SUO: Re: Lattices
Jon,
I was not talking loosely. I was talking formally.
The lattice of all possible theories (in any given
version of logic) indeed does form a true lattice.
And the lattice of all possible models for each
of those theories forms a dual lattice. They are
both true lattices, and all of the standard
lattice operators apply to them.
JS> The theories are the intentions and the models are
> the extensions -- they are two separate structures.
> There is a duality between intensions and extensions,
> which has been recognized in various forms since
> the time of Aristotle.
JA> Yes, but, ..., that fearful symmetry was broken
> by Peirce's theory of information.
I agree that the traditional talk about intention
(or comprehension) and extension do not capture the
full sense of the term "meaning", and I also agree
that Peirce had the first and, so far at least,
the only serious theory of "meaning" (which is
the major reason why I continue to insist that
any modern philosopher who tries to mouth words
about "meaning" without coming to grips with
what Peirce said on the subject is incompetent).
On the other hand, Peirce did on various occasions
cite the traditional duality of comprehension and
intension approvingly -- that is, citing it as a
useful distinction. But of course, he would never
use the term "meaning" as a synonym for comprehension
(or intention). And neither do I.
I also did not say that the lattice of theories or
the lattice of models captures the full notion of
what Peirce was trying to say with the phrase
"meanings grow". However, I do claim that those
lattices provide very useful formal structures
that enable us to make a start toward implementing
those open-ended "microsenses" that are manifestations
of what Peirce was talking about. But I also admit
that they are only a start.
John