Embedding Quantum Mechanics into a Broader Noncontextual Theory

Abstract

Scholars concerned with the foundations of quantum mechanics (QM) usually think that contextuality (hence nonobjectivity of physical properties, which implies numerous problems and paradoxes) is an unavoidable feature of QM which directly follows from the mathematical apparatus of QM. Based on some previous papers on this issue, we criticize this view and supply a new informal presentation of the extended semantic realism (ESR) model which embodies the formalism of QM into a broader mathematical formalism and reinterprets quantum probabilities as conditional on detection rather than absolute. Because of this reinterpretation a hidden variables theory can be constructed which justifies the assumptions introduced in the ESR model and proves its objectivity. When applied to special cases the ESR model settles long-standing conflicts (it reconciles Bell’s inequalities with QM), provides a general framework in which previous results obtained by other authors (as local interpretations of the GHZ experiment) are recovered and explained, and supports an interpretation of quantum logic which avoids the introduction of the problematic notion of quantum truth.

1 Introduction

According to the statistical interpretation of quantum mechanical probabilities, quantum mechanics (QM) refers only to ensembles of measurement outcomes or to ensembles of physical systems, and does not say anything about properties of individual systems (Ballentine 1970). We instead accept in this paper an interpretation (that we call realistic,¹ following Busch et al. 1991) according to which QM deals with individual examples of physical systems (briefly, physical objects) and their properties. But, then, QM turns out to be a nonobjective theory. As Mermin writes, “It is a fundamental quantum doctrine that a measurement does not, in general, reveal a preexisting value of the measured property.” (Mermin 1993). More generally, QM is nonobjective in the sense that, for every physical object, there are physical properties whose values may be brought into existence by a measurement but do not preexist to it (Busch et al. 1991).

Nonobjectivity has upsetting consequences that have both physical and philosophical relevance. From the point of view of the quantum theory of measurement nonobjectivity raises the fundamental objectification problem (see, e.g., Busch et al. 1991; Busch and Shimony 1996; Busch 1998), i.e., intuitively, how it may happen that some non-preexisting physical properties become real after a measurement (this problem is often illustrated by famous quantum paradoxes, as Schrödinger’s cat, Wigner’s friend, etc). Moreover, nonobjectivity is the deep root of some difficulties in the interpretation of quantum mixtures (Timpson and Brown 2005). From the point of view of the theory of probability nonobjectivity implies that the usual epistemic interpretation of probability cannot be maintained in the case of quantum probability, which is necessarily nonepistemic, or ontic (Beltrametti and Cassinelli 1981; Busch et al. 1991). It follows, in particular, that the deterministic principle of causality breaks down and that no description of the microscopic world more detailed than the description provided by QM is in principle possible. From the point of view of logic nonobjectivity implies that the classical notion of truth as correspondence cannot be maintained in the language of QM, so that many authors uphold that a nonclassical logic (quantum logic) must be adopted in it (see, e.g., Rédei 1998; Dalla Chiara et al. 2004). Finally, nonobjectivity implies that models that would entail objectivity of physical properties cannot be constructed for QM (wave-particle duality), which is counterintuitive.

The consequences of nonobjectivity resumed above have been widely discussed from the very beginning of QM. Many scholars have wondered whether there are possible alternative interpretations of QM avoiding nonobjectivity without introducing contradictions. This perspective has given rise, in particular, to a series of attempts at constructing hidden variables (h.v.) theories for QM trying to restore a somewhat classical perspective of the physical world underlying the probabilistic description of QM. Such attempts, however, must face several theorems, the most famous of which are Bell’s (Bell 1964) and Bell–Kochen–Specker’s (Bell 1966; Kochen and Specker 1967), which do not prohibit that h.v. theories for QM exist, as exemplified by Bohm’s theory (Bohm 1952), but imply that such theories must necessarily be contextual and nonlocal: hence, nonobjective.

The above theorems (sometimes dubbed “no-go” theorems) seemingly are solidly founded and many alternative proofs of them have been given (see, e.g., Greenberger et al. 1990; Hardy 1991, 1993; Mermin 1993; Calude et al. 1999). Moreover, a strong experimental support to nonlocality has been provided by Aspect’s and similar experiments (see, e.g., Genovese 2005, for a broad bibliography on this topic). Therefore most scholars think that contextuality and nonlocality (hence nonobjectivity) are unavoidable consequences of the mathematical formalism of QM, and reject every attempt at recovering objectivity (it may be interesting to note that also some important generalizations of QM, as unsharp QM, do not solve the objectification problem, see Busch and Shimony 1996). Rather, there are recent efforts to obtain the structure of QM basing on general principles and notions inspired by quantum information theory, as Zeilinger’s foundational principle (Zeilinger 1999), CBH theorem (Clifton et al. 2003), quantum Bayesianism (Caves et al. 2002a, b; Fuchs and Schack 2004) and D’Ariano’s proposal for deriving QM as the mathematical representation of a fair operational framework (D’Ariano 2010; D’Ariano and Tosini 2010). These approaches (some of which are still incomplete or are exposed to serious objections, see, e.g., Timpson 2008) either are not realistic in the sense specified above, or do not question contextuality and nonlocality, which are instead considered as basic features and resources for quantum information processing.

At variance with the positions quoted above, we maintain that a subtler analysis of the h.v. programs, the “no-go” theorems and the experimental support to contextuality may open new theoretical possibilities. Indeed one of us, together with some collaborators, has shown in a series of papers (Garola 1999, 2000, 2002; Garola and Pykacz 2004; Garola and Solombrino 1996a, b; Garola and Sozzo 2010a) that the proofs of Bell’s and Bell–Kochen–Specker’s theorems introduce an implicit assumption about the simultaneous validity of empirical physical laws (metatheoretical classical principle, or MCP) that is suitable for classical mechanics but may be questioned in QM. If this assumption is weakened, so as to fit in better with the operational attitude of QM (metatheoretical generalized principle, or MGP), the aforesaid proofs cannot be completed. This result implies that the possibility of providing noncontextual (hence local) interpretations of the mathematical formalism of QM cannot be excluded, at variance with standard beliefs. To show that interpretations of this kind are actually possible, a semantic realism (SR) interpretation of QM has been supplied in some of the papers quoted above (Garola 1999, 2000, 2002; Garola and Solombrino 1996a, b) which recovers objectivity of QM at a semantic level, replacing MCP with MGP and avoiding ontological commitments. Successively the SR interpretation has been modified and generalized, and a new theoretical framework has been produced, called extended semantic realism (ESR) model (Garola 2002, 2003; Garola and Pykacz 2004; Garola and Sozzo 2009, 2011a, b, c, 2012; Sozzo and Garola 2010). This model recovers objectivity embodying the formalism of QM into a broader mathematical formalism and substituting the assumptions introduced in the SR interpretation with a reinterpretation of quantum probabilities as conditional on detection rather than absolute (which implies replacing MCP with MGP in the quantum part of the model). This proposal seems to us interesting not only from a physical but also from a philosophical point of view. It avoids indeed the problematic consequences of objectivity resumed above, settles long-standing conflicts and provides a general framework in which some previous results in the literature are recovered and explained. We therefore intend to illustrate the basic ideas and consequences of the ESR model in an informal way in the present paper. Our presentation offers two novelties. Firstly, the ESR model is introduced as a general theory at a macroscopic level without mentioning h.v., at variance with previous presentations (Sects. 2, 3, 4, 5). A h.v. theory for the ESR model is then sketched afterwards, to prove that the ESR model is an objective theory and to provide a set-theoretical picture of the microscopic world underlying the ESR model, justifying on this basis the assumptions introduced in the model (Sect. 6). Secondly, we collect and summarize some recent results obtained by considering relevant physical problems from the viewpoint of the ESR model (Sect. 7). We can thus explicitly show how the ESR model reconciles Bell’s inequalities with QM (Garola and Sozzo 2010b, 2011a, c), recovers some local interpretations of the Greenberger–Horne–Zeilinger experiment proposed by other authors (Garola et al. 2013) and provides a suitable background for introducing quantum logic avoiding the problematic notion of quantum truth (Garola and Sozzo 2013).

