September 13, 1998

Determinism and the Semidecidability of a Free Choice

Dennis L. Feucht

Innovatia Laboratories
14554 Maplewood Road
Townville, PA 16360 USA

Donald MacKay advanced an argument that for a free agent, physically determinate predictions about the agent were logically indeterminate for the agent. He concluded that free will is not excluded in a physically determinate world. Others have objected that MacKay's argument fails to treat the predictions as truly deterministic. The argument presented here, unlike MacKay's, begins with full determinism instead of free will, and concludes that free will is not logically indeterminate but semidecidable. Like MacKay's argument, it concludes that deterministic predictions about agents cannot exclude free will.

Donald M. MacKay was a Scottish physicist, brain researcher, and contributor to the understanding of the relationship between science and Christian faith. He championed an argument that physical determinism does not negate freedom of the will. Just as Heisenberg's uncertainty principle reveals a basic physical limitation on the determinacy of the physical world, MacKay's argument has value in that it makes explicit a basic logical limitation on determinacy as it relates to self-conscious beings, or agents.

MacKay's Argument from Free Will

MacKay argues[1] that even if we, as free-choosing agents, are subject to physically determinate prediction, free choice is nevertheless logically indeterminate. As usually presented, the argument makes a minimal assumption about the relationship between brain (or cognitive mechanism) and mind: brain and mental activities are correlates. Physical activity in the brain is related (somehow) to conscious events. He avoids assignment of causality; all that is necessary is a correspondence. From this single physical assumption, the rest of his argument is purely logical.

The argument then proceeds in an unusual way in that it involves self‑referencing logic. Our minds are culturally conditioned to propositional logic. We think in it without thinking about it. MacKay's use of self‑referencing logic[2] (such as that found in Gödel's Incompleteness Theorem) can take some getting used to. A simple example of the logical indeterminacy that arises due to self-reference is the truth of the statement: "This statement is false." If true, then it is false; but if false, then it must be true. It is not possible to conclude its truth value, which is logically indeterminate.

MacKay argues that for an agent, A, about whom a physically determinate prediction, P1, is made, that P1 is logically indeterminate for A. Once A is offered P1, A is affected by P1 and is no longer the (unaffected) agent described by P1. Thus, P1 cannot be determinate for A because it fails to account for its effect on A. To be determinate, all effects must be accounted for.

To remedy this, a second kind of prediction, P2, can be offered instead which takes into account the effect P1 otherwise would have on A. But if P2 does not affect A when it is offered to A, then A is not the agent assumed in P2. That is, P2 assumes that A will be affected by it when offered, but if A is not, then the "A" of P2 is not an A who is not affected.

In MacKay's form of the argument, the effect is belief, the essential quality of an agent. Therefore, no prediction can exist that the agent would be both correct to believe (P1) and incorrect to disbelieve (P2). Which of the two predictions is true, MacKay argues, is up to the agent. MacKay thus concludes that a physically determinate prediction for the predictor (a noninteracting observer) is not inevitable for the agent to whom it is offered. In this sense, the agent is free to choose the actual outcome, though it could have been predicted deterministically by an observer who does not interact with A.

MacKay's argument might appear to assume deterministic predictions and conclude free will for the agent. But it instead argues from free will by allowing the agent, who is assumed free to choose, to respond to the predictions. It is up to the agent to believe a prediction or not, and such an act of free will determines the truth-value of the prediction. The physical outcome depends on the choice of the agent.

An Argument from Determinism Instead

MacKay's argument can be turned around and argued from deterministic predictions instead. Then P predicts what A will believe about P. If A is physically determinate, then a truly deterministic P accounts for its effect on A before it is offered. P1 is not complete because it fails to take its own effect on A into account. And P2 is not comprehensive in accounting for its effect on the agent; it only assumes the agent will believe it. But a determinate prediction will be comprehensive in its account of its own effect.

A fully deterministic prediction also has two cases. For case 1, P3 predicts that A will believe it. In this case, when presented to A, because P3 is deterministic, A believes it. P3 is consistent in predicting A's response and A's belief in P3 is true.

For case 2, P4 predicts that A will not believe P4 when it is offered to A. P4 is offered, and, because P4 is deterministic, A does not believe it. A's belief about P4 is false, but in not believing P4, A believes P4, for that is the negation of P4. For case 2, the truth-value of P4 is indeterminate.

The vacuity of a prediction that only asserts whether the agent will believe it or not can be given more content by adding a conjunctive statement to it that is true: P AND {true statement}. The truth-value of the compound statement still depends on P, though the prediction contains an additional claim. This addition, however, is not critical to the argument here.

This result is common in the logic of truth and proof. Automated theorem provers can prove true statements in finite time, given a finite database of true statements. But even for finite databases, false statements take an infinite time to disprove.[3] This semidecidability is also evident in the above argument from deterministic prediction. P3 is true, but P4, is logically indeterminate, and no assumption was made that A had free will. Even if the ability of the agent to choose breaks down, these behaviors are also predicted. This case is irrelevant to MacKay's argument because he assumes that A chooses to believe or disbelieve.

Such free-will defense rests, as does MacKay's argument, on Gödelian limitations of logic. Our understanding of physical events rests on a logic of causality that precludes self-reference. But in such cases, even deterministic predictions have limitations that allow for agency. One could then argue for the semidecidability of a free choice.

Heisenberg Uncertainty and Logical Indeterminacy

Though the self‑referencing aspect of the argument from determinism is similar to the arguments from quantum mechanics, MacKay rejected quantum indeterminacy as a way of accounting for free will because quantum effects are too small in scale to directly affect any known neural mechanisms, and consequently show their effect only statistically.

Heisenberg's uncertainty principle anticipates self-reference in that the measurer interacts with what is being measured, affecting the physical quantities of position or speed. In the above arguments, P affects the state of mind of A. While this correlates with brain states of A, the effect is not necessarily accountable in terms of Heisenberg uncertainty. Others have argued that Heisenberg uncertainty fluctuations are adequate to explain the independent existence of mind.[4]

Propositional logic - the classical logic of Aristotle - is fully decidable because its universe of possible propositions is finite, and all of them can be tested for truth in finite time. The logic underlying modern physics is, at the least, first-order predicate logic, which contains universal quantifiers ("x, "for all x"), where x can be a variable over an unbounded range of terms. For example, a negative universal is unprovable (in finite time). Predicate logic is consequently semidecidable. In the development of physical concepts such as causality or determinism, the limitations of  this logic also apply to the physical concepts it supports.


MacKay recognized that we think and behave as though we are free and argued that physical determinism does not deny this basic fact of our personal experience. The kind of physical determinism that MacKay's argument allows is limited to what can be predicted about A without interacting with A. If a more comprehensive determinism is assumed instead, the result is similar in that even fully deterministic predictions are semidecidable. From this, we can conclude that either the logic we employ in our understanding of determinism is inadequate to describe the world in (at least) the case of self-conscious agents, or the world is itself limited in ways that we recognize through the logical indeterminacies in our understanding of it. In neither case can we conclude that our understanding of physical determinism invalidates our experience as free agents.


[1]For example, see The Clockwork Image, Inter-Varsity Press, London, ISBN 0-87784-557-3.

[2]See Raymond Smullyan's book, Forever Undecided, for an entertaining introduction to higher‑order, and increasingly self-aware, systems of logic. Another light introduction is Gödel, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter.

[3] Logic for Computer Science, Steve Reeves, Michael Clarke, Addison-Wesley, 1990, pp. 83-88 introduces the idea of semidecidability.

[4]In The Emperor's New Mind, Roger Penrose argues for quantum effects as the basis for free will.