SUO: Re: Propositions
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
John F. Sowa wrote:
>
> Jon,
>
> No, no, no!!!! Absolutely not:
>
> > > but it seems to me like, on the output side of things,
> > > anyway, that if something is indeed a theorem, then the
> > > canonical representative of its equivalence class ought
> > > to be just the value true.
>
> Some logicians made the mistake of identifying a proposition with
> the set of all possible worlds in which it happens to be true.
> But that identification collapses all theorems to just T.
> Such an idiotic collapse would cause Fermat's last theorem
> and "2+2=4" to be identified as "the same proposition".
> That definition utterly fails to capture the informal
> notion of a proposition as the "meaning" of a sentence.
> No mathematician in his or her right mind would ever
> say that Fermat's last theorem and "2+2=4" have
> the "same meaning".
>
> The definition I gave makes a much finer distinction
> of meaning, which I claim is much closer to the
> informal notion. Please reread the excerpt
> from Ch 5 that I put on the web:
>
> http://www.bestweb.net/~sowa/logic/meaning.htm
>
> John Sowa
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
John,
Good, finally something we can disagree about --
I was beginning to worry! -- and to wonder if
we would ever have a real conversation here.
First of all I have hedged my claims by saying
that this is merely a matter of interpreting
and implementing calculi for the object domain
that we find discussed under Peirce's Alpha,
propositional calculus, or sentential logic.
I do believe that there are broader lessons
to be drawn, but I am content to be careful
and ploddingly methodical in drawing them.
Second, I am only talking about classical logic here --
all sorts of other folks, working from a diversity
of basic axioms and different intuitions, draw all
sorts of other "fine distinctions" that will be no
concern of mine so long as I continue to work from
a classical stance.
Third, this is one of those places where a two-pronged approach
to the meaning of that very ambivalent word "meaning" comes to
one of its chief points. To wit, it really is true, after all,
at least, for the mean time, that we are forced to distinguish
denotation from connotation, reference from sense, semantics
from semiotics, or pick your own favorite labels for the axes.
Fourth, I do not concern myself very much with this "all possible universes" stuff --
since I can scarcely deal with one circumscribed "universe of discourse" at a time --
I genuinely doubt if anybody, any fallible and mortal finite information creature,
can really even imagine anything approaching anything like the putative reference
of this "all possible universes" (APU) phrase, and I suspect that most folks stay
pretty darn close to those paltry few worlds that are recognizably found to fall
within "yet another variorum on the erratic themes of this one" (YAVOTETOTO).
All right now, staying within the frame and the scope of these considerations and
principles, I really do mean to say that this "splendid sentence" from Leibniz is
just a splendidly obscure way of alluding to a constant truth-value, or a constant
truth-function of suitable type, otherwise known as "1". And the fine distinctions
that mediately stood in the way of our recognizing this immediately are the sorts
of fine distinctions that -- though we need to maintain them in the mean time for
the practical purpose of linking together our various impressions of the various
other occasions of obscurity, past, present, future, with which we are bound by
our nature to find ourselves afflicted -- we could, after all, really do without.
Now, there are some "fine distinctions" that I draw among propositions,
understood by me as functions of type f : X -> B, for an appropriate X,
that is to say, distinctions among propostions that, at some degree of
abstraction, might otherwise be identified, and these refinements fall
across the board, applying in particular to the proposition 1 : X -> B.
But these distinctions are all summed up in the choice of the domain X.
As far as "theorems" that we have not yet proved,
they remain in their exiles of markedness quotes.
There is a big difference, a pragmatic difference,
between a sign or a sentence that we know to be true
and sign or a sentence that we do not know about yet.
Many Regards,
Jon Awbrey
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