2 The Issue of Nonobjectivity

The meaning of the terms that occur in the definition of nonobjectivity provided at the beginning of Sect. 1 is well known, because the issue of nonobjectivity of QM has been extensively discussed in the literature. Nevertheless, we intend to comment briefly on it in this section, to reconsider some important details and establish the basis for the position that will be upheld in this paper.

First of all, let us note that the merely intuitive notion of physical property is rather vague. In the definition of nonobjectivity, instead, the term “physical property” has a precise meaning, for it denotes a pair $E=(A,\varSigma )$, with $A$ an observable of a given physical system and $\varSigma$ a set of possible (real) outcomes of $A$(from the point of view of unsharp QM we consider here only sharp physical observables; the extension of our reasoning to unsharp observables is possible but it would introduce unnecessary complications in this paper). Every observable is empirically interpreted on a (probabilistic) equivalence class of exact (efficiency 1) measuring devices, which are obviously only approximated by real devices. Hence every physical property is empirically interpreted on an equivalence class of exact dichotomic measuring devices (or yes-no experiments, or registration apparatuses, see, e.g., Beltrametti and Cassinelli 1981; Ludwig 1983). Then, measuring $E$ means checking whether the outcome of an exact measurement of $A$ belongs to $\varSigma$, which can be done by means of an exact dichotomic measuring device whose outcomes are usually labeled yes and no. Moreover, a typical sentence assigning a physical property takes the form

• (a) “the physical object $\alpha$ has the physical property $E$”

(briefly, $E(\alpha )$ in the following), and the outcome of an exact measurement of $E$ establishes whether $E(\alpha )$ is true (if the outcome yes occurs) or false (if the outcome no occurs). A sentence of the form

• (b) “the physical object $\alpha$ has the physical property $E$ with probability $p$ whenever $\alpha$ is in the state $S$”

does not assign instead a physical property in the sense specified above.²

Secondly, let us observe that also the intuitive notion of state must be specified to acquire a precise meaning. For instance, Aerts writes (by using the same symbols introduced above with a different interpretation):

“The physical entity $S$ ‘is’ at each moment in a certain state $p$. In our approach the states describe the reality of the entity and the structure of the set of states expresses the main part of the ‘calculus of being’ ” (Aerts 1999a).

According to other scholars, a state $S$ can be empirically interpreted as an equivalence class of preparation procedures, each of which, when activated, prepares physical objects in the state $S$ (see again Beltrametti and Cassinelli 1981; Ludwig 1983).³

Thirdly, let us recall that, in the realistic perspective mentioned above, a physical property $E$ is usually maintained to be objective in a given physical state $S$ if and only if either all physical objects in the state $S$ possess $E$ or they possess its negation (Busch et al. 1991; equivalently, one can define $E$ as objective in a state $S$ iff either $E$ or its negation is actual in the state $S$, see Aerts 1999a, b). This definition, however, is still unsatisfactory in our opinion, because the word “possess” is only intuitively defined. To make it more precise, let us note that it has some relevant consequences. Indeed, if $E$ is objective in the state $S$, then for every measurement context and physical object $\alpha$ in the state $S$ a value (true/false) of $E(\alpha )$ is defined (though it may remain unknown) which does not depend on $\alpha$ (value definiteness of E in S, or, briefly, VD$(E,S)$; by abuse of language we directly attribute this value to $E$ in the following). Moreover the value of E is independent of the measurement context (noncontextuality of E in S, or, briefly $\lnot C(E,S))$, even if a measurement may perturb it. Hence the following implication holds,

$$\begin{aligned} \hbox {O}(E,S)\Rightarrow \hbox {VD}(E,S)\wedge \lnot \hbox {C}(E,S)\,, \end{aligned}$$

(2.1)

where O($E,S)$ denotes objectivity of $E$ in $S$, and $\wedge$ and $\lnot$ denote classical logical conjunction and negation, respectively. We assume that this implication can be reversed, that is,

$$\begin{aligned} \hbox {O}(E,S) \Longleftrightarrow \hbox {VD}(E,S)\wedge \lnot \hbox {C}(E,S), \end{aligned}$$

(2.2)

which makes the definition of objectivity more precise, as required. We will then say that an objective physical property $E$ is possessed by a physical object $\alpha$ if its value is true, not possessed if its value is false.

Based on (2.2) we say that a physical theory is objective (briefly, O) if for every physical system $\Omega$ considered by the theory and for all states of $\Omega$ all physical properties are objective, i.e., value definiteness and noncontextuality hold in general. Hence we briefly write

$$\begin{aligned} \hbox {O}\Longleftrightarrow \hbox {VD}\wedge \lnot \hbox {C}. \end{aligned}$$

(2.3)

Definition (2.3) implies that classical mechanics is an objective theory. According to the standard interpretation, QM is instead nonobjective (briefly, $\lnot \hbox {O}$). This assumption, that introduced a revolutionary epistemological perspective in physics and has the upsetting consequences mentioned in Sect. 1, was debated for decades. Since we want to show that it represents a deep philosophical option but not a logical necessity, let us comment briefly on it.

Formerly, Heisenberg’s uncertainty principle was maintained to support the assumption of nonobjectivity of QM. But, as Mermin writes, “the uncertainty principle only prohibits the possibility of preparing an ensemble of systems in which all those properties are sharply defined”, while “the question is whether properties of individual systems possess values prior to the measurements that reveal them” (Mermin 1993).

Successively, however, the “no-go” theorems mentioned in Sect. 1 were demonstrated which imply nonobjectivity. Indeed, let us schematize Bell’s and Bell–Kochen–Specker’s theorems by using the symbols introduced above and adding the symbol LOC to denote the assumption of locality of a theory (we recall that a theory is local if the result of a measurement on a part of a physical system is not affected by simultaneous measurements on parts of the system which are far away in the space). We obtain

$$\begin{aligned} \hbox {QM}\Rightarrow \lnot \hbox {LOC} \end{aligned}$$

(2.4)

(Bell’s theorem) and

$$\begin{aligned} \hbox {QM}\Rightarrow \hbox {C} \end{aligned}$$

(2.5)

(Bell–Kochen–Specker’s theorem). Let us observe then that

$$\begin{aligned} \lnot \hbox {LOC}\Rightarrow \hbox {C}\,, \end{aligned}$$

(2.6)

let us add the symbol $\vee$ to denote classical logical disjunction and let us note that (2.3) implies

$$\begin{aligned} \lnot \hbox {VD}\vee \hbox {C}\Rightarrow \lnot \hbox {O}\,. \end{aligned}$$

(2.7)

Putting together (2.4)–(2.7) we conclude that both Bell’s and Bell–Kochen–Specker’s theorems imply nonobjectivity, as stated.

The question whether an objective interpretation of QM is possible seems thus to be definitively answered in the negative. As we have anticipated in Sect. 1, however, the proofs of the above theorems require an implicit assumption whose consistency with QM is questionable (Garola 1999, 2000, 2002; Garola and Pykacz 2004; Garola and Solombrino 1996a, b; Garola and Sozzo 2010a). Since we want to uphold the thesis that an objective reinterpretation of QM is actually possible, we discuss our criticism of Bell’s and Bell–Kochen–Specker’s theorems in the next section.

