A Formalizable Proof of the No-Supervenience Theorem: A Diagonal Limitation on the Viability of Physicalist Theories of Consciousness

The gist: The proof demonstrates that any physicalist theory describing a self-aware, intelligent system must be internally inconsistent by showing that it implies an infinite number of distinguishable brain states, which no physical system can accommodate.

Introduction and Motivation

The no-supervenience theorem establishes constraints on theories of consciousness by examining the general properties of sets of theories without taking into account the metaphysical qualities associated with individual members of those sets. The proof is difficult to formalize because it relies on both alethic and epistemic notions of possibility, necessitating an adaptation of an existing mathematical formalism, the inference device (Wolpert, 2008). This approach allows epistemic modal inferences to be codified formally in predicate modal logic without requiring an explicitly epistemic interpretation of modal logic.

Intuitive Summary of the Proof

The intuitive argument suggests that if every mental process corresponds to a brain process, then the thought I think therefore I am corresponds to a statement about the NCC corresponding to that thought working correctly. If physicalism is true, this leads to an NCC which must correspond to another state, NCC∗, and so on. This chain necessitates an infinite number of distinguishable brain states, which implies that any mind capable of the Cartesian Cogito cannot be a physical system.

Formal Framework and Postulates

The proof is expressed using a restricted axiomatization of first-order modal predicate logic. A physical system is defined as some set of states, together with a set of dynamical rules linking those states. An observer is represented as a Wolpert inference device that evolves to a special state W(p) only if p is true. A Wolpert state is accurate and reliable for some proposition p using modal logic if  (W(p)  p). The realism postulate defines real processes as those where WA(x) and WB(x) both have the value TRUE, and A  B.

The Principle of Empiricism and Contradiction

The principle of empiricism states that no single observer can infer a priori, from the restricted axioms of predicate modal logic, whether any given real process is a process of type P or of type R. This is expressed in modal logic as  (W(p)  p). The system M, assuming it can reason using predicate modal logic, must satisfy WM( (WM(p)  p) = TRUE. This leads to the conclusion that no real physical system which operates as a Wolpert inference device can accurately infer p.

Conclusion and Limitations

The diagonal contradiction arises because the principle of empiricism induces a non-terminating sequence of physical processes due to the requirement that real processes be objective. If M is capable of reasoning in the restricted axiomatization, it can prove that no physicalist theory can describe any system which is both omega-capable and capable of reasoning using predicate modal logic without being inconsistent. The theorem applies to theoretical models, showing that no physicalist model of a system can be omega-capable. This result rules out viable physicalist theories of consciousness. The proof relies on a diagonal method, which is not common in analytic philosophy of mind.

References

Caplain, G., Is consciousness a computational property?, Informatica, 19, pp 615-619.

Caplain, G., Is consciousness not a computational property? - Reply to Bojadziev., Informatica, 24, pp 79-81.

Kripke, S., Naming and Necessity., Oxford: Blackwell.

Reason, C. M., A No-Go Theorem for the Mind-Body Problem: An Informal Proof that No Purely Physical System Can Exhibit all the Properties of Human Consciousness., Journal of Mind and Behavior, 40 (2), pp 95-120.

Reason, C. M., Consciousness is Not a Physically Provable Property., Journal of Mind and Behavior, 37 (1), pp 31-46.

Reason, C. M. and Shah, K., Conscious Macrostates Do Not Supervene on Physical Microstates., Journal of Consciousness Studies, 28 (5-6), pp 102-120.

Wolpert, D., H., Physical Limits of Inference., Physica D, Nonlinear Phenomena, 237 (9), pp 1257-1281.

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The no-supervenience theorem limits the capacity of physicalist theories to provide a comprehensive account of human consciousness. A Formalizable Proof of the No-Supervenience Theorem: A Diagonal Limitation on the Viability of Physicalist Theories of Consciousness.

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The proof we shall present is mathematical in nature, and does not rely on any of the methodology usually found within the philosophy of mind. In particular, the proof is independent of any metaphysical interpretation (since any sound mathematical formalism must be valid under all interpretations). Since the mathematical derivation is necessarily rather dense and difficult to follow, here is a simply intuitive summary of the proof.

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A Note on the Use of Modal Logic

The proof uses modal logic, and in particular relies on the following equivalence: -p   p. Afficionados of modal logic will doubtless recognize that this equivalence -- which asserts that no proposition can be necessarily true in any language unless it is a theorem of that language -- is not generally true. However it is true in any axiomatization of modal logic where only the axioms of modal logic are allowed as axioms -- or what we shall henceforth call a restricted axiomatization. For example, a model M comprising two possible worlds: w1: P and Q, w2: P and Q does not support the inference: (P → Q) → ( Q →  P). But this is only true because the model implicitly assumes  P as an axiom, and since this axiom is not a theorem of predicate modal logic, it will not be an axiom in any restricted axiomatization of predicate modal logic. One can express this by saying that if the axiom or axioms which define a model are not necessary, then any implication contingent on those axioms is not generally necessary either, or alternatively, that if, for some axiom P which defines a model: P → Q then:  P → Q. It therefore necessarily follows that the equivalence: -p   p can only be invalidated if additional axioms or postulates are added to the restricted axiomatization. The effect of this restriction is to limit the application of modal reasoning to a priori inferences <ref:2307.

1. No physicalist theory can describe a system which is both omega-capable and capable of reasoning using predicate modal logic without being inconsistent. This limits AI systems from claiming to possess consciousness if their underlying architecture requires reasoning within this formal framework.

2. AI systems cannot establish any empirical proposition p with any quantifiable degree of confidence without making additional axiomatic assumptions which are provably incompatible with physicalism. This means the system cannot reliably assert its own knowledge or make probabilistic claims about external reality based solely on its physical substrate and logical capabilities.

3. The system's ability to perform a function like the Cartesian cogito, such as determining they are not in a dreamless sleep at some time t, is inconsistent with being a physicalist model of that system. This implies that an AI cannot be definitively characterized as having human-like self-awareness based on current physicalist constraints.

The no-supervenience theorem limits the capacity of physicalist theories to provide a comprehensive account of human consciousness. The proof of the theorem is difficult to formalize because it relies on both alethic and epistemic notions of possibility. This article outlines a formalizable proof using predicate modal logic in which the epistemic inferences are expressed in terms of an existing mathematical formalism, the inference device (Wolpert, 2008). The resulting proof shows definitely that any physicalist theory which describes a self-aware, intelligent system must be internally inconsistent.