Consider the
function represented with (1) and the function represented with (2),

(1) F(x) = | x - 1 |

(2) F(x) =

^{+}√(x^{2}- 2x + 1)
letting ‘x’
range over the integers. Have we represented the same function twice? Or have
we represented two functions that share an “input-output profile,” gestured at
with (3)?

(3) { ... , <-2, 3>, <-1, 2>, <0, 1>, <1, 0>,
<2, 1>, <3, 2>... }

This question
bears, directly and historically, on the I-language/E-language distinction that
Chomsky introduced in

*Knowledge of Language*(1986).
For any integer,
the absolute value of its predecessor

x

*is*the positive square root of the successor of the result of subtracting twice that number from its square. There’s no magic here: (x - 1)^{2}=x

^{2}- 2x + 1. So the set of ordered pairs specified with (4)*is*the set specified with (5).
(4) {<x, y> : y = | x - 1 | }

(5) {<x, y> : y =

^{+}√(x^{2}- 2x + 1)}
In this respect,
(4) and (5) are like ‘Hesperus’/‘Phosphorus’, ‘George Orwell’/‘Eric Blair’,
‘the smallest prime number’/‘the second positive integer’, etc. The morning
star is the evening star, regardless of what people do or don’t know. Likewise,
the set of ordered pairs <x, y> such that y = | x - 1 | is the set of ordered pairs <x,
y> such that y =

^{+}√(x^{2}- 2x + 1). So getting back to the initial question, do we have one function or two? Frege and Church, who knew something about functions, held that it depends on what you mean by ‘function’. But they also held that an important sense, (1) and (2) should be understood as representing*different functions*that have the*same extension*.
Frege contrasted
Functions with their Courses of Values. But he also said that Functions are
“unsaturated,” as reflected with the variable ‘x’ in (1) and (2),

*and*that this precludes referring to Functions. If you can’t imagine why Frege said this, count yourself lucky. It leads to claims like ‘The successor function is not a Function’. Yuck. In 1941, Church made the point clearer in*On the Calculi of Lambda Conversion*: we can talk about functions as*procedures*that map inputs onto outputs (functions “in intension”), or as*sets*of input-output pairs (functions “in extension”). But when a set has infinitely many elements, specifying it—as opposed to just giving hints and using ellipses, as in (3)—requires procedural description. In this sense, the procedural notion is primary, as Frege had noted; and in this sense, (2) specifies a different function than (1).
Church also wanted
to ask questions about

*computability*. So he invented a notation for specifying procedures and their input-output profiles. Expressions of his lambda calculus can be construed*intensionally*so that (6) is true, or construed*extensionally*so that (7) is true.
(6)
λx . | x - 1 | ≠ λx .

^{+}√(x^{2}- 2x + 1)
But as Church
stressed, while extensional interpretation is adequate for some mathematical
purposes—viz., when it doesn’t matter

*how*outputs are paired with inputs—you need the intensional interpretation if you want to talk about*algorithms*(i.e., ways of computing outputs given inputs). In retrospect, this all seems pretty obvious. Eventually, I’ll discuss some ironies regarding how lambdas ended up being used in semantics. Today, the important point is that the ‘I’ in ‘I’-language connotes (among other things) ‘Intensional’ in Church’s procedural sense.
Chomsky also took
I-languages to be

*internal*(“some element of the mind”) and concerned with*individuals*as opposed to social artifacts. One might add that I-languages are biologically*implemented*and*innately*constrained. Alliteration is mnemonic. But Chomsky clearly viewed I-languages as generative procedures (see p. 23). So one crucial contrast is with*extensional*conceptions of language, according to which the “same language” can be determined by different procedures. Here it’s worth recalling Quine’s obsession with extensionality—and as Chomsky mentions, Lewis’ characterization of*language*as a social practice (“ruled by convention”) and*languages*as sets of pairs <s, W> where s is a*string*of sounds or marks, and W is a set of possible worlds.
My next post
will focus on Lewis, who said, “A grammar uniquely determines the language it
generates. But a language does not uniquely determine the grammar that
generates it.” He added, “I know of no promising way to make objective sense of
the assertion that a grammar Γ is used by a population

*P*, whereas another grammar Γ', which generates the same language as Γ, is not.” Really? No way to even make sense of the idea that people use procedure (1) rather than procedure (2)? And then he said, “I think it makes sense to say that languages might be used by populations even if there were no internally represented grammars.” Whatever sense one makes of Lewis, there was room for a procedural alternative to extensional conceptions of languages.
But there are other
conceptions: strings of a corpus; Quinean complexes of “dispositions to verbal
behavior;” etc. Davidson said that a “radical” interpreter would ascribe languages
to speakers; yet it was unclear what this implied for speaker psychology. So
Chomsky introduced ‘E-language’ as a cover term for any language, in whatever
sense, that is not an I-language. There’s no serious question about whether humans
acquire E-languages. If we use ‘acquire’ so that (it comes out true that) kids
acquire dispositions, social practices, sets, and corpora, then anyone who
acquires English acquires

*many*things that count as E-languages. And there’s no serious question about whether humans acquire I-languages, since there is no alternative account of how we can connect so many articulations with so many meanings as we do; see my earlier post on unambiguity. The interesting question, for purposes of scientific inquiry, is what explains what. Regarding the various “things” that count as languages, we want to know what they*are*, and which ones are good candidates for being*explanatorily primary*.
One can imagine
discovering that each speaker of French has acquired the

*same*I-language, which is kept in a glass case, guarded by L'Académie française. French children may have a kind of telepathy that lets them access this shared procedure, I-French; where such access is, like cell phone service, imperfect and subject to individual variation. In which case, I-French isn’t internal or individualistic in Chomsky’s sense. (It’s not*analytic*that I-languages have these features.) Less fancifully, Michael Dummett held that each speaker grasps her native language imperfectly and partially. One is free to posit procedures that connect articulations and meanings that are communally determined. But that raises the question of how kids in a community acquire the alleged public procedures. To be sure, we speak of acquiring citizenship by birth, and acquiring the age of majority. So we can say that each child in Lyon acquires—by participating in social practices (i.e., by growing up and talking)—a thing kept under glass in Paris. But we also speak of adolescents acquiring secondary sexual characteristics, caterpillars acquiring wings, etc. And one might suspect that when a child acquires a capacity to connect articulations with meanings, she does so by*implementing her own*I-language, where this procedure is relevantly (and deeply) like those her parents/peers use to connect articulations with meanings. More on this next week.
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