its extension.
I don’t think that’s 100% correct, nor is it totally off-base either. I am pretty sure he really is referring to the uses of the terms in logic:
Intensional logics are systems that distinguish an expression’s
intension (roughly, its sense or meaning) from its extension
(reference, denotation). The purpose of bringing intensions into logic
is to explain the logical behaviour of so-called intensional
expressions. Intensional expressions create contexts which violate a
cluster of standard principles of logic, the most notable of which is
the law of substitution of identities – the law that from a = b and
P(a) it follows that P(b). For example, ‘obviously’ is intensional
because the following instance of the law of substitution is invalid
(at least on one reading): Scott = the author of Waverley; obviously
Scott = Scott; so, obviously Scott = the author of Waverley. By
providing an analysis of meaning, intensional logics attempt to
explain the logical behaviour of expressions such as ‘obviously’. On
the assumption that it is intensions and not extensions which matter
in intensional contexts, the failure of substitution and related
anomalies can be understood.
A lot of “languages” can be seen to be fundamentally based on the association of one general domain with some other. A language is often a method of convenience, because the thing itself cannot be used as a representational system to exchange with another person. For example, if we only ever showed people physical objects as a way to talk about them, we would have to transport physical objects constantly. Because we can arbitrarily pair something easier to transport, like random sounds emitted with the mouth, with the idea of this or that thing, we can relay representations of those things more practically.
Maybe there is flexibility in how you conceive of it, but an extension is often something outside the rules of the system - like how words ultimately point to things, but when you play with words, you can forget what their meanings are, for a bit - the extension could be what the language is structured to communicate about. But an intension is inherent to the system itself. It’s part of the actual language system that generates the rules of that game.
I think a cool simple way to think about this is defining some rules of play which have zero ulterior motive, like the game of chess. There are just rules about what pieces can be moved where and when. I would say chess is identical to a language. Except chess is a game played for its own sake. It doesn’t have a second layer of meaning, which could be anything, like, you associate each square on the board with a musical tone. If you did, maybe you would find that while chess can be played on its own, once you see correspondences between game play and sounds, it’s like you have a system for creating music, a system for designing sequences of sounds.
I also need to think about it more but I think that’s the basic idea. Chomsky just wants to define an intrinsic logical system which operates purely by rules. He is probably rejecting a theory that human language could be understood so abstractly without a subjective understanding of the connotations and qualities and experience of words.
For instance, "a set of round squares" and "a set of unicorns" are different sets in I-language, but both empty sets in E-language.
I do not know enough about minimalism to know if that is correct but I don’t think so. I don’t think E-language is a part of the language system portrayed in MP. It’s just a general introductory concept to distinguish between the cognitive rules of language (the “language faculty”) as opposed to how humans culturally think about languages like “English” and “Romanian”. Chomsky says in that text that there are only I-languages: each unique human brain has a system within it that determines the sentences it will produce. He also says later on I think that in principle every I-language could be unique or that in reality I-languages are observed to have features they could have but never do, so I-language presumably has some common governing features to explain why I-languages have similarities and don’t just vary in all imaginable directions.
More succinctly: there is a video interview of Chomsky saying “In the modern theory of linguistics, there is no such thing as ‘the English language’”. And, in the very beginning of his work, people were more taken aback by his approach to linguistics, and there is one interview where someone says “It sounds to me like your theory of linguistics is more of a theory of mathematical psychology”, and Chomsky replies, “That is language.”
In addition, one of his early works was actually published in a “Handbook of Mathematical Psychology”.
Maybe it is hard for us to grasp now, as with certain paradigm shifts in the history of thought - there are many, many examples - but we live in the aftermath of Chomskian linguistics, so we might have difficulty understanding how people thought about language before Chomsky.
After checking out his paper “The Algebraic Theory of Syntax,” and a little bit of the “Principles and Parameters” approach, I have come to wonder if Chomsky is best understood as almost like a “logician of language”, he is so inclined to extremely formal systematization of language, and good at it, that he both provides very original, novel and compelling logical models for how a system producing expressions like natural language could be defined, but it may also be a blindspot in that I think it sometimes seems that he is not receptive to non-logical paradigms, sort of. That’s a claim that could be developed, anyway.