Date: Jun 14, 2013 6:06 AM
Subject: The Comprehension Principle for Concepts & Relations: Minimal Restriction.

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') 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.

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.