By "unitarity" I mean that the sums of the probabilities in the given cases would be 1. Non-unitarity would, I assume (for now!), allow for final negative probabilities as well as imaginary-/complex-numbered ones, or ones greater than 1. An even more exotic option would be probabilities that were evolving so as to never "permanently" sum to any specific number.

I've read of non-unitarity only very briefly, relative to I believe something Feynman wondered about re: physics. But since people love to try to connect various notions of free will (usually moderately strong notions, it seems) with probabilistic indeterminism, has any work been done on such connections in probability theories that deny unitarity? How essential is unitarity to probability theory anyway?

Free will and probability

One objection to indeterministic free will that I've seen is, "But it would be random, not intentional, and free will has to be intentional." Besides conflating qualitative with quantitative randomness, this objection seems to conflate chance and randomness, which seem distinct on reflection. "If it's intentional, there's no chance it will happen," seems often false; "If it's intentional, there's a small chance it will happen," depends on the intent (intending to destroy the One Ring in the real world has a vanishingly small chance of being satisfied, however ardently one wishes for such).

But then, "If I intend A, then other things being equal, there's a strong positive probability that I'll do A," can be a plausible design; even more acutely, strongly intending to do something seems to increase the probability that one will try to do it (even were we talking about trying to go to Mount Doom, say).

  • Why would free will involve probabilities not summing up to 1? If anything, probabilities should be undefined in the first place. One of the objections to "quantum free will" is that it is unclear how stable probabilities that quantum mechanics predicts and that we observe can be so inexorably reproduced when the choices are "free". Btw, Feynman did consider negative probabilities, but his probabilities still summed up to 1.
    – Conifold
    Oct 13, 2023 at 23:31
  • If your underlying logic is non-classical, then it is possible for the probability of the partitions of a set of possibilities to sum to something other than unity. In the case of intuitionistic logic, and other paracomplete logics, the probabilities may sum to less than one; with paraconsistent logics, they may sum to more than one.
    – Bumble
    Oct 13, 2023 at 23:35
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    You should define what you mean by soft/hard free will. There are no commonly accepted definitions. Then you should explain what is the connection between free will and probability. I don't see any. Oct 14, 2023 at 4:19
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    @KristianBerry for your questions to prompt citation-based answers would require there to be many paper-writing philosophers who think about the world in the way that you do, and, I say this with huge admiration and respect, you seem to have such a unique outlook that the probability of a large number of philosophers having already considered the points you raise is vanishingly low! Oct 14, 2023 at 6:02
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    Search semimeasure decision theories with non unitary probability measures which are often used under inherent uncertainty or incomputability, your OP about free will sounds like strongly related to such decision or prediction theories… Oct 14, 2023 at 7:17

1 Answer 1


In evaluating what is needed for real libertarian free will to be valid, I have identified two things:

  1. The physical world must be causally open to conscious causation acting on it.

  2. Causation must include more options than "caused vs random".

For point 1, There have been multiple attempts to use the indeterminacy to QM to leverage up to an indeterminacy in the universe. There was another recent question asking about these efforts from the last several decades, which referenced Koch and Hepp's paper Is quantum mechanics relevant/irrelevant to explain conscious processes?. I agree with Koch and Hepp that the approaches they evaluated look to be unsuccessful.

But I suggested in my answer that the leveraging up of QM indeterminacy to macro scale effects, such as all of life depending on photosynthesis, or the catastrophic effects of a single gamma ray on an organism's survival if the ray triggers cancer, provides an alternative path for this indeterminacy to be effectuated. Koch and Hepp did not evaluate chaotic leveraging of QM indeterminacy in their paper, and I believe if they had, they would have been forced to a different conclusion. This leveraging is mathematically and empirically verified.

For point 2, this is where your speculation about probability summation may be needed. I am not a logician, nor probability theorist, so I am not sure how these matrix into the problem.

