## 1. Introduction: The Relation between Logic and Mechanics

## 2. Quantum Logic in a Nutshell

An orthogonal projector $\mathbf{P}$ onto a linear subspace $P\subset \mathcal{H}$ is indeed the operator associated with an observable that can take only the values one (and always one if the state $\psi \in P$ is in the subspace P) or zero (and always zero if the state $\psi \in {P}^{\perp}$ belongs to the orthogonal subspace to P). Thus, we can consider that measuring the observable $\mathbf{P}$ is equivalent to performing a test on the system or checking the validity of a logical proposition p on the system.

- The negation of a proposition p is defined by the authors as follows: “since all operators of quantum mechanics are Hermitian, the mathematical representative of the negative of any experimental proposition is the orthogonal complement of the mathematical representative of the proposition itself” ([1], pp. 826–827). The orthogonal complement ${P}^{\perp}$ of the subspace P is the set whose elements are all the vectors orthogonal to the elements of P. From a physical perspective, such an orthogonal complement satisfies the following property: given a subset $P\subset \mathcal{H}$ and a pure state $\psi $, $\psi \left(P\right)=1$ iff $\psi \left({P}^{\perp}\right)=0$ and $\psi \left(P\right)=0$ iff $\psi \left({P}^{\perp}\right)=1$. As Dalla Chiara and Giuntini underlined, “$\psi $ assigns to an event probability 1 (0, respectively) iff $\psi $ assigns to the orthocomplement of P [notation adapted] probability 0 (1, respectively). As a consequence, one is dealing with an operation that inverts the two extreme probability-values, which naturally correspond to the truth-values truth and falsity (similarly to the classical truth-table of negation)” ([18], p. 132).
- Concerning the conjunction, Birkhoff and von Neumann noticed that one can retain the very same set-theoretical interpretation as the classical conjunction, since the intersection of two closed subspaces $P,Q\subset \mathcal{H}$ is still a closed subspace. Thus, one maintains the usual meaning for the conjunction: the pure state $\psi $ verifies $P\cap Q$ iff $\psi $ verifies both P and Q. Thus, quantum logic does not introduce a new logical operator for the conjunction.
- Contrary to the previous case, the logical operator for disjunction cannot be represented by the set-theoretic union, since the set-theoretical union of the two subspaces $P,Q$ will not be in general a subspace; thus, it is not an experimental proposition. Therefore, in QL, one introduces the quantum logical disjunction as the the closed span of the subspaces $P,Q$, which is an experimental proposition. Such a statement corresponds to the “smallest closed subspace containing both P and Q” ([13], p. 55).
- The logical implication is defined by set theoretical inclusion: given two experimental propositions p and q, p implies q means that whenever one predicts p with certainty, one can predict also q with certainty, and this is equivalent to state that p is a subset of q. This fact is particularly important since the authors showed that “it is algebraically reasonable to try to correlate physical qualities with subsets of phase-space” and thus, “physical qualities attributable to any physical system form a partially ordered system” ([1], p. 828, notation adapted).

## 3. Bohmian Mechanics, Observables, and Quantum Measurements

#### 3.1. The Bohmian Treatment of the Measurement Process

- In this theory, there are no superpositions of particles in physical space; therefore, the physical situation described in (6) is avoided by construction. Consequently, macroscopic superpositions are forbidden;
- Measurement results are functions of the primitive ontology and its dynamical evolution provided by the Schrödinger equation and the guiding equation. Thus, the individual physical processes responsible for the macro-objectification of measurement outcomes are independent from external observers;
- Wave functions are not subjected to stochastic jumps: the wave function’s collapse loses its fundamental role in the dynamics of the theory, being an effect of the interaction between subsystem and environment. Hence, in BM, observers never cause the result of a given measurement (cf. [21,26,33,34,37]).

