SUO: Re: Lattices
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John F. Sowa wrote:
>
> 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.
When I talk effectively, I have to pay very close attention to the
distinctions in roles between objects and signs, numbers and numerals,
that which cannot be, generally speaking, in the mind or in the computer,
and that which can be.
When I talk formally, the word "theory" means "set of sentences".
Sentences are signs. In fact, they are finite signs. Therefore
sentences are the sorts of things that we can regard, literally
enough for our present purposes, as residing in the states of
finite information and control devices.
By extension, a finite set of sentences can be regarded in all of the same
ways that we would regard a single sentence, and so a finite axiom set can
be regarded, almost literally, as being laid up in the states or the stores
of our favorite finite information and control device.
But theories in general, arbitrary sets of sentences, cannot be so regarded.
An infinite set of sentences has to be regarded as a mathematical object,
and then we encounter all of the usual sorts of problems as to how one
keeps signs in the intended relationships to their intended objects.
As long as we stick to the idyllic preserve of "pure" mathematics, all of these
issues of practical semiotics and computational pragmatics tend to be dismissed
as "matters of mere notation", and so the actual problems of doing it for real
have largely gone untouched. We will not be able to do that in this business,
not if we want to keep it honest.
A phrase like "lattice of all possible theories" has at least two possible meanings.
So a person who uses the phrase "'the' lattice of all possible theories" is using
either the word "theory" or the word "the" in a loose fashion.
If S c A*, for suitable A, is the pertinent set of all possible syntactic sentences,
then Pow(S) is one fairly direct candidate for a "lattice of theories". It exists
independently of semantics, and no matter what we decide to call it at the end of
the day, it is a thing that we will have to cope with, in some virtual or partial
manner, if we want to deal with theories, arbitrary sets of sentences, for real.
The lattice that you speak of is a very nice mathematical object,
assuming that all of the proper parameters are in place to make
it well-defined in relative terms. But I always try to think
ahead to the day when some synderella of programmer is going
to have to relate the bits and pieces that she has on hand
to all of these beatific visions out of Plato's heaven.
That job is some kind of quotient computation, relating
concrete signs to their referential equivalence classes.
This consideration places all of this talk of lattices
back within a sign-relational framework, which is the
only place where we can do much more than wave our
hands at the actual objects and the actual task.
Jon Awbrey
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