Date: May 22, 2013 5:54 PM
Author: fom
Subject: Re: Extensional Logic

On 5/22/2013 2:16 PM, Zuhair wrote:
> I think it is plausible to study extensions of n_th order predicates:
>
> A zero order predicate is an "object"
> A first order predicate is a predicate that only hold of objects.
> A second order predicate is a predicate that hold of zero or first
> order predicates.
> In general and n_th order predicate is a predicate that holds of 0 or
> 1st or ... or (n-1)_th order predicates.
>
> Now lets add a monadic symbol e to second order logic, and stipulate
> the following rule:
>
> if P is an n_th order predicate then eP is a term.
>
> eP is read as: "extension of P"
>
> And axiomatize the following:
>
> If P is an n_th order predicate and Q is an m_th order predicate then:
>
> eP=eQ iff (for all x. P(x) <-> Q(x))
>
> Of course one can define a general membership relation in the
> following manner:
>
> x E y iff Exist G. G(x) & eG=y
>
> However it is clear that E is not an n_th predicate, so eE is not a
> term! thus avoiding Russell's paradox!


Yes. Something similar occurs when Rosser examined
New Foundations and the axiom of choice. A certain
relation involving individuals could not be represented
in the theory as a class.

In my own deliberations, I formulated the definitions,

AxAy(xcy <-> (Az(ycz -> xcz) /\ Ez(xcz /\ -ycz)))

AxAy(xey <-> (Az(ycz -> xez) /\ Ez(xez /\ -ycz)))

With the thought of

T :=> 'c'

F :=> 'e'

since the "relations-as-objects" could not
sensibly be members of the universe (I admit
a denotation of V in the theory). Thus, the
"relations-as-objects" became proxies for the
truth values.

In any case, when "general variables" or, as in
your case, an untyped membership relation is
considered there is necessarily some class that
is not represented as an object in the theory.

>
> This way can enable us to define every natural number
>
> 0 = eP
> 1 = eP*
> 2 = eP**
>
> were P, P*, P** are defined as:
>
> For all x. P(x) iff ~x=x
>
> For all x. P*(x) iff x=eP
>
> For all x. P**(x) iff x=eP*
>
> Successor relation of x can be defined as {x} after relation E.
>
> Should we have infinitely long formulas then definitely we can define
> Number and thus prove PA in logic.
>
> Anyhow I'm not really sure of this method yet. But it does show that
> some mathematics does follow from pure logical theories like the
> extensional logic defined above.
>
> Zuhair
>
>