#### 3.2. Contextuality in Bohmian Mechanics

## 4. Classical Logic in the Quantum Context

- Classicality of ∧: Given two atomic propositions $p,q$, the complex proposition $p\wedge q$ is true in BM iff p and q are both true; this statement is false otherwise. Consequently, a proposition with n conjuncts ${p}_{1}\wedge {p}_{2}\wedge \cdots \wedge {p}_{n}$ is true if and only if every ${p}_{i}$ is true, and false otherwise (i.e., there must be a state of affairs that makes every conjunct true). Hence, the logical operator ∧ retains its classical meaning. For instance, the sentence “the particle k has position ${q}_{k}$ and velocity ${v}_{k}$” is true in BM iff it is the case that the particle k is actually located in space at the position ${q}_{k}$ and if it has velocity ${v}_{k}$. If one of these two statements is false, the conjunction will be false as well. This case is not particularly interesting, since also in standard QL, the conjunction is defined as in classical logic, so the meaning of this connective remain unaltered from the classical to the quantum transition.
- Classicality of ∨: Given two atomic propositions $p,q$, the complex proposition $p\vee q$ is true in BM iff at least either p or q is true, i.e., there must be a state of affairs that makes p or q true; this statement is false in the case in which both p and q are not verified. Consequently, a proposition with n disjuncts ${p}_{1}\vee {p}_{2}\vee \cdots \vee {p}_{n}$ is true if and only if at least one ${p}_{i}$ is true, and false otherwise. Hence, the logical operator ∨ retains its classical meaning. Contrary to QL, in Bohmian mechanics, one does not have to introduce a new operator for the quantum disjunction, since the primitive ontology of BM is always exclusively located in one specific support of the quantum mechanical wave function, as already underlined in the previous section. This fact ensures that its evolution will evolve in only one of the macroscopically disjoint possible measurement outcomes. As Bacciagaluppi clearly underlines “the configuration of the system is located only in one of these different components, and this is already a matter of classical logic. The cat is (classically) either alive or dead, because the particles that compose it are (classically) either in the live component or the dead component of the quantum state.” ([13], p. 72). Similarly, taking into account the previous example of the spin measurement along the z-axis, the particle will have spin up or spin down in the case in which it will be located above or under the symmetry line, respectively. Hence, one can model this experimental situation with a classical disjunction. This marks a notable difference between BM and standard QM.
- Classicality of ¬: Given a proposition p, p is false in those cases in which it is not verified, i.e., in cases in which there is a physical state of affairs that makes $\neg p$ true. Thus, also the logical operator “¬” maintains its classical meaning. For instance, the negations of the sentences “the velocity of the particle i is v” and “the position of the particle i is q” are expressed by the statements “the velocity of the particle i is not v” and “the position of the particle i is not q” respectively, having positions and velocity of this particle values different from v and q. In addition, let us consider how BM handles the case of two-valued operators such as spin observables—recalling that these do not represent genuine properties of Bohmian particles. In QL, one models the possible outcomes of a spin measurement as two different and complementary tests, suppose T for spin-up and ${T}^{\perp}$ for spin-down along a given axis, say z. Therefore, the negation of the proposition “a certain system passed the test T with probability 1”, corresponds to the following statement “the system passed with probability 1 the test ${T}^{\perp}$, the orthogonal complement of the test T”. Such an interpretation can be retained in BM as well. In the context of this theory, to state that a particle i does not have z-spin-up is equivalent to asserting that i has not been found in a certain position in space corresponding to the location of the detector associated with a certain state of the measured operator. Hence, the negation of the proposition “the particle i has z-spin up” is translated into the statement “the particle i is not at q”, where q is the position where the particle would have been found if it had z-spin up. Furthermore, knowing that in spin measurements, a particle will be necessarily found either in the state spin-up or spin-down, in BM, one will add the information that the particle i has been found in another position corresponding to the location of the detector associated with a the state “spin-down”. Thus, one can recover the usual interpretation of negation as (ortho-)complementation as in the the logic of classical and quantum mechanics. As a consequence, negation has the following properties also in BM: (i) $\neg \neg p=p$, (ii) $p\wedge \neg p=0$, and (iii) $p\vee \neg p=1$.

