Howard, list,
Two statements of the kind that Peirce _/describes/_ with "There is some
one individual" etc., etc., are equivalent. You can see this in the
example of *A*: 'There is something round or blue' and *B*: 'There is
something round or there is something blue' by recognizing that their
respective negations are obviously equivalent: *~A*: 'There is nothing
round or blue' is obviously equivalent to *~B*: 'Neither is there
something round nor is there something blue' which is obviously
equivalent to 'There is nothing round and there is nothing blue'. Two
statements' negations are equivalent *if and only if* their affirmations
are equivalent.
What's tricky here is that there are two alternatives, not one, in
'there is something round or blue': the alternative represented by
'something' and the alternative represented by 'round or blue'. The
items in these two alternatives get, so to speak, mixed together at the
same level. Watch and see:
*Pretend that the universe consists of three individuals: Alf, Beth, and
Cam.*
*Something is round or blue.*
/Therefore, and equivalently (in this pretend-universe),/
*Alf is round or blue, or Beth is round or blue, or Cam is round or blue. *
/Therefore, and equivalently, by mere expansion, /
*Alf is round or Alf is blue, or Beth is round or Beth is blue, or Cam
is round or Cam is blue.*
/Therefore, and equivalently, by mere re-ordering,/
*Alf is round or Beth is round or Cam is round, or Alf is blue or Beth
is blue or Cam is blue. *
/Therefore, and equivalently,/
*Something is round or something is blue.*
Like I said, in an idea like that of 'something', we condense the idea
of an alternative so much that it gets blurry, we forget some of the
structure of the alternative.
How does one keep the alternative represented by 'something' and the
alternative represented by 'round or blue' from mixing together so
thoroughly?
Here's one way: modal logic (some first-order version, I think):
*Something is {necessarily {round or blue}}.**
**Alf is {necessarily {round or blue}} or Beth is {necessarily {round or
blue}} or Cam is {necessarily {round or blue}}.*
As long as *'Something is {necessarily {R or B}}'* doesn't entail
*'Something is {necessarily R} or {necessarily B}'*, then the two
alternatives won't thoroughly mix, and we won't end up with *'Something
is {necessarily R} or something is {necessarily B}'*.
Another way to get past the 'strange rule' is by saying 'a certain' and
interpreting it not as the variable 'some' but instead as a veiled or
unknown constant. People won't generally regard this as a solution
although they may occasionally do it unconsciously. Anyway, in that
case, in logic one will want to be able to distinguish among veiled
constants, in the sense of distinguishing _/a certain one/_ from _/a
certain possibly other one/_; this will lead to the assignment of
letters, and it will quickly become apparent that such letters are
pretty much the same as the familiar dummy letters abbreviating proper
names in schemata, except that instead they'll be newly assigned proper
names or pseudonyms at least. Sometimes a person's initials might as
well be a pseudonym: "New Miracle Phosphor-Wax makes my floors glow all
night long! - A.J. in northern Alaska." Now, a new name or a pseudonym,
logically, is not a dummy letter but simply a new name, in one sense or
another, for some individual. So there seems something a bit gratuitous
about the whole idea of a veiled or unknown constant in first-order
logic, but cases where the utterer knows its identity and the listener
doesn't, suggest that there could be some use for it.
Apparently another way to get past the 'strange rule' is through
branching-quantifiers logic, but I don't know how that would work.
On your other remarks:
Mathematical logic is not always very useful for pure mathematics,
though it is useful for pure mathematics when infinities get involved
(for those interested, Peirce did at least once allow of help here from
logic to pure math, see CP 7.525, undated). I don't think that
mathematical or deductive logic should be confined to being the
vestibule to pure math, or (by earlier tradition) the vestibule to
philosophy, for that matter, any more than probability theory and other
such 'applied' yet highly general and significant maths are.
The 1's and 0's in computer programs are ways of translating icons,
indices, and symbols into a specialized symbol-expression system for
efficient transmission to a place where they will be translated back
into normal icons, indices, and symbols. Otherwise it wouldn't be
meaningful or even occur. To call them "meaningless" as you sometimes to
is to say that they can't be fairly translated into normal icons,
indices, and symbols. Of course they can, although very probably not
personally by you or me simply glancing at arrays of 1's and 0's. That's
an aspect of why it's called encoding; it depends on something's knowing
the code in order to be able to interpret them. The level of the
specialized binary symbolic-expression system doesn't seem fundamental
to logic.
Best, Ben
On 2/16/2015 9:17 AM, Howard Pattee wrote:
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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