Non-Boolean Locality and Realism

• Non-Boolean locality and realism is a domain in quantum foundations that rethinks traditional Boolean hidden-variable theories by rejecting globally pre-assigned values.
• It employs methodologies such as operational realism, many-worlds reconstructions, and Heisenberg-picture approaches to maintain locality despite contextual measurements.
• Recent analyses demonstrate that non-Boolean formulations can reconcile quantum correlations with locality and extend to multipartite systems beyond conventional Bell nonlocality.

Non-Boolean locality and realism denotes a family of approaches in quantum foundations that disputes the identification of locality with “local realism” understood as a Boolean hidden-variable scheme with globally pre-assigned values. In this literature, the issue is not only whether quantum theory is local or realistic, but whether locality and realism must be expressed through a single classical event algebra at all. The main alternatives either reinterpret realism operationally and contextually, relocate realism to a deeper ontology, or preserve locality by abandoning the Boolean/Kolmogorovian structure presupposed by standard Bell models (Laudisa, 2022, Gomes et al., 2017, Kuypers, 10 Jan 2026).

1. Bell locality, “local realism,” and the classical background

Bell’s original 1964 analysis centered on locality, not on an independent postulate of realism. In the historical reconstruction, Bell’s key point was that “it is the requirement of locality” that creates the essential difficulty, while determinism in the EPR setting is inferred from locality plus the perfect correlations, not assumed at the outset (Laudisa, 2022). In modern notation, the local-causal template is

$$p(a,b|A,B)=\sum_{\lambda} p_{\lambda}\,p(a|A,\lambda)\,p(b|B,\lambda),$$

or, in a purely probabilistic form,

$P(A,B|a,b,\lambda)=P(A|a,\lambda)P(B|b,\lambda).$

The later expression “local realism” arose historically through a gradual redescription of Bell’s locality condition. Between Bell’s 1964 theorem, the CHSH formulation, the Clauser-Horne analysis, and the Clauser-Shimony review, locality was increasingly presented as one part of a composite “local-realistic” worldview, and “realism” itself was often glossed as the existence of definite properties whether or not they are observed (Laudisa, 2022). The historical point is not merely terminological. In these formulations the hidden state space $\Lambda$, the probability measure over $\lambda$, and the event calculus are classical: the framework is Boolean and Kolmogorovian.

This background matters because Bell’s theorem excludes local hidden-variable theories formulated on that classical structure. A central implication of the historical analysis is that the popular expression “local realism” often obscures which assumptions are formal and which are rhetorical. The theorem directly constrains locality as expressed inside a classical probability model; it does not by itself settle every possible non-Boolean or non-classical conception of locality or realism (Laudisa, 2022).

2. EPR reality, Weak Realism, and the dispute over counterfactuals

The classical point of departure is the EPR criterion of reality: “If, without in any way disturbing a system, we can predict with certainty the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity” (Moldoveanu, 2012, Nisticò et al., 2010). In one line of interpretation, this criterion encodes a classical or Boolean intuition: physical systems possess definite values independent of measurement, and counterfactual measurement outcomes can be treated as meaningful truth-valued propositions. Suárez explicitly links this to the contrast between set-theoretic logic and “the logic of projective spaces,” arguing that quantum superposition undermines the classical disjunction structure presupposed by EPR-style realism (Moldoveanu, 2012).

On that basis, one diagnosis is that quantum mechanics is both non-local and non-realistic. Suárez argues that under locality, EPR realism for spin leads to an inequality contradiction, whereas allowing parameter and outcome dependence can reproduce singlet correlations only at the cost of frame-dependent existence claims, so that EPR-realism ceases to be Lorentz invariant (Moldoveanu, 2012). A different diagnosis shifts the burden away from locality. Tresser replaces Bell locality by the weaker Effect After Cause Principle and shows that Bell-type contradictions still go through provided Weak Realism is retained. In that analysis, Weak Realism becomes “the only hypothesis common to the contradictions obtained in all versions of Bell’s Theory,” and nonlocality is not the common source of the contradiction (Tresser, 2010).

A further dispute concerns how far the EPR criterion extends. Nisticò and Sestito distinguish a strict interpretation, under which reality can be ascribed only when an actual measurement on a separated system licenses a certainty prediction, from a wide interpretation, under which the mere possibility of such a measurement suffices. Under the strict reading, their weak extension of quantum correlations blocks the standard GHSZ, Hardy, and Bell arguments from proving an incompatibility between quantum mechanics and the principle of locality and reality (Nisticò et al., 2010). Laudisa, by contrast, argues that counterfactual reasoning in EPR and Bell does not smuggle in an extra “strong realism”; once locality is made explicit, the counterfactual claims are licensed by locality itself rather than by an independent metaphysical postulate (Laudisa, 2017).