- $p=$ the particle i has x-spin-up;
- $q=$ the particle i has y-spin-up;
- $r=$ the particle i has y-spin-down,

“the particle i has been found after a x-spin measurement at the position q in the spatial region $\Delta $, where the detector associated with the state x-spin-up is located”.

and “the particle i has y-spin-down” with:“the particle i has been found after a y-spin measurement at the position ${q}^{\prime}$ in the spatial region ${\Delta}^{\prime}$ , where the detector associated with the state y-spin-up is located”,

“the particle has been found after a y-spin measurement at the position ${q}^{\prime \prime}$ in the region ${\Delta}^{\prime \prime}$, where the detector associated with the state y-spin-down is located”.

## 5. Conclusions

## Funding

## Conflicts of Interest

## References

- Birkhoff, G.; von Neumann, J. The Logic of Quantum Mechanics. Ann. Math.
**1936**, 37, 823–843. [Google Scholar] [CrossRef] - Reichenbach, H. Philosophic Foundations of Quantum Mechanics; University of California Press: Berkeley, CA, USA, 1944. [Google Scholar]
- Mackey, G.W. Quantum Mechanics and Hilbert Space. Am. Math. Mon.
**1957**, 64, 45–57. [Google Scholar] [CrossRef] - Finkelstein, D. The Logic of Quantum Physics. Trans. N. Y. Acad. Sci.
**1963**, 25, 621–637. [Google Scholar] [CrossRef] - Kochen, S.; Specker, P. Logical Structures Arising in Quantum Theory. In Symposium on the Theory of Models: Proceedings of the 1963 International Symposium on the Theory of Models; Addison, J., Henkin, L., Tarski, A., Eds.; North-Holland: Amsterdam, The Netherlands, 1965; pp. 177–189. [Google Scholar]
- Jauch, J.; Piron, C. On the Structure of Quantal Propositional Systems. Helv. Phys. Acta
**1969**, 42, 842–848. [Google Scholar] - Jammer, M. The Philosophy of Quantum Mechanics: The Interpretations of QM in Historical Perspective; John Wiley and Sons: Hoboken, NJ, USA, 1974. [Google Scholar]
- Quine, W.V.O. Two Dogmas of Empiricism. Philos. Rev.
**1951**, 60, 20–43. [Google Scholar] [CrossRef] - Putnam, H. Is Logic Empirical? In Boston Studies in the Philosophy of Science; Cohen, R.S., Wartofsky, M.W., Eds.; D. Reidel: Dordrecht, The Netherlands, 1968; Volume 5, pp. 216–241. [Google Scholar]
- Dummett, M. Is Logic Empirical? In Contemporary British Philosophy; Lewis, D., Ed.; George Allen and Unwin: Crows Nest, Australia, 1976; pp. 45–68. [Google Scholar]
- Bell, J.; Hallett, M. Logic, Quantum Logic and Empiricism. Philos. Sci.
**1982**, 49, 355–379. [Google Scholar] [CrossRef] - Weingartner, P. (Ed.) Alternative Logics. Do Sciences Need Them? Springer: Berlin/Heidelberg, Germany, 2004. [Google Scholar]
- Bacciagaluppi, G. Is Logic Empirical? In Handbook of Quantum Logic and Quantum Structures; Engesser, K., Gabbay, D., Lehmann, D., Eds.; Elsevier: Amsterdam, The Netherlands, 2009; pp. 49–78. [Google Scholar]
- David, F. The Formalisms of Quantum Mechanics; Springer: Berlin/Heidelberg, Germany, 2015. [Google Scholar]
- Beltrametti, E.G. Quantum Logic and Quantum Probability. In Alternative Logics. Do Sciences Need Them? Weingartner, P., Ed.; Springer: Berlin/Heidelberg, Germany, 2004; pp. 339–348. [Google Scholar]
- Jaeger, G. Entanglement, Information and the Interpretation of Quantum Mechanics; Springer: Berlin/Heidelberg, Germany, 2009. [Google Scholar]
