ONT Re: Logic Of Relatives
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LOR. Note 60
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There is a comment that I ought to make on the concept of
a "structure preserving map", including as a special case
the idea of an "order-preserving map". It seems to be a
peculiarity of mathematical usage in general -- at least,
I don't think it's just me -- that "preserving structure"
always means "preserving 'some', not of necessity 'all',
of the structure in question". People sometimes express
this by speaking of "structure preservation in measure",
the implication being that any property that is amenable
to being qualified in manner is potentially amenable to
being quantified in degree, perhaps in such a way as to
answer questions like "How structure-preserving is it?".
Let's see how this remark applies to the order-preserving property of
the "number of" mapping 'v' : S -> R. For any pair of absolute terms
x and y in the syntactic domain S, we have the following implications,
where "-<" denotes the logical subsumption relation on terms and "=<"
is the "less than or equal to" relation on the real number domain R.
x -< y => 'v'x =< 'v'y
Equivalently:
x -< y => [x] =< [y]
It is easy to see that nowhere near all of the distinctions that make up
the structure of the ordering on the left hand side will be preserved as
one passes to the right hand side of these implication statements, but
that is not required in order to call the map 'v' "order-preserving",
or what is also known as an "order morphism".
Jon Awbrey
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