{"id":"3121703a-5b78-4e4b-b6f9-751962c79622","arxiv_id":"2501.15226","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Quantum indeterminacy is modeled in bivalent quantum logic via non-truth-functional disjunction, where 'up or down' can be true while both disjuncts are false.","lead":"This paper argues that quantum indeterminacy can be explained without abandoning the principle of bivalence: a quantum disjunction can be true even when both parts are false. The authors propose the determinable-based account as an illustration, and claim that so-called Aristotelian accounts that drop bivalence are based on a confusion.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 5's defense of bivalence conflates lattice join with set-theoretic union; span(p,¬p)=H only makes p∨¬p true, not p or ¬p true, so the argument against truth-value gaps does not follow.","rationale":"The paper's central thesis is that dropping Bivalence is unnecessary because non-truth-functional ortho-disjunction can express quantum indeterminacy within a bivalent logic. The main obstacle to this thesis is the Fletcher–Taylor argument that the structure of closed subspaces forces truth-value gaps. Section 5 is the only direct rebuttal of that argument, and it is the step on which the rest of the paper depends. The rebuttal replaces the set-theoretic union of a subspace and its orthocomplement with their lattice join, notes that the join is the whole space, and infers that no state fails bivalence. That inference is unavailable precisely because the paper rejects truth-functionality: the truth of p∨¬p does not put the state in h(p) or in h(p)^⊥. The state can be in the span while p and ¬p are both false, which is exactly the phenomenon the paper uses in Section 6 to model gappy indeterminacy. To call this a bivalent account, one must adopt a non-standard notion of falsity as 'not true' rather than the standard quantum-logical 'state lies in the orthocomplement'. The paper never states or defends that choice, and the formal bivalent semantics is delegated to a companion paper. So the strongest claim is not established as it stands. This is an internal gap in the argument, not a disagreement with the program; if the companion paper supplies a semantics that makes the assignment principled, the concern would be resolved. The reader's weakest assumption identifies the same Section 5 conflation, and the verdict of REJECT is accordingly unchanged.","tokens_in":14714,"tokens_out":15583,"duration_ms":147892,"concrete_test":"Formalize the Section 5 semantics in a two-dimensional Hilbert space. Let h(p) be the x-spin-up ray, h(¬p) the x-spin-down ray, and ψ=(|↑x⟩+|↓x⟩)/√2. Define the state-membership valuation v_ψ(q)=T iff ψ∈h(q). Then v_ψ(p)=F, v_ψ(¬p)=F, and v_ψ(p∨¬p)=T because span(h(p),h(¬p)) is the whole space. This single instance shows that the Section 5 inference from 'join is the whole space' to 'bivalence is preserved' fails; the only way to retain bivalence is to adopt the non-standard claim that false means 'not true' rather than 'ψ∈h(¬p)', which the paper does not explicitly define or defend. If the authors supply the promised bivalent semantics from Horvat & Toader 2023 and show it assigns T to p or ¬p in this state, the objection would be answered.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central rebuttal to Fletcher and Taylor in Section 5 is invalid. F&T argue that bivalence fails because ran(P) ∪ ran(P)^⊥ is not the whole space, leaving states in neither subspace. The authors reply that the lattice operation is join, not set-theoretic union, and that the span of a subspace and its orthocomplement is the whole Hilbert space. This does not establish bivalence. Bivalence, in the sense F&T use, requires that every state lie in h(p) or in h(¬p)=h(p)^⊥ — i.e., the set-theoretic union of the two subspaces exhausts the space. The span being the whole space only shows that the ortho-disjunction p∨¬p is true in every state. Since the paper's own thesis is that ortho-disjunction is non-truth-functional, the truth of p∨¬p does not imply that p or ¬p is true. Indeed, for a nontrivial subspace P, a state in the span but in neither h(p) nor h(p)^⊥ is exactly the case where both p and ¬p receive F under the natural state-membership valuation. Whether this is compatible with bivalence depends on stipulating that falsity means 'not true' rather than 'state lies in the orthocomplement'. The paper never states or defends this stipulation, and the promised bivalent semantics is only cited to a companion paper, not presented here. Thus the central claim that Bivalence need not be dropped is unsupported at its load-bearing step.