Abstract
We implement a recent characterization of metaphysical indeterminacy in the context of orthodox quantum theory, developing the syntax and semantics of two propositional logics equipped with determinacy and indeterminacy operators. These logics, which extend a novel semantics for standard quantum logic that accounts for Hilbert spaces with superselection sectors, preserve different desirable features of quantum logic and logics of indeterminacy. In addition to comparing the relative advantages of the two, we also explain how each logic answers Williamson’s challenge to any substantive account of (in)determinacy: For any proposition p, what could the difference between “p” and “it’s determinate that p” ever amount to?
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Notes
Not all agree that EEL is orthodox doctrine (Wallace, 2012a, p. 4850, 2012b, p. 108, 2013, p. 215, 2019, secs. 6–7). But there is historical evidence of the at least implicit acceptance of this doctrine through analysis of early quantum mechanics textbooks (Gilton 2016), including that of von von Neumann (1932), as well as of Einstein at the 1927 Solvay conference (Bacciagaluppi and Valentini, 2009, p. 486).
Reichenbach (1944, secs. 30–34) made an early attempt at this sort of project through his three-valued logic for quantum mechanics. Specifically, he introduces “indeterminate” as a third truth value and shows how a sentence of the form “A is indeterminate” can be expressed using certain combinations of logical operators applied to the propositional variable A (1944, p. 153). However, his logic is truth-functional and so does not reflect the underlying structure of how subspaces of a quantum system’s Hilbert state space, representing the extensions of its possible properties according to EEL, fit together. Quantum logic does just this, but as our examples of the non-truth-functionality of quantum logic in Sec. 2.2.3 demonstrate, to do so it must be non-truth-functional.
We intend this and what follows as a brief synopsis of some well-known formalism for those already familiar with quantum mechanics. See, e.g., Myrvold (2018, sec. 2) and Ismael (2020) for further development and many references to the literature. Our restriction to finite-dimensional Hilbert spaces simplifies many technicalities, but we do not believe that this restriction is essential: any substantive claims we make should be retained under an appropriate extension to the infinite-dimensional case.
This expression is of the form required by the spectral decomposition because at most one of P and P⊥ can be the zero operator on their common domain.
Linear combinations of states in different superselection sectors are mixed states rather than the pure states to which we have confined attention here. We return to further considerations regarding mixed quantum states in the concluding section.
Dishkant’s semantics were actually for a more general class of logics, called orthologics, which at the time were sometimes known as “minimal” quantum logic. As Dalla Chiara and Giuntini (2002, sec. 2) point out, however, the latter name is now misleading as, since that time, even more sparing structures related to quantum logic have been considered and developed.
Dalla Chiara and Giuntini (2002, sec. 2) instead select the accessibility relation to be reflexive and symmetric but not transitive, so that it holds between any two states that are not orthogonal in the Hilbert space. This allows them to define important structure for a Hilbert space, such as the sets of states that form subspaces, from the accessibility relation, without first assuming that the states arise as unit vectors in a Hilbert space. Indeed, this allows them to characterize the semantics for the more general class of orthologics, not just those that arise from a “Hilbert lattice.” However, in the process of doing this they do implicitly assume that the Hilbert space so reconstructed (or the analog thereof for general orthologics) has only a single superselection sector. While this is a common and adequate assumption for many applications, because our characterization of quantum metaphysical indeterminacy refers to Hilbert spaces that may have more than one superselection sector, we must take a different approach. This is why we have allowed ourselves to replace the bare set of nodes (“possible worlds”) usually introduced for Kripke semantics with a Hilbert space. That said, it may be possible to proceed with their reconstruction strategy by defining two accessibility relations, one for selection rules (non-orthogonality) and one for superselection rules (same sector), or by defining the latter in terms of the transitive closure of the former. In either case we might use known explicit formulas for the join and meet of two projection operators in terms of their pseudoinverses (Piziak et al., 1999). We leave this possibility for future investigation.
Interested readers may note an analogy between how the interpretation functions are constrained to act on conjunctions and disjunctions and the truth-value tables for the same in Kleene’s weak three-valued logic (Kleene, 1952). This is not so surprising considering Kleene’s objectives concerning partial recursive functions compared with our own concerning operators defined on only subspaces of a given Hilbert space.
If both α and β are true, then the state ψ lies in the domain and range of each of the projection operators associated with α and β, hence lies in the range of the projection operator with their common domain and common range. If at least one of α or β is false, then the state ψ lies in the domain of the associated projection operator—say, that for α— and in the range of its orthogonal complement. That guarantees that it will never lie in the common range of that operator and the one associated with the other—say, that for β. If that latter operator has the same domain as the former, then by definition the intersection of their ranges will be a subset of that for α, hence the orthogonal complement of that intersection will be a superset of that for α.
We adapt this list and the following comparison with the distributive law from de Ronde et al. (n.d., sec. 2).
Those familiar with Boolean algebra may recognize that the right-hand side of these equivalences are more complicated than the usual absorption laws. The conjunction with 1β is necessary, however, for if V(α) and V(β) do not share the same domain, then V((α \(\wedge\) β)) and V((α \(\vee\) β)) are defined only on the intersection of the domains of V(α) and V(β). If V(α) and V(β) do share the same domain, then V((α \(\wedge\) 1β)) = V(α).
There are also other differences with Torza’s account. For instance, his version of condition 1 is in terms of sentential truth-value gaps, rather than propositional ones. See Fletcher and Taylor (2021, sec. 4.4) for further discussion and critique of this position.
As suggested by their difference in grammar, these operators are not logical duals of one another. Instead, if Δ is understood as analogous to necessity, then ∇ would be understood as analogous to contingency (rather than possibility).
The proofs for these are generally analogous to their proofs in QL. The only complication is for disjunctive associativity when not all of the formulas are either property ascriptions or non-property ascriptions. In each of these cases, one can apply the inset formula below, or a variation on it, to expand each of the two formulas in question.
Semantically, the determinacy operator formally implements in the object language something close to what Randall and Foulis (1983, p. 843) call the “canonical map,” which embeds the lattice of subspaces into the lattice of subsets of states. However, they interpret the subspaces as operationally testable propositions and the subsets as property ascriptions. See also Foulis et al., (1983, sec. 4) for more details on how this map is constructed in their framework.
The proofs for these are generally analogous to their proofs in QL. The only complication is for disjunctive associativity when not all of the formulas are either property ascriptions or non-property ascriptions. In these cases, one can apply the inset formulas for the interpretation of a disjunction of a property ascription with a non-property ascription.
This semantic version of modus tollens does hold when both sides of the entailment relation are either both property ascriptions, or both non-property ascriptions. It only fails to hold when one side is a property ascription and the other is a non-property ascription.
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Fletcher, S.C., Taylor, D.E. Two quantum logics of indeterminacy. Synthese 199, 13247–13281 (2021). https://doi.org/10.1007/s11229-021-03375-2
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DOI: https://doi.org/10.1007/s11229-021-03375-2