Truth values of quantum phenomena

Truth values of quantum phenomena Arkady Bolotin∗ Ben-Gurion University of the Negev, Beersheba (Israel)

arXiv:1708.00396v1 [quant-ph] 1 Aug 2017

August 2, 2017

Abstract In the paper, the idea of describing not-yet-verified (i.e., pre-existing) properties of quantum objects with logical many-valuedness is scrutinized. As it is argued, to promote such an idea one must first decide how to deal with bivalence of the everyday macrophysical experience in manyvalued semantics and, correspondingly, clear up the tradeoff between logical 2-valuedness and many-valuedness. The conceptual difficulties accompanying a two-valued reduction of manyvalued logics are discussed. Keywords: Quantum mechanics; Many-valued logics; Bivalence, Truth-functionality; Truth values; Quantum logic.

1

Introducing logical many-valuedness in quantum mechanics

The argument claiming that quantum theory could not be comprehended on the grounds of classical two-valued logic is rather straightforward and goes like this. Let us consider a typical quantum interference experiment where a quantum particle being released from a source is absorbed by a screen after passing through a two-slit barrier 1 . Suppose that immediately behind that barrier are placed two which-way detectors able to verify (e.g., by way of clicking) the particle’s passage through a corresponding slit. Let X1 denote the proposition of the click of the detector placed behind slit 1 such that X1 is true (denoted by “1”) if the detector clicks and X1 is false (denoted by “0”) if the detector does not. Let X2 in an analogous manner denote the proposition of the signal from the detector placed behind slit 2 and X12 stand for the exclusive disjunction on the truth values of the propositions X1 and X2 , explicitly, X12 ≡ X1 ∨ X2 ≡ (X1 ∨ X2 ) ∧ ¬(X1 ∧ X2 )

∗

.

(1)

Email : [email protected]

1

In the present paper, rather than being strictly restricted to spatially arranged slits, quantum interference is considered generally for any set of perfectly distinguishable alternatives.

1

The compound proposition X12 outputs true if either X1 or X2 is true, that is, X12 corresponds to the assertion that the particle passes through exactly one slit. Assume that the propositions X1 and X2 are in possession of pre-existing truth values – i.e., ones existing before the detectors can click – that are either revealed or transformed by the act of verification of the particle’s passage. In agreement with the given assumption, let us suppose that the pre-existing truth values of the propositions X1 and X2 are such that X12 is true. Provided that P is the probability function mapping propositions Y, Z, . . . to the unit interval [0, 1] such that P[Y ] = 1 if Y is true, P[Y ] = 0 if Y is false, and P[Y ∨ Z] = P [Y ] + P [Z] − P [Y ∧ Z], the probability of finding the particle at a certain region R on the screen would be given by the sum of the patterns P[R|X1 ] and P[R|X2 ] emerging in setups with a one-slit barrier, namely, P[R|X1 ∨ X2 ] = 21 P[R|X1 ] + 12 P[R|X2 ] (on condition that P[X1 ] = P[X2 ]). It would mean that in the case in which the pre-existing truth value of the compound proposition X12 is 1, quantum interference would be nonexistent even with none of the detectors present at the slits. So, by contrast, let us suppose that before the verification the truth values of X1 and X2 are both 1 and therefore the pre-existing truth value of the compound proposition X12 is 0. But then – in contradiction to the quantum collapse postulate – one would find that it is not true that only one of the detectors will signal if the particle’s passage through the two-slit barrier is observed. Thus, for the assumption of pre-existing truth values to be consistent with the occurrence of quantum interference and quantum collapse, the pre-existing truth value of the compound proposition X12 must be neither 1 nor 0. Such could be only if prior to the verification X12 does not obey the principle “a proposition is either true or false”, i.e., the principle of bivalence 2 . From the violation of this principle, one can infer that results of future non-certain events can be described using many-valued logics. For example, consider a 3-valued logic {0, 12 , 1} which includes only one additional truth value 21 besides the classical ones 0 and 1 3 . If pre-existing truth values of the propositions X1 and X2 are both 12 , where the truth value 12 is interpreted as “possible” and the valuations ¬ 12 = 12 , 21 ∨ 21 = 12 , 12 ∧ 21 = 12 hold, then the truth value of the compound proposition X12 will be 12 , i.e., also “possible”. In this case, before the verification one can only assert that the both statements – the particle passes through exactly one slit and the particle passes through more than one slit – are possible. Attractive as it might seem, the idea of describing not-yet-verified (i.e., pre-existing) properties of quantum objects with many-valued logics suffers from at least two major problems. 2

