Predictionism: From Cogito to Logic and Computation

Abstract

Predictionism grounds binary verification and Boolean operations in the epistemic certainty of Descartes' (1996) cogito ergo sum. It is an interpretive proposal whose theorem-level content is the finite-protocol results below. Traditional approaches to classical logic often assume the principle of bivalence axiomatically. Predictionism asks how bivalence can arise from self-verifying thought. By formalizing the Cogito as a cyclical prediction and verification process, the framework explains why verification has a true/false structure under a specified test, context, and threshold. This supports the fundamental Boolean operations (NOT, AND, OR). Together with sequencing, iteration, and memory, these operations can support universal (Turing-complete) computation. By connecting the self-evident certainty of the Cogito to the core mechanisms of logic and computation, Predictionism offers a framework for grounding these systems in predictive verification.

Predictionism in Plain Terms

A referee's flag is either up or down. It can go up hesitantly, after a long look, or in the wrong call, and none of that changes the shape of the outcome: the play stands or it does not. Whatever deliberation happens beforehand, the record that is kept has two values.

Classical logic assumes that statements are true or false and builds everything on that. The question here is where the assumption comes from, and the answer is the moment of checking. A prediction meets an outcome, and what gets written down is that it held or it did not. Two values, because the record is the result of a comparison and a comparison either passes or fails.

From that one move the rest follows in a way you can watch. Expecting the opposite of what you expected is negation. Needing both of two things to hold is conjunction. Being ready for either of two things is disjunction. These three build every truth table there is, and with a way to run steps in order and remember results between them, they build any computation.

The starting case is the one that cannot go the other way. Try to check whether you are thinking, and the checking is thinking, so the flag can only go one way. Descartes found that; what this account adds is that the finding has the form of a verification, which is why it can carry the weight of a logical foundation and not just a psychological observation.

One boundary belongs here. The claim covers what gets retained at the moment of checking, on protocols where a check is defined. It does not say that every judgment a mind makes is two-valued, or that vagueness and degrees of belief are illusions.

1. Introduction

Classical logic underpins modern mathematics, philosophy, and computer science, yet its fundamental principle of bivalence is often accepted without deeper justification. Predictionism grounds bivalence in the binary outcome of verification: a prediction is confirmed or disconfirmed relative to a specified test, context, and threshold.

When Descartes attempted to doubt everything, he arrived at a singular certainty: the act of doubting itself confirmed the presence of thinking, revealing a binary verification process, which Hintikka (1962) analysed as performative rather than inferential - the proposition "I exist" must be true (as the very act of doubting demonstrates existence) and cannot possibly be false (as attempting to verify "I do not exist" is self-contradictory) - thus establishing that consciousness, at the most fundamental level, operates through a true/false verification structure that provides the foundation of bivalence in logic.

Formalizing the insight as a cycle of doubt → prediction → verification → renewed doubt, we illustrate how bivalent verification naturally emerges. We then extend this to show how Boolean operations (NOT, AND, OR) arise from such a binary verification framework, and how universal computation follows when those operations are combined with sequencing, iteration, and memory.

The account has three parts:

• Bivalence: Grounded in the retained outcome of verification, showing how the principle of bivalence is grounded in the self-verifying loop of consciousness, rather than simply taken as an axiom.
• The predictive cycle: Specified by mathematical definitions for the state space, prediction function, verification function, and the cyclical doubt function. These clarify how self-reflection enforces a binary outcome.
• Boolean operations and computation: The core Boolean operators arise from different facets of the predictive cycle, and that universal computation follows from their functional completeness.

Universe 00110000

2. The Cyclical Predictive Structure of Consciousness

2.1 The Fundamental Predictive Loop and the Cogito

Consciousness is modelled here as an iterative cycle:

1. Doubt: The mind questions any presumed certainty.
2. Prediction: The mind projects a state or outcome.
3. Verification: The mind confirms or disconfirms this prediction through direct experience or introspection.
4. Renewed Doubt: The cycle restarts, questioning the result of verification.

In Descartes’ cogito, ergo sum, the verification step locks in a unique way: the act of denying or doubting one’s own thinking only reaffirms that one is, indeed, thinking. Thus, the verification cannot be neutral or partial; it is inevitably confirmed as true. This special instance of the predictive cycle embodies the fundamental certainty: “I think, therefore I am.”

2.2 Formal Definitions and Notation

The cyclical model uses four objects:

• State Space (ℛ): The set of all possible internal states of the system (the mind).
• Predicted State Space (ℛ′): A subset of ℛ capturing states that the system explicitly anticipates (i.e., outcomes under consideration). Formally, ℛ′ ⊆ ℛ.
• Predictive Function (P): A total function P: ℛ → ℛ′. For each state r, P(r) represents the prediction about the next or anticipated state.
• Verification Function (V): A mapping V: ℛ → {0,1}. For each state r, V(r)=1 if the prediction made in or about r is verified, and V(r)=0 otherwise.
• Doubt Function (D): A mapping D: ℛ → ℛ that moves the system from a (verified) state back to a state of uncertainty, thus restarting the cycle.

The notation δ(r) is used in place of V(r). This is simply a notational convenience to emphasize the binary output, δ(r) ∈ {0,1}.

2.3 Binary Verification in the Cogito

Theorem 1 (Binary Verification in the Cogito):
In the Cogito, the act of self-awareness necessarily produces a binary verification outcome, offering a non-arbitrary grounding for classical bivalence.

