Usually free-will is assumed by most faith traditions as a prerequisite for moral responsiblity in order to justify eternal punishment. The argument goes as "you are truly responsible for your immoral deeds because you were perfectly capable of acting morally, but chose not to, and therefore you are guilty and deserve punishment". In other words, there is an assumption of a self capable of having chosen otherwise. But this definition sounds kind of handwavy to me. What do they really mean by "capable of having chosen otherwise"? Does this mean that if we somehow go back in time and press the "univere's play button" again, a different decision would have come out of the same self? But how would that be different from randomness?

I would appreciate a formal (hopefully, mathematical if you can) definition of free-will that clearly differentiates it from determinism, randomness and any kind of determinism-randomness hybrid. Furthermore, I would love to read a logical discussion of how the formal definition provided would support the existence of true moral responsibility. Let me explain why. If the only options available were determinism, randomness and determinism-randomness hybrid, I cannot see any hope for true moral responsibility:

  1. Determinism: your actions, either moral or immoral, are just the ripple effect of past events, you cannot help but do what the laws of Physics make you do, you had no choice, it's not your fault, therefore there is no true moral responsibility on your part.
  2. Randomness: your actions are random, chaotic, there is no control, you are lucky/unlucky that you behave morally/immorally by random chance (e.g. because of weird quantum randomness in the brain, for example). In other words, it's just a matter of luck, a matter of winning the morality lottery. If you behave immorally, it's not really your fault, it's just your bad luck. Therefore there is no true moral responsibility on your part either.
  3. Determinism-Randomness Hybrid: your actions are the result of a combination of deterministic rules applied to past events combined with random quantum noise or something along those lines. Depending on how strong are the causal ripple effects from past events and how strong are the random noises altering them, you end up acting either morally or immorally. In part you are unlucky, in part you have no choice. Anyway, it's not your fault. Therefore there is no true moral responsibility on your part either.

So, somehow, free-will is supposed to have a mysterious formal definition that distinguishes it from determinism, randomness and determinism-randomness hybrid that allows for true moral responsibility in a way that these other concepts can't. I would really appreciate such a formal definition.

EDIT: my question was tagged as possible duplicate of Is free will a third option aside from chance and necessity?, but that's clearly not the case, because there is no formal/mathematical definition of free-will on that post, leaving aside any discussion of moral responsibility.

EDIT2: with respect to what I mean by "formal/mathematical definition of free will", this question might be of interest.

  • 2
    What you are asking for does not exist: if one had a model of free will that satisfies libertarian wish list and is consistent with the laws of physics (as currently understood) controversies surrounding it would likely cease. All we know is that determinism, pure chance and (classical) mixtures of the two, featured in most traditional arguments, do not exhaust the possibilities because quantum mechanical randomness is unsplittable due to the Bell inequalities. Just as we know that odd perfect numbers are not ruled out by known theorems, but have no explicit example of one.
    – Conifold
    May 12, 2018 at 4:23
  • 1
    free will and eternal punishment is accepted by monotheistic religions that assert an extra-cosmic God. It is not accepted by 'many' religions. Eternal barbecuing is not accepted by all monotheistic religions. Free will is a dilemma that all monotheistic religions face. Monastic traditions face no free will controversy and have no eternal barbecuing. May 12, 2018 at 4:49
  • 2
    Possible duplicate of Is free will a third option aside from chance and necessity?
    – viuser
    May 12, 2018 at 14:36
  • 2
    It is hard, if at all possible, to set free-will or liberty of consciousness in math or logic formalizations because those formalizations are themselves based on the preassumption that there is only determinism in the end.
    – ttnphns
    May 13, 2018 at 6:44
  • 1
    @ttnphns, I would not be so sure, Quantum Mechanics is formalized in mathematics and as far as I understand Quantum Mechanics supports indeterminism.
    – xwb
    May 13, 2018 at 14:37

7 Answers 7


First, I'm not sure exactly what you expect by a "formal" definition of free will.

However, I will try to report about a conception of free will that "clearly differentiates it from determinism, randomness and any kind of determinism-randomness hybrid."

This sort of conception of free will, which tends to be called "libertarian" free will, often means that a person's actions are caused only by that person's "self", with that self being decoupled from the world's causality. The actions are therefore not determined by physical events in the world. But the actions are also not random. The actions are, instead, chosen by the self. As such, that self becomes morally responsible for those actions.

