Bivalent Quantum Indeterminacy

Claudio Calosi; Iulian D. Toader

REVIEW 3 major objections 3 minor 6 references

Bivalent Quantum Indeterminacy

T0 review · 3 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A bivalent quantum logic with a non-truth-functional ortho-disjunction is sufficient, the paper argues, to express quantum indeterminacy without dropping bivalence.

desk verdict A promising but flawed defense of bivalence in quantum logic: Section 5 conflates the truth of p∨¬p with bivalence, and the main thesis is unsupported as it stands. read the letter →

arxiv 2501.15226 v1 pith:VQM6EJUE submitted 2025-01-25 physics.hist-ph

Claudio Calosi , Iulian D. Toader This is my paper · ORCID

classification physics.hist-ph

keywords quantumindeterminacybivalencelogicortho-disjunctiontruth-functionalitydeterminable-basedaccountorthomodularlatticevalueindefiniteness

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that quantum indeterminacy, understood as the value indefiniteness of observables, does not force one to drop the principle of bivalence. Its central claim is that the non-truth-functionality of quantum disjunction, by itself, can express indeterminate states of affairs: a disjunction can be true even though each disjunct is false, as when a spin-z system is in a superposition of spin-x up and spin-x down. Such a disjunction models the possession of a determinable without any determinate, and it does so inside a bivalent quantum logic based on orthomodular lattices. If this is right, the common "Aristotelian" move of rejecting bivalence to make room for quantum indeterminacy is unnecessary, and indeterminacy need not be a gap in logical space.

What carries the argument

The load-bearing mechanism is the orthomodular lattice $L_A$ of closed subspaces of a Hilbert space, with orthocomplementation as negation, intersection as meet, and the span operation as join, called the ortho-disjunction. Because join is not set-theoretic union, $h(p) \cup h(q)$ can be the whole space while neither $h(p)$ nor $h(q)$ contains the state; this is what lets an ortho-disjunction be true with all disjuncts false. The authors use this non-truth-functional disjunction to analyze quantum determinables as ortho-disjunctions of their maximally specific determinates, so that having a determinable does not entail having any determinate.

What would settle it

Prepare a spin-1/2 system in the state $|+z\rangle$ and let $p$ say "spin x up," so $h(p)$ is the line spanned by $|+x\rangle$. The state vector lies in the span of $h(p)$ and $h(p)^\perp$, which is the whole space, but in neither subspace; assigning truth by state membership makes $p$ false and $\neg p$ false. If bivalence is read as "for every $p$, $p$ is true or $\neg p$ is true," this one valuation refutes the paper's claim; if bivalence is read merely as "every sentence gets either T or F," the example illustrates the paper's non-truth-functional picture. The settling question is which reading of bivalence is at stake.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the value indefiniteness displayed by quantum observables can be accounted for without any truth-value gap. In an orthomodular lattice of closed subspaces, negation is orthocomplementation and disjunction is the join/span operation, which is not truth-functional; consequently a sentence like "spin z" can be represented as an ortho-disjunction of "spin z up" and "spin z down" that comes out true even when each disjunct is false. The authors argue that this is exactly what the determinable-based account of metaphysical indeterminacy needs: a quantum system can have a determinable but no unique determinate, and this is expressed objectually by a true non-truth-functional disjunction of false determinate ascriptions. Bivalence is retained, on their view, because every sentence receives one of the two truth values; what fails is not bivalence but truth-functionality, the idea that the value of a compound is fixed by the values of its parts.

Load-bearing premise

The argument depends on the assumption that if a subspace and its orthocomplement together span the whole space, then bivalence is safe; but a quantum state can be a superposition that lies in neither the subspace nor its orthocomplement, making both a proposition and its negation false.

