SUO: Re: Semes To Be The Truth
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.
>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.
>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.
>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.
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". 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
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. 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.
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.
John Sowa