|
|
xilog wrote:
> Newberry wrote:
> > >
> > > However, if you ask "what is a higher order logic?" then one reasonable
> > > answer to that might be "a logic in which it is possible to quantify
> > > over sets or functions or predicates (not just over individuals)" and
> > > there are many logical systems in which this can be done but which are
> > > not typed (though they are not normally called higher order logics).
> > > The best known is set theory
> >
> > This is what confuses me. Is set theory a multi-order logic done in a
> > first order logic?
Yes, many set theories presume a cumlative stucture of sets:
sets, sets of sets, sets of sets of sets, on and on. See: Quine.
Clearly , there is no logic that requires this necessity.
>
> Its a bit like that, but the ontology is more liberal.
> All the entities which exist in w-order logic at any type exist also in
> the domain of Zermelo set theory.
> However, there also exist additional sets whose elements do not all
> have the same type.
> Since the domain of set theory is not separated into types, you run the
> risk of your set theory being inconsistent, and this is what lead to
> the infamous "Russell's Paradox".
> Russell's "Theory of Types", the precursor of modern higher order
> logics, and Zermelo's axiomatisation of set theory, were both published
> in 1908 and supplied two different ways of resolving the paradoxes, the
> first based on types, the second based on the restriction of
> comprehension to separation.
>
> Roger Jones
|
|