Editorial extensions

If this is right

Metaphysical accounts of quantum indeterminacy that reject bivalence are not necessary, because bivalent quantum-logic object-level accounts are sufficient.
Quantum indeterminacy should not be understood as a gap in logical space, which shifts support from meta-level gap accounts toward object-level accounts.
The determinable-based account can answer Torza's objection by analyzing determinables as ortho-disjunctions rather than classical disjunctions.
Bivalence and value indefiniteness can coexist within the same quantum logical framework.
Non-bivalent quantum logic is presented as a worse option overall for modeling quantum indeterminacy.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper's ortho-disjunction analysis could be exported to other debates that invoke a "gap in logical space," such as the open future, yielding a non-gappy semantics for future contingents.
Because the paper ties bivalence to "every sentence has one of two values" rather than to "p or not-p is true," its account suggests a general template: indeterminacy can be modeled by non-truth-functional connectives inside a two-valued semantics.
A natural next step, not taken here, is to extend the bivalent semantics to the full quantum logical language, including conjunction and the Sasaki hook, and to check whether the determinable-as-ortho-disjunction analysis remains conservative over the Hilbert lattice.
The distinction between failure of bivalence and failure of truth-functionality might also be applied to the sorites paradox, giving vagueness and quantum indeterminacy a common logical form.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's my take on Calosi and Toader, 'Bivalent Quantum Indeterminacy.' The paper argues that bivalent quantum logic can account for quantum indeterminacy without dropping bivalence, relying on the non-truth-functional character of ortho-disjunction. The genuinely new move is the application to the determinable-based account: a quantum determinable is identified with an ortho-disjunction of its determinates, which can be true even when all disjuncts are false. The replies to Torza and to Fletcher and Taylor's reduction objection are clever and worth engaging with.

The central argument against dropping bivalence, however, does not work. In Section 5 the authors claim that Fletcher and Taylor's bivalence argument fails because the join of two subspaces is their span, and the span of P and P⊥ is the entire Hilbert space. That only establishes that p∨¬p is true in every state. It does not establish that p or ¬p is true in every state. A state in the span but in neither P nor P⊥ is precisely one where p is neither true nor false under the standard state-membership semantics. The authors even describe such cases: in their spin example, both 'spin x up' and 'spin x down' are false while their ortho-disjunction is true. That is exactly a truth-value gap. Pointing to the span does nothing to remove it.

This is load-bearing. The paper's headline claim is that bivalence never needs to be dropped, and Section 5 is the only direct defense against the strongest argument that it does. The promised bivalent semantics is not presented here but cited to the authors' own earlier work (Horvat and Toader 2023). The reader cannot check whether that semantics actually delivers bivalence without a gap. The philosophical ambition is clear and the paper is lucidly written, but the main thesis is unsupported at its decisive step.

Who is this for? Philosophers working on metaphysical indeterminacy and quantum logic. It is a serious contribution to a narrow debate, and the determinable-as-ortho-disjunction idea may survive a rewrite. If this lands on your desk, I would send it to referees; the Section 5 issue is substantive and referees could help the authors repair it or kill it. I would not accept it in its current form.

Referee Report

3 major / 3 minor

Summary. The paper argues that quantum indeterminacy (value indefiniteness) does not require dropping the principle of bivalence. The authors claim that a bivalent quantum logic with a non-truth-functional ortho-disjunction can express indeterminate states of affairs, because a disjunction can be true even when all its disjuncts are false. They distinguish bivalence from truth-functionality, criticize 'Aristotelian' accounts that reject bivalence, and suggest that the determinable-based account of quantum indeterminacy illustrates the bivalent approach. The paper defends this view against objections from Fletcher and Taylor (2021) and Torza (2021), and concludes that bivalent quantum logical object-level accounts are the 'front-runner' for understanding quantum indeterminacy.

Significance. If the paper's thesis were successfully established, it would offer a genuinely new metametaphysical option: quantum indeterminacy need not be modeled as a truth-value gap, but can be accommodated within a bivalent semantics by exploiting the non-truth-functionality of quantum disjunction. This would shift the debate away from bivalence and toward the proper understanding of quantum logical connectives, with consequences for the classification of accounts of quantum indeterminacy. The paper also performs a useful service by insisting on the distinction between failure of bivalence and failure of truth-functionality, a distinction that is often blurred. However, the technical and philosophical support for the central claim is substantially underdeveloped, and the main argument against the leading non-bivalent position is invalid as formulated. The paper is best viewed as a programmatic sketch rather than a completed defense.

major comments (3)

