4/16/2013

numerals, and language with an infinite lexicon

Here is an idea that I've seen in several places: "it is plain that if a language is to be learnable, the number of basic significant elements (words) has to be finite" (Tim Crane, The Mechanical Mind, 2nd ed. p.140).  (Is the locus classicus for this Davidson's "Theories of Meaning and Learnable Languages?")

But how does this square with numerals (words for numbers)?  There are an infinite number of numerals.  And arithmetical language is learnable. 

So how could/should Crane, Davidson, et al. deal with this apparent counterexample?  I'm not sure... perhaps (despite appearences) numerals are  not themselves genuine words; rather only the digits are genuine words, and the higher numerals are complex supra-word symbols. 

5 Comments:

At 16/4/13, Blogger Jonathan Livengood said...

Obviously, they should just deny that arithmetic is learnable.

 
At 18/4/13, Blogger Greg Frost-Arnold said...

Note to self:
Upon further thought, it seems to me that '9,471' is not really different from 'SSSSS0' or '|||||' (as representations of the number 5). All three are singular terms, but none are proper names: they are rather function symbols (applied to proper names).
(Right?)

 
At 18/4/13, Anonymous Kai von Fintel said...

The basic significant elements of natural languages are morphemes, not words. A word like “screwdriver” is made up of at least three basic significant elements (screw, drive, -er).

In natural language, numerals are definitely complex expressions (words or phrases) made from more basic significant elements. The grammar of numerals in natural languages is actually quite interesting. One place to start: http://dx.doi.org/10.1093/jos/ffl006.
The grammar of numerals in arithmetic is also quite interesting, at least as a toy example for how compositional systems work. This is in fact the traditional opening example of introductions to the semantics of programming languages. See for example:

Slonneger, Kenneth & Barry L. Kurtz. 1995. Formal syntax and semantics of programming languages: a laboratory approach. Addison-Wesley. Chapter 9: Denotational Semantics. The book is online at http://homepage.cs.uiowa.edu/~slonnegr/plf/Book/. Chapter 9 is at http://homepage.cs.uiowa.edu/~slonnegr/plf/Book/Chapter9.pdf.

The example is rarely used in introductions to natural language semantics. But I once tried. Here’s a handout from that class: http://mit.edu/fintel/fintel-2008-numerals.pdf. There’s a lot of expository lecture going with that handout but I have no very good record of that; there are raw notes at http://mit.edu/fintel/numerals-notes.txt.

 
At 18/4/13, Blogger Greg Frost-Arnold said...

Wow -- Thanks for all of those resources, Kai! This looks very helpful.

 
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