```Date: Jun 14, 2013 6:06 AM
Author: Zaljohar@gmail.com
Subject: The Comprehension Principle for Concepts & Relations: Minimal Restriction.

I'm thinking of a kind of second order logic which has minimalrestriction 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 canbe substituted by ONE value only.Lower case letters represent variable symbols.Objects shall be represented by Starred symbols, so X* is a constantobject symbol, while x* is a variable object symbol (i.e. x* rangeover the whole of domain of discourse of OBJECTS)Predicates shall be left non starred, so P is a constant predicatesymbol denoting a particular predicate, while p is a variablepredicate symbol (i.e. a symbol ranging over ALL predicates)If a symbol is primed then it might stand for an object or a predicateNow 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') 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') tonaturals can be defined such that no variable predicate symbol isassigned the same value of an argument of it that is a variablepredicate symbol.-------------------------------------------------------------------Now how this prevent's Russell's paradox.Russell's paradox on second order logic isExist g. for all p. g(p) iff ~p(p)Clearly any function stipulated on ~p(p) will assign the same value top (because it is a function) thus violating the above condition.This is a minimal kind of restriction on the Comprehension Principlefor Concepts & Relations.I'm not sure if this is enough to prevent paradoxes in second orderlogic, but if so, then this is more than enough to interpret secondorder 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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