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Topic: The Comprehension Principle for Concepts & Relations: Minimal Restriction.
Replies: 1   Last Post: Jun 14, 2013 8:25 AM

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Zaljohar@gmail.com

Posts: 2,665
Registered: 6/29/07
Re: The Comprehension Principle for Concepts & Relations: Minimal Restriction.
Posted: Jun 14, 2013 8:25 AM
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On Jun 14, 1:09 pm, Zuhair <zaljo...@gmail.com> wrote:
> I'm thinking of a kind of second order logic which has minimal
> restriction on the Comprehension Principle of Concepts and Relations.
>
> Before I go to that I'll outline some notations.
>
> Upper case letters represent constant symbols, i.e. symbols that can
> be substituted by ONE value only.
>
> Lower case letters represent variable symbols.
>
> Objects shall be represented by Starred symbols, so X* is a constant
> object symbol, while x* is a variable object symbol (i.e. x* range
> over the whole of domain of discourse of OBJECTS)
>
> Predicates shall be left non starred, so P is a constant predicate
> symbol denoting a particular predicate, while p is a variable
> predicate symbol (i.e. a symbol ranging over ALL predicates)
>
> If a symbol is primed then it might stand for an object or a predicate
>
> Now we outline the:
>
> ---------------------------------------------------------------
> Modified Comprehension Principle of Concepts and Relations:
>
> If phi(p',..,q') is a modified formula then:
>
> Exist g. for all p',..,q'. g(p',..,q') iff phi(p',..,q')
>
> is an axiom.
>
> where: phi(p',..,q') is a modified formula iff
>  a function from all object and predicate symbols in phi(p',..,q') to
> naturals can be defined such that no variable predicate symbol is
> assigned the same value of an argument of it that is a variable
> predicate symbol.



No this fails, the function must assign to predicate variables one
step higher value than any assignment given to any of it's arguments
that are predicate variables.


> -------------------------------------------------------------------
>
> Now how this prevent's Russell's paradox.
>
> Russell's paradox on second order logic is
>
> Exist g. for all p. g(p) iff ~p(p)
>
> Clearly any function stipulated on ~p(p) will assign the same value to
> p (because it is a function) thus violating the above condition.
>
> This is a minimal kind of restriction on the Comprehension Principle
> for Concepts & Relations.
>
> I'm not sure if this is enough to prevent paradoxes in second order
> logic, but if so, then this is more than enough to interpret second
> order arithmetic, and even possibly any n_th order arithmetic.
>
> Should anyone find a clear paradox, then may she/he present it please.
>
> Zuhair





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