Re: Question about Basic Law V


lugita15@xxxxxxxxx wrote:
Frege's Basic Law V, also known as the Principle of Unrestricted
Comprehension, states that for all concepts F and G, the extension of F
is equal to the extension of G iff F and G or coextensional, i.e. the
same elements fall under both. As is now known paradoxes like Russel's
Paradox and the Burali-Forti Paradox prove that this axiom is
inconsistent with full second order logic. However, notice that both
paradoxes arise from the supposition that some concept F has an
extension. That is why I propose the following:

Let Extension(F) denote the assertion that F has an extension. Then
Basic Law V may be modified to state that for all concepts F and G, if
Extension(F) or Extension of G, then the extension of F is equal to the
extension of G iff F and G are coextensional. (This is similar in
form, although not identical, to Boolos' Small V. Nevertheless, I am
not equating Extension(F) with Small(F) and the forms of our statements
are a bit different.) It seems to be that we may explicitly state that
some concepts do not have extensions, and the rest do. Two such
concepts should be the concept of all ordinals and the Russel concept.


My question is, that if we consider coextional concepts to be the same
concept, then exactly which concepts should be declared as not having
an extension? I obviously do not want to give extensions to any
concepts which will cause contradictions. Also, I do not want to deny
extensions to concepts which will not cause contradiction.

Let us call concepts which if given extensions would cause
contradiction paradoxical concepts. How many concepts are paradoxical?
If there are only finite number of them, I would like a list of them.
If there are a countable number of them, are they recursively
enumerable? In the worst case in which they are either countable but
not recursively enumerable or they are uncountably infinte, then do
paradoxical concepts have some common property?

If you define "paradoxical concept" to be one such that you can derive
a contradiction from the hypothesis that the concept has an extension,
then the set of such concepts would of course be recursively
enumerable. However, it is not obvious how you would set up axioms
ruling out all such concepts and permitting all the rest, given that
the language cannot talk about its own semantics. And even if you could
work out how to do this, it is not obvious that the resulting system
would be consistent.

Please note that I am
not looking for an intensional property, like being impredicative,
since if there are two concepts which are coextenional, one can have it
and the other cannot. Rather, I am looking for an extensional
property, like being equinumerous with the universal concept.(Although
not this particular property, since it disallows the Frege-Russel
definition of cardinal numbers, which is as far as is known consistent.
For instance, Quine's NF uses the Frege-Russell definition)

Any help would be greatly appreciated.
Thank You in Advance.

P.S. This has absoultely nothing to do with my original question, but
does anyone know of a concept that can only be impredicatively
defined? An impredicative definition of a term x is a definition that
refers to a set of which x is a member.

.

Relevant Pages

  • Re: Question about Basic Law V
    ... without extension. ... As is now known paradoxes like Russel's ... extensions to concepts which will not cause contradiction. ... This has absoultely nothing to do with my original question, ...
    (sci.logic)
  • Re: Question about Basic Law V
    ... Comprehension, states that for all concepts F and G, the extension of F ... As is now known paradoxes like Russel's ... extension of G iff F and G are coextensional. ... extensions to concepts which will not cause contradiction. ...
    (sci.logic)
  • Re: Question about Basic Law V
    ... Comprehension, states that for all concepts F and G, the extension of F ... As is now known paradoxes like Russel's ... extension of G iff F and G are coextensional. ... Given a predicate, ...
    (sci.logic)
  • Re: Question about Basic Law V
    ... Comprehension, states that for all concepts F and G, the extension of F ... As is now known paradoxes like Russell's ... deny extensions to concepts which will not cause contradiction. ...
    (sci.logic)
  • Question about Basic Law V
    ... Frege's Basic Law V, also known as the Principle of Unrestricted ... Comprehension, states that for all concepts F and G, the extension of F ... As is now known paradoxes like Russel's ... extensions to concepts which will not cause contradiction. ...
    (sci.logic)
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