ONT Re: Logic Of Relatives
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LOR. Discussion Note 22
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HC = Howard Callaway
HC: You quote the following passage from a prior posting of mine:
HC: What remains relatively unclear is why we should need a system of notation
in which teridentity appears or is needed as against one in which it seems
not to be needed -- since assertion of identity can be made for any number
of terms in the standard predicate calculus.
HC: You comment as follows:
JA: This sort of statement totally non-plusses me.
It seems like a complete non-sequitur or even
a contradiction in terms to me.
JA: The question is about the minimal adequate resource base for
defining, deriving, or generating all of the concepts that we
need for a given but very general type of application that we
conventinally but equivocally refer to as "logic". You seem
to be saying something like this: We don't need 3-identity
because we have 4-identity, 5-identity, 6-identity, ..., in
the "standard predicate calculus". The question is not what
concepts are generated in all the generations that follow the
establishment of the conceptual resource base (axiom system),
but what is the minimal set of concepts that we can use to
generate the needed collection of concepts. And there the
answer is, in a way that is subject to the usual sorts of
mathematical proof, that 3-identity is the minimum while
2-identity is not big enough to do the job we want to do.
HC: I have fallen a bit behind on this thread while attending to some other
matters, but in this reply, you do seem to me to be coming around to an
understanding of the issues involved, as I see them. You put the matter
this way, "We don't need 3-identity because we have 4-identity, 5-identity,
6-identity, ..., in the 'standard predicate calculus'". Actually, as I think
you must know, there is no such thing as "4-identity", "5-identity", etc., in
the standard predicate calculus. It is more that such concepts are not needed,
just as teridentity is not needed, since the general apparatus of the predicate
calculus allows us to express identity among any number of terms without special
provision beyond "=".
No, that is not the case. Standard predicate calculus allows the expression
of predicates I_k, for k = 2, 3, 4, ..., such that I_k (x_1, ..., x_k) holds
if and only if all x_j, for j = 1 to k, are identical. So predicate calculus
contains a k-identity predicate for all such k. So whether "they're in there"
is not an issue. The question is whether these or any other predicates can be
constructed or defined in terms of 2-adic relations alone. And the answer is
that they cannot. The reasons for the misconception otherwise appear to be
about as various a virus as the common cold, and about as resistant to cure.
I have taken the trouble to enumerate some of the more prevalent strains,
but most of them appear to go back to the 'Principia Mathematica', and
the variety of nominalism called "syntacticism" -- Ges-und-heit! --
that was spread by it, however unwittedly by most of its carriers.
Jon Awbrey
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