Howard,
It's not just my instinct, it's Peirce's that there's something strange
about an equivalence between where one _/seems/_ to formulate talk of
one thing and where one _/seems/_ to formulate talk of possibly two things.
The problem is that symbolic logic requires us to contract ideas into
concentrated forms where we tend to forget some of the structure of
alternatives in them. Deductive logic and related fields such as
probability theory can be peculiarly tricky, a bit _Twilight Zone_-ish
at times, because they are specially about abstract structures of
alternatives, alternatives among universes in a sense, rather than about
structures of, /per se/, space, measure, group, or order. Also,
conveniences of logical verbal expression can require a little getting
used to (e.g., 'some' is usually taken in the sense of 'at least some';
and, very notably, 'if /p/ then /q/' just means 'not both /p/ and not
/q/' because logicians don't like to invent funny expressions like
"/p/ nandn't /q/" or "/p/ nayor /q/" to say it instead).
First-order logic with identity and with axiomatic completeness can
distinguish 'at least one,' 'at most one,' 'exactly one,' 'at least
two,' 'at most two,' 'exactly two,' etc., and I've read that it allows
time expressions (e.g., where one treats "Jack at /t/₁" as an object, or
something like that). I'm no logician, but I think that one could allow
numeric predicates as in '5/xyzwu/'; I don't know whether they can be
indefinite (even if still finite) in valence without losing axiomatic
completeness (i.e., moving us into 2nd-order logic), and to treat them
as variables would require 2nd-order logic at least, I think.
If you want to make good on the instinctive sense that the 'strange
rule' should be avoidable, other than by thinking of veiled or unknown
constants (amounting to 2nd-order variables much like Quine's "dummy
letters" in schemata for 1st-order logic), then you'll need some sort of
modal logic (of the 2nd-order kind, I think), or branching quantifiers
logic (2nd-order), or some other area in 2nd-order logic. More
generally, if that which Poincaré desired was really mathematics, then
why did he complain? He already has lots of it. Axiomatically complete,
first-order logic with identity is about all things in a sense, but its
completeness of axioms keeps it from being powerful enough to be all
things useful to theory. Think of it as a sandbox.
Best, Ben
On 2/14/2015 11:19 PM, Howard Pattee wrote:
At 09:30 PM 2/14/2015, Benjamin Udell wrote:
The strange rule really isn't so strange. In CP 4.569 Peirce (without
calling it the 'strange rule') 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: This "strange rule" illustrates Poincaré's criticism of logic as
an impoverishment of natural language that can neither count nor tell
time.
BU: In other words, 'there is something round or blue' is equivalent
to 'there is something round or there is something blue'. It seems
kind of strange because in the first case one seems to mention one
individual thing, while in the second case one seems to mention
possibly two things.
HP: I think Ben's instinct is correct. It is indeed strange that the
difference between two different numbers, like one and two, are
ignored. Only using logic rules is such indifference possible. The
past and future succumb to the same logical indifference.
Howard
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