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I've been reading Chomsky's The Logical Structure of Linguistic Theory, and have come up rather suddenly against a stumbling block. On page 133, he uses without explanation some notation that's unfamiliar to me. I've included an image because I couldn't figure out how to format the superscript & multiple subscripts together. I'm willing to do the work to understand this: I don't need someone to spell it out for me. I've figured out that ∈ and ⊂ are from set theory, and I've begun doing some reading on set theory. (The meaning of those two symbols, specifically, is easy enough, & it doesn't seem like I need much set theory to understand what he's saying here.) But I can't figure out what's going on with the italic, non-cursive capital C with both the superscript and subscript letters. How do I figure out the relationship between n and nᵢ? What should I be reading? I'm assuming that the lowercase and capital n/N have some standard relationship. What should I be looking into to understand that?

image of a segment of p 133 of LSLT

Once again, I'm very willing to do the reading myself. I've been Googling things like (Chomsky "set theory" symbols) and (Chomsky logic "set theory"), but no dice, as yet. I just need to be set in the right direction.

Much thanks for any help.

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  • I suppose I should add that I have a speculative reading of this, in which it's not a prior symbolic system but Chomsky's shorthand which is a little underexplained for a reader like me: Each 𝒞ⁿ is an order; each nᵢ is a category within that order. N is the total number of orders. aₙ is the total number of categories in any order n. Thus a₁ > a₂ > … > a<sub>N</sub> makes sense, as there should be fewer categories on higher orders. The arrows ⇒ just point to explanations of symbols in the previous comments. I'm dissatisfied with this explanation because I'm basing it entirely on speculation. Jan 29 '16 at 11:16
  • Capital C with superscript a and subscript b is the number of combinations of a things taken b at a time.
    – Greg Lee
    Jan 29 '16 at 19:35
  • Looking at the quote, I can't make sense of much. Consider condition (iv). It seems to say that a certain set is a proper subset of a certain other set only when the two sets are the same. But a set is never a proper subset of itself. So, ...??
    – Greg Lee
    Jan 29 '16 at 19:53
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    Historically, $\subset$ was not consistently used for proper subset, and Chomsky for sure didn't follow that convention in the 70s. So it just means modern $\subseteq$ here.
    – kgr
    Feb 2 '16 at 21:58
  • The arrows are almost certainly logical implication; (iii) just says all elements of $C^n_n_i$ are words, (iv) implies that no class is a subset of another with the same $n$. Word classes ought to refer to parts of speech I think, but I have no idea what he's doing with $n$, $n_i$, and $a_n$.
    – Joseph
    Feb 2 '16 at 23:25

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