[FOM] expressive power of natural languages

Richard Heck rgheck at brown.edu
Wed Nov 30 21:31:22 EST 2011

On 11/30/2011 02:36 PM, americanmcgeesfr at gmx.net wrote:
> Hello FOMers,
> I was wondering if there is any (at least semi-)conclusive view about
> the expressive power of a natural language like english resulting in a
> statement like "whatever it is, it is a language of at least 2nd
> order". Of course, I know of Tarski's comment suspecting natural
> languages to be somehow (semantically) universal. But what I'm
> interested in is a hint pointing me in a direction what to look for,
> i.e. is the fact that one quantifies over classes in a natural
> language enough to label it higher order? Can there be anything wrong
> to take it to be at least a many-sorted first-order language?
Most linguists nowadays take it to be patently obvious that natural
language contains the resources for so-called "plural" quantification,
as in the so-called Geach-Kaplan sentence, "Some critics admire only one
another", which can be shown not to be first-order expressible. And, as
George Boolos observed, plural quantification is inter-translatable with
monadic second-order quantification (his own interest being in
second-order set theories). In that sense, it is clear that natural
languages can express monadic second-order quantification.

In some sense, it's of course also obvious that any second-order
language is inter-translatable with a many-sorted first-order language.
Issues in that vicinity are not likely to be empirically resolvable,
however. That said, the difference between these perspectives presumably
comes down to one's attitude towards the comprehension axioms: whether
one regards them, as in the second-order setting, as logically true, or,
as in the first-order setting, as non-logical axioms with no special
claim on our credence. Here again, Boolos thinks it is just obvious, and
logically so, that, e.g., there are some sets that are not members of
themselves, where that is meant to be a plural comprehension axiom
(\exists xx \forall y[y is among xx iff y \notin y). Here "is among" is
a logical relation among pluralities and the objects that constitute them.

A good place to start with this is the Stanford Encyclopedia article on
plural quantification:
Boolos's papers, collected in his /Logic, Logic, and Logic/, are the
source for much of this literature. An early work in linguistics is
Barry Shein's /Plurals and Events/.

Richard Heck

Richard G Heck Jr
Romeo Elton Professor of Natural Theology
Brown University

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