The basic solution to point 2 is Agency Causation. That causation has three options, not two, and one of the three is "caused by an agent's will". I found a good recent book that articulates a version of agency causation, by Helen Steward: https://global.oup.com/academic/product/a-metaphysics-for-freedom-9780198706465?cc=es&lang=en&#. I did not find any discussion of probability summing in her work, but she is not the only agent causation theorist, so you may want to dig into the writings in this community.

Note, I did not find Steward's book convincing in itself. And this is related to a logic issue, which your probability summing may apply to.

I had several concerns with Steward's thinking.

A) she stayed with a basic physicalist model, while for an agent to even exist, she needed to at least accept Popperian emergent dualism with a world 2, for there to be a place for an agent to BE to influence matter. She also had apparently not explored the problems physicalism has with abstract objects, which include information. Physics itself has basically become a world 1/3 dualism, as information is treated as real, but it has no mass nor energy. If she had adopted Popperian 3 worlds, which is a naturalist triplism, then I would have found her model to be more self-coherent. This is not a fatal problem, as she already accepted strong emergence for consciousness, so her approach can be recast in triplist terms that explicitly admit to the causal openness of physics.

B) The bigger concern I had with her approach is that she really does not deal with "what causes agents to choose" effectively. This is the problem that recursion answers face in the Munchausen Trilemma -- "and then what is the justification for THAT answer/reason/cause". The general approach of science is to pursue the 3rd leg -- and to accept as an interim working solution a regularity we discover, while then trying to find a reason why that regularity holds. But agent causation seems to have to stop at the first leg -- "this is just the way the world works and we don't have an explanation".

One of the reasons I don't bother digging into logic is that naturalism and logic themselves have trouble with the trilemma. My empirical/pragmatic approach, and all of science, is only justifiable per leg 2, and that is a recognized fallacy of circularity. Pluralism of logic leaves us with an infinity of disparate logic evaluations, and no method to evaluate these diverse evaluations. Here is my summary of the Trilemma, and the problems it provides for rationality: Is the Münchhausen trilemma really a trilemma?

The question I have for you, is not whether free will can be resolved using nonunitary probability, but can nonunitary probability rescue logic from the trilemma? I have serious doubts it would be useful at all, and if you can't resolve the trilemma, then you can't solve my concern B) for agent causation.

  • My solution to the trilemma is to assume that each subsolution, besides the empty one, is correct "somewhere," as are combinations like infinite-cycle coherentism and foundherentism (Susan Haack), all the way to an epistemic-graph structure embodying all positive solutions in an erotetic logic (so that some questions are their own partly non-assertoric answers, e.g. "What is the first question in this comment?" is the first question in this comment). Oct 14, 2023 at 16:53
  • As an aside, I will be willing(!) to accept this answer for approaching the subject with relevant references, although I will wait a little to finalize my acceptance. I wish I could figure out how to write better new questions, here, but then I've found that even when I provide comprehensive answers to questions about Kant, like the one about whether he was skeptical of the external world ever, my answers are often not sufficient for whoever posted the questions :/ Oct 14, 2023 at 16:55
  • @KristianBerry -- if you accept the first leg -- unjustified brute fact -- then Agent Causation solves the free will problem, and no, you don't need to consider probabilities that sum to something other than 1.
    – Dcleve
    Oct 14, 2023 at 17:31
  • I tend to link logical and mathematical pluralism with the indeterminacy of the world such as allows for multiple-possibilities-of-action, so I would associate strong free will, then, not with foundationalism, nor coherentism nor infinitism, nor even the subsolutions of the trilemma composed of any two of those, but only the solution composed of all three (foundationalism + infinitism + coherentism together). I suppose it would be even easier to think this if we added skepticism as the empty solution to the "equation," though. Oct 14, 2023 at 17:52
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    @ScottRowe The Trilemma teaches us a lot about the limits to rationalism.
    – Dcleve
    Oct 15, 2023 at 4:15

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