To close this section we must still add a remark that will be useful in Sect. 6. It is indeed important to observe that the above notion of objectivity implies that a theory is nonobjective if there are in it, for some state $S$, physical properties that are not objective. It does not imply instead that a nonobjective theory must be such that, for every state $S$, all physical properties are nonobjective. Hence, in the case of QM many scholars associate both objective and nonobjective physical properties with every quantum state. In particular, some authors consider a property $E$ as objective in a state $S$ if and only if the probability $p(E,S)$ that a physical object $\alpha$ in the state $S$ displays the property $E$ when an exact measurement of $E$ is performed on it is 0 or 1 (Busch et al. 1991; equivalent though formally different definitions can be found in Aerts 1999a, b and Piron 1976). It is then easy to see that the standard interpretation of quantum probabilities implies that this definition of objectivity is equivalent in QM to the general definition provided by (2.2) (which is independent of the physical theory that is considered). Indeed, let the physical property $E$ be objective in the state $S$ according to the definition of objectivity in terms of probability. Then the value of $E$ can be considered as defined and independent of the measurement context. Hence objectivity of $E$ in $S$ in the sense established by (2.2) holds. Conversely, let us observe that the empirical interpretation of physical properties implies that, if value definiteness of $E$ in S holds, then an exact measurement of $E$ on any physical object $\alpha$ in the state $S$ must yield the value of $E$ in the given measurement context as outcome (adequacy condition for exact measurements).⁴ Hence, if $E$ is objective in $S$ in the sense established by (2.2), then $E$ either is displayed by all physical objects in $S$ or by none of them whenever an exact measurement of it is performed. Therefore $E$ is objective in $S$ according to the definition of objectivity in terms of probability.

Because of the above equivalence one can say that $E$ is possessed by the physical object $\alpha$ in the state $S$ if $p(E,S)$ is 1, not possessed if it is 0, which is intuitively satisfactory.

3 The Criticism of the No-go Theorems

To understand the core of the critical analysis of Bell’s and Bell–Kochen–Specker’s theorems that we want to resume in this section a preliminary remark is needed.

According to a known epistemological perspective (received viewpoint: see Braithwaite 1953; Carnap 1966; Hempel 1965), any theory T states general theoretical laws by means of its theoretical language (i.e., the formal apparatus of the theory). These laws may have no direct empirical interpretation, but can be used to obtain derived theoretical laws and then, by means of correspondence rules, empirical laws which are stated in the observational language⁵ of T and establish experimentally controllable relationships between physical properties (by abuse of language we will call empirical laws in the following also the derived theoretical laws that correspond to empirical laws stated in the observational language of T). This perspective, though criticized by several scholars (Hanson 1958; Kuhn 1962; Feyerhabend 1975) still provides, in our opinion, a basic framework for understanding the structure of scientific theories, but it must be refined in the case of QM. Indeed, in classical theories there is no theoretical limit to the possibility of confirming or falsifying an empirical law. In QM, instead, this possibility is restricted because incompatible observables exist in QM whose values can be neither measured nor predicted simultaneously. Hence an empirical physical law cannot be checked, in principle, in a physical situation that is characterized by values of observables that are not compatible with the observables that occur in the law. This raises several problems concerning the range of validity of empirical physical laws, which are well exemplified by the proofs of the theorems mentioned above.

Let us consider the Bell–Kochen–Specker’s theorem. All its proofs proceed ab absurdo and can be schematized as follows.

• (i) One assumes that for every physical object in a given state each observable has a prefixed value which is independent of any measurement but is obtained as outcome if an exact measurement is performed. Bearing in mind the definitions introduced in Sect. 2 this amounts to assume value definiteness and noncontextuality, that is objectivity, together with the adequacy condition for exact measurements.

• (ii) One assumes that all predictions of QM must be preserved.

• (iii) One observes that assumption (ii) implies the following Kochen-Specker condition. KS. Let be $f$ a function and let $A, B, C,\ldots ,$ be mutually compatible observables represented in QM by the operators ${\hat{A}}, {\hat{B}}, {\hat{C}},\ldots ,$ respectively. If

  $$\begin{aligned} f\left( {\hat{A}},{\hat{B}},{\hat{C}},\ldots \right) =0 \end{aligned}$$

  (3.1)

  is an empirical law of QM and simultaneous exact measurements of $A, B, C,\ldots$, are performed, then the outcomes a, b, c, ..., of $A, B, C,\ldots$, respectively, must satisfy the following equation

  $$\begin{aligned} f\left( a,b,c, \ldots \right) =0. \end{aligned}$$

  (3.2)

• (iv) One considers a set of empirical quantum laws

  $$\begin{aligned} \left\{ {{\begin{array}{l} {f\left( {\hat{A}},{\hat{B}},{\hat{C}},\ldots \right) =0} \\ {g\Big ({\widehat{{A}^{\prime }}},{\widehat{{B}^{\prime }}},{\widehat{{C}^{\prime }}},\ldots \Big )=0} \\ {\ldots \,\ldots \,\ldots \,\ldots } \\ \end{array} }} \right. \end{aligned}$$

  (3.3)

  and deduces from (iii) that the corresponding equations $f(a, b, c,\ldots )=0$, $g(a^{\prime },b^{\prime }, c^{\prime },\ldots )=0$, $\ldots$ must hold simultaneously.

• (v) One observes that, because of (i), the outcomes $a, b, c, {\ldots }, a^{\prime }, b^{\prime }, c^{\prime }, {\ldots },$ represent preexisting values of the observables $A, B, C, {\ldots }, A^{\prime }, B^{\prime }, C^{\prime }, {\ldots },$ respectively, which are defined independently of any measurement procedure for any physical object.

• (vi) One shows that the empirical laws in (3.3) can be chosen in such a way that the values $a, b, c, {\ldots }, a^{\prime }, b^{\prime }, c^{\prime }, {\ldots },$ cannot simultaneously satisfy the equations $f(a, b, c,\ldots )=0,\,g(a^{\prime }, b^{\prime }, c^{\prime }, \ldots ) = 0, {\ldots }$

• (vii) One deduces from (vi) that the simultaneous assumption of QM and noncontextuality leads to a contradiction. Since, of course, one does not want to refute QM, one is obliged to reject noncontextuality (hence objectivity).

The problematic step in the above scheme is item (iv), because it introduces physical situations that can be described in the metalanguage of QM but are not epistemically accessible. Indeed, applying the KS condition to many empirical laws at the same time implies that one assumes the simultaneous validity of these laws. But, if the laws are chosen in such a way to obtain the contradiction pointed out in (vii), then for every pair of laws there are observables in one of the laws that are not compatible with some observables in the others. Therefore, the physical situation in which all observables in the aforesaid laws have simultaneously defined values can be described in the metalanguage of QM but is not epistemically accessible because only one (at choice) of the empirical laws can be checked. The set of empirical laws in (3.3) cannot be simultaneously confirmed or falsified in this kind of physical situations. Hence one can choose between the following alternative ways of assigning a truth value to a sentence which states an empirical law of the set.

MCP (metatheoretical classical principle). A sentence stating an empirical physical law is true in every physical situation, be it epistemically accessible or not.

MGP (metatheoretical generalized principle). A sentence stating an empirical physical law is true in every epistemically accessible physical situation, but it can be true as well as false in a physical situation that is not epistemically accessible.

When proving Bell–Kochen–Specker’s theorem the repeated application of the KS condition implies postulating the unrestricted simultaneous validity of all empirical quantum laws in (3.3) (item (iv)). Hence one assumes the truth of all sentences which state these laws in a physical situation that is not epistemically accessible, that is, one implicitly adopts MCP. This principle looks natural if one believes that the mathematical apparatus of QM reflects some kind of physical reality and that the theoretical laws of QM correspond, with some unavoidable approximations, to the laws of physical reality (realism of theories: see Hacking 1983). In this view it seems obvious to assume that the empirical laws which are deduced from theoretical laws of QM necessarily hold in any physical situation, that is, to assume the unrestricted validity of the empirical laws deduced from the general formalism of QM. MGP is instead consistent with a more cautious perspective that considers the general theoretical laws of QM as formal statements, which are useful to produce empirical laws (by deduction and correspondence rules), but have no truth value, for the theoretical language of the theory has no direct and complete semantic interpretation (footnote 5). Therefore MGP seems more appropriate to the empiristic and “anti-metaphysical” demands of QM because it avoids any extension of the validity of the empirical laws beyond the domain in which they are verifiable.

