Equivalence & Equality.
From: Bill Taylor (w.taylor_at_math.canterbury.ac.nz)
Date: 09/11/04
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Date: 11 Sep 2004 00:28:44 -0700
Part 1 EQUIVALENCE
""""""
This is just a recalling of that classic text-book puzzle: "For relations,
why don't symmetry and transitivity together imply reflexiveness?"...
...because... a ~ b --> b ~ a by symmetry
and then a ~ b & b ~ a --> a ~ a by transitivity!
As is well known, the answer is, yes, this works PROVIDED there is any
such b for a to be equivalent to. But there may be no such b at all!
There may be a relation with various equivalence classes, and the odd one
or two (or many) individuals that are related to nothing, not even themselves.
A moderately natural example is on the domain of the reals,
with the relation being
a ~ b iff ab > 0.
Now all the positives are related, and all the negatives are related,
but 0 isn't related to anything.
Still, the near-proof is a very cool one, a bit like getting something for 0.
Part 2 EQUALITY
""""""
There is another cool case in logic, where one can get something for
almost nothing, in a remarkably similar way. But for some reason most
text books don't do it this way, in spite of their general preference
for slick reductions to the most minimal assumptions possible.
When passing from FOL to FOL(=), 1st-order logic with equality,
the equality concept is almost always introduced via...
(a) "=" is an equivalence relation, with
(b) unlimited substitutivity, i.e. P(x)
"" and x = y (or y = x)
"" implies P(y)
To conform to minimalist slickness, the disjunction (y = x) is usually
dispensed with, as symmetry of equality does the job.
HOWEVER: if one keeps that disjunct, (as seems necessary), one can dispense
with BOTH symmetry and transitivity for "=", as these can be proved from
the substitution rules! (We adopt FOL with natural deduction.)
introduce a = b (I)
thus b = b (II) by substition of I into itself,
thus b = a by substitution of I into II,
extract a = b --> b = a Q.E.D.
introduce a = b (I)
introduce b = c (II)
thus a = c by substitution of I into II,
extract a=b & b=c --> a=c Q.E.D.
Quod Easily Done, as we say. Transitivity is cool enough,
but substituting an equation into itself twice, for symmetry,
is the ultimate in Alice-in-Wonderland slickness!
Perhaps the reason it is not done is that it does use natural deduction,
which books seem to try to avoid when possible; it does seem to require
the two forms of substitutivity as mentioned above; and that anyway,
it is still required to define equality as a REFLEXIVE relation with
unlimited substitutivity.
This last, is almost identical to the similar proviso in part 1. (!)
Kyewt.
------------------------------------------------------------------------------
Bill Taylor W.Taylor@math.canterbury.ac.nz
------------------------------------------------------------------------------
I do not claim my argument is logical, but simply that I'm right.
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