SUO: Re: Comments on Whitten's Starter Ontology
Chris Menzel wrote:
>
> Jon wrote:
> > From this POV, propositional expressions are just notations
> > for boolean-valued functions f : X -> B, where B = {0, 1}.
>
> Jon, intuitively, what is X here? I can't help but notice the formal
> similarity of your suggestion to the standard account of propositions
> (i.e., semantic values of propositional variables) in the possible
> worlds semantics for propositional logic, where X is just the set of
> possible worlds (or, more intuitively perhaps, possible contexts).
> The idea is that a sentence expresses a function from contexts to
> truth values. Intuitively, then, an assertion of a sentence in a
> given context yields truth or falsity, depending on the context.
> Is this your idea? Sorta?
>
> Regards,
>
> -chris
>
> --
>
> Christopher Menzel # web: philebus.tamu.edu/~cmenzel
> Philosophy, Texas A&M University # net: chris.menzel@tamu.edu
> College Station, TX 77843-4237 # vox: (979) 845-8764
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Chris,
I probably should explain some things about my own "scope and limits" here,
in other words, the scope of my present work and the limits of my previous
experience. I have spent the better part of the last twenty years just
trying to work out a better way of doing propositional calculus (ZOL)
in software, using some old ideas of C.S. Peirce plus a few extensions,
in the same spirit, of my own invention. I tell myself that this is the
drosophila melanogaster, e-coli, or lab rat model for most of the rest
of our hard computing problems, and some days I actually believe that.
But it means that I will almost always be working with minimal models
of realistic situations, and just hoping that there is some analogy
or morphism between the reality and the model that will capture
a sufficient number of the features of the former to make the
whole exercise useful.
Okay, so there you have the size and shape of my own swimming pool,
the one in which I am still learning to swim, against the day when
I hope to be up to taking on the Channel.
There are, of course, no sharks in this pool, although there is
one heck of a "Nasty Piranha" (NP), of the NP-complete variety.
All in all, it's enough for me, at least, for a while.
So, with that pre-ramble, X is just the rectangular area of
an ordinary venn diagram, a universe of discourse that can
be filled with whate'er you choose. And f : X -> B is just
the shading in of a part of its area, where the functional
value 1 indicates the elements that it means to indicate.
Now, what will quickly develop from this picture is that
a person will soon pick out a finite number of his or her
favorite propositions -- "Come here often?" is a perodically
recurrent if not the perennial favorite. These propositions
are optimally chosen to be "independent" of each other, that
is, "orthogonal" in a logical sense, and are commonly dubbed
as one's "basic propositions" or singled out by referring to
them as "coordinate projections" of the form x<j> : X -> B,
for j = 1 to n. I usually picture these as the n "circles"
of the venn diagram. After that, if a given system of basic
propositions is moderately adequate to the task of describing,
more or less approximately, every other area of an "arbitrary"
shape that one needs to cover in the universe of discourse X,
then one will find it convenient to "factor" any "arbitrary"
proposition f : X -> B through the "cartesian product" B^n,
as in the following diagram:
f
X -----> B
\ ^
<x<1>, ..., x<n>> \ / f'
\ /
v
B^n
This says that f(x) = f'(x<1>(x), ..., x<n>(x)), where we
can think of the bit-list <x<1>(x), ..., x<n>(x)> € B^n
as the binary coding of the element x € X, and where
f' is the "derived mapping" from codes to B.
Given this sort of set-up, we can proceed to work with
derived propositions f' : B^n -> B, using truth tables
or something equivalent.
As far as your very charitable and generous interpretation goes,
granting me dominion over all possible worlds to roam around in,
I wish I had some claim to it -- it sounds so spectacular! --
but I, for one, have all that I can do to deal with this one
little universe that I am currently wandering through.
Now I do somewhat dimly appreciate that one way of interpreting
venn diagrams, especially if the basic propositions are things
like "it is raining here" (R) and "it is snowing here" (S),
is to call each cell a "possible universe", but that strikes
me as gilding the dandelion just a little, and I have enough
trouble working up the nerve just to describe this business of
bussing truth tables around as a brand of "model theory", even
though it is sorta model-theoretic in a very impoverished sense.
But, on second thought, it does seem like this thing that you are
describing as a space of contexts -- if I can understand each one of
these "contexts" as a sufficiently determinate existential situation,
one that is non-concomitant, non-co-occurrent, pre-emptive, or preclusive
of every other such context -- might just be an especially interesting case
of the general sorts of domains that we ought to keep in mind for X. Okay,
so I am working up the courage to think about that some more. Wish me luck!
Jon
P.S.
On reading your comment again, I do notice a fine point
that bears on this statement:
> Intuitively, then, an assertion of a sentence
> in a given context yields truth or falsity,
> depending on the context.
One of the things that I am trying to avoid here is Quine's
and others way of doing what almost seems like two different
theories of ZOL, one with non-assserted forms of connection
like "conditionals" and bi-conditionals" and another pass
for asserted forms like "implications" and "equivalences",
when all the time it's the very same forms of underlying
truth-functions, the only difference being whether you are
trying to "apply" them to arguments or obtain truth-values
or trying to "invert" them or "solve" them, say at a value
of "truth", in order to elicit the arguments in the domain
that "satisfy", that is, the models or their code surrogates.
So, in view of all this, I probably would prefer to say that
it is the "application" of a sentence (or of the proposition
that it denotes) to a given context that yields true or false.
When somebody asserts a sentence (or the proposition that it
denotes) then I take them to be indicating the collection of
situations, states of affairs, or things that is covered by
that statement. In geometry, for f : X -> Y, this would be
called the "fiber" at a given functional value y, that is,
(f^(-1))(y).
J.A.
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