The above analysis has important consequences. First, it shows that, contrary to common beliefs, one cannot prove that nonobjectivity follows from the mathematical formalism of QM without introducing an epistemological assumption that it is problematic from the point of view of QM itself. Second, it entails that the existing proofs of Bell–Kochen–Specker’s theorem cannot be completed if one adopts MGP instead of MCP. MGP, in fact, implies that there may be sentences stating empirical laws that are false in physical situations that are not epistemically accessible. Hence the KS condition cannot be applied simultaneously to all laws that are introduced in the proof of Bell–Kochen–Specker’s theorem.

Analogous conclusions can be achieved in the case of Bell’s theorem. Indeed, it can be shown that also the proofs of nonlocality require the implicit adoption of MCP (Garola and Pykacz 2004).

Both results reported above suggest that it is possible to contrive interpretations of QM that avoid the problems following from nonobjectivity by replacing MCP with MGP. We therefore discuss this possibility in the next sections. Before closing this section, however, we would like to mention an argument of non-ontological nature that can be used to support the adoption of MCP (a similar argument is often used against the upholders of “local realism”, see footnote 4). One can consider indeed an ensemble of identically prepared physical objects, partition it into identical subensembles, and then test one of the laws in (3.3) in each subensemble, choosing different laws in different subensembles. If QM is correct, each test will confirm the law that has been chosen. Hence one cannot understand how a law could not hold in a subensemble in which one has chosen to test another law. Accepting such a possibility would imply assuming a kind of conspiracy of nature to prevent us from discovering the breakdowns of the selected laws, which is obviously paradoxical.

The above argument is seemingly sound but it depends on the implicit premises that each physical property which occurs in the laws of QM has an empirical interpretation and that all physical objects are detected when exact measurements are performed. If one or both these premises are given up, the argument loses its validity. We shall see in the next section that our proposed alternatives just question the first or the second premise.

4 Objective Alternatives

The criticism of Bell’s and Bell–Kochen–Specker’s theorems resumed in Sect. 3 is pointed at the epistemological perspective that underlies the premises of the proofs (given the premises, the proofs are obviously correct). As we have anticipated at the end of Sect. 3 it suggests that one should renounce MCP in favour of the weaker MGP, and that in this case alternative interpretations of QM restoring objectivity are not a priori impossible. It does not guarantee, of course, that interpretations of this kind exist. In particular, one can object that Aspect’s and similar experiments have shown that Bell’s inequalities are violated, as predicted by QM, which proves nonlocality, hence nonobjectivity.⁶ This objection is not insurmountable, because it is well known that every set of experimental results can be predicted, in principle, by a plurality of theories, which provide different explanations of them (Chalmers 1999). In other words, the experimental confirmation of a physical theory is not sufficient to select it as the unique theory explaining the obtained results. Nevertheless, we think that the above objection must be taken into due account, and that every attempt at reinterpreting and generalizing QM must satisfy at least two basic conditions.

• (i) It must explain the success of QM and reproduce (within the limits of the experimental errors) the predictions of QM that have been experimentally confirmed.

• (ii) It must recover in some way the basic mathematical formalism of QM that led to such successful predictions.

Bearing in mind the conditions above, one of us, together with some collaborators, has provided two different approaches for recovering objectivity preserving the mathematical structure of QM. To understand the basic ideas underlying these approaches let us note preliminarily that the “no-go” theorems and most h.v. theories share some common premises besides implicitly adopting MCP. In particular, they accept the standard assumption in QM that, in absence of superselection rules, the representation of the physical observables of a physical system $\Omega$ by means of self-adjoint operators on the Hilbert space associated with $\Omega$ is surjective. Moreover, they interpret the quantum probability provided by the Born rule (that is, the probability that a physical object $\alpha$ in the state $S$ displays the physical property $E$ when $E$ is measured) as absolute (intuitively, as the large numbers limit of the frequency of E in the set of all physical objects in the state $S)$. Both these premises are relevant to our aims. Indeed, our first approach, besides replacing MCP with MGP, also weakens the former premise above, assuming that the support of a pure entangled state $S$ (that is, the physical property represented by the same one-dimensional projection operator that represents $S$ in QM) is purely theoretical and has no empirical counterpart. In this way, objectivity can be recovered at a semantic level, in the sense that a truth value (true/false) can be defined for every statement that assigns a property to a physical object, avoiding ontological commitments. As anticipated in Sect. 1 this approach was called semantic realism (SR) interpretation (Garola 1999, 2000, 2002; Garola and Solombrino 1996a, b). Successively this interpretation was modified and generalized, giving rise to the simpler and more intuitive extended semantic realism (ESR) model (Garola 2002, 2003; Garola and Pykacz 2004; Garola and Sozzo 2009, 2010b, 2011a, b, c, 2012; Sozzo and Garola 2010). This model does not replace a priori MCP with MGP and does not renounce the surjective mapping of the set of physical observables on the set of self-adjoint operators. It reinterprets instead quantum probabilities as conditional on detection rather than absolute, modifying the second premise of the “no-go” theorems and standard h.v. theories mentioned above (by referring to the empirical laws of QM one can then show that MGP holds while MCP fails to hold in the ESR model). This reinterpretation leads to embody the mathematical formalism of QM into a broader formalism which provides also absolute probabilities and admits an objective interpretation. We maintain that this result is interesting for the reasons exposed in Sect. 1 and therefore devote the next section to supply an informal exposition of the basic ideas of the ESR model.

5 The ESR Model

In the ESR model the weak form of realism specified at the beginning of Sect. 1 is adopted. This model therefore deals with physical objects and their properties. As in QM, a physical system $\Omega$ is characterized by a set of observables or, equivalently, by a set of physical properties (each observable is associated indeed with a family of physical properties). Moreover, the ESR model does not assume a priori that all physical properties characterizing $\Omega$ are objective (in the sense specified in Sect. 2). It can instead be proven a posteriori that an objective interpretation of the ESR model is possible.

Intuitively, the basic idea underlying the ESR model is that the set of physical properties possessed (i.e. objective with value true, see Sect. 2) by a physical object $\alpha$ may be such that $\alpha$ has a nonzero probability of remaining undetected whenever a physical property E is measured on it, even if no cause of inefficiency occurs in the measuring apparatus that is used to perform the measurement (of course such a probability depends on $E)$. Hence $\alpha$ may remain undetected even if an idealized measurement of $E$ is performed.⁷ This assumption radically changes the standard perspective and has long-ranging consequences. Indeed, it is a fundamental assumption of QM that an exact measurement of a physical property always establishes whether this property is possessed or not possessed by a physical object after the measurement, even if it may not preexist to the measurement. Nondetection may only occur because of inefficiency of the real measuring device.

