Existence as a Predicate

Question

In Predicate logic if I wanna say, Atoms exist, I don't/*can't (?) use Ex = x exists (make existence a predicate) and state Ea, where a = Atoms. The correct way to express Atoms exist is Ex(Ax), Ax = x is an atom.

This leads to problems e.g. how do we predicate logicize Descartes' cogito, ergo sum?
If I think then I exist
Existence is a predicate here.
Ax(Tx --> Ex). Not permitted.
Is it then, Ax(Tx --> Ex(Tx))? For a particular constant, e.g. x = Smith = s, instantiating gets us nowhere e.g. Ts --> Ts.

Another issue is with the statement nothing exists. We would be required to say Ex(Ox) = There exists an x such that x is nothing. We also can't say Ax(~Ex) = For all x, x doesn't exist.

We can, of course, use a hybrid system (sentential logic + predicate logic)

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Attempt #1 for a solution
Define a hypothetical/postulate a predicate/property B such that B is both sufficient and necessary for existence.

So, If I think then I exist = Ax(Tx --> Bx). This appears to be a good solution, but the problem is Existence = B, they're logically equivalent, i.e. Existence <--> B and so we're back to square one.

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Attempt #2 for a solution
Postulate 2 properties/predicates

1. S a property sufficient but not necessary for existence. So If I think then I exist = Ax(Tx --> Sx)
2. N a property necessary but not sufficient for existence. Then, nothing exists = Ax(~Nx)

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Is this solution ok? Was there even a problem in the first place? Can we improve the situation?

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EDIT 0 INIZIO

1. Tx [x thinks]
2. Ex(Tx) [There exists an x that thinks, Existential Generalization (EG)]
3. Td [Descartes' thinks]
4. Ex(Tx) [EG]

We may not say of Winnie the Pooh (w) that Tw, it's meaningless or is false for the reason that Winnie has to exist to think; Descartes was right, esse (being, sum) is a necessary consequence of cogito i.e. cogito, ergo sum is a good premise.

EDIT 0 FINITO

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EDIT 1 INIZIO

IF r THEN s is a conditional in sentential logic.
Conditionals in predicate logic become universal statements, for all x blah, blah, blah = Ax(Px --> Qx), where P and Q are predicates.

IF I think THEN I exist = Ax(Tx --> ?). It's worth noting here that universal statements lack existential import and so an existence predicate here, to replace ?, is square peg in a round hole.

To make claims about the esse (being/existence) of some x, we must first find a predicate that applies to x e.g.

1. Ax(Bx --> Fx) [premise]
2. Bw --> Fb [1 UI]
3. Bw [premise]
4. Fw [2, 3 MP]
5. Ex(Fx) [4 EG]

Ax(Bx --> Fx) = IF x can birth children THEN xis a female. w = Winnie Mandela.
Line 5, Ex(Fx) is an existential claim, females exist.

We could take a page out of this proof of existence and apply it to the cogito.

1. Ax(Sx --> Tx) [Premise]
2. Sd --> Td [1 UI]
3. Sd [Premise]
4. Td [2, 3 MP]
5. Ex(Tx & x = d) [4 EG]

Ax(Sx --> Tx) = IF x is skeptical (dubito) THEN x is thinking (cogito), d = Descartes.
Line 4, Ex(Tx & x = d) is there exists an x that thinks and that x = Descartes.

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To capture the essence of the cogito (a reductio ad absurdum), to do justice to Descartes ...

1. Ax(Sx --> Tx) [premise]
2. Sd --> Td [1 UI]
3. Sd [premise]
4. ~Ex(x = d) [assume for reductio ad absurdum]
5. Td [2, 3 MP]
6. Ex(Tx & x = d) [5 EG]
7. Ex(x = d) [From 6, I'm cluless as to what rule applies]
8. Ex(x = d) & ~Ex(x = d) [4, 7 Conj]
9. Ex(x = d) [4 - 8 reductio ad absurdum]

EDIT 1 FINITO

Answer 1

Free logics allow you to introduce a name for something that is not in the domain. There is a predicate to say that something is in the domain.

Various logics inspired by Meinongian metaphysics also treat existence as a normal predicate. I've seen two broad classes of these logics formalized. Edward Zalta has developed a logic in which there are two forms of predication. Other authors have developed Modal Meinongian logics which push the question of existence onto something similar to possible worlds.