- Bub, J. Quantum Information and Computing. In Philosophy of Physics; Butterfiled, J., Earman, J., Eds.; Elsevier: Amsterdam, The Netherlands, 2007; pp. 555–660. [Google Scholar]
- Dalla Chiara, M.; Giuntini, R. Quantum Logics. In Handbook of Philosophical Logic, 2nd ed.; Gabbay, D., Guenthner, F., Eds.; Springer: Berlin/Heidelberg, Germany, 2002; Volume 6, pp. 129–228. [Google Scholar]
- Bell, J.S. Speakable and Unspeakable in Quantum Mechanics; Cambridge University Press: Cambridge, UK, 1987. [Google Scholar]
- Maudlin, T. Three measurement problems. Topoi
**1995**, 14, 7–15. [Google Scholar] [CrossRef] - Bricmont, J. Making Sense of Quantum Mechanics; Springer: Berlin/Heidelberg, Germany, 2016. [Google Scholar]
- Bassi, A.; Ghirardi, G.C. Dynamical Reduction Models. Phys. Rep.
**2003**, 379, 257–426. [Google Scholar] [CrossRef][Green Version] - Everett, H. “Relative state” formulation of quantum mechanics. Rev. Mod. Phys.
**1957**, 29, 454–462, Reprinted in DeWitt, B.S.; Graham, N. (Eds.) The Many-Worlds Interpretation of Quantum Mechanics; Princeton University Press: Princeton, NJ, USA, 1973; pp. 141–149. [Google Scholar] [CrossRef][Green Version] - Wallace, D. The Emergent Multiverse. Quantum Theory According to the Everett Interpretation; Oxford University Press: Oxford, UK, 2012. [Google Scholar]
- Rovelli, C. Relational quantum mechanics. Int. J. Theor. Phys.
**1996**, 35, 1637–1678. [Google Scholar] [CrossRef] - Dürr, D.; Goldstein, S.; Zanghì, N. Quantum Physics without Quantum Philosophy; Springer: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
- Dalla Chiara, M.; Giuntini, R.; Greechie, R. Trends in Logic. In Reasoning in Quantum Theory. Sharp and Unsharp Quantum Logics; Springer: Berlin/Heidelberg, Germany, 2004; Volume 22. [Google Scholar]
- Engesser, K.; Gabbay, D.; Lehmann, D. (Eds.) Handbook of Quantum Logic and Quantum Structures; Elsevier: Amsterdam, The Netherlands, 2009. [Google Scholar]
- Sakurai, J.J. Modern Quantum Mechanics; Addison-Wesley Publishing Company: Boston, MA, USA, 1994. [Google Scholar]
- Griffiths, D.J. Introduction to Quantum Mechanics, 2nd ed.; Pearson Education Limited: London, UK, 2014. [Google Scholar]
- Allori, V. Primitive Ontology and the Structures of Fundamental Physical Theories. In The Wave-Function. Essays on the Metaphysics of Quantum Mechanics; Albert, D.Z., Ney, A., Eds.; Oxford University Press: Oxford, UK, 2013. [Google Scholar]
- Esfeld, M. The primitive ontology of quantum physics: Guidelines for an assessment of the proposals. Stud. Hist. Philos. Mod. Phys.
**2014**, 47, 99–106. [Google Scholar] [CrossRef][Green Version] - Dürr, D.; Teufel, S. Bohmian Mechanics: The Physics and Mathematics of Quantum Theory; Springer: Berlin/Heidelberg, Germany, 2009. [Google Scholar]
- Bohm, D. A suggested interpretation of the quantum theory in terms of “hidden” variables. I. Phys. Rev.
**1952**, 85, 166. [Google Scholar] [CrossRef] - Holland, P.R. The Quantum Teory of Motion. An Account of the de Broglie-Bohm Causal INterpretation of Quantum Mechanics; Cambridge University Press: Cambridge, UK, 1993. [Google Scholar]
- Hubert, M.; Romano, D. The wave-function as a multi-field. Eur. J. Philos. Sci.