Based on the idea exposed above, one is led to replace each quantum observable $A$ of $\Omega$ with a generalized observable $A_{0}$, whose set $\varXi _{0}$ of all possible outcomes is obtained by adding a no-registration outcome $a_{0}$ to the set $\varXi$ of all possible values of $A$. Every generalized observable $A_{0}$ of $\Omega$ is then associated with a family of physical properties, each of which is a pair ($A_{0},\,\varSigma$), with $\varSigma$ a subset of $\varXi _{0}$ and the correspondence is one-to-one. Therefore the no-registration outcome is considered as a possible result of an idealized measurement of $A_{0}$, providing information about the set of physical properties possessed by $\upalpha$.⁸ The set $\fancyscript{F}_{0}$ of all physical properties contains in particular the subset $\fancyscript{F}$ of all pairs of the form ($A_{0},\,\varSigma$) such that $A_{0}$ is a generalized observable and $a_{0}$ does not belong to $\varSigma$. This subset is in one-to-one correspondence with the set of all physical properties characterizing $\Omega$ according to QM: hence the two subsets can be identified, and we make this identification in the following for the sake of simplicity. Coming to states, the notion of state is directly transferred from QM to the ESR model, together with the standard distinction between pure states and (proper and improper) mixtures. The set of all states will then be denoted by $\fancyscript{S}$. Whenever a physical property $F\in \fancyscript{F}$ is measured on a physical object $\alpha$ in a state $S\in \fancyscript{S}$ (we assume that each measurement is idealized from now on) it may occur that the physical properties possessed by $\alpha$ are such that $\alpha$ is not detected. Hence one is led to introduce a probability $p_S^d (F)$ that $\alpha$ is detected in the measurement (detection probability, or intrinsic detection efficiency), which depends on $F$ and $S$ but not on the measuring apparatus. Therefore the overall probability $p_S^t (F)$ that $\alpha$ displays $F$ is given by

$$\begin{aligned} p_S^t (F) = p_S^d (F)p_S(F), \end{aligned}$$

(5.1)

where $p_S(F)$ is the conditional probability that $\alpha$ displays $F$ whenever it is detected in the measurement. Furthermore, it is easy to see that, if a physical property $F=(A_{0},\,\varSigma )\in \fancyscript{F}_{0}\backslash \fancyscript{F}$ (hence $a_{0}$ belongs to $\varSigma$) is measured on $\alpha$, then the overall probability $p_S^t(F)$ is given by

$$\begin{aligned} p_S^t (F)= 1-p_S^t (F^{c}) \end{aligned}$$

(5.2)

with $F^{c}=(A_{0},\,\varXi _{0}\backslash \varSigma )\in \fancyscript{F}$. We therefore only deal with physical properties in $\fancyscript{F}$ in the following.

Let us come now to the fundamental assumption of the ESR model. We state it as follows.

AX. If $S$ is a pure state and $F\in \fancyscript{F}$, then the probability $p_S (F)$ coincides with the quantum probability that a physical object $\alpha$ in the state $S$ displays F when $F$ is measured on $\alpha$.

Assumption AX allows one to recover the basic mathematical formalism of QM within the ESR model, as required by condition (ii) in Sect. 4. Yet, it substantially modifies its interpretation. Consider in fact an ensemble extS of physical objects prepared in the macroscopic state $S$. According to QM, whenever an exact (or, more restrictively, a first kind, ideal) measurement of a physical property $F\in \fancyscript{F}$ is performed, all physical objects in extS are detected and the quantum rules yield the probability that a physical object $\upalpha \in$ extS displays $F$ (absolute probability). According to the ESR model, instead, whenever an idealized measurement of $F$ is performed, only the physical objects in a subset $(extS)^{d}\,\subset \,extS$ are detected, and the quantum rules yield the probability that a physical object $\alpha$ displays $F$ if it belongs to the subset $(extS)^{d}$ (conditional on detection probability).

The reinterpretation above has some important consequences. First of all, it entails that the mathematical formalism of QM must be extended if one wants to predict absolute probabilities. But we have as yet no theory which allows us to evaluate the detection probability $p_S^d (F)$. Therefore we must consider this probability as a numerical parameter (depending on $S$ and $F$) which can be determined experimentally, at least in principle. Moreover, bearing in mind that $F = (A_{0},\varSigma$), with $A_{0}$ a generalized observable and $\varSigma$ a subset of the real axis $\mathbb {R}$, we can introduce a detection probability density defined on $\mathbb {R}$. This function can be combined with the mathematical formalisms of QM to obtain the mathematical representations of states, generalized observables and physical properties which must be used in the ESR model for evaluating absolute and conditional on detection probabilities. The details of this procedure can be found in the papers listed at the end of Sect. 4. We limit ourselves here to report its final results and briefly comment on them.

(i) The mathematical representation of pure states.

Assumption AX implies that pure states are represented by unit vectors of Hilbert space $\fancyscript{H}$ associated with $\Omega$, or, equivalently, by one-dimensional (orthogonal) projection operators on $\fancyscript{H}$, as in QM. We assume in the following that no superselection rule exists. Hence the mapping of the set $\fancyscript{V}$ of all unit vectors of $\fancyscript{H}$ on the set of pure states of $\Omega$ is surjective, while the correspondence between pure states and one-dimensional projection operators is bijective.

(ii) The mathematical representation of physical properties.

As far as only pure states are concerned, the mathematical representation of a physical property $F=(A_{0},\varSigma )\in \fancyscript{F}$ is provided by the pair $(P^{\hat{A}}(X),\,\{T_\psi ^{\hat{A}}(X)\}_|\psi >\in \fancyscript{V})$. In this representation $\hat{A}$ denotes the self-adjoint operator that represents the quantum observable $A$ from which $A_{0}$ is obtained and $X$ is any Borel set on the real line $\mathbb {R}$ such that $X\cap \varXi _{0}=\varSigma$. Then, $P^{\hat{A}}(X)$ is the projection operator associated with $X$ by the spectral decomposition of $\hat{A}$: hence it coincides with the standard representation of $F$ in QM. $T_\psi ^{\hat{A}} (X)$ is instead a linear, bounded, positive operator such that $\Vert T_\psi ^{\hat{A}} (X)\Vert <1$ (effect), which is defined via the detection probability density and the spectral decomposition of $\hat{A}$. By using the projection operator $P^{\hat{A}}(X)$ and the family $\{T_\psi ^{\hat{A}} (X)\}_|\psi >\, \in \fancyscript{V}$ one can provide mathematical expressions for the conditional on detection and the overall probability, respectively, that a physical object $\alpha$ in the state $S$ represented by $|\psi$ > displays $F$ when $F$ is measured on $\upalpha$. The conditional on detection probability is obtained following standard quantum rules, in accordance with assumption AX. The overall probability is obtained by following rules similar to standard quantum rules, with $T_\psi ^{\hat{A}} (X)$ in place of $P^{\hat{A}}(X)$, but in this case the representation of $F$ depends on the state $S$ of $\upalpha$. This is a distinguishing feature of the ESR model, which does not occur in previous generalizations of QM, as unsharp QM.

(iii) The mathematical representation of generalized observables.

Still restricting to pure states, the mathematical representation of a generalized observable $A_{0}$ is provided by the pair $(\hat{A},\,\{T_\psi ^{\hat{A}}\}_|\psi >\, \in \fancyscript{V} )$. In this representation $T_\psi ^{\hat{A}}$ denotes the positive operator valued (POV) measure on ${\mathbb {R}}$

$$\begin{aligned} T_\psi ^{\hat{A}}:X\in {\mathbb {B}}({\mathbb {R}})\longrightarrow \, T_\psi ^{\hat{A}} (X)\in {\mathfrak {B}}(\fancyscript{H}), \end{aligned}$$

(5.3)

where ${\mathbb {B}}({\mathbb {R}})$ is the set of all Borel sets on ${\mathbb {R}}$ and ${\mathfrak {B}}(\fancyscript{H})$ is the set of all linear bounded positive operators on $\fancyscript{H}$. The first element of the pair coincides with the representation of the quantum observable $A$ from which $A_{0}$ is obtained and must be used to calculate conditional on detection probabilities, in accordance with assumption AX. The second element of the pair allows one to evaluate overall probabilities, but the representation of $A_{0}$ that must be used depends on the state $S$ of the physical object on which measurements are performed, as in the case of physical properties.

(iv) The mathematical representation of mixtures.