Philosopher Robert Kane has attempted to say this is possible in a purely physicalist view of persons, but I have yet to understand how that could be possible. It seems to me that in order for this "self"-caused concept of free will to even have a chance to be possible, the self has to be supernatural, something like a soul, and that the causal nature of self-causing agents has to be (at least at this point, possibly forever) entirely mysterious. Some Christian philosophers, such as philosopher/theologian J.P. Moreland, put forth just this view. This is also a doctrine of, for example, the Catholic Church.

  • By formal, I mean something ideally as close to a mathematical definition as possible. Anyways, do you know / can you elaborate more on how the "choosing by the self" works? How does that "choosing" operate in a way that is neither random, deterministic nor a combination of both?
    – xwb
    May 12, 2018 at 19:10
  • @xwb I'm having trouble imagining redescribing what I wrote above in language that is similar to mathematical definitions without it seeming gratuitous ("Call an act, A, an act of free will, if and only if a world, B, at time t,", etc.). As far as elaborating on how the choosing by the self works, I already wrote that this remains "entirely mysterious". But if that is unsatisfying, so is the origin of the universe/multiverse, or why causality holds at all (cf. Hume).
    – Chelonian
    May 12, 2018 at 19:54
  • Regarding your trouble redescribing what you said in mathematical terms, I think this question I just posted might be relevant: philosophy.stackexchange.com/questions/52107/…
    – xwb
    May 12, 2018 at 23:57
  • Agree with the "soul decoupled from causality". However, if that supernatural "soul" has to be (exist) as an absolute, i.e. on its own grounds, it must be God or its emanation. God will not tolerate another absolute to exist besides himself. So, if one wants to secure the "soul" from God or to remove God from the scene they should say the soul does not be (is not a being) - it is not a "substance". It is possible: consciousness exists by the mode of non-being.
    – ttnphns
    May 13, 2018 at 9:43
  • @xwb Papers that use that sort of language abound in philosophy. Some quick searches pulled up this: eprints.lse.ac.uk/46931/1/… as well as this: andrewmbailey.com/pvi/FormalApproach.pdf You will have to scan through them to find the symbol-heavy parts (obviously, they aren't entirely symbolized, but either are math papers).
    – Chelonian
    May 13, 2018 at 11:56

What you are looking for does not exist.

First off, there is no one formal definition for freewill. Not everyone agrees on what free will is beyond generally agreeing that humans have it. There are myriad variants, each with their own little twists.

One variant that will give you trouble is the metaphysical freewill approach, which explicitly states that freewill is something beyond the physical world. When you make that statement, you skip past all of physics, by necessity. These arguments are distinguished from determinisim/chaos/etc by simply not being connected to them in any way. If you find a layer of physics in their definition of freewill, they will redefine it to not include that physics. Because they value this freedom to redefine freewill, they must only provide it descriptive phrasings.

Another variant that will give you trouble is the compatibilists, who argue that there is no fundamental need to differentiate metaphysical freewill from physical processes. They argue that our "self" may be one, or the other, but that there is no way to discern the difference. These individuals, by necessity, will not provide you a distinction because they don't believe one exists.

One thing I have found helpful, myself, is to break free of the shackles of the statistical definition of randomness. As you noticed in your question, statistical randomness requires the idea of being able to create more than one universe, where things play out differently but it all stems from the same random variable. These arguments are enormously difficult to work with because we don't really know anything about other universes. We don't even know whether they exist at all, and certainly don't know anything about them. We can only conjecture.

If one recognizes the existence of an "inside" and an "outside," such as we often apply at the edges of our skin, it does open the door for something to be "unpredictable" by anything outside, even if it isn't statistically random. If the information needed to predict the action only exists inside, the outside simply cannot make any useful predictions.

As a compatabalist myself, I find that construction convenient. If there is no metaphysical self, this pattern still is useful for defining morality in a subjective sense: its related to what the "outside" is permitted to predict about what will happen inside. This boundary can be shrunk however arbitrarily small, yielding constructs which look more and more like a metaphysical self (more formally, looks like a P-Zombie). If there is a metaphysical self, then the construction naturally permits things like morality in an objective sense. Thus, in all reasonable situations, the two behave identically, even though they come to markedly different conclusions. I find that is close enough to a formal definition for me to work with.