[Section 5] The central rebuttal to Fletcher and Taylor is logically invalid. Fletcher and Taylor argue that bivalence fails because the set-theoretic union ran(P) ∪ ran(P)^⊥ does not exhaust the Hilbert space, leaving states for which a property ascription p is neither true nor false. The authors reply that the lattice operation ∪ is not set-theoretic union but the join (span), and that span(ran(P), ran(P)^⊥) is the entire space. But this only establishes that the ortho-disjunction p ∨ ¬p is true in every state; since the paper itself emphasizes that ortho-disjunction is non-truth-functional (Section 3), the truth of p ∨ ¬p does not imply that p is true or that ¬p is true. The existence of states outside the set-theoretic union still yields states where, under the standard subspace semantics, p is not true and ¬p is not true. The paper's assertion that 'Bivalence fails if and only if the operation ∪ between ran(P) and ran(P)^⊥ = ker(P) is set-theoretic union' is false: bivalence concerns the assignment of truth values to sentences, not the algebraic operation used to represent disjunction. Thus the paper has not undermined Fletcher and Taylor's argument against bivalence.
[Section 6] The paper's central claim that a bivalent QL can account for quantum indeterminacy is not supported within the manuscript. The only explicit reference to a bivalent semantics is to Horvat and Toader (2023), a companion paper by one of the authors, which is neither reproduced nor summarized here. No truth conditions are given for atomic sentences, negation, or disjunction; without such a semantics, the key assertion that an ortho-disjunction can be true while all disjuncts are false is simply stipulated. This is not a minor omission: the entire positive account rests on the availability of a bivalent but non-truth-functional valuation. The manuscript must either present the semantics in detail or prove the existence of the required valuations.
[Section 6] The treatment of gappy indeterminacy requires a stipulation about the meaning of 'false' that is neither stated nor defended. In the example of the state (|↑⟩x − |↓⟩x)/√2, the paper says that both 'e is spin x down' and 'e is not spin x down' are false. This is compatible with bivalence only if falsity is understood as 'not true' (i.e., non-membership in the relevant subspace), rather than as 'the negation is true'. Without this stipulation, the situation can equally be described as assigning a third, indeterminate truth value to both sentences. Since the distinction between bivalence and truth-functionality is the heart of the paper, the truth conditions for negation and the meaning of falsity must be made explicit and defended against the natural alternative reading.

minor comments (3)

[Section 3] The displayed definition of orthocomplementation, 'h(p) = h(q)⊥ iff {x : x ⊆ h(p)} = {x : x⊥h(q)}', appears to be ill-formed or a typo. The right-hand side should presumably involve the orthogonality relation on both sides (e.g., {x : x⊥h(p)} = {x : x ∈ h(q)}). Please clarify.
[Section 5] The symbol ∪ is used inconsistently for both set-theoretic union and the lattice join (span). This ambiguity contributes to the confusion in the argument against Fletcher and Taylor. Recommend using distinct symbols (e.g., ∪ for set-theoretic union and ∨ or ⊔ for the join) throughout.
[Various] The references to 'Horvat and Toader 2023' and to 'a companion paper' are too vague. Please specify which claims are established in those works and how they support the present argument, ideally by summarizing the relevant results.

Circularity Check

1 steps flagged · score 4.0 of 10

Load-bearing bivalent semantics is cited to the authors' own companion paper, but the central philosophical claim does not reduce to that citation.

self citation load bearing [§4, footnote 32 (Bivalence or Non-Bivalence?)]
"For the explicit construction of a bivalent semantics for QL, see again Horvat and Toader 2023."

The paper's claim that a Bivalent QL account of QI is available presupposes that QL admits a bivalent semantics. That presupposition is not derived in the paper; the only support offered is a companion paper by co-author Toader. Because the cited semantics is not reproduced and no machine-checked or independently verified version is cited, the load-bearing premise reduces to an unverified self-citation rather than to an argument given here. The rest of the paper's discussion (e.g., §6) depends on this premise when it describes ortho-disjunction as true with false disjuncts under Bivalent QL.

full rationale

The paper's main argument—that non-truth-functional ortho-disjunction can express quantum indeterminacy without dropping bivalence—does not, on its face, reduce to its inputs: the non-truth-functionality is illustrated by the f=g∘h valuation in §3 and by the spin-x example in §6, and the Torza/Fletcher-Taylor objections are answered by invoking the join semantics. The only genuinely load-bearing external support is the promised 'explicit construction of a bivalent semantics for QL,' which is not presented here but cited to Horvat and Toader (2023), a companion paper by co-author Toader. That citation supplies the technical existence claim on which the viability of Bivalent QL rests; it is not machine-checked, not reproduced, and its assumptions are not stated, so under the rules it does not count as independent evidence. The §5 reply to Fletcher and Taylor ('Bivalence fails iff ... set-theoretic union') is a substantive (and arguably invalid) interpretive move rather than a circular reduction: the conclusion is not identical to the cited semantics, and if the move fails it is a correctness problem, not a derivation-by-construction. Hence the paper has partial self-citation load-bearing, but its central thesis retains independent content.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper does not introduce new free parameters or entities. It relies on standard quantum logical semantics and on several domain assumptions about the nature of quantum indeterminacy and the equivalence of logical gap and truth-value gap. The most load-bearing premise is the unstated assumption in Section 5 that the join operation's spanning the whole space is sufficient for bivalence, which is false.