The representations of physical properties and generalized observables provided above refer to pure states only, but it can be proven that they allow one to provide mathematical expressions of conditional on detection and overall probabilities also in the case of measurements on physical objects in mixed states (or mixtures). The physical and mathematical reasonings leading to this result also supply mathematical representations of mixtures in the ESR model (Garola and Sozzo 2011a, b, 2012). To be precise, an improper mixture $N$ is represented by the same density operator $\rho _N$ that represents it in QM. A proper mixture $M$ is represented instead by a family of pairs $\{(\rho _M (F),p_{M}^{d} (F))\}_{F\in \fancyscript{F}}$, where $\rho _M (F)$ and $p_M^d (F)$ are a density operator and a detection probability, respectively, which depend on $F$. Both elements of the pair are needed if one wants to evaluate the conditional on detection probability $p_M (F)$ and the overall probability $p_M^t (F)$ that a physical object $\alpha$ in the state $M$ displays $F$ when $F$ is measured on it.⁹

By using the representations provided above one can show that assumption AX can be straightforwardly extended to improper mixtures because such mixtures are represented as in QM (Garola and Sozzo 2012). Moreover, one can state suitable postulates that take the place of the Lüders postulate of QM in the ESR model (Garola and Sozzo 2009, 2011a, b, 2012; two postulates are actually needed, one applying to pure states and improper mixtures, one applying to proper mixtures). Finally, a general treatment of time evolution in the ESR model can be worked out. For the sake of simplicity we will not discuss all these topics in the present paper.

To conclude this section it remains to discuss whether the ESR model satisfies condition (i) in Sect. 4, that is, whether it can explain the success of QM and reproduce (within the limits of the experimental errors) the predictions of QM that have been experimentally confirmed. To this aim, we must distinguish two different kinds of physical experiments.

Firstly, let us consider experiments checking overall probabilities. Then, Eq. (5.1) together with assumption AX shows that the predictions of the ESR model are theoretically different from the predictions of QM. But the difference depends on the values of the detection probabilities. If these parameters (to be experimentally determined, as we stated above) are close to 1, such a difference may be negligible and remain unnoticed. Moreover, in the case of real measurements it may be appreciable but concealed by inefficiencies of the measuring apparatuses. We thus get a possible explanation of the success of QM from the point of view of the ESR model. There are, however, physical situations in which the predictions of the ESR model may diverge from the predictions of QM. This occurs, for example, in the case of proper mixtures, because of the difference between the mathematical representations of this kind of mixtures in the two theories (Garola and Sozzo 2012). In addition, the ESR model predicts upper limits of the efficiencies of measuring apparatuses in some typical EPR-like physical situation, which also may be experimentally relevant (Garola and Sozzo 2010b, 2011c). Thus, there are several possibilities of checking the ESR model, suggested by the model itself. No check of this kind, however, has been performed up to now.

Secondly, there are many experiments, as Aspect’s and successive experiments aiming to test Bell’s inequalities (Genovese 2005), which seemingly test absolute probabilities but actually test conditional on detection probabilities from the point of view of the ESR model. These experiments indeed take into account only detected physical objects, not all objects that are actually prepared. In this case, the predictions of the ESR model coincide with the predictions of QM.

6 The Issue of Objectivity

As we have observed in Sect. 1, the ESR model was conceived with the aim of avoiding the problematic consequences of nonobjectivity of QM. Therefore a noncontextual set-theoretical picture of the microscopic world underlying the macroscopic structure described in Sect. 5 was introduced as a constitutive part of the ESR model (its microscopic part) in all former presentations of the model. This picture can be considered as a h.v. theory for the ESR model, showing that it admits an objective interpretation. In the description of the ESR model provided in Sect. 5, however, no reference to h.v. theories was done, because we intended to stress that our generalization and reinterpretation of QM can be worked out without a priori referring to h.v. The microscopic part of the ESR model can then be introduced afterwards to show that the ESR model can be provided with an objective interpretation, and we devote the present section to a brief illustration of this topic.

Our h.v. theory for the ESR model introduces a set ${\fancyscript{E}}$ of purely theoretical microscopic properties as h.v. Whenever a physical object $\alpha$ is considered, every microscopic property either takes value 1 (it is possessed by $\upalpha )$ or 0 (it is not possessed by $\upalpha )$. The set of all microscopic properties possessed by $\upalpha$ is then called the microscopic state of $\upalpha$. Hence the set ${\fancyscript{S}}_{\mu }$ of all microscopic states is a subset of the power set ${\fancyscript{P}}({\fancyscript{E}})$ of ${\fancyscript{E}}$.

The connection between the microscopic and the macroscopic part of the ESR model is established by introducing the following assumptions.

(i) Every microscopic property $f\in {\fancyscript{E}}$ corresponds to a physical property $\varphi \left( f \right) =F\in {\fancyscript{F}}$, and the mapping $\varphi :{\fancyscript{E}}\longrightarrow {\fancyscript{F}}$ is bijective.

(ii) For every microscopic state $S_{\mu }\,\in {\fancyscript{S} }_{\mu }$ and state $S\,\in {\fancyscript{S}}$ (sometimes called macroscopic state, to stress the difference between $S$ and $S_{\mu })$ a conditional probability $p(S_{\mu }{\vert }S)$ is defined, interpreted as the probability that $S_{\mu }$ is the microscopic state of a physical object $\alpha$ in the macroscopic state $S$.

(iii) Whenever an (idealized) measurement of a physical property $F$is performed on $\alpha$, the microscopic state $S_{\mu }$ of $\alpha$ induces a probability $p_{S_\mu }^d (F)$ that $\alpha$ is detected. If the detection occurs and the yes (no) outcome is obtained, then $\alpha$ possesses (does not possess) the physical property $F$ from the macroscopic point of view illustrated in Sects. 2 and 5, while from the point of view of the microscopic picture illustrated above it possesses the microscopic property $f=\varphi ^{-1}(F)$. If no detection occurs, $F$ is not possessed by $\alpha$ according to the former point of view (indeed, if $F=(A_{0},\Sigma )\,\in {\fancyscript{F}}$ the outcome of the measurement is $a_{0}$, which does not belong to $\Sigma )$, while $f$ can be possessed as well as not possessed by $\alpha$ according to the latter point of view, because the measurement does not provide information on $f$ in this case.

Based on (iii) one can write down the equation

$$\begin{aligned} p_{S_\mu }^t (F) = p_{S_\mu }^d (F)p_{S_\mu } (F). \end{aligned}$$

(6.1)

The factor $p_{S_\mu }^t (F)$ in Eq. (6.1) denotes the overall probability that a physical object $\alpha$ in the microscopic state $S_{\mu }$ turns out to possess the physical property $F$ when $F$is measured on it. The factor $p_{S_\mu } (F)$ denotes instead the conditional probability that $\alpha$ displays $F$ when it is detected. We stress that the value of this factor is either 0 or 1 (if $f=\varphi ^{-1}(F)$ does not belong or belongs to $S_{\mu }$, respectively). It is then easy to see that Eq. (5.1) can be deduced from Eq. (6.1) by using assumption (ii) (Garola 2003; Garola and Pykacz 2004; Garola and Sozzo 2010b). Thus, the h.v. theory sketched above justifies the basic equation of the ESR model.

Let us come to objectivity. Let us firstly consider the special case of a deterministic ESR model in which, for every $S_{\mu } \in {\fancyscript{S}}_{\mu }$ and $F\in {\fancyscript{F}}$, the probability $p_{S_\mu }^d (F)$ is either 0 or 1 (we note that this restriction does not imply a similar restriction on the value of $p_S^d (F)$ in Eq. (5.1)). In this case the microscopic state $S_{\mu }$ of a physical object $\upalpha$ determines whether $\upalpha$ is detected or not whenever a measurement of $F$ is performed. If $\upalpha$ is detected, then $F$ either is possessed by $\upalpha$ (if $p_{S_\mu } (F)$ = 1) or it is not possessed by $\upalpha$ (if $p_{S_\mu } (F) = 0$). If $\upalpha$ is not detected, then $F$ is not possessed by $\upalpha$, as we have seen above. Hence the result of a measurement of $F$ depends only on the microscopic state $S_{\mu }$ of $\alpha$, not on the measurement context. Therefore a noncontextual h.v. theory can be constructed in the case of a deterministic ESR model. Moreover, value definiteness is assured by the h.v. theory. Bearing in mind the definition of objectivity provided in Sect. 2 we conclude that a deterministic ESR model can be provided with an objective interpretation.