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":14977,"tokens_out":11041,"duration_ms":102501,"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":[{"comment":"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":"Section 5"},{"comment":"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":"Section 6"},{"comment":"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.","section":"Section 6"}],"minor_comments":[{"comment":"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":"Section 3"},{"comment":"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.","section":"Section 5"},{"comment":"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.","section":"Various"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses an important question in the metaphysics of quantum mechanics and contains a useful distinction between bivalence and truth-functionality. However, the key argument in Section 5 is invalid, and the promised bivalent semantics is only cited to a companion paper by one of the authors. The manuscript is not self-contained and does not, as it stands, establish its central claim. I believe the thesis could be salvaged by a substantial revision that provides the missing semantics and repairs the argument, but this would require more than minor adjustments. The heavy reliance on the authors' own unpublished work is a concern for the editor to weigh."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThis 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.\n\nWho 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.","headline":"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.","tokens_in":15537,"tokens_out":4932,"would_cite":false,"duration_ms":43568,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A bivalent quantum logic with a non-truth-functional ortho-disjunction is sufficient, the paper argues, to express quantum indeterminacy without dropping bivalence.","keywords":["quantum indeterminacy","bivalence","quantum logic","ortho-disjunction","truth-functionality","determinable-based account","orthomodular lattice","value indefiniteness"],"falsifier":"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.","tokens_in":14475,"feed_emoji":"⚛️","tokens_out":10372,"duration_ms":90402,"temperature":0.7,"pith_summary":"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.","feed_headline":"Drop bivalence? Not needed for quantum indeterminacy","feed_subtitle":"A disjunction can be true while each disjunct is false—enough to model quantum indeterminacy inside bivalent logic.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the target argument that orthocomplementation entails failure of bivalence; Section 5 of the paper is a direct reply to it.","marker":"Fletcher and Taylor (2021)"},{"why":"Establishes the distinction between absence of two-valued homomorphisms and rejection of bivalence, which the paper's Conflation charge relies on.","marker":"Demopoulos (1976)"},{"why":"Provides the original quantum logic calculus in terms of closed linear subspaces, cited to show that the founders did not reject bivalence when reading the lattice structure.","marker":"Birkhoff and von Neumann (1936)"},{"why":"Constructs an explicit bivalent semantics for quantum logic, supporting the possibility of a bivalent QL.","marker":"Horvat and Toader (2023)"},{"why":"Formulates the logical-gap account of quantum indeterminacy and the objection that determinables are analyzable into determinates, both of which the paper addresses.","marker":"Torza (2021)"},{"why":"Defines the determinable-based account of metaphysical indeterminacy that the paper argues is best illustrated by bivalent quantum logic.","marker":"Wilson (2013)"},{"why":"Develops the determinable-based account of quantum indeterminacy in a classical setting, providing the object-level view the paper extends to quantum logic.","marker":"Calosi and Wilson (2019)"},{"why":"Further develops the determinable-based account of quantum indeterminacy, including the double-slit discussion that the paper draws on.","marker":"Calosi and Wilson (2021)"},{"why":"Shows that non-truth-functional disjunction can dissolve the sorites paradox, used by the paper to indicate that bivalence need not be dropped for indeterminacy.","marker":"Rumfitt (2018)"}],"fun_headline_variants":["Bivalence holds: disjunction handles quantum fuzziness","Quantum indeterminacy without breaking bivalence","Non-truth-functional disjunction saves bivalence","True disjunction, false parts: quantum indeterminacy","Keep bivalence, drop truth-functionality for quanta"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Bivalence holds: disjunction handles quantum fuzziness","Quantum indeterminacy without breaking bivalence","Non-truth-functional disjunction saves bivalence","True disjunction, false parts: quantum indeterminacy","Keep bivalence, drop truth-functionality for quanta"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000727,"raw_usage":{"total_tokens":3175,"prompt_tokens":784,"completion_tokens":2391,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":400,"completion_tokens_details":{"reasoning_tokens":2315}},"tokens_in":400,"tokens_out":2391,"duration_ms":17173,"temperature":1.0,"reasoning_tokens":2315,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T14:28:51.062287+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}