assumptions (4)

domain assumption Quantum logic is correctly represented by orthomodular lattice semantics, with the Hilbert lattice as the standard model.
Section 3 takes orthomodular lattices as the standard algebraic semantics for quantum logic, ignoring alternatives such as Kochen-Specker partial Boolean algebras.
domain assumption A gap in logical space is equivalent to a truth-value gap under the stated assumptions.
Section 2 cites Torza (2022) for the equivalence between IndeterminacyLG and IndeterminacyTVG, assuming no irreferential terms and expressibility of all facts.
domain assumption Quantum indeterminacy (value indefiniteness) is metaphysical in nature.
Section 1 states this assumption explicitly, acknowledging it is not uncontroversial.
domain assumption The determinable-based account is an object-level account of metaphysical indeterminacy.
Section 2 classifies determinable-based accounts as object-level, following Wilson and Calosi and Wilson.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Bivalent Quantum Indeterminacy." pith.science (2026). https://pith.science/paper/VQM6EJUE

@misc{pith2026250115226,
  author       = {Pith},
  title        = {Pith review of: Bivalent Quantum Indeterminacy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VQM6EJUE}},
  note         = {Machine review of arXiv:2501.15226}
}

read the original abstract

This paper provides a novel metametaphysical approach to quantum indeterminacy. More specifically, it argues that bivalent quantum logic can successfully account for this kind of indeterminacy, given the non-truth-functional character of its disjunction. Furthermore, it suggests that the determinable-based account of quantum indeterminacy illustrates precisely this possibility.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

6 extracted references · 5 canonical work pages

[1]

Is logic empirical?

Bacciagaluppi, G. (2009) “Is logic empirical?”, in Gabbay D ., D. Lehmann, and K. Engesser (eds.) Handbook of Quantum Logic , Elsevier, Amsterdam, 49–78. Barnes Williams, R. (2011). A Theory of Metaphysical Indete rminacy. In Bennett, K. & Zimmerman, D. W. (eds.), Oxford Studies in Metaphysics volume 6, Oxford : Oxford University Press, 103-148. Bell, J. ...

work page 2009
[5]

Could logic be empirical? The Putnam-Kr ipke debate

Quantum Metaphysical Indeterminacy an d the Ontological Foundations of Orthodoxy. Studies in History and Philosophy of Science Part A 90: 235-246. Skow, B. (2010). Deep metaphysical indeterminacy. Philoso phical Quarterly, 60(241), 851-858. doi:10.1111/j.1467-9213.2010.672.x . Stairs, A. (2016). “Could logic be empirical? The Putnam-Kr ipke debate” in Chu...

arXiv 2010
[41]

Toader, I. D. (2023). Distribution can be Dropped: Reply to R umﬁtt. Analysis, https://doi.org/10.1093/analys/a Torza, A. (2020). Quantum metaphysical indeterminacy and w orldly incompleteness. Synthese, 197 (10): 4251-4264. Torza, A. (2021). Quantum metametaphysics. Synthese 199 (3-4):1-25. Torza, A. (2022). Derivative metaphysical indeterminacy a nd qua...

work page doi:10.1093/analys/a 2023
[1978]

Quantum logic and meaning

Dunn, J. M., & Hardegree, G. (2001). Algebraic methods in philosophical logic , Oxford University Press. Fletcher, S. and Taylor, D. (2021a) Quantum indeterminacy a nd the eigenstate-eigenvalue link. Synthese 199: 11181-11212. French, S., & Krause, D. (2006). Identity in Physics: A Histo rical, Philosophical, and Formal Analysis. Oxford: Oxford University...

work page doi:10.48550/arxiv.2304.08450 2001
[2021]

