Hey Ron and Kevin!!! This has been a wonderful discussion thanks to
both of you guys. I would like to contribute and I would like to give
the following formalization that maybe would help us. But it is just a
maybe. Just to let you guys know Ax= for all x, Ex = for at least one
x. I don't have the backward E or the upside down A. Anyways, here we go.

1. Ax[Px <--> Ey(Q(y,x,P) & Dy)] That is

Px = x has the property of P (or a name P)
Q(y,x,P) = y qualifies x as p
Dy = y is a definition (or a account).

Lets assume that "P" is the name "animal" and "s" for Socrates and "m"
for the picture of a monkey and "r" for a rational animal and "t" for
the figure of an animal. So we have the following

2. Ps <--> Ey(Q(y,s,p) & Dy) UI from 1

So since it is Socrates, we would have to choose "r".

3. Ps <-->(Q(r,s,P) & Dr) EI from 2

Here is where I have to stop because I cannot choose "t", since the
rule of EI says that I cannot choose another object. Like we all know.
So let me start over for "m".

1.Ax[Px <--> Ey(Q(y,x,P) & Dy)]
2. Pm <--> Ey(Q(y,m,P) & Dy) UI from 1
3. Pm <--> (Q(t,m,P) & Dt) EI from 2


Of course, like everyone knows, this is not Aristotelian Scholar
talking here--just a janitor who find this stuff fascinating. It seems
to me that I would have to make two separate arguments for me to know
what s and m are. In other words, P is not as important as to what
type of thing s or m is since the corresponding definition has to be,
by their nature, different. But if two things have the same nature (or
definition) then I can keep using the same argument. That is the
following: Let us say that "b" is Plato and "a" is Aristotle.

1.Ax[Px <--> Ey(Q(y,x,P) & Dy)] Assumption
2.Ax[Px <-->(Q(r,x,P) & Dr)] EI from 1.
3. [Ps <-->(Q(r,s,P) & Dr)] UI from 2
4. [Pa <-->(Q(r,a,P) & Dr)] UI from 2
5. [Pb <-->(Q(r,b,P) & Dr)] UI from 2

And so on. That is whatever individual that fits in the definition. So
for me to know when two things are Homonymous, I must have a separate
argument and when they are synonymous I only need one argument. I
guess the main problem is that a syllogism would not help us because I
am always using the same middle term for two different conclusions. So
homonymous and synonymous are proper for Categories although there are
in some way remove from simple apprehension or that is:

1. All P are r
2. s is P
Therefore 3. s is r

1. All P are t
2. m is P
There fore 3. m is t.

Or I guess I could be more respectful to Aristotle and W-R by
formalizing in the following way.

1. AxAz{H(x,z) <--> En ED1 ED2[ C(nx,D1) & C(nz,D2) & ~(D1 = D2)]} where

H(x,z) = x is homonymous with z.
C(nx,D1) = x's name corresponds to definition 1 (D1)
C(nz,D2) = z's name corresponds to definition 2 (D2)
~(D1 = D2) = D1 is not identical with D2.

I will continue later. I have to clean some toilets. LOL.