[FOM] free logic

[Mario Carneiro]

On Sun, Oct 25, 2015 at 9:24 PM, Mitchell Spector <spector at alum.mit.edu>
wrote:

> It appears that you are taking <= to be a relation symbol of your
> language, and then applying the following principle:
> if R is an n-place relation symbol, if t1, ..., tn are terms, and if at
> least one of t1, ..., tn is undefined, then R(t1, ...., tn) is false.
>
> But if this is what you're doing, it has an unfortunate consequence.
> Specifically, consider the formula 1/0 - 1/0 > 1/0 - 1/0.
>
> If > is taken to be a relation symbol of the language also, then 1/0 - 1/0
> > 1/0 - 1/0 is false.
>
> But if x > y is taken to be an abbreviation for NOT(x <= y), then 1/0 -
> 1/0 > 1/0 - 1/0 is true.
>
>
> This seems like an undesirable state of affairs.
>


I would put forward that this state of affairs is not as undesirable as you
suggest. This issue also comes up in the plain ZFC approach to relations:
Suppose that < is defined as a relation on R, that is, a subset of R x R.
and similarly for <=. Then x > y <-> NOT(x <= y) is true when x and y are
in R, but not generally. Is i < i (where i is the imaginary unit)? I would
suggest that i < i be false, not undefined, and similarly for i <= i.

With this interpretation of relations in untyped set theory, the Harvey's
suggestion seems to fit naturally. (Consider also the possibility of what I
call "reverse closure": If x R y for some relational predicate R, we can
deduce that x and y are defined and in the domain of the relation, which
simplifies the antecedents to theorems of the form x R y => A.)

Mario
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