**2018**, 8, 521–537. [Google Scholar] [CrossRef][Green Version] - Passon, O. No-Collapse Interpretations of Quantum Theory. In The Philosophy of Quantum Physics; Friebe, C., Kuhlmann, M., Lyre, H., Näger, P., Passon, O., Stöckler, M., Eds.; Springer: Berlin/Heidelberg, Germany, 2018. [Google Scholar]
- Maudlin, T. Why Bohm’s theory solves the measurement problem. Philos. Sci.
**1995**, 62, 479–483. [Google Scholar] [CrossRef][Green Version] - Goldstein, S.; Tumulka, R.; Zanghì, N. The Quantum Formalism and the GRW Formalism. J. Stat. Phys.
**2012**, 149, 142–201. [Google Scholar] [CrossRef][Green Version] - Bell, J.S. On the impossible pilot-wave. Found. Phys.
**1982**, 12, 989–999. [Google Scholar] [CrossRef][Green Version] - Dürr, D.; Goldstein, S.; Zanghì, N. Quantum Equilibrium and the Role of Operators as Observables in Quantum Theory. J. Stat. Phys.
**2004**, 116, 959–1055. [Google Scholar] [CrossRef][Green Version] - Lazarovici, D.; Oldofredi, A.; Esfeld, M. Observables and Unobservables in Quantum Mechanics: How the No-Hidden-Variables Theorems Support the Bohmian Particle Ontology. Entropy
**2018**, 20, 381. [Google Scholar] [CrossRef][Green Version] - Bricmont, J. The de Broglie-Bohm theor as a rational completion of quantum mechanics. Can. J. Phys.
**2018**, 96, 379–390. [Google Scholar] [CrossRef] - Goldstein, S.; Taylor, J.; Tumulka, R.; Zanghì, N. Are all particles real? Stud. Hist. Philos. Sci. Part B Stud. Hist. Philos. Mod. Phys.
**2005**, 36, 103–112. [Google Scholar] [CrossRef][Green Version] - Goldstein, S.; Taylor, J.; Tumulka, R.; Zanghì, N. Are all particles identical? J. Phys. A Math. Gen.
**2005**, 38, 1567–1576. [Google Scholar] [CrossRef][Green Version] - Esfeld, M.; Lazarovici, D.; Lam, V.; Hubert, M. The physics and metaphysics of primitive stuff. Br. J. Philos. Science. Publ.
**2017**, 68, 133–161. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**Schematic representation of the three different possible directions of the pointer in physical space (

**left**) and the relative supports of the pointer’s wave function in configuration space (

**right**).

**Figure 2.**Schematic representation of the pointer pointing in the neutral direction in physical space before the measurement, solid line. Dashed lines represent physically possible, but not actualized measurement outcomes (

**left**). The relative support of the pointer’s wave function and particle configuration describing the experimental situation before the measurement’s performance (

**right**).

**Figure 3.**Schematic representation of the pointer pointing in the right direction in physical space, meaning that the outcome R has been obtained, solid line (

**left**). The relative support of the pointer’s wave function and particle configuration describing the experimental result are shown on the (

**right**). For practical purposes, the empty branch of the wave function can be neglected.

**Figure 4.**Schematic representation of the contextual nature of the property of spin in Bohmian mechanics. These pictures are taken from [43]. (

**a**) An idealized spin measurement; (

**b**) An idealized spin measurement with the direction of the field reversed with respect to the one of (

**a**).

**Figure 5.**Truth table of the distributivity law. The red line represents the truth value of the statements discussed in the above example.

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