Relational categorical propositions

I know Sommers and Englebretsen and others have treated these matters, but I'd like to think it over myself.

There are three reasons why term logic is regarded inadequate:

1. Inability to treat reflexive inferences, eg.
(a) Everybody killed himself
(b) Cato killed Cato
(c) Someone killed Cato

2. Inability to treat propositions with multiple quantification, eg.
(d) All horses are animals
(e) All horses' heads are animals' heads

3. Inability to treat transitions between term logic and propositional logic, eg.
(f) Every inhabitant of the island is either Asiatic or European
(g) Either every inhabitant of the island is Asiatic or s/he is European


Relational problems

First and second problem belong both to relational difficulties. Normal logical form for (a) is
Everybody is someone who killed himself
or, in more general form
Everybody is someone who is somehow related to himself
and in existential form
Everybody is himself
or in its most general form
Everything is itself
which might be called the axiom of identity.

This axiom can be weakened in several ways:
One way to weaken it is to go back by one step and to consider only one class of beings as in
Everybody is himself
or
Every blog is itself...
for, as everything is itself, everything in every class of things is also. This is the way of constricting the subject.

The other way is to loosen the subject as in
Something is itself
for, as everything is itself and there is something, something is itself. Constricting and loosening of the subject may be taken together:
Somebody is himself.

Third way is to loosen the predicate as in
Everything is something
for, as it is itself, it is something. This is strong form of traditional categorical proposition.

Fourth way is to loosen the copula, ie. to generalize the way how subject is related to predicate:
Everything is somehow related to itself
for, as it is itself, it is somehow related to itself.

All this may be taken together in strong form as in
Everything is somehow related to something
which might be called the axiom of relatedness, or in weak form as in
Something is somehow related to something
which may be considered a general form of relational categorical proposition in contrast to simple categorical proposition in form
Something is something.

From these considerations we can resolve the first difficulty:

Everybody killed himself, (assertion)
therefore Cato killed himself. (weakening)
Everybody is himself, (axiom)
therefore Cato is himself. (weakening)
Cato killed himself,
and Cato is himself,
therefore Cato killed Cato.
Cato killed Cato,
and Cato is somebody,
therefore somebody killed Cato
(and Cato killed somebody).