# Existence as a Predicate

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)

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.

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)

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

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

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.

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

• Perhaps this goes to show that existence should not be inferred as such. Alternatively, reasoning in which existence is inferred is circular (for better or worse). As far as Descartes goes, we might want to look more for a derivation, "I exist as a thinking substance," which is more contentious (but for sort-of-Kantian reasons is perhaps doable, though not to the effect that thought is the matter of said substance). Commented Dec 5, 2023 at 18:47
• See Free Logic for an approach. Commented Dec 5, 2023 at 19:52
• There have been many attempts to formulate the cogito formally. It is not as straightforward as it seems. Your current suggestion is not good. The cogito can really only be expressed in the first person. It is supposed to be an indubitable proof that I exist since I cannot ccherently doubt that I doubt. Commented Dec 6, 2023 at 8:23
• Why are you trying to prove that Descartes exists? That is not the point of the cogito. The point is to establish indubitably that I exist. According to Descartes, I cannot coherently doubt my own existence since to do so would be to doubt that I am doubting. One may object as many have done that it is unreasonable to assume that I is a coherent concept. But I can easily doubt that Descartes exists, or anybody else for that matter. Descartes' purpose is to establish an indubitable starting point for his epistemology. Commented Dec 6, 2023 at 10:28
• See the post Can Cogito, ergo sum be formalized?: the most reasonable reading of D's argument is that it is not an inference but an intuition. 'When someone says “I am thinking, therefore I am, or I exist,” he does not deduce existence from thought by means of a syllogism, but recognizes it as something self-evident by a simple intuition of the mind. (Replies 2, AT 7:140).' " Commented Dec 6, 2023 at 11:14