[FOM] Weak logic axioms

[Michael Lee Finney]

While I don't recall the exact post, I believe that Harvey did exactly
that recently.

However, empty domains are an issue relating to quantification. Until
you have a workable replacement for propositional logic, considering
the issues relating to quantification is premature unless you simply
want empty domains in classical logic. After all, there is frequently
significant impact on quantification in weaker logics. Also, there are
"free" logics which do not assume that the domain is empty.

I have always believed that extensional assumptions due to
quantification are a problem. Why can't I talk about Unicorns?
Perhaps they don't exist now - but it only takes some trivial genetic
mods to make them exist (you just need a horn bud, already people
"create" unicorns by grafting or fusing horn buds).

When I define a mathematical object, I don't know if one exists or
not or if such an object is inconsistent. All of my reasoning should
be valid in all possible cases. Nonetheless, there are significant
difficulties with empty domains. At the very least, you need to be
able to specify existence independently of quantification so that the
appropriate conditions can be specified.

There is Hilbert's e (epsilon) operator. I don't think that it every
quite worked right, but I am not sure about that.

Additionally, why are you restricting the question to first-order
predicate logic? That is not sufficient even to define finite numbers.
You need at least partial second order logic. How do empty domains
work for second order logic?


Michael Lee Finney


>           With all this contention about logics, I’ve wondered why
> no one has expressed the view that empty domains seem more
> mathematically appropriate, rather than assuming first-order logic
> domains must be non-empty.  (Especially since the topic of the empty
> set being included in any set has been bandied about.)

> Charlie Silver