Contingent Free Choice

• Contingent free choice is defined as a framework where the freedom of a decision depends on an underlying structure such as causal order, contextual dependence, or pragmatic enrichment.
• It applies in quantum foundations and related fields by contrasting strong independence assumptions with weaker conditions that allow counterfactual and contextual dependencies in Bell scenarios and hidden-variable models.
• The concept spans disciplines—ranging from experimental psychology to modal semantics and deontic logic—demonstrating that free choices are meaningful only when specific epistemic or methodological structures are fixed.

Searching arXiv for the cited literature and closely related work on contingent free choice, free choice, and related formalisms. Contingent free choice denotes a family of positions in which “free choice” is not treated as an absolute, context-free primitive, but as a notion whose content depends on an underlying structure: causal order, hidden-variable representation, pragmatic-semantic enrichment, deontic guards, or observer-relative physical description. In quantum foundations, the central contrast is between a strong independence-based notion of free choice and weaker notions that allow counterfactual or contextual dependence while preserving alternative possibilities (Colbeck et al., 2013, Fourny, 2019). In adjacent literatures, the term also appears in analyses of free choice inference in modal semantics, free choice permission in deontic logic, and the free-choice paradigm in experimental psychology, where the common issue is how much structure must be fixed before a choice can be called free, informative, or admissible (Aloni et al., 2023, Governatori et al., 2019, Selinger et al., 2018).

1. Causal and conceptual foundations

A prominent causal analysis defines free choice relative to a causal order on a set of random variables $\Gamma$. On this view, a choice $A$ is free if it is uncorrelated with all variables not in its causal future; equivalently, it may be correlated only with variables in its causal future (Colbeck et al., 2013). The paper formalizes causal order as a preorder relation and emphasizes that the direction of the condition is essential: one should not replace “$A W$” by “$W A$,” because that would allow counterintuitive cases in which two settings are perfectly correlated yet both count as free.

The canonical example is a preparation variable $Z$, a setting $A$, and an outcome $X$. Independence of $A$ from $Z$, written $P_{A|Z}=P_A$, is natural if $A$0 is earlier than $A$1. By contrast, demanding $A$2 is too strong, because $A$3 is expected to depend on $A$4 rather than conversely (Colbeck et al., 2013). This establishes the paper’s central claim that free choice is not an isolated probabilistic condition but a statement about causal structure.

This causal account is explicitly contingent in the sense that the admissible correlation structure depends on the specified causal order. Once the causal order is fixed, the freedom condition determines exactly which variables a choice must be independent of. A plausible implication is that “contingent” here does not mean arbitrary; it means indexed to a physically or formally motivated background structure.

The same paper relates this definition to Bell’s own formulation, quoting the idea that free variables have implications only in their future light cones (Colbeck et al., 2013). That connection matters because it anchors the notion of freedom in spacetime compatibility rather than in an unconstrained demand for universal statistical independence.

2. Bell scenarios and the strong independence assumption

In Bell-type scenarios, two distant settings $A$5 and $A$6 are arranged so that neither lies in the causal future of the other. With outcomes $A$7 and $A$8 and preparation $A$9, the causal-future account yields the conditional-independence condition

$A W$0

because $A W$1, $A W$2, and $A W$3 are not in the causal future of $A W$4 in the Bell setup (Colbeck et al., 2013). Analogous conditions apply to $A W$5.

This condition is stronger than mere independence from preparation. It excludes correlations with spacelike-separated settings and outcomes when those are not causally downstream of the choice. The rationale is that if an event is not in the future of $A W$6, then a free $A W$7 cannot be statistically tied to it. The paper presents this as the natural Bell-type freedom-of-choice assumption, not as an “exotic extra postulate” (Colbeck et al., 2013).

A different line of work makes the strength of this assumption especially explicit by distinguishing independent free choice from contingent free choice. Independent free choice is defined as the condition that a decision event $A W$8 is independent of every event $A W$9 outside its future light cone both probabilistically and counterfactually: $W A$0 and

$W A$1

Here the counterfactual clause means that if the choice had been $W A$2, then $W A$3 would still have been true (Fourny, 2019). On this formulation, the usual no-go results against improved predictive power depend not simply on free choice, but on a much stronger doctrine that forbids both statistical and counterfactual dependence between the choice and anything outside its future light cone.

Contingent free choice, by contrast, requires only that alternative choices remain possible across some counterfactual worlds. The rest of the world need not remain fixed across those worlds (Fourny, 2019). This suggests a sharp distinction between two questions often run together in the literature: whether an agent could have done otherwise, and whether the rest of the world would have remained unchanged had the agent done otherwise.

3. Contingent free choice in extension proposals for quantum theory

A direct use of the term appears in a proposal to extend quantum theory to a deterministic, contextual theory with improved predictive power by weakening independent free choice to contingent free choice (Fourny, 2019). The paper argues that strong no-go theorems—Bell, Kochen-Specker, Conway-Kochen, and Renner-Colbeck in the author’s reading—derive their force from the independent-free-choice premise. Once that premise is weakened, a different class of theories becomes available.

