[FOM] On definitions of terms, and free logic
Tennant, Neil
tennant.9 at osu.edu
Tue Oct 27 07:43:03 EDT 2015
Harvey wrote of Rob's example
{x | x = 1/0 or x = 2} = {1/0, 2}
that this identity statement
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"... is false. The reason is that what is on the left side is
defined, but what is on the right side is undefined. This is because
1/0 is undefined, and for a term to be defined, all component terms
must be defined. That's true about any operation whatsoever. The
following is correct:
{x | x = 1/0 or x = 2} = {2}"
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I might not be alone in having always understood the notation {t1,...,tn}
to be merely a definitional abbreviation for the properly formed set-abstraction term
{x| x=t1 v ... v x=tn}.
If one takes this view, then one can say that Rob's example is true by definition,
and indeed that
{1/0, 2} = {2}.
For Harvey to regard {t1,...,tn} as the output of an operation on the inputs t1,...,tn
(hence, undefined if any of the input-terms is undefined) he must be countenancing
a different n-ary function-sign {...} for each value of n. The definitional abbreviationist
does not have to follow him in this.
Neil Tennant
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