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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

Agent Smith
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  • 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. – Conifold Oct 28 '23 at 09:28
  • Gracias @Conifold. What about categorical logic? Do you see any issues? – Agent Smith Oct 28 '23 at 09:46
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    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. – Mauro ALLEGRANZA Oct 28 '23 at 12:02
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    Does this answer your question? Can Cogito, ergo sum be formalized? – Kristian Berry Oct 28 '23 at 15:10
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    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). – Kristian Berry Oct 28 '23 at 15:12
  • Moreover, we should keep in mind Descartes' radically unclear notion of modality, wherein a truth can be eternal and necessary even if contingent on the hypermodality of the divine will. (God holds fast the necessity of the eternal truths, making them undeniable for us; but for God, well...) Howso this affects the Cartesian notion of actuality, as a modality, is something to be considered squarely in interpreting, "I doubt things, I actually doubt them; I am actual," as a paraphrase of, "I exist as doubting, thinking being." – Kristian Berry Oct 28 '23 at 15:14
  • RETRACTION: Apparently the cogito as such shows up in the Discourse, which predates the Meditations by about ten years. – Kristian Berry Oct 28 '23 at 22:05
  • @KristianBerry, spasibo for the link. Quite interesting. – Agent Smith Oct 29 '23 at 04:51
  • @KristianBerry, how would you formalize the critical statement If x is the greatest being then x exists in Anselm's ontological argument? – Agent Smith Oct 29 '23 at 05:12
  • The comment by @MauroALLEGRANZA is a good one. To capture Descartes' argument we might need something like: there are thoughts (observation); if there are thoughts there exists a thing that is having these thoughts (additional assumption); therefore there exists a thing that is having these thoughts (MP); this thing I choose to call I (existential instantiation); so, I exist. Of course, the argument may easily be challenged, and Kant does claim that it is question-begging. – Bumble Oct 29 '23 at 05:56
  • @AgentSmith to get comparative quantification, we'd need generalized quantifiers. I suspect a fruitful reformulation of Anselm's argument could involve relations between those (rather than e.g. existence "in" the understanding being a hodgepodge of mere existential quantification and a predication of that interiority, we would look for a relation between two quantifier types). – Kristian Berry Oct 29 '23 at 12:11
  • @KristianBerry, looked up Gödel's Ontological Proof. Nothing there that may clarify the matter for me. There is this sentence though, the conclusion: Ex(Gx) = There exists x such that x is God. No conditional to go with that; perhaps implicit, but the problem remains. How do we express a conditional If A then something exists, in predicate logic? – Agent Smith Oct 29 '23 at 13:58
  • I will go post some more comments about this in the main chat @AgentSmith – Kristian Berry Oct 29 '23 at 17:46
  • @KristianBerry, thank you. If the problem is resolved, can you please post an answer to this thread. Arigato gozaimus. – Agent Smith Oct 30 '23 at 02:22

1 Answers1

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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!

Speakpigeon
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