  • 1
    I've found that chaos theory solves the issue for me, in much the way you're describing. Choices are a product of people's minds, and the choices are made in full accordance with the deterministic (or random) laws of nature, but the chaotic structure of the brain means that they cannot be predicted. The decision may be deterministic in the grand sense, but it cannot be determined except by having the individual actually make the decision. A simulation complex enough to predict accurately what the decision would be would be a person in it's own right. May 14, 2018 at 18:13
  • Good insight on Chaos Theory. The recent discovery that some macro-level events are Chaotic, not just Quantum events, has fairly effectively refuted Determinism as a global worldview. There are aspects of physics that are constrained, but the final outcome is underdetermined. While this opens the door to dualist interactionism, it STILL does not solve the problem for free will. The "caused/uncaused-and-therefore-random" dichotomy remains unaddressed by physics having uncaused/random features, and the possibility of caused agency from minds. Minds would still have to be caused or random.
    – Dcleve
    Oct 9, 2021 at 17:34

Mark Balaguer (Free Will, MIT, 2014 (FW)) claims that “we can distinguish many kinds of free will” (FW 50), that is, there are many different ways to define free will. He looks at two very different definitions: Hume-style free will used by compatibilists and not-predetermined free will which he wants to use within a materialist context like that of the OP’s question.

He claims that compatibilists define free will as Hume-style free will, that is, you have free will if “You did want you wanted to do”. (FW 46) This sets up your desires as the cause of your choices which allows them to be determined. Since they are caused or predetermined by something they aren’t free.

To get around that predetermination, he defines free will as explicitly not-predetermined free will (FW 75-6):

For a decision to be a product of my free will (in the sense of not-predetermined free will, as opposed to Hume-style free will), two things need to be true. First, it needs to have been me who made the decision; and second, my choice needs to have not been predetermined by prior events. In other words, it needs to be the case that (a) I did it, and (b) nothing made me do it.

This second definition of free will does not allow determinism to provide a cause. Balaguer wants to fit this definition into a materialist view of reality where we are the neural processes of our brains. He has to find a way to make his view of free will fit in with the “determinism, randomness and determinism-randomness hybrid” mentioned by the OP. He thinks this can be done by allowing free will to happen intermittently during periods where we make “torn decisions” (FW 76):

In particular, we only exercise free will (if we have it at all) when we make torn decisions--when we’re in situations where we’re confronted with multiple options that seem equally good to us, and we stop and think for at least a brief moment about what we should do, and then we settle the matter with a conscious choosing.

Balaguer claims that neuroscience “leaves open the possibility of indeterminism”. (FW 94) However we are "nowhere near ready" (FW 122) to say that we have this kind of free will or not.

Balaguer provides a definition of free will that fits it into a materialist perspective distinguishing it from the "determinism, randomness and determinism-randomness hybrid" mentioned in the OP. He leaves it open for future neuroscience to decide if we have free will or not. His perspective is not a religious or spiritual one.

Robert Kane, whom Chelonian mentioned in his answer, (see “Free Will: New Foundations for an Ancient Problem” in Free Will (Hackett Readings in Philosophy)) uses "self-forming actions" to construct an "event-causal libertarian view of free will" that may offer something similar.

  • Good answer, but if one accepts that desires and what we "want" is caused, it remains a compatibilist free will. Balaguer's model has to assume that intuitions are not caused, and Kahneman's extensive studies on System 1 intuitions, show they have pretty clear patterns, which sure looks like causation. I am an advocate of libertarian free will, but have yet to find a libertarian who is able to define free will in ways that are not actually compatibilism, or randomness.
    – Dcleve
    Oct 9, 2021 at 17:12

You have identified a conceptual problem that those who want to believe in libertarian free will suffer from. While physics itself appears to be underdetermined, based on the last century of both Chaos and Quantum Mechanics insights, this does not help libertarians, as you note. There IS no "formal definition" that allows a mind to be both a non-random agent of one's character, but not also CAUSED by one's character. All definitions provided for libertarian free will, if probed deeply enough, will boil down to caused or random, or some bounded/chaotic combo of the two.

