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Re: The Comprehension Principle for Concepts & Relations: Minimal Restriction.
Posted:
Jun 14, 2013 8:25 AM


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