Let us come to the general case. If the ESR model is not deterministic, then the detection probability $p_{S_\mu }^d (F)$ is not bounded to be either 0 or 1. In this case one must add an assumption in the h.v. theory for the ESR model. To be precise, one must assume that the detection probability is epistemic. This means that a further hidden variable $g$ can be introduced (which does not correspond to a macroscopic property because it does not belong to the domain of $\varphi )$ such that assigning the value $g(\upalpha )$ of $g$ on a physical object $\upalpha$ together with the microscopic state $S_{\mu }$ of $\upalpha$ determines whether $\upalpha$ is detected or not whenever a measurement of $F$ is performed. The proof that the ESR model admits an objective interpretation can now be carried on as the proof sketched above in the specific case of the deterministic ESR model, substituting the microscopic state $S_{\mu }$ of $\upalpha$ with the pair ($S_{\mu },g(\upalpha ))$ (Garola and Sozzo 2010b).

7 Bell’s Inequalities, GHZ Experiment and Quantum Logic within the ESR Model

We intend to show in this section that the ESR model throws a new light on some typical issues in the foundations of QM, solving some old problems and reconciling seemingly incompatible positions. Our discussion will be based on the results obtained in several previous papers and will avoid technical details, mainly concentrating on the methodological and epistemological meaning of these results.

(i) Bell’s inequalities.

By using the terminology introduced in Sect. 2 one can say that the original proofs of Bell’s theorem (Bell 1964; Clauser et al. 1969; Clauser and Horne 1974) show that introducing in QM the assumption of value definiteness together with the assumption of locality implies inequalities that are violated in QM. Some different proofs were successively produced that avoid resorting to inequalities (Greenberger et al. 1990; Hardy 1991, 1993; Mermin 1993; Calude et al. 1999). While also the latter proofs can be questioned from the point of view of the ESR model (Sect. 3) we concentrate here on Bell’s inequalities because they provide theoretical predictions that can be, and actually have been, experimentally checked (Genovese 2005).

The standard procedures leading to Bell’s inequalities can be resumed as follows. Let $\Omega$ be a composite system made up by two far-away subsystems $\Omega _{1}$ and $\Omega _{2}$. Let $A(a)$ and $B(b)$ be dichotomic observables of $\Omega _{1}$ and $\Omega _{2}$, depending on the parameters $a$ and $b$, respectively. Furthermore, let +1 and $-$1 be the possible values of $A(a)$ and $B(b)$, and let us consider a great number of exact measurements of $A(a)$ and $B(b)$ on items of $\Omega$ in the state $S$. Let each measurement be characterized by the value $\lambda$ of a random hidden variable defined on the set of all measurements and ranging over the measurable space $\Lambda$, and let $\rho (\lambda )$ be the probability density on $\Lambda$ (which depends on $S)$. If we assume value definiteness, then for every $\lambda$ the values $A(\lambda ,a)$ and $B(\lambda ,b)$ of $A(a)$ and $B(b)$, respectively, are defined, and the expectation value of the product of the dichotomic observables $A(a)$ and $B(b)$ in a state $S$ is

$$\begin{aligned} E(a,b) =\int \limits _\Lambda \,d\lambda \rho (\lambda )\,A(\lambda ,a)\,B(\lambda ,b). \end{aligned}$$

(7.1)

By considering different values of the parameters $a$ and $b$, one obtains the original Bell inequality

$$\begin{aligned} {\vert } E(a,b)-E(a,c) {\vert }\le 1+E(b,c) \end{aligned}$$

(7.2)

or the Clauser-Horne-Shimony-Holt inequality

$$\begin{aligned} {\vert } E(a,b)-E(a,b^{\prime }) {\vert } + {\vert } E(a^{\prime },b)+E(a^{\prime },b^{\prime }) {\vert }\le 2. \end{aligned}$$

(7.3)

It is then well known that there are examples (e.g., pairs of electrons in a singlet state) in which Eqs. (7.2) and (7.3) are violated by the expectation values predicted by QM for suitable choices of the parameters. According to the standard interpretation of QM this implies that value definiteness and locality cannot both hold in QM, and some further reasonings show that the former assumption can be maintained but locality must be dropped. Hence one concludes that QM is a nonlocal theory. By assuming the adequacy condition for exact measurements (Sect. 2) one can then check whether the equations above or the predictions of QM are fulfilled. It is a widespread belief that the experiments that have actually been performed show that Bell’s inequalities are violated while QM is confirmed.¹⁰

From the point of view of the ESR model the theoretical predictions for Aspect-like experiments coincide with the predictions of QM, as we have seen at the end of Sect. 5. Thus we expect that these predictions will be confirmed by the experimental data. However, the ESR model is an objective theory, as we have seen in Sect. 6, which means that value definiteness and noncontextuality (hence locality) hold in it. Hence one may wonder whether Bell’s inequalities also hold, possibly bearing a new interpretation because of the reinterpretation of quantum probabilities. It is easy to see that the answer is negative, and that different inequalities hold in this model. Indeed, the dichotomic observables $A(a),\,B(b),\,{\ldots }$ must be substituted by the trichotomic generalized observables $A_{0}(a),\,B_{0}(b),\,{\ldots },$ respectively, in each of which a no-registration outcome is adjoined to the outcomes +1 and $-$1. By calculating the expectation values of these generalized observables using overall probabilities, and restricting to generalized observables whose no-registration outcomes are 0, one can show that the following modified Bell’s inequalities must replace the inequalities (7.2) and (7.3), respectively,

$$\begin{aligned} {\vert } E(A_{0}(a),B_{0}(b)) - E(A_{0}(a),B_{0}(c)) {\vert }\le 1+E(A_{0}(b),B_{0}(c)) \end{aligned}$$

(7.4)

and

$$\begin{aligned} {\vert } E(A_{0}(a),B_{0}(b)) \!-\!E(A_{0}(a),B_{0}(b^{\prime })) {\vert }\!+\! {\vert } E(A_{0}(a^{\prime }),B_{0}(b)) \!+\!E(A_{0}(a^{\prime }),B_{0}(b^{\prime })) {\vert }\le 2,\qquad \quad \end{aligned}$$

(7.5)

where $E(A_{0}(a),B_{0}(b))$ denotes the expectation value of the product of the trichotomic observables $A_{0}(a)$ and $B_{0}(b)$, etc. (Garola and Sozzo 2010b, 2011a, c).

Equations (7.4) and (7.5) do not imply a priori any contradiction with QM because the expectation values that occur in them are calculated by using overall probabilities and do not coincide with quantum expectation values. However, to grasp the difference between Eqs. (7.2) and (7.3), on one side, and Eqs. (7.4) and (7.5), on the other side, it is expedient to consider a special case. We therefore assume that, for every $A_{0}\,\in \{A_{0}(a),\, B_{0}(b),\,{\ldots }\}$, the detection probability $p_S^d (A_{0},\Sigma )$ depends on $A_{0}$ but not on $\Sigma$, so that we can write $p_S^d (A_{0},\Sigma )=p_S^d (A_{0})$. In this case we obtain from Eqs. (7.4) and (7.5)

$$\begin{aligned} p_S^d (A_{0}(a)) {\vert } p_S^d (B_{0}(b)) E(a,b)-p_S^d (B_{0}(c))E(a,c) {\vert }\le 1+ \,p_S^d (A_{0}(b)) p_S^d (B_{0}(c))E(b,c)\nonumber \\ \end{aligned}$$

(7.6)

and

$$\begin{aligned}&p_S^d (A_{0}(a)) {\vert } p_S^d (B_{0}(b))E(a,b) -p_S^d (B_{0}(b^{\prime }))E(a,b^{\prime }) {\vert } \nonumber \\&\quad +\, p_S^d (A_{0}(a^{\prime })) {\vert } p_S^d (B_{0}(b)) E(a^{\prime },b) + p_S^d (B_{0}(b^{\prime })) E(a^{\prime },b^{\prime }) {\vert }\le 2\,. \end{aligned}$$

(7.7)

In Eqs. (7.6) and (7.7) the terms $E(a,b),\,E(a,c),\,{\ldots }$ are formally identical to the terms denoted by the same symbols in Eqs.  (7.2) and (7.3) but are now interpreted as conditional expectation values, that is, expectation values obtained by considering detected physical objects only. They coincide, because of assumption AX in Sect. 5, with standard quantum expectation values. But Eqs. (7.6) and (7.7) do not imply any contradiction with QM if the detection probabilities take suitable values. Hence these equations can be considered as constraints imposed by physics on the values of the detection probabilities that occur in them.