Bibliography on quantum logics and related structures

Emergent Quantum Indeterminacy. Ratio. At: https://onlinelibrary.wiley.com/doi/10.1111 Mariani, C., Michels, R., & Torrengo, G. (2021) Plural metap hysical supervaluationism. Inquiry, 67(6), 2005–2042. https://www.tandfonline.com/doi/full/10.1080/0020174X.2021.1982404. Paviˇ ci´ c, M. (1992) “Bibliography on quantum logics and related structures”, in Inte...

work page arXiv 2021
[5643]

The Possibility Structure of Physic al Systems

Dalla Chiara, M., R. Giuntini, and R. Greechie (2004) Reasoning in Quantum Theory. Sharp and Unsharp Quantum Logics , Kluwer, Dordrecht. Darby, G. (2010). Quantum Mechanics and Metaphysical Indet erminacy. Australasian Journal of Philosophy, 88(2), 227-245. doi: 10.1080/00048400903097786. 17 Darby, G. (2014) Vague Objects in Quantum Mechanics? In: Aki ba,...

work page doi:10.1080/00048400903097786 2004

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Is logic empirical?

Bacciagaluppi, G. (2009) “Is logic empirical?”, in Gabbay D ., D. Lehmann, and K. Engesser (eds.) Handbook of Quantum Logic , Elsevier, Amsterdam, 49–78. Barnes Williams, R. (2011). A Theory of Metaphysical Indete rminacy. In Bennett, K. & Zimmerman, D. W. (eds.), Oxford Studies in Metaphysics volume 6, Oxford : Oxford University Press, 103-148. Bell, J. ...

work page 2009

[2] [5]

Could logic be empirical? The Putnam-Kr ipke debate

Quantum Metaphysical Indeterminacy an d the Ontological Foundations of Orthodoxy. Studies in History and Philosophy of Science Part A 90: 235-246. Skow, B. (2010). Deep metaphysical indeterminacy. Philoso phical Quarterly, 60(241), 851-858. doi:10.1111/j.1467-9213.2010.672.x . Stairs, A. (2016). “Could logic be empirical? The Putnam-Kr ipke debate” in Chu...

arXiv 2010

[3] [41]

Toader, I. D. (2023). Distribution can be Dropped: Reply to R umﬁtt. Analysis, https://doi.org/10.1093/analys/a Torza, A. (2020). Quantum metaphysical indeterminacy and w orldly incompleteness. Synthese, 197 (10): 4251-4264. Torza, A. (2021). Quantum metametaphysics. Synthese 199 (3-4):1-25. Torza, A. (2022). Derivative metaphysical indeterminacy a nd qua...

work page doi:10.1093/analys/a 2023

[4] [1978]

Quantum logic and meaning

Dunn, J. M., & Hardegree, G. (2001). Algebraic methods in philosophical logic , Oxford University Press. Fletcher, S. and Taylor, D. (2021a) Quantum indeterminacy a nd the eigenstate-eigenvalue link. Synthese 199: 11181-11212. French, S., & Krause, D. (2006). Identity in Physics: A Histo rical, Philosophical, and Formal Analysis. Oxford: Oxford University...

work page doi:10.48550/arxiv.2304.08450 2001

[5] [2021]

Bibliography on quantum logics and related structures

Emergent Quantum Indeterminacy. Ratio. At: https://onlinelibrary.wiley.com/doi/10.1111 Mariani, C., Michels, R., & Torrengo, G. (2021) Plural metap hysical supervaluationism. Inquiry, 67(6), 2005–2042. https://www.tandfonline.com/doi/full/10.1080/0020174X.2021.1982404. Paviˇ ci´ c, M. (1992) “Bibliography on quantum logics and related structures”, in Inte...

work page arXiv 2021

[6] [5643]

The Possibility Structure of Physic al Systems

Dalla Chiara, M., R. Giuntini, and R. Greechie (2004) Reasoning in Quantum Theory. Sharp and Unsharp Quantum Logics , Kluwer, Dordrecht. Darby, G. (2010). Quantum Mechanics and Metaphysical Indet erminacy. Australasian Journal of Philosophy, 88(2), 227-245. doi: 10.1080/00048400903097786. 17 Darby, G. (2014) Vague Objects in Quantum Mechanics? In: Aki ba,...

work page doi:10.1080/00048400903097786 2004