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*Subject*: PROPOSAL: Lambda Notation For Dummies (and & Rosta) & Lojban*From*: Logical Language Group <lojbab>*Date*: Tue, 15 Nov 1994 11:49:30 -0500 (EST)*Cc*: lojbab@access.digex.net (Logical Language Group)

Those who are geniuses (i.e. already understand lambda notation) should skip the first half of this, but look at the second half, after the "======". Lambda notation is essentially a gimmick for writing down functions without using names. Consider the mathematical function "squared": 2 squared is 4, 11 squared is 121, etc. The usual way of writing the definition of this function is: 1) x squared = x * x where "*" substitutes for the multiplication sign, as usual in ASCII. There is an implication that the "x" on both sides of the equation means the same thing, and further more that it has no special significance except this sameness. It would be equally to the purpose to write: 2) y squared = y * y and this definition of "squared" is exactly the same as Example 1. By introducing lambda, we manage to get the uses of the variable all on one side of the equation, thus: 3) squared = \lambda(x) x * x or, with the same meaning: 4) squared = \lambda(y) y * y but not of course 5) squared' = \lambda(x) y * y In Example 5, "squared-prime" is a function which returns the square of y, whatever y may be (presumably a free variable defined externally to this paper), and totally ignores its argument. So we see that \lambda(x) behaves formally like (Ex) or (Ax), in that it binds a variable locally. Semantically, however, it is quite different: it produces not a quantification over an open sentence, but a function defined by an open sentence. (That's all that Lojban needs: you can skip down to "======" if you want.) Suppose we want to talk about functions of two arguments? Consider "sum", defined as: 6) sum(x)(y) = x + y (The motivation for the notation "sum(x)(y)" rather than the usual "sum(x,y)" will become apparent in a moment.) The lambda-fication of Example 6 is: 7) sum = \lambda(x) \lambda(y) x + y How to understand Example 7? It says that "sum" is a function with one argument, called "x", whose value is >another function<. This other function, which has no name, takes one argument called "y" whose value is the sum of the argument and "x". Local scoping guarantees that the "x" in question is that constructed by the outer lambda. Let us take a concrete example: What is the value of 8) sum(2)(3) By expanding the inner part, "sum(2)", we get: 9) (\lambda(y) 2 + y)(3) ------------------ The underlined part is a function without a name, the value of "sum(2)". I have parenthesized it for clarity. This nameless function, which could be called "increase-by-two", is then applied to the argument 3 to produce: 10) 2 + 3 which is 11) 5 So we get the expected answer in the end. "Lambda calculus", which is not explained here, is a branch of mathematics that treats everything as a function, including even constants ("2" is rewritten as "\lambda(x) 2", i.e. a function which ignores its argument and always returns 2). There are various rules of inference, including the idea that lambda-expressions may have their variables changed as long as the change is consistent and doesn't interfere with any inner lambda expressions ("alpha-conversion"). ====== The Lojban analogue of lambda functions shows up with properties. Consider the selbri "mamta", with place structure "x1 is the mother of x2". The most obvious property here is 11) the property of being a mother focused on the x1 place; however, there is also 12) the property of having a mother focused on the x2 place. Both of these are "le ka mamta". If we want to look at more specific properties, like "being the mother of Jack" or "having Jill as a mother", we can disambiguate: 13) le ka mamta la djek. the property-of being-a-mother of-Jack 14) le ka le djil. mamta the property-of Jill being-the-mother [of-someone]. In Examples 13 and 14, the sumti on which the property is focused is indicated by elision, but if the question "What is elided?" is asked, no answer is forthcoming: "zo'e" just begs the question, and "da" introduces unwanted existential quantification. In the past, proposals have been made to overload the use of "kau" (currently used for indirect questions, and proposed for indirect commands as well). This proposal involves creating an explicit "lambda quantifier", which would formally belong to selma'o PA but would be attached only to da-series KOhA or BY cmavo. Call it "xa'e" temporarily. Using it, Example 11 can be translated: 15) le ka xa'eda mamta [zo'e] and Example 12: 16) le ka [zo'e] mamta xa'eda Likewise, in the mekso domain, we can express Example 9 as: 17) li na'u xa'e da zo'u nu'a pi'i da da te'u ci the-number (the-operator \lambda (X) : the-product-of X and-X) of-3 (This formulation is off the cuff, and may contain errors.) -- John Cowan sharing account <lojbab@access.digex.net> for now e'osai ko sarji la lojban.

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