The proposed framework recasts quantum experiments as dynamic games with imperfect information. In the EPR case, the physicists’ measurement settings become decision nodes, and the universe’s measurement outcomes become later decision nodes. The universe is treated as a player as well, under the claim that “the physicist picking a measurement axis and the universe picking a measurement outcome are two faces of the same physical contingency phenomenon” (Fourny, 2019). Information sets encode spacelike ignorance, and the experiment’s causal structure determines the game tree.

The associated solution concept is the Perfectly Transparent Equilibrium (PTE), together with its extensive-form counterpart, the Perfect Prediction Equilibrium (Fourny, 2019). The framework assumes agents are rational in all possible worlds and perfect predictors or omniscient in all possible worlds. Worlds that are inconsistent with this joint rationality-and-prediction requirement are eliminated by an iterated procedure described as elimination of “impossible possible worlds” or inconsistent timelines. The intended result is an at-most unique self-consistent world.

In that framework, contingent free choice is formally weaker than independent free choice. For each available choice $W A$4, there exists at least one counterfactual world in which the agent makes choice $W A$5, written in terms of a closest-world function $W A$6 as

$W A$7

The perfect-prediction condition is

$W A$8

Thus a different choice would have been predicted differently, but still correctly (Fourny, 2019).

The paper contrasts this with Nash equilibrium and local hidden-variable theories. Nash-style resolution is described as noncontextual because it assigns strategies off the equilibrium path, whereas the PTE is contextual because only the actual path is assigned and unreached nodes remain undefined (Fourny, 2019). This is presented as analogous to the contrast between preassigned outcomes for all possible measurements and contextual assignment only to the measurements actually performed.

A plausible implication is that the proposal does not deny alternative possibilities; rather, it relocates them into a counterfactual space in which background conditions may vary with the choice. The paper also states important caveats: the framework is still underdeveloped, utilities are parameterizing ingredients, empirical Born-rule statistics remain to be recovered, and the proposal is offered as an example of what becomes possible once independent free choice is weakened, not as a finished physical theory (Fourny, 2019).

4. Equivalence with context-independent mapping in hidden-variable models

A separate but closely related result concerns hidden-variable models. For an empirical system of random variables

$W A$9

where $Z$0 is a measurement property or setting and $Z$1 a context, the paper distinguishes three model forms: a general context-by-context HVM, a context-independent mapping form, and a free-choice form (Dzhafarov, 2021).

The general HVM is

$Z$2

The CI-mapping form is

$Z$3

The free-choice form is

$Z$4

If both assumptions hold, the model becomes

$Z$5

The main theorem states that HVMs (3), (6), and (8) are pairwise equivalent (Dzhafarov, 2021). The proof uses two probabilistic facts: any family can be coupled into a jointly distributed family, and any jointly distributed family can be treated as a single random variable with components as measurable functions of it. From this, explicit context dependence in the hidden variable can be absorbed into the response function, and explicit context dependence in the response function can be absorbed into a family of context-dependent hidden variables.

This result matters for contingent free choice because it undermines any sharp structural distinction between “outcomes depend on context” and “hidden variables depend on settings.” The paper explicitly states that what may look like direct dependence on context versus indirect dependence via hidden variables is mathematically interchangeable (Dzhafarov, 2021). Consequently, a Bell-type violation cannot be attributed in a logically unique way either to lack of free choice or to failure of local causality or CI mapping.

The theorem also holds with or without disturbance or signaling (Dzhafarov, 2021). This broadens its scope beyond non-signaling Bell scenarios and suggests that contingency in the free-choice assumption may reflect representational freedom in model construction as much as any uniquely isolable physical resource.

5. Non-absoluteness of free choices in extended Wigner’s friend scenarios

A more recent development argues against the absoluteness of free choices once they are internalized inside a Wigner’s-friend-style protocol (Walleghem, 14 May 2026). The setup contains Alice and Bob as “friends” inside sealed labs, Wigner as superobserver, and Charlie and Debbie as observers of outgoing qubits. Alice and Bob each make a binary free choice $Z$6, which is encoded into qubits $Z$7. Wigner can either ask for the choices directly ($Z$8) or perform a joint entangled measurement on the labs ($Z$9). Charlie and Debbie choose settings $A$0.

The locality principle used is Local Agency: $A$1 The paper combines this with an Absoluteness of Observed Events style principle applied to the free-choice variables themselves (Walleghem, 14 May 2026). The protocol uses a PBR-based measurement structure, with four product preparations,

$A$2

and four orthogonal measurement outcomes $A$3, each orthogonal to one product preparation. This yields several zero-probability relations, including

$A$4

$A$5

$A$6

$A$7

These exclusions are then propagated across contexts using locality (Walleghem, 14 May 2026).