Before you conclude that we cannot have free will, note this is actually a problem for almost every concept we work with. This was one of the key insights of Socrates, and he is famous for his dialogs that reveal the assumed, but invalid conception behind much of what we think.

Hume, continuing Socrates example, showed that "causation" has this same problem. One cannot show logical necessity behind something being caused, as causation applies to contingent things. Hume settled for "causation == correlation", which doesn't allow causation to do the things that you will need it to do in your coding/logical approach.

Popper's efforts to repair empiricism, post Hume, still run into the logic problems behind the meaning of causation. "Causation" is an inferred nature of indirectly postulated realty, where some aspect of the "laws" inferred for physics, logic, or psychology act in a controlling/constraining way to effectively necessitate a particular outcome. If one accepts that empiricism is a judgement call inference as to the nature of reality (indirect realism) based on the reliability/trustworthiness of boldly tested inferences, you STILL have multiple problems.

  • "Judgement" is explicitly in this.
  • the inference process is uncertain itself and needs to be adopted with hesitation
  • plus the "reality" inferred to is itself contingent, not necessary
  • AND -- the justification for this methodology is ITS EMPIRICAL SUCCESS -- IE its justification is explicitly circular

so as a logician/coder, you can't use Popper's empiricism to get to causation either.

These sorts of problems bedevil every aspect of the Analytic effort to logicize/mathematicise philosophy. Quine, for instance, in ""Two Dogmas of Empiricism", noted that NO language, either formal or natural, can ever have sufficient precision in its definitions to actually support analyticity!!!! And Godel showed there are logical limits of any logical method actually being able to establish "truth", even within its own arbitrary abstract framework, much less in the world. And meanwhile, the multiple geometries, and multiple logics that both mathematicians and logicians have discovered, show that there is nothing specials about either Cartesian geometry, or classical formal logic. They are just useful MODELS, which are in many cases not applicable to "reality".

The cumulative problems with logic/analytics have convinced me, at least, that formal "truth" is an impossible standard. We seem to be limited, forever, to pragmatic "good enough working model to trust within this local/bounded problem" judgment call approximation to "true". Pragmatic "truth" is, I believe, all we can ever hope to have.

So where does that leave one with "free will" and moral responsibility? What this discussion shows, is that logic frameworks are not definitive, constraining, or globally valid. Causation, in its ill-defined "strong" non Humean sense, is a super useful assumption for our understanding of the world. So is formal logic. But neither are definitive, so just because the combo of formal logic and strong ill-defined 'causation" exclude free will, does not mean that we cannot have free will. Because, after all, assuming "free will" is ITSELF a super useful assumption in doing morality, and morality is very important for us to be able to live in this world.

So we have pragmatic justification for multiple different and incompatible useful frameworks. That is actually a common situation in pragmatic and logically incomplete fields. These include the fields of science and engineering, literary interpretation and criticism, and theories of aesthetics. The world, and our pragmatic efforts to understand it, cannot be constrained by your effort to logicize or code our understanding, because none of our models can be more than local and pragmatically useful.

So -- I at least am comfortable using free will, and moral responsibility, despite my own inability to explicitly define them.

  • Can't one just take "the pattern of physical events in the universe follows some mathematical rule" as a brute fact (or a fact that derives from another untestable metaphysical assumption, like the will of God), rather than trying to show that causality is a matter of necessity? This does lead to the epistemological problem of how you justify belief in the truth of this claim (Hume's problem of induction), but it's certainly a coherent logical possibility that it's true. The problem of whether "free will is neither random nor determined" is even a coherent possibility isn't epistemological.
    – Hypnosifl
    May 16, 2022 at 19:10
  • @Hypnosifl -- if our physics were causally closed and deterministic, then yes one could do that. I inferred that xwb was asking for something MORE though, based on several passages, here is one: "I would appreciate a formal (hopefully, mathematical if you can) definition of free-will". In my experience, logical/analytic oriented philosophers are not presenting analytics as A POSSIBLE way to do philosophy, among many, but as THE way, or at least the BEST way of many. And if one's assumptions needed to do analytics are arbitrary, then it is not the only, and not justified as the best way.
    – Dcleve
    May 16, 2022 at 19:51
  • This was admittedly an inference about the askers motivation. But also, the "if" above does not hold, so it isn't an option anyway.
    – Dcleve
    May 16, 2022 at 19:53
  • I don't see this as limited to determinism since we can define stochastic mathematical functions. And sure, I agree they were asking for a formal definition, but I think that would fit with my idea that they were asking if free will is a coherent possibility, since for an analytic thinker a precise definition of something may be needed to assert it as a coherent possibility. Even once some idea is established as a coherent possibility, there is still the epistemological question of what would justify believing that possibility obtains in the real world, but that's a separate question.
    – Hypnosifl
    May 16, 2022 at 20:53
  • @Hypnosifl -- I agree that one can create a mathematical model where free willing is replaced with a randomizer function, and it would be a logically coherent model, at least so long as one does not examine the concept of caution too deeply. And that Free Will does not satisfy the analytic requirement of a precise definition. But DO that examination, and causation fails as well. Analytics is useful, but can never actually satisfy its own pre-requisites. Hence not meeting them is not grounds to reject free will.
    – Dcleve
    May 18, 2022 at 18:14

