To be anachronistic: imagine formulating the cogito in the flavor of modern first-order logic (not "real" FOL altgoether, though; as I'm looking over what I've written here, my mind is screaming at me that the translations I've given have the same flavor but aren't really the same dish, so to speak). Let i be the (to-be-bound) variable and T the predicate of thought. Then, in FOL, we would not merely write Ti, but ∃i(Ti). By A&B → A (not in a strictly valid manner, though) we can then generate the following line of reasoning:
- ∃i(Ti) → ∃i
- ∴ ∃i
A somewhat different way of saying it:
- ∃x((x = i) & Ti)
- ∃i → ∃x
- ∴ ∃x
(2) in the first reformulation of the cogito is not strictly valid, or perhaps is not strictly well-formed (regardless of validity) in that we're not FOL-wise meant to write down quantified expressions "just like that." I'm not actually sure how the equals sign works in FOL, so I'm not sure about if (1) in the second reformulation is licit. Anyway, the whole bit can be convoluted some by consideration of the theory of indexicals as a whole, since i is meant as an indexical variable.
Again, there seems to be something off about these reformulations. The SEP article on Descartes' epistemology mentions competing interpretations of the cogito as an axiom vs. as a theorem. The presence of a "therefore" in some versions of the cogito makes it look theorematic, but as the above shows, it's hard to make it look theorematic in anything more than an, "A → A," kind of way.
Alternatively, I'm actually sympathetic to the idea that the cogito falls apart on a deeper level, in the sense that I'm not sure that the concept of existence is ultimately coherent, in which case "inferring" the existence of something will be an invalid inference by the by. So one would be able to say, "I think, but I don't exist; I don't anti-exist, either; indeed, nothing exists or anti-exists." Then the cogito would become, "I think; therefore I think."
Addenda. Here are some other "deviant" options, with their problems mentioned in passing:
Work in a modal logic with an actuality operator on propositions, where, "⚬A," reads, "It is actual/actually true that A." See Bumble's answer to this PhilosophySE question for what seems to be an indirect counterexample to, "A → ⚬A," but otherwise assume that A → ⚬A. Then say:
- Ti → ⚬Ti
- Alternatively, ∃i(Ti) → ⚬∃i(Ti)
Still, not really anything better than A → A, it seems. —Background problem (in Descartes hermeneutics, anyway): Cartesian modal logic seems to encode for contingently necessary truths, so one would like to be careful about how one might read an actuality operator into that logic (contingently necessary truths fly in the face of standard modal logic, wherein ◊□A goes to □A; so since Cartesian modal logic is nonstandard by the by, one is reluctant to haphazardly throw an actuality operator into that mix).
Even more deviously:
Work in a logic with an existence predicate E!, like a Zalta (or pseudo-Zalta) logic for abstract objects. Then write (for any generic variable x and any generic predicate F):
- ∃x(Fx) → ∃x(E!x)
- ∴ ∃i(E!i)
Background problem: again, as far as interpreting Descartes goes, one wishes to be careful about introducing existential quantifiers and predicates together, especially in the form of a conditional like (1) that seems as if it would have to be as fundamentally knowable (if knowable at all) as the cogito is itself supposed to be.
One more option, here:
Work in a weak epistemic (propositional) logic ("weak" as in "not thought out in much detail"), where, "kA," reads, "It is known that A." For present purposes, also use E! instead of ∃. Say that:
- k(Fx) → k(E!x)
- ∴ k(E!i)
This seems theorematic enough to license a "therefore" in its wording, though again, now, isn't (2) a principle that has to be as fundamental as the cogito is supposed to be?
A parallel interpretation problem. Another way to reformulate it could be, "I perform the act of thought; therefore, I am a substantial being." Not just, "I perform an action; therefore, I am an active being," but a substantial one, and Descartes inherited the waning of scholasticism, especially considering his ethnicity/nationality, so even so did he mean by "substance" what the scholastics attributed to Aristotle, viz. the doctrine that the things worthy to be called by the name of "substance" were those things which "are always a subject and never a predicate only" in the objective ordering of form and matter.
So this is a stronger claim than, "I think, and my thinking implies my existence." He is specifically claiming that he knows, from the fact that his thoughts are his own actions, the deeper fact that he is such as to be a "subject that can never only be predicate." In this, he is significantly prefiguring Immanuel Kant's use of what Kant calls "the" I-think, a fundamental propositional force in empirical cognition, a proposition-forming operator that is constantly taking operands and yielding much of our cognition thereby. Recall that Descartes at one point (not necessarily in the Meditations, or I mean I don't recall where exactly) goes over the difference between adventitious ideas and innate ideas. This inner capacity to differentiate ideas both formally and materially establishes that these ideas are predicates of us by the by, are predicates of the I-think even, whereas the I-think can become its own predicate, as I-think-that-I-think, but only then in such a way that it remains a subject of itself. So it satisfies the postscholastic sense Descartes had about the concept of subsistence (not necessarily the wording he used, granted, but more like his culturally conditioned propensity to use the kinds of concepts that are affiliated with the word often enough, and especially in philosophy).
Accordingly, "I think," would go to, "I perform the act of thought, which means that I am a substance of which thoughts are predicates."
One last reformulation, in erotetic logic. Usually, erotetic logic is understood as the art of inferring questions from assertions, or from other questions. Inferring assertions from questions is not often explicitly spoken of, though G. E. Moore's open-question argument might be styled such an act of reasoning. Alternatively, tracking the alleged presuppositions of a question could be thought of as transcendentally arguing from questions to assertions. So now consider:
- Do I exist?
- ∴ I do exist.
This will perhaps not be a universally valid argument scheme, i.e., "Does x exist?" will not always go to, "x does exist." At least, we might rephrase the inference so:
- Is there an x such that I am x and x is asking this very question?
- ∴ There is an x such that I am x and x is asking this very question.
Then the inference to me as existing is given through the indexical for "this very question," i.e. that impersonal indexical nevertheless is convertible into the personal one, "and we are done." QED