[FOM] On definitions of terms, and free logic---a quick follow-up
Tennant, Neil
tennant.9 at osu.edu
Tue Oct 27 08:22:39 EDT 2015
The definitional abbreviationist treats {t1,...,tn} as a mere abbreviation for the set-abstraction term {x|x=t1v...vx=tn}. The curly-parentheses operationalist thinks that, for each value of n, there is a distinct n-ary
operation sign { ,..., } that, when supplied with the n argument terms t1,...,tn, forms the functional term {t1,...,tn}.
In my last post I pointed out that the definitional abbreviationst can say
{1/0,2} = {x|x=1/0 v x=2} = {2}
whereas the curly-parentheses operationalist says any identity statement involving {0/1,2} is false, while the identity statement {x|x=1/0 v x=2} = {2} is true.
The follow-up point of this post is to observe that the definitional abbreviationist can also assert the identity claim
{1/0} is the empty set, i.e. {1/0} = {x|~x=x}.
Another nice thing is that this last identity is a theorem of core set theory in free logic.
By way of corollary, there is an embarrassment for the reductive set-theorist who wishes to identify 0 as the empty set, and identity 1 as the singleton of the empty set. S/he would get
{{emptyset}/emptyset} = emptyset
and would then have some explaining to do.
There is also an embarrassment for the Fregean who tries to handle non-denoting (i.e., undefined) singular terms by making them (arbitrarily) denote the empty set. For then the empty set would be its own singleton (if s/he is also a definitional abbreviationist). Also, talking of the world more widely, s/he would assert that Santa Claus is 1/0.
Neil Tennant
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