SUO: Re: Mapping from one notation to another
Tom,
The difference between (a) and (b) is typical of examples
from mathematics (a) and examples from physical reality (b):
> a) in the context of plane geometry, "equiangular triangle"
> and "equilateral triangle" have different intensions (rules)
> but identical extensions;
>
> b) Quine's distinction between "creature with a heart" and
> "creature with a kidney": different intensions (meanings)
> but identical extensions.
>
> When Church speaks of "functions in intension", is he doing
> anything more than giving a name to examples like these?
> Or, at least, to examples like (a)?
In the excerpt from Church's book on the lambda calculus,
he was addressing mathematical and logical issues.
Example (a) is a rather simple one, in which the equivalence
could be proved very easily, but Church would consider
examples in which the equivalence might be difficult or
even impossible to prove (such as some of Goedel's
true, but unprovable statements). So it's conceivable
that some mathematical examples could get into the murky
issues that are typical of examples about the world.
In any case, Church was interested in the broader
philosophical issues, as illustrated by his talk
on the ontological status of women.
> So I offer this, along with several of my earlier diatribes,
> as cashing in my claim, back in June, I think, to explain
> in my own words why I believe that Quine demolished the
> analytic/synthetic distinction, replacing it with an
> analytic/synthetic continuum.
I agree that Quine put one more nail in the a/s coffin,
but quite a few people before him had already fastened
the lid rather securely, including some of my favorite
philosophers -- Peirce, Whitehead, and Wittgenstein.
I also agree that you can introduce a relativized
a/s distinction, in which the borderline is determined
by which propositions are assumed as axioms (or laws)
and which are merely contingent facts. That gets into
issues related to my paper on multimodal reasoning:
http://www.jfsowa.com/pubs/laws.htm
Laws, Facts, and Contexts
The laws are statements that are taken to be
necessarily true according to different modalities
with different degrees of "entrenchment".
And indeed, in Section 7 of that paper, I discuss
Peirce's five levels of necessity and possibility:
* Logical possibility. A dark blue context, Peirce's
equivalent of ?p, would mean that p is consistent
or not provably false.
* Subjective possibility. In light blue, ?p would mean
that p is believable or not known to be false.
* Objective possibility. In red, ?p would mean that p
is physically possible. As an example, Peirce noted
that it was physically possible for him to raise his
arm, even when he was not at the moment doing so.
* Interrogative mood. In green, ?p would mean that p
is questioned.
* Freedom. In purple, ?p would mean that p is free or
permissible.
(Depending on what software you're using, the diamonds
in the original paper might not appear above.)
This classification gives 5 levels rather than a continuum,
but it would be possible to break it down into many more
sublevels. Subjective possibility for one person, for
example, would not have the same laws or axioms as
subjective possibility for another. And even in physics,
you could have two theories that are identical in every
prediction, but they might differ in which propositions
were considered synthetic (i.e., laws of nature) and
which were considered analytic (i.e., definitions).
I completely agree with the following:
> But the bottom line is this: the semantics of natural languages,
> and specialized dialects of it (manufacturing, health care,
> law, physics, etc.), is constantly changing. Formalizations of
> any piece of any of these languages distill out a subset of
> the semantics of that piece of language, and freeze it.
However, I would quibble about your use of the term "the
formalism". I would prefer to replace the word "formalism"
with something that implies the axioms and definitions of
the ontology, independent of any particular notation:
> The formalism is frozen, but the real living language,
> in which we think and talk to others, is never frozen.
> Formalizations of any piece of any of these languages distill
> out a subset of the semantics of that piece of language,
> and freeze it. The formalism, therefore, becomes more and
> more of an anachronism.
If you change the word "formalism" to "formal ontology", then
I would completely agree. That is why I have been recommending
a lattice or hierarchy of theories with an open-ended number
of options. The process of moving from one theory to another
corresponds to a belief revision step.
> Formalisms of linguistic focii near the center of our
> conceptual sphere, become anachronisms very, very slowly.
> I am talking here about logic and mathematics.
No, I would distinguish mathematics and logic as structural
mechanisms for organizing and formulating the theories. By
themselves, logic and mathematics make no statements about
the world itself, and they can never be true, false, or
"anachronistic". The only question you can ask about any
theory of math or logic is whether it is appropriate to a
particular subject. The result of applying an inappropriate
mathematical theory to some subject might be false, but the
math itself is not false.
> This has been pure stream of consciousness for the last 45 minutes.
> If it's off base, by being either irrelevant or incorrect, I'd like
> to hear the argument. If it's not, what should we do about such
> reflections to materialize any practical value they might have?
The direction you're taking is quite compatible with the approach
I have been recommending for organizing the "knowledge soup".
For a statement of the problem, see the following slides:
http://www.jfsowa.com/talks/souprepr.htm
Representing Knowledge Soup
The following talk says more about the knowledge soup, but
it then gets into issues about analogies (which are related
to the approach I recommend, but I need to say a lot more
about how they are used):
http://www.jfsowa.com/talks/tosi.htm
Task-Oriented Semantic Interpretation
Section 7 of the following paper says more about the knowledge
soup:
http://www.jfsowa.com/pubs/signproc.htm
Signs, Processes, and Language Games
The first six sections say a lot more about Peirce, Whitehead,
and Wittgenstein and about some very important issues they
addressed, but most 20th-century analytic philosophers ignored.
I wouldn't say that P., W., & W. were ahead of their time --
but that they were unfashionable because of some unfortunate
historical accidents.
John