Taken together, these positions show that “realism” is not a single invariant premise. It may mean EPR elements of reality, Weak Realism, counterfactual definiteness, measurement realism, or a broader metaphysical commitment. The non-Boolean literature exploits precisely this instability: once the classical reading of value definiteness is weakened, locality and realism can be reformulated rather than jointly discarded.

3. Non-Boolean local realist reconstructions

Several proposals attempt to preserve locality by changing the ontology or the measurement postulates rather than by accepting Bell-style nonlocality. Bednorz’s “Local Realism in Quantum Many Worlds” introduces a single global POVM over all possible free choices and readouts, together with a fixed number $N$ of interacting copies or worlds. The central requirement is that the theory provide one joint POVM $K[\{\alpha\},\{\beta\},\{\gamma\},\dots]$ respecting the causal past of each readout, so that local realism is maintained without superdeterminism or retrocausality. In that framework the empirical event structure is no longer the Boolean algebra of one world; it is realized through POVMs distributed over many copies in the same spacetime (Bednorz, 2015).

A different reconstruction uses the Heisenberg picture. Kuypers, building on Deutsch-Hayden and Raymond-Robichaud, identifies the local noumenal state of a qubit with the triple of Heisenberg descriptors together with the fixed Heisenberg state, and shows that these local states satisfy formal separability and no-action-at-a-distance conditions: $\pi_A(N^{AB}) \odot \pi_B(N^{AB}) = N^{AB},$

$$(V \times W)\star (N^A \odot N^B) = (V\star N^A)\odot(W\star N^B).$$

In this account the local beables are non-commuting operator-valued descriptors, not classical random variables, and branching is represented by local relative descriptors rather than by a global collapse of the Schrödinger state (Kuypers, 10 Jan 2026).

Other many-worlds proposals preserve Einstein locality by altering realism rather than dynamics. One model begins with a classical Local Realistic Model and then adds a final local branching when Alice’s and Bob’s outcomes meet; this step changes direct realism into modal realism while leaving locality intact, and the multiplicity of branches is adjusted to reproduce the quantum probabilities $\sin^2\theta$ and $\cos^2\theta$ (Vongehr, 2011). Khrennikov’s pre-quantum classical statistical field theory takes yet another route: the ontology is classical and field-like, but quantum observables are nonobjective because detector clicks arise from threshold interactions with random classical fields rather than from the revelation of pre-existing values. The model is explicitly classical, contextual, and local, while denying measurement realism for quantum observables (Khrennikov, 2011).

These reconstructions share no single ontology, but they share a structural move. Locality is preserved by rejecting the assumption that all relevant events and properties belong to one Boolean space of jointly definite values. What changes is either the logic of events, the ontology of worlds, the status of measurement outcomes, or the representation of local states.

4. Operational reality and realism-based nonlocality

A different non-Boolean program does not try to save Bell-local hidden variables at all. Instead it redefines physical reality operationally within quantum theory. Bilobran and Angelo’s criterion takes an observable $P(A,B|a,b,\lambda)=P(A|a,\lambda)P(B|b,\lambda).$0 and the unrevealed-measurement channel

$P(A,B|a,b,\lambda)=P(A|a,\lambda)P(B|b,\lambda).$1

An observable is “real” for a preparation $P(A,B|a,b,\lambda)=P(A|a,\lambda)P(B|b,\lambda).$2 when $P(A,B|a,b,\lambda)=P(A|a,\lambda)P(B|b,\lambda).$3, and its irreality is quantified by

$P(A,B|a,b,\lambda)=P(A|a,\lambda)P(B|b,\lambda).$4

Reality is therefore basis-dependent, state-dependent, and explicitly non-Boolean: it is not a global hidden-variable assignment but invariance under an unrevealed measurement (Gomes et al., 2017).

From this starting point the paper defines contextual realism-based nonlocality by the variation of irreality under a remote unrevealed measurement: $P(A,B|a,b,\lambda)=P(A|a,\lambda)P(B|b,\lambda).$5 or, equivalently,

$P(A,B|a,b,\lambda)=P(A|a,\lambda)P(B|b,\lambda).$6

The associated state measure is

$P(A,B|a,b,\lambda)=P(A|a,\lambda)P(B|b,\lambda).$7

This notion is explicitly different from Bell nonlocality. For pure bipartite states $P(A,B|a,b,\lambda)=P(A|a,\lambda)P(B|b,\lambda).$8, the measure coincides with the entanglement entropy,

$P(A,B|a,b,\lambda)=P(A|a,\lambda)P(B|b,\lambda).$9

so maximally entangled states are maximally nonlocal in arbitrary dimension and the measure is free of the usual Bell-nonlocality anomalies (Gomes et al., 2017). For Werner states

$\Lambda$0

which are separable for $\Lambda$1 and violate CHSH only for $\Lambda$2, the realism-based nonlocality remains strictly positive for every $\Lambda$3. The paper gives the closed form

$\Lambda$4

and contrasts its asymptotic decay under noise with the sudden death of Bell nonlocality (Gomes et al., 2017).