The contradiction arises from assuming both that $A$8 are absolute facts and that Local Agency holds. The relevant locality conditions are written as

$A$9

and

$X$0

Together with the zero-probability relations, this implies

$X$1

for the common event

$X$2

Since the $X$3 are exhaustive Wigner outcomes, one gets

$X$4

But Local Agency also implies independence from whether $X$5 or $X$6, while the protocol gives

$X$7

Hence contradiction (Walleghem, 14 May 2026).

The conclusion is not that free choice is impossible, but that free choices cannot be maintained as absolute, observer-independent single facts across all contexts in the protocol if locality is retained (Walleghem, 14 May 2026). In that sense, free choice becomes contingent on perspective or measurement context. This is a distinct meaning of “contingent free choice” from the game-theoretic weakening of independence, but both reject an unconditional, observer-independent treatment of choice.

6. Semantic and logical extensions beyond quantum foundations

Outside quantum foundations, “free choice” and its contingent variants are studied in modal semantics and deontic logic, where contingency concerns witness structure, emptiness conditions, or normative guards rather than spacetime locality.

In bilateral state-based modal logic (BSML), free choice inference is analyzed over information states rather than single worlds. Formulas are evaluated with support and anti-support relations,

$X$8

and the central technical device is the nonemptiness atom

$X$9

The paper’s account of contingent free choice centers on pragmatic enrichment that eliminates zero-models, namely states satisfying a formula only via empty-witness configurations (Aloni et al., 2023).

For modal disjunctions, the canonical pattern is

$A$0

The enrichment $A$1 appends $A$2 so that only genuine, nonempty witnesses remain. The paper presents this as the formal core of narrow-scope free choice (Aloni et al., 2023). It also studies two extensions: BSMLI, adding global disjunction $A$3, and BSMLO, adding the emptiness operator $A$4, where

$A$5

BSMLO is particularly relevant because $A$6 can cancel the effect of $A$7, formalizing reversibility of free-choice strengthening (Aloni et al., 2023).

In classical deontic logic, the issue is free choice permission (FCP), formalized as

$A$8

Unrestricted FCP leads to Permission Explosion under classical implication closure and monotonicity. The paper shows, for example, that from $A$9 one can derive $Z$0 for arbitrary $Z$1 if FCP is combined with suitable monotonicity principles (Governatori et al., 2019). To avoid this, it distinguishes weak permission $Z$2 from strong permission $Z$3, rejects full monotonicity, and introduces guarded forms such as

$Z$4

It then develops six Hilbert-style systems admitting guarded FCP (Governatori et al., 2019).

These literatures differ substantially in subject matter, but they share a structural point: free choice effects are not treated as unconditional consequences of disjunction or permission alone. They arise only under added constraints—nonempty witness states, anti-support conditions, or guards blocking prohibited disjuncts. This suggests a general encyclopedic characterization: contingent free choice is a family of doctrines in which the force of a free-choice claim depends on an explicit background condition.

7. Experimental and methodological usage in the free-choice paradigm

A further usage appears in experimental psychology’s free-choice paradigm, used to study choice-induced attitude change. The standard design has three stages: initial ranking, choice between two items, and final ranking. The key dependent variable is the spread, defined from the change in positions of the chosen and rejected items between stages one and three (Selinger et al., 2018).

The paper argues that the standard paradigm contains a logical flaw identified by Chen and Risen: stage-two choice is itself informative about the subject’s latent preference ordering. Hence positive average spread can arise even if preferences do not change. Under the null model, a subject has a ranking distribution $Z$5, and stages one and three are independent samples from the same distribution. Even in that stationary-preference model, the stage-two choice acts as extra evidence about latent ranking, creating a probabilistic artifact (Selinger et al., 2018).

Selinger and Tapp propose mathematically corrected designs for which the expected average spread is zero under the no-change null. Proposition 1 states that this holds in any of the following three designs: every subject chooses between the same pre-selected pair; each subject chooses between positions $Z$6 chosen uniformly at random; or every possible pair $Z$7 is used exactly once across subjects (Selinger et al., 2018). The symmetry argument is that swapping the first and third rankings changes the sign of the spread while leaving the null distribution unchanged, forcing zero expectation.

The paper explicitly connects this to contingent free choice analysis by stating that the free-choice paradigm supports such analyses only when the design is mathematically balanced so that, under a stationary-preference null, the expected spread is truly zero (Selinger et al., 2018). Here contingency does not concern causal order or locality, but the dependence of interpretability on design symmetry. Positive spread alone is not evidence of dissonance, because it may reflect “choice reveals preference” rather than “choice changes preference” (Selinger et al., 2018).

This methodological usage reinforces a broader theme across the literature. Whether the domain is Bell experiments, hidden-variable theory, modal semantics, deontic logic, or attitude-change experiments, “free choice” becomes epistemically usable only relative to a specified structure. The structure may be a causal preorder, a hidden-variable representation, a state-based semantics, a guarded permission system, or a null experimental design. Contingent free choice is therefore best understood not as a single doctrine, but as a cluster of technically distinct attempts to replace absolute free-choice assumptions with explicitly structured ones.