Deconstructing the 'Cogito' - ChatGPT

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The Cogito as a Self-Verifying Performance: A Formalisation

The Cartesian cogito, ergo sum is often treated rather superficially: a mere sentence - "I think, therefore I am" - representing an irrefutable proof that something, at least, actually exists (me!).

Properly understood, it is a first-person (and possibly interior) speech-act that appears to secure its own truth. The aim here is to present a compact, stand-alone account that treats the cogito as a self-verifying performance, and to show by formal means why attempts to deny it even in its most minimal formulation still fail within a coherent logical framework.

1. Why Formalise?

Informal discussions of the Cogito tend to conflate three issues:

• what “I” refers to,
• what “thinking” amounts to,
• the difference between uttering words and undergoing a mental act.

A formal treatment lets us separate grammar from ontology, state a minimal bridge from occurrent properties to existence, and model the cogito as a dynamic update of the reasoner’s epistemic state.

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2. Language and Reading Guide

Agent: a single reasoner a.

Atomic predicates: E(a) = “a exists”; T(a) = “a is thinking now.”

Epistemic modality: K_a φ = “a knows φ.” Assume the standard S5 axiom system for the modal logic of knowledge.

Dynamic operator: [δ] φ = “after performing the private act δ of radical doubt, φ holds.”

Avowal functor: Avow_a(φ) = “a now sincerely avers φ about their own occurrent state.” Avowals target self-presenting predicates such as T(a).

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3. Axioms and Principles

Epistemic base (S5). Propositional tautologies; Modus Ponens; Necessitation for K_a; and axioms K, T, 4, 5 for knowledge. This is standard and neutral.

Existence Bridge. No property without a bearer (thinking implies existence):

(EB) T(a) → E(a)

This is analytic: occurrent properties presuppose something that has them.

First-person authority (immunity to error through misidentification). For self-presenting mental predicates (pain, thinking, doubting), sincere avowal yields knowledge (if you avow something you know it):

(IEM) Avow_a(ψ) → K_a ψ

for ψ in the self-presenting class (including T(a)).

Doubt as a private dynamic event. The Cartesian move is an act. Let δ be “I now doubt everything doubtable.” Its postconditions capture that doubting is occurrent and self-intimating when avowed:

(D1) [δ] T(a)  - 'I'm doubting everything; I'm thinking that thought

(D2) [δ] Avow_a(T(a))  - and now I also avow that thought.

Knowledge under update. A standard dynamic-epistemic inference schema:

(DEL) If ⊨ [δ](φ → ψ) and ⊨ [δ] φ, then ⊨ [δ] ψ

 - in each case, if you doubt things and the implication (φ → ψ) holds, and the antecedent φ, then so does the consequence ψ; doubting doesn't affect the underlying logic.

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4. The Cogito Theorem

Claim. After performing radical doubt, the agent knows they exist:

⊨ [δ] K_a E(a)

Proof sketch with commentary.

(1) From axiom (D2), [δ] Avow_a(T(a)). By (IEM), avowing an occurrent self-presenting state yields knowledge of it. With (DEL), infer [δ] K_a T(a). 

Narrative: in doubting, you thereby think; your sincere avowal confers knowledge of that thinking.

(2) From axiom (EB), T(a) → E(a).

By N (Necessitation rule), K_a(T(a) → E(a)) holds.

(3) Combine (1) and (2) by epistemic Modus Ponens inside K_ato obtain:

 [δ] K_a E(a).

Narrative: you know you are thinking; you know that thinking entails existence; therefore you know you exist. QED.

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5. Semantics in Plain Terms

To make the formal proof precise, we need a model of how knowledge works in terms of possible worlds.

5.1. The epistemic model.

We begin with a standard S5 epistemic model. This is just a structure with:

- a set of possible worlds W;

- an accessibility relation R_a which, for agent a, connects each world to the worlds they consider possible;

- a valuation function V that tells us which atomic facts are true in each world.

In this kind of model, the statement K_a φ (“agent a knows φ”) is true at a world if φ is true in all the worlds accessible from that world via R_a. In other words, if φ holds in every situation the agent takes to be possible, then the agent knows φ.

5.2. The doubt event.

Now add an event called δ, which represents the act of radical doubt. To describe events, we give them:

- a precondition: what must already be true for the event to occur;

- a postcondition: what the event guarantees to be true afterwards.

For δ, the precondition is trivial (it can always occur - you can always doubt). The postcondition is twofold:

- it ensures that after the event, T(a) (“a is thinking”) is true;

- it also ensures that the agent has made a sincere avowal of this fact (I assert “I am thinking”).

5.3. Updating the model.

When the event δ happens, the epistemic model is updated. This means we look at each world that was possible before and extend it to include the new facts that the event makes true. After this update, the resulting “successor” worlds all have T(a) true and also contain the avowal that T(a) is occurring.

This procedure is what specialists call a “product update,” but you don’t need the technical term. The important point is: the event reshapes the space of possibilities so that, in all futures consistent with the event, the agent is indeed both thinking (cogito) and has expressed that fact ('I think') if only to themselves.

5.4. From avowal to knowledge.

The principle (IEM - immunity to error through misidentification) states that for self-presenting mental states, a sincere avowal (“I am thinking now”) guarantees knowledge of that state. Since δ forces such an avowal, it follows that after the event, the agent knows that they are thinking. Symbolically, [δ] K_a T(a).

5.5. From thinking to existence.

The analytic bridge (EB) is always valid: if an agent is thinking, that agent exists. Because this holds in all worlds by definition, the agent also knows it. Combining this with knowledge of thinking yields knowledge of existence: [δ] K_a E(a).

5.6. Privacy of the event.

Finally, note that the event δ is private. Only the agent who performs the doubt gains this knowledge - the update in possible world structure is indexed to that agent alone. Observers outside do not acquire any knowledge just because the event occurred. This is why Descartes’ inference is strictly first-personal: it secures the doubter’s own existence to themselves, but not to anyone else.

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6. Why This is Genuinely Cartesian

The account assumes neither a robust personal ego nor any claims about an external world. It requires only: (i) occurrent thinking entails existence; (ii) sincere first-person avowals of occurrent states yield knowledge; (iii) radical doubt is an occurrent state that can be avowed. The cogito is thus a dynamic self-verification, not a syllogism.

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7. Pressure Points and Alternatives

7.1 Deny IEM - immunity to error through misidentification? So replace

(IEM) Avow_a(ψ) → K_a ψ   - (if you avow something you know it)

with a stronger transparency principle for occurrent thought:

(TR) T(a) → K_a T(a)        - (if you're thinking, you know it).

Then (D1) yields [δ] K_a T(a), and the remainder proceeds. The cost is a heavier assumption about introspective transparency.

7.2 Deny the existence bridge? Dropping (EB) abandons a bearer-property semantics. Talk of “thinking with no thing that thinks” no longer targets the Cartesian claim; it changes the subject into fantasy.

7.3 Model “empty appearances”? To make: 

“it seems that I am aware, yet nothing exists”

hold, you would need a successor where T(a) is true while E(a) is false, and that violates (EB). If you also drop (EB), you lose the subject needed for first-person knowledge entirely. Either way, the attempt does not defeat the theorem within any coherent semantics.

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8. Scope and Limits

This formalisation does not reassure those who want a thick, well-defined ego, diachronic personal identity, or knowledge of the external world. It isolates the minimal Cartesian kernel: perform radical doubt, gain knowledge of thinking, infer existence.

Importantly, third-person scepticism remains: nothing here licenses, for observers, “any system that utters ‘I think’ has to exist.”

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