The same work places realism-based nonlocality in a hierarchy of quantumness measures: $\Lambda$5 The hierarchy is significant because it shows that nonlocal modulation of “reality” can occur below entanglement and well below Bell nonlocality. Here “non-Boolean” does not mean an alternative hidden-variable algebra; it means that reality is a contextual quantum property extracted from entropy and decoherence rather than a classical truth assignment.

5. Multipartite extensions and the hierarchy of quantumness

Fucci and Angelo extend the same operational program to tripartite systems. For local observables $\Lambda$6, $\Lambda$7, and $\Lambda$8, the tripartite contextual quantity is

$\Lambda$9

and the bipartition measure is

$\lambda$0

The genuine tripartite version is then

$\lambda$1

For pure states admitting a tripartite Schmidt decomposition,

$\lambda$2

the measure reduces to

$\lambda$3

so it coincides with genuine tripartite entanglement for that class (Fucci et al., 2019).

The mixed-state behavior is more striking. For fully classical states of the form

$\lambda$4

with commuting local projectors, the paper shows that $\lambda$5 for suitably incompatible measurement contexts, even though such states are fully separable and have no Bell nonlocality, no steering, and no discord (Fucci et al., 2019). For noisy GHZ and W states,

$\lambda$6

the tripartite realism-based nonlocality is monotonically decreasing in $\lambda$7 but remains strictly positive for every $\lambda$8, vanishing only for the fully mixed state. The same study shows that the measure is not monogamous in general—pure GHZ already violates monogamy—but becomes monogamous in some noisy GHZ and W regimes after taking powers $\lambda$9 of the measure (Fucci et al., 2019).

These results extend the non-Boolean theme in a precise way. Multipartite nonlocality need not be identified with Bell inequality violation or even with entanglement. Remote operations can modify local degrees of reality across all bipartitions even in correlated mixed states that are classical in the Bell, steering, and discord senses.

6. Competing diagnoses and open directions

The current landscape is defined less by a single doctrine than by incompatible diagnoses. One line maintains that Bell-type contradictions reveal the failure of locality once one combines it with any suitable realist completion; another argues that locality can be preserved if realism is weakened, redefined, or moved to a deeper ontology. Historical analysis reinforces the point that the phrase “local realism” often overstates what Bell actually assumed and tends to import a specifically classical notion of realism into the theorem’s interpretation (Laudisa, 2022).

Contemporary proposals split accordingly. Some explicitly aim to restore locality: by adopting a strict interpretation of the EPR criterion and blocking the wide extension of correlations (Nisticò et al., 2010), by using many copies and a single causally constrained joint POVM (Bednorz, 2015), by replacing Schrödinger states with Heisenberg-picture noumenal states and local branching (Kuypers, 10 Jan 2026), or by arguing that separated experiments should be described only through marginals rather than through a joint quantum distribution (Graft, 2013). Other analyses continue to locate the obstruction in realism rather than locality: Weak Realism remains the common assumption in Bell contradictions even when locality is weakened to the Effect After Cause Principle (Tresser, 2010), and EPR-realism for spin leads either to inequality violations under locality or to Lorentz-noninvariant existence claims under nonlocal rescue strategies (Moldoveanu, 2012).

A further refinement separates statistical Bell nonlocality from counterfactual nonlocality in outcome series. Hnilo argues that the statistical violation of Bell inequalities can be reproduced by a simple non-Boolean local and realist model, but that a distinct “Sica’s non-Locality” persists at the level of counterfactual detection series; he then relates the corresponding contextual instruction to the Hellwig-Kraus postulate of covariant quantum state collapse (Hnilo, 9 Mar 2026). Another line introduces ergodicity as a third foundational option: loophole-free Bell tests may force the rejection not only of locality or realism but also of ergodicity at the hidden-variable level, and the proposed discriminator is the randomness structure of outcome time series (Hnilo, 2020).

The common thread is that non-Boolean locality and realism is not a single interpretation. It is a research domain organized around one recurring claim: Bell-local hidden variables formulated on a Boolean event space do not exhaust the conceptual possibilities. Whether locality is preserved by operational notions of reality, by contextual nonobjectivity, by many-world structures, by Heisenberg-picture separability, or by alternative probabilistic assumptions remains contested. What is no longer tenable in this body of work is the idea that the only alternatives are straightforward Bell nonlocality or the abandonment of realism in every sense.