Cogito Ergo Sum in Predicate Logic

Descartes' famously declared "cogito ergo sum (I think, thus I exist).

How do you translate this into predicate logic?

If T = I think and E = I exist, propositional logic has no problems (vide infra):

1. T -> E
2. T
Ergo,
3. E

But, the catch (re Kant), existence is not a predicate. As aligned with Kant's pronouncement (well-known from his criticism of St. Anselm's Ontological Proof of God), predicate logic doesn't accommodate E (I exist). I could be mistaken and hope to be shown where.

EDIT 1 START
Borrowing from Kurt Gödel's ontological proof (for God), where the apropos expression Ex(Gx) = there exists an x such that x is God, I propose the following formalization:

Ax(Tx --> Ex(Rx)) = For all x, if x thinks then x exists such that x is a thinker.

EDIT 1 END

• Predicate logic does accommodate "I exist" even without the existence predicate, for example ∃x(x=I). So cogito would be Th(I) → ∃x(x=I), where Th(x) stands for "x thinks". Moreover, some modern logicians do introduce the existence predicate E!, Kant notwithstanding, see SEP. Oct 28, 2023 at 9:28
• Gracias @Conifold. What about categorical logic? Do you see any issues? Oct 28, 2023 at 9:46
• If you use standard predicate logic, nanes are always referring. Thus, the simple fact that we use "I" begs the question wrt original Descartes intuition. Oct 28, 2023 at 12:02
• Does this answer your question? Can Cogito, ergo sum be formalized? Oct 28, 2023 at 15:10
• Originally, the cogito was not a "therefore" set of statements, it was more like, "I am, I exist thinking," just a stream-of-consciousness pertaining to the hyperbolic doubts Descartes was entertaining in the Meditations. The Cartesian circle is not the loop of the "ergo sum" there but the appeal to God to certify even the cogito (which God is certified by what It certifies in turn). Oct 28, 2023 at 15:12

Descartes' famously declared "cogito ergo sum (I think, thus I exist). How do you translate this into predicate logic?

The Cogito is an enthymematic expression for:

If I think, then I exist;

I think;

I exist;

If x stands for "I", Tx for "I think" and Ex for "I exist", the logic of the Cogito is just:

(Tx → Ex) ∧ Tx ⊢ Ex

There is no difficulty in using "exist" as predicate. We all understand what it means, and that something exists is presumably either true or false.

But, the catch (re Kant), existence is not a predicate.

Yet, it is usually accepted as true that Kant does not exist:

Kx → ¬Ex

We can even infer by Modus Tollens that he doesn't think!

• ???? Kant does not exist? Yes, obviously not now. Kant is dead and therefore he does not think. So what? Oct 28, 2023 at 12:00
• 🙂 Jokes are always fun, oui? Oct 29, 2023 at 5:02
• I guess the follow-up question is why did Gottlob Frege (inventor of predicate logic) take a Kantian approach to logic, one in which Ex = x exists (existence as a predicate) is awkward to express? Oct 29, 2023 at 14:06
• @AgentSmith Because there is no good reason to believe that every logician should be able to understand logic. Oct 30, 2023 at 9:08