1. Definition of the Cogito State: Let r₀ ∈ ℛ represent the state of direct self-awareness. In this state, the system makes the prediction P(r₀) = "I am thinking."
2. Possible Verification Values: V(r₀) ∈ {0,1}.
   • Case 1: V(r₀)=1. This outcome verifies the proposition "I am thinking," which directly implies that "I exist" is true.
   • Case 2: V(r₀)=0. Denying the act of thinking is self-refuting because any attempt to verify “not thinking” presupposes the very act of thought that is being denied. Thus, this case is impossible.
3. Conclusion: The only consistent outcome is V(r₀)=1. This binary check confirms that the proposition "I exist" is true and "I do not exist" is false, thereby aligning with classical bivalence.

Universe 00110000

3. Derivation of Boolean Logic and Turing-Completeness

3.1 The Emergence of Boolean Operations from the Predictive Loop

The canonical Boolean operations arise from the verification scheme δ(r) ∈ {0,1}. By extension, all of classical Boolean algebra can be generated.

Theorem 2 (Boolean Operations from Binary Verification):

Given a system where any state r has a binary verification outcome δ(r), the essential logical operations (NOT, AND, OR) follow naturally from distinct aspects of the cyclical predictive process.

Proof (Sketch):

1. Negation (NOT) from Distinction:

   For any predicate S defined on states in ℛ, we must be able to distinguish S from ¬S. If δ(S(r)) = 1, then δ(¬S(r)) = 0, and vice versa. Formally:

   δ(¬S(r)) = 1 - δ(S(r))

   This basic ability to differentiate between a statement and its negation underlies the idea of logical NOT.

2. Conjunction (AND) from Sequential Conditions:

   Consider two state predicates S₁ and S₂. Moving from a state satisfying S₁ to a subsequent state that also satisfies S₂ requires both conditions to hold. If we introduce a transition predicate T(S₁, S₂) that indicates the system can move from S₁ to S₂, then:

   δ(S₁(r)) ∧ δ(T(S₁(r), S₂(r))) = min(δ(S₁(r)), δ(T(S₁(r), S₂(r))))

   This expresses the logical AND operation: both S₁ and T must evaluate to 1 for the system to move on.

3. Disjunction (OR) from Predictive Branching:

   When predicting multiple possible outcomes (e.g. V₁, V₂, …, Vₙ), at least one might occur. The system effectively encodes an OR among these outcomes:

   δ(V₁(r)) ∨ δ(V₂(r)) ∨ … ∨ δ(Vₙ(r)) = max(δ(V₁(r)), δ(V₂(r)), …, δ(Vₙ(r)))

   Thus, logical OR arises from branching anticipations of possible verifications.

These three operations are functionally complete, so they generate every Boolean function, a fact established for two-valued logic by Sheffer (1913) and classified in full by Post (1941). Boole (1854) supplied the algebra, and Shannon (1938) showed that circuits of switches realize it.

3.2 From Boolean Logic to Universal Computation

Theorem 3 (Universal Computation from Binary Verification):
The progression from Boolean algebra to universal computation is well-established in the foundations of computer science. With the Boolean operations established on the retained verification values, the path to universal computation is short:

Proof (Sketch):

1. Boolean Completeness: As shown in Theorem 2, NOT, AND, OR are sufficient to express any Boolean function. It is a standard result in logic and computer science that functional completeness is a stepping stone to universal machines.
2. From Logic Gates to State Machines: Composing these basic Boolean operations allows the construction of digital circuits (or abstract state machines) capable of storing bits, running algorithms, and branching on conditions.
3. Turing completeness: Functional completeness alone does not suffice, and four further conditions scope the result: closure under composition of protocols, memory that retains verification outcomes across steps, uniform specification of the protocols, and finite resources that scale with the computation. With those, the system simulates a universal Turing machine in the sense Turing (1936) defined and Kleene (1952) developed.

In short, the self-verifying structure of consciousness, with its binary checks, naturally supports full classical logic. Once we have classical logic, the path to general computation is well established in theoretical computer science.

4. Conclusion

Predictionism proposes that the basis of classical logic - and therefore of universal computation - lies in the binary verification structure of consciousness. The Cogito, as the paradigmatic example of self-verification, provides a foundational instance where bivalence is enforced and cannot be rationally denied. From this focal point:

• Bivalence is groundedbivalence in the binary dynamics of doubt and verification, showing that the Cogito’s self-awareness is an unassailable example of how a verification step cannot be other than 0 or 1.
• The Boolean operations (NOT, AND, OR) emerge from structural features of a predictive cycle. This formal correspondence highlights the non-arbitrary nature of classical logic.
• These logical operations support Turing-completeness when combined with sequencing, iteration, and memory. The binary checks at the root of verification can therefore serve as the basis for general computation.

This approach suggests that classical logic and computation reflect an inherent feature of self-conscious thought rather than merely convenient human constructs.

References

• Boole, George. An Investigation of the Laws of Thought. Walton and Maberly, 1854.
• Descartes, René. Meditations on First Philosophy. Translated by John Cottingham, revised ed., Cambridge University Press, 1996.
• Hintikka, Jaakko. "Cogito, Ergo Sum: Inference or Performance?" The Philosophical Review 71, no. 1, 1962, 3–32.
• Kleene, Stephen Cole. Introduction to Metamathematics. North-Holland, 1952.
• Post, Emil L. The Two-Valued Iterative Systems of Mathematical Logic. Princeton University Press, 1941.
• Shannon, Claude E. "A Symbolic Analysis of Relay and Switching Circuits." Transactions of the American Institute of Electrical Engineers 57, no. 12, 1938, 713–723.
• Sheffer, H. M. "A Set of Five Independent Postulates for Boolean Algebras, with Application to Logical Constants." Transactions of the American Mathematical Society 14, no. 4, 1913, 481–488.
• Turing, A. M. "On Computable Numbers, with an Application to the Entscheidungsproblem." Proceedings of the London Mathematical Society s2-42, no. 1, 1936, 230–265.