The result resumed above constitutes in our opinion an interesting achievement of the ESR model. But we can go a little further. Indeed, the h.v. theory for the ESR model sketched in Sect. 6 can be developed by introducing microscopic observables and their expectation values. One can then show that standard Bell’s inequalities hold at a microscopic level (which is purely theoretical and cannot be experimentally checked). Moreover, we have just stated that modified Bell’s inequalities hold at a macroscopic level (which can be experimentally checked). Finally, it follows from assumption AX in Sect. 5 that the quantum inequalities deduced by using standard QM rules hold at a macroscopic level whenever only detected physical objects are considered (which also can be experimentally checked). We can thus conclude that Bell’s inequalities, modified Bell’s inequalities and quantum inequalities do not conflict, but rather pertain to different parts of the picture provided by the ESR model. This “conciliatory” result is important in our opinion both from a physical and from an epistemological point of view (Garola and Sozzo 2010b).

(ii) The GHZ experiment.

More than twenty years ago, Greenberger, Horne and Zeilinger proved Bell’s theorem without introducing inequalities (Greenberger et al. 1990), thus avoiding several objections raised against the original proofs of this theorem (see footnotes 6,10). Their argument was based on a thought experiment which could be transformed into a real experiment (GHZ experiment). Twelve years later Szabó and Fine supplied a local h.v. theory for this experiment, thus showing that also explanations of it avoiding nonlocality were possible. To attain this result they introduced the assumption that “the detection efficiency is not (only) the effect of random errors in the detection equipment, but it is a more fundamental phenomenon, the manifestation of a predetermined hidden property of the particles” (Szabó and Fine 2002). In addition, their theory also requires that the probability measure which has to be defined on the space of the h.v. must satisfy some constraints following from QM. One can then easily see that introducing these constraints in the Szabó and Fine framework implies modifying the standard interpretation of quantum probabilities, considering them as conditional on detection rather than absolute. Szabó and Fine, however, did not provide a general theory of quantum phenomena which avoids nonlocality, and their assumption and reinterpretation refer to the specific case of the GHZ experiment only. A theory of this kind is instead supplied by the ESR model, based on assumptions that restate and generalize Szabó and Fine’s. Indeed, the general idea that intrinsic detection efficiencies exist which do not depend on flaws or lack of efficiency of the measuring apparatuses corresponds in the ESR model to the Szabó and Fine assumption reported above. Furthermore, the reinterpretation of quantum probabilities as conditional on detection in the ESR model corresponds to the aforesaid introduction of constraints in Szabó and Fine’s theory for the GHZ experiment. It is then natural to think that Szabó and Fine’s results should be recovered within the ESR model whenever the specific physical situation envisaged by Greenberger, Horne and Zeilinger is considered. This conjecture has been recently proven correct, at least in the case of the “toy” models supplied by Szabó and Fine for the GHZ experiment. Indeed a family of local finite models for this experiment which contains Szabó and Fine “toy” models can be singled out by considering different detection probabilities in the ESR model (Garola et al. 2013). This achievement seems interesting to us because it connects the ESR model with previous relevant but very specific results in the literature.

(iii) Quantum Logic.

It is well known that several nonstandard logical systems are interpreted as formalizing the features of notions of truth different from the classical notion of truth as correspondence, and that many logicians uphold that the plurality of logics is an important achievement which parallels the plurality of geometries that constitutes one of the revolutionary results of the mathematical research in the XIX century. Other logicians and philosophers instead maintain that nonstandard logics can be recovered as fragments of a suitable extension of classical logic (CL), in a unified view (global pluralism) which restores the unity of logic and avoids controversies about the notion of truth.

In the realm of physics many nonstandard logics were proposed after the birth of QM (Jammer 1974). In particular, Birkhoff and von Neumann’s quantum logic (QL), now often called standard (sharp) QL, seems to spring out directly from the mathematical formalism of QM (Birkhoff and Neumann 1936). Therefore many scholars maintain that QM introduces a nonclassical notion of truth (quantum truth) whose features are formalized by standard QL (Dalla Chiara et al. 2004). One may then wonder whether a perspective of global pluralism can be adopted also in the specific case of standard QL, so that this logical structure can be recovered within an extended classical framework. A first step in this direction was done in some previous papers, where it was shown that standard QL could be seen as the mathematical structure resulting from selecting a subset of sentences that are testable according to QM in the set of all sentences of a suitable classical language: QL would then formalize the features of the metalinguistic notion of testability in QM rather than a notion of quantum truth (Garola 1992, 2008). This view has been generalized and implemented in a recent paper (Garola and Sozzo 2013). In this paper a general procedure is worked out for obtaining a concrete (theory-dependent) logic associated with a physical theory ${\mathcal {T}}$ expressed by means of a classical language ${\mathcal {L}}$ with a notion of truth as correspondence (we use the term concrete logic here to indicate a logical structure that is closely related to the mathematical formalism of a physical theory, as in Dalla Chiara et al. 2004). This procedure consists of four steps.

• 1. An observational (pre-theoretical) sublanguage $L$ of ${\mathcal {L}}$ is considered, and a derived three-valued notion of C-truth (true/false with certainty, or indeterminate) is introduced in $L$, defined in terms of classical truth but depending on ${\mathcal {T}}$.

• 2. A physical preorder $\prec$, induced by the notion of C-truth, is defined in $L$.

• 3. A notion of verification is introduced in $L$ by selecting a subset $\Phi _{V}$ of sentences of $L$ that are verifiable, or testable, according to ${\mathcal {T}}$.

• 4. A weak complementation $\bot$ induced by ${\mathcal {T}}$ is defined on ($\Phi _{V},\prec )$.

The structure ($\Phi _{V},\prec ,\bot )$ is then the required concrete logic.

The above procedure is philosophically relevant. Indeed one can obtain a variety of concrete logics by changing some of its three basic elements, that is, ${\mathcal {T}}$, $L$ and the notion of verification. In particular, one can apply it by selecting a simple observational language $L(x)$ which is suitable for expressing basic notions and relations in a wide class of physical theories and considering QM. One then recovers standard QL as the concrete logic associated with QM (up to an equivalence relation) if a standard notion of verification according to QM is adopted. This realizes the aforesaid unified view in the case of QL, avoiding the problematic notion of quantum truth.

The result resumed above, however, has to face an important objection. Indeed, an orthodox quantum physicist would observe that it is obtained by associating a set with every property of a physical system (the extension of the property) whose elements are interpreted as physical objects possessing the property. But such a set cannot be defined according to the standard interpretation of QM because of nonobjectivity. It follows indeed from our definitions in Sect. 2 that nonobjectivity implies that the set of physical objects in a given state $S$ which display a given property $E$ whenever exact measurements of $E$ are performed may depend on the measurement context and is not prefixed. It is then apparent that this objection vanishes if one accepts the point of view of the ESR model, in which objectivity of physical properties is recovered. Thus, the ESR model provides a support to a philosophical position, that is global pluralism, which avoids in our opinion many problems in logic and related fields.¹¹