[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


"... 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}"

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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