Ben and list,
I agree that Poincaré's complaints about logic
were excessive, probably because he was irritated
more by Russell's attitude than by logic itself;
but I'm still missing something about that strange theorem.
Peirce says: "The logical Principle is that to
say that there is some one individual of which
one or other of two predicates is true is no more
than to say that there either is some individual
of which one is true or else there is some
individual of which the other is true."
HP: The way I interpret it there are two
statements here that are not equivalent:
(1) There is some one individual of which one or
the other of two predicates is true.
(2) There either is some individual of which one
predicate is true or else there is some
individual of which the other predicate is true.
Statement (1) is explicit that the two
predicates refer to just one individual.
Statement (2) is ambiguous about the number of
individuals to which the two predicates refer, so
that "some individual" in statement (2) could
reside in a non-empty set of a number of
individuals and that one or the other predicate
could apply to one or another individual in the set.
Another topic: I also understand how one might
develop higher order logics as you suggest, but I
do not see what would be gained by more complex
logics (except fun for logicians). As I mentioned
earlier, what I still find amazing (and what in
their time neither Peirce nor Poincaré could have
appreciated) is that all our distance
communication is today accomplished by just two
symbol vehicles manipulated physically by the
simplest two-variable Boolean logic. This appears
to ultimate extreme of semiotic reduction or
simplification. For example, the moment I typed
"Ben" it became a sequence of 24 0s and 1s, and
such sequences are all that is ever manipulated
and transmitted throughout all our discussions
(and everyone else's). Of course there are
complex hierarchies of coding and parsing of
these sequences, but all of this is still done by
physical circuitry executing two-variable Boolean truth tables.
At this level I see no reason to introduce indices and icons.
Howard
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