Subject:
Metalanguage and higher-order logic (was Lost in Translation)
From:
"John F. Sowa" <sowa@bestweb.net>
Date:
Tue, 13 Apr 2004 22:39:32 -0400
To:
Jon Awbrey <jawbrey@ATT.NET>
CC:
SUO <standard-upper-ontology@IEEE.ORG>, Ontology <ontology@IEEE.ORG>, cg@cs.uah.edu

Jon,

Your note addresses an important distinction
that is often confused:  The difference between
higher-order logic and metalanguage.  In his
development of logic, Peirce explored both of these
notions, but it is not clear whether he himself
clearly distinguished them.

Following is the ending of the passage you quoted
from Peirce:

CSP>  By 'logical' reflexion, I mean the observation
> of thoughts n their expressions.  Aquinas remarked
> that this sort of reflexion is requisite to furnish
> us with those ideas which, from lack of contrast,
> ordinary external experience fails to bring into
> prominence.  He called such ideas 'second intentions'.
>
> It is by means of 'relatives of second intention'
> that the general method of logical representation
> is to find completion.
>
> C.S. Peirce, 'Collected Papers', CP 3.488-490,
> "The Logic of Relatives", 'The Monist', vol. 7,
> pp. 161-217, 1897.

In this passage, it seems that Aquinas and Peirce are
using the term "second intention" for language about
language, which is now called metalanguage.  But
in his paper of 1885, "On the Algebra of Logic II",
Peirce used the term "first-intentional logic" for
quantifcation over individuals, which is now called
first-order logic.  In that same paper, he used the
term "second-intentional logic" for quantification
over relations, and he gave some formulas that are
clear examples of second-order logic.

JA> Another way of stating my concern is to say that something
> has evidently been lost in the translation from classical
> discussions about higher intentional logic to epi-Fregean
> ramblings about the orders of logical formalism.  I begin
> to suspect that the current fuss about "order" is similar
> to those pre-Hausdorff confusions about dimensionality.

There certainly is a lot of confusion about these two notions,
and it is important to realize that they are orthogonal:

 1. You can use pure first-order logic as a metalanguage to
    talk about statements in FOL.  (Tarski did a lot of
    work on this topic in connection with model theory.)

 2. You can use higher-order logic to quantify over
    relations without using metalanguage.

Many people use the term "higher-order" when they are
actually using metalanguage.  And it is possible to get
an enormous increase in expressive power by using
metalanguage without using higher-order logic.

For more about metalanguage and its use in supporting
versions of modal, temporal, and intentional logics,
see the following paper and the references cited there:

   http://www.jfsowa.com/pubs/laws.htm
   Laws, Facts, and Contexts

John Sowa