I don't usually reply to older questions even when the Community bot bumps them, on account of me not knowing whether the original poster will ever notice the reply. That being said, I would like to contribute to this question for external reasons, so this time, I'll bite.

"Preamble": to help refine the problem's formulation, one might check out the SEP article on chance and randomness, and see whether the distinction between these, and then secondary distinctions internal to either, can figure in a "way out" such as the OP was asking after.

Besides that, let us suppose a "volitional logic." I am nowhere near as competent at predicate logic as I am at propositional logic, so I will style this logic as having a basic operator on propositions. Let us say:

S = "It is willed that S," e.g., "I willed that the door would open," and then, "It was willed that the door would open." This might sound ungrammatical but sometimes in English we do say things like, "Would that I had never been born!" so it does look like would/will can take propositions. At any rate, even if we could never make this sound normal in English, the notion of a will operator on propositions could be allowed formally, which is what we are looking for here.

A basic alternation over such an operator is deciding whether it is factive. In this case:


However, we seem capable of willing things that we fail to accomplish. For atomic actions, like (perhaps) blinking, I would say that the will operator is factive, or at least closer to factive, than with respect to "molecular" actions. So maybe we'd go with:

S → ◊S

And then, to go in the direction we're looking for:

S → ◊S

I am not really sure that we can iterate and permute these will and possibility operators to exactly land on an axiom or theorem expressive of, "There is free will," or, "Free will exists" (understanding free will in strong terms, modulo the OP's reference to the alternative-possibilities principle). However, consider a choice operation:


I would be hard pressed to say that this is intelligible without something like strong free will modulo "alternative possibilities." Eventually, my own argument turns into something like, "Strong free will is implicitly presupposed by the concept of disjunctive knowledge overall," i.e. unless the existence of strong free will is presupposed, skepticism would dominate our mental strategies inescapably (or we would never even have the concept of disjunctive knowledge at all). Whether this solves or circumvents the issue of defining strong free will relative to determinism, randomness, etc. remains to be seen.

EDITAs far as determinism vs. causal chains goes, let's ask the question: "((AB))?" That is, when choosing between A and B, do I have to choose to choose between them? And so on and on.


Indeed a quite complicated question spanning many concepts... I will address one fact only, but I think is helpfull towards your goal. Suppose you made a bad choice in the past. If you understand your error, then in a similar case in the future, you can/will make a better choice. So, this is free-will, in the sense that from a random past event (bad choise) you "broke" determinism and the whole "determinism-randomness hybrid" cycle by understanding your error or changing your behaviour. And this is what moral responsibility is. Hope this helps.


There are many definitions for free will, none of which has any official status. The best definition I have found is simply: The ability to make decisions.

This definition sets free will as a concept completely outside physics and within psychology instead. There we will find also the concept of responsibility.

Trying to find responsibility in physics, determinism or randomness is quite pointless.

  • @tkruse Why do you have such a hostile attitude towards me? Your comment about the purpose of this forum is both false and out of context. Do you have anything to say about my answer? May 17, 2022 at 12:36
  • @tkruse There are no such rules. Besides, I am not pushing any personal philosophies. My opinion about the best definition for free will is not a philosophy. It is a necessary step that has to be taken to ensure that people will understand what I am saying. You cannot discuss free will if you don't define it first. May 17, 2022 at 13:22

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .