SUO: Re: Semes To Be The Truth
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John F. Sowa wrote:
>
> Jon,
>
> I have no quarrel with the claim that "truth" is the
> denotation of a true proposition. But that is very
> different from the claim that there exists one and
> only one true proposition.
>
> > | Finally, and in particular, we get a Seme of that
> > | highest of all Universes which is regarded as the
> > | Object of every true Proposition, and which, if we
> > | name it [at] all, we call by the somewhat misleading
> > | title of "The Truth".
> > |
> > | C.S. Peirce, 'Collected Papers', CP 4.539.
>
> Note that Peirce says "every true Proposition" -- a phrase
> which is consistent with the assumption that there exist
> more than one true proposition, even infinitely many.
Yes, but here he is using the word "proposition" in a way
that we no longer do, to mean the actual sign or sentence.
Anyway, that is the way that I have to interpret his usage,
considering this standard portion of his sign classification:
| Sign
| Index
| Icon
| Symbol
| Term
| Proposition
| Argument
| Deduction
| Induction
| Abduction
Believe me, I tried to continue this usage as long as I could,
but finally had to give it up because I encountered too many
rather insistent authorities who insisted on using either one
or both of the words "proposition" and "statement" -- please
don't ask me why, but they do! -- to mean either one or all
of these things: (1) the abstract, formal, logical object
that is denoted by the now-called "propositional expression"
or "sentence", or (2) the equivalence class of sentences that
serves as a stand-in for those dispositions of thinkers who do
not wish to acknowledge the existence of anything falling under
the description of ostensibly "abstract objects" and their ilk.
So I gave it up and now use the term "expression" or "sentence".
Let's not worry about empirical statements just yet,
as it seems like a good idea to try to get clear on
purely logical statements before tackling the world.
This following sort of picture always comes to mind here:
| Imagine that one picks out a finite collection of one's
| favorite propositions for describing an object domain X.
| The 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 k. I usually picture
| these as the k "circles" of a venn diagram for the universe X.
| If a given system of basic propositions is moderately adequate
| to the demands of describing, more or less approximately, every
| other region of a "relatively arbitrary" shape that one needs to
| cover in the universe X, then one finds it basically convenient
| to "factor" any "arbitrary" proposition f : X -> B through the
| "cartesian power space" B^k, as in the following diagram:
|
| f
| X o------>o B
| \ ^
| c = <x<1>, ..., x<k>> \ / f'
| v /
| o
| B^k
|
| This says that f(x) = f'(c(x)) = f'(x<1>(x), ..., x<k>(x)), where
|
| c(x) = the "code" of x = the bit-list <x<1>(x), ..., x<n>(x)> in B^k
|
| is the binary coding of the element x in 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^k -> B, using truth tables
| or something equivalent.
What's the point, you ask? Well, I think of the vertex X as being
the point where the otherwise pure logic gets applied, and this is
a species of referential meaning that can vary from application to
application, a "run time parameter", so to speak. But the logical
functions themselves, enjoying types like f' : B^k -> B, I cannot
see any way to classify these with any more pretense of refinement
than to sort them into "logical equivalence classes" (LEC's) based
on, what else, logical equivalence. And that puts all theorems in
the same pot, all absurdities in another, and all contingencies to
gather with birds of variegate and sundry like-continged feathers.
> > Now, if every true proposition denotes the same object --
> > and here I think that Peirce ('Monist', 1906?) concurs
> > with what Frege also says, though I think that this is
> > just the very model of a typical mathematical attitude --
> > then all true propositions have ultimately one and the
> > same referential meaning, at least.
>
> Perhaps, but that does not imply that they are the same
> proposition, nor that they "have the same meaning" in
> any other than that one very narrow sense.
I do not think that you want to admit connotations of arbitrary sorts
into a matter of logical semiotics, and short of sorting them out by
interpretive type, say the way that 1 : B, 1 : B -> B, 1 : B^2 -> B,
and so on are distinguished, I just do not see a way to do this that
makes logical sense. Otherwise you will find youself forced back to
allowing that "(X (Y))" and "X => Y" really do mean different things,
just because somebody thinks they do. One can distinguish, and I do
this in my own work, definitions of "referential equiv classes" (REC's),
of which LEC's are a special case, and "semiotic equiv classes" (SEC's),
but the end of logic requires a close coordination among all of these,
and I think, except for changing X, as close a congruence to identity
as we can get. (It cannot be perfect, since REC's are model-theoretic
and SEC's are proof-theoretic.
> > Since Peirce uses the term "proposition", unless he has in mind
> > a design to play it equivocally, for the "syntactic expression",
> > or the "sentence" as we more often prefer to say, then
>
> What do you mean by "we", Kimosabe? -- as Tonto said
> to the Lone Ranger when they were surrounded by Indians.
Well, he only said that because a comic book writer used a phonetic
spelling for "¿quién sabe?", in this context the Spanish equivalent
for "Nemo" or "Nescio". So who is worse off, I wonder: Tonto, who
is twice over linguistically oppressed in a lone dominant narrative,
or the true identity of the Masked Man, one who lies forever buried
in a hell of a large equivalence cairn of unidentified persons?
Like I said above, me and all the folks have forced me,
much against my druthers, to speak their language, and
so now I am stuck with having to talk this way. Tonto,
the Lone Ranger, and I will all of us have our revenge --
but all in good time.
> > I would have to say that this propositional expression,
> > say, "e", denotes a function e : X -> B, with the type
> > of e being left indefinite for the present moment, not
> > yet run time, nor even compile time, but only IOU time.
> > This semes to suggest that the type of the proposition,
> > to be e-nunciate, is a co-notation that e-fects itself
> > not in the mediate but only in the ultimate denotation.
> > I belive that Peirce would fairly call that a "symbol".
>
> Peirce said many things about propositions during his career.
> Following is a statement that I quoted from the Monist (1905),
> "What Pragmatism Is" as reprinted in CP 5.411-436:
>
> | The meaning of a proposition is itself a proposition.
> | Indeed, it is no other than the very proposition of
> | which it is the meaning: it is a translation of it.
Sure, but folks hereabouts will misunderstand what he's saying here
unless you trouble to add after each "proposition" "-al expression".
> This is one of my inspirations for the definition of "proposition" as
> an equivalence class of sentences that are said to have "the same meaning".
Sure, I recognize the sign up ahead, and I see the zone
that it betides, but you have just crossed over into that
other way of talking about propositions, not as sentences,
as before you crossed the line, but now as those abstract
twilight entities called equibbling classes -- see how
easily that happens?
> Another inspiration is Alonzo Church's definition of "intension"
> as an equivalence class of expressions according to some specified
> equivalence relation:
>
> http://www.bestweb.net/~sowa/logic/alonzo.htm
Yes, but what shall be the application
of the 2nd intension to who's on 1st?
That is, shall the intension be strained
any finer than its normative quality?
> I combined these two notions in my definition of
> a "meaning-preserving translation", which determines
> which sentences are to be considered as expressions
> of "the same proposition":
>
> http://www.bestweb.net/~sowa/logic/meaning.htm
>
> A grouping of sentences by their denotations would
> collapse them into just two equivalence classes.
Oh no, not one bit!
Never again, I say,
once we get past B.
|{B^k->B}|=2^(2^k).
> A grouping by truth in all possible worlds would be somewhat finer,
> but still too coarse, since it would treat "2+2=4" as synonymous
> with Fermat's Last Theorem.
I do not see that as a problem, at least, not any more than saying
that both denote the same logical object. A logical fact, once we
find it, does not change a thing about the pragmatic difficulty of
finding it. The difficulty of getting that dogie into that corral
is a property of the dogie, not of the corral. I would be willing
to entertain the notion that "2+2=4" and Fermat's Last Theorem are
statements that refer to different underlying domains, but without
a better grasp of the complete domain to which the latter refers --
for example, do the properties of the proof that detours through
a wild open range of many intermediate domains, like algebraic
varieties, elliptic functions, and modular forms, just for
starters, also accrue to the statement of the theorem on
the grounds from which it initially took off? Beats me.
Or what happens when somebody finds a whole nuther proof?
> My recommended definition is to require
> any meaning-preserving translation to have
> four properties: proof-preserving, invertible,
> vocabulary preserving, and structure preserving.
> Even these requirements leave a lot of room
> for further refinement, as I discuss in
> the meaning.htm excerpt.
Yes, I can see the potential for indefinite refinements --
I just do not see them distilling the logical substance.
Jon Awbrey
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