Obviously, it is possible to avoid this conclusion merely by accepting nonlocal realism (i.e., an interpretation of quantum theory in terms of ‘hidden variables’ such as Bohmian mechanics [1, 2, 3]). But in doing so one would confront with additional deficiencies that plague the ‘hidden variables’ approach (the analysis of those deficiencies can be found, e.g., in [4, 5]). 3

That might be such three-valued logical systems as the Kleene (strong) logic K3 or the 3-valued Lukasiewicz system [6, 7].

2

The first is the problem of choosing a proper system of many-valued logic. It concerns with the following question: Because there are infinitely many systems of many-valued logic, which of them should be chosen for the quantum mechanical description? How can a specific system be decided on to avoid a charge of arbitrariness? Notwithstanding the significance of the first problem, the next one seems even more serious: When the act of the verification is finished in the macrophysical domain (and so the path the particle has taken is known with certainty), the propositions X1 and X2 conform to the principle of bivalence. So, unless one can demonstrate that a many-valued logic has a proper application to the objects of everyday macrophysical experience (and for this reason our logic needs revising), the following question has to be answered: How do pre-existing multivalent truth values become classical bivalent truth values? As in the literature dealing with many-valued logics in quantum mechanics little is said about the process of passing from a many-valued to a 2-valued logic, the discussion of some aspects of this problem is offered in the present paper.

2

Preliminaries

We will start by briefly introducing a few necessary preliminaries. Let us consider a complete lattice L = (L, ⊔, ⊓) containing any set L where each two-element subset {y, z} ⊆ L has a join (i.e., a least upper bound) and a meet (i.e., a greatest lower bound) defined by y ⊔ z ≡ l.u.b.(y, z) and y ⊓ z ≡ g.l.b.(y, z), correspondingly. In addition to the binary operations ⊔ and ⊓, let the lattice L contain a unary operation ∼ defined in a manner that L is closed under this operation and ∼ is an involution, explicitly, ∼y ∈ L if y ∈ L and ∼(∼y) = y. Let §VN (⋄) = [[⋄]]v where the symbol ⋄ can be replaced by any proposition associated with a property of a physical system refer to a valuation, i.e., a mapping from a set of propositions being as stated denoted by S = {⋄} to a set VN = {v} where v are truth-values ranging from 0 to 1 and N is the cardinality of the set {v}: §VN : S → VN

(2)

.

At the same time, let us assume a homomorphism f : L → S such that there is a truth-function v that maps each lattice element denoted by the symbol ∗ to the truth value of the corresponding proposition, namely, v(∗) = [[⋄]]v

(3)

,

basing on the following principles: v(y) = 0

if 3

y = 0L

,

(4)

v(y) = 1

if

y = 1L

(5)

,

where 0L and 1L are the least and the greatest elements of the lattice, correspondingly. These principles imply that the least and the greatest lattice elements are identified with always false and always true propositions. Then, allowing that the following valuational axioms hold v(y ⊔ z) = [[Y ∨Z]]v

,

(6)

v(y ⊓ z) = [[Y ∧Z]]v

,

(7)

v(∼y) = [[¬Y ]]v

(8)

,

the truth value of the compound proposition X12 can be determined by the expression v(x12 ) = v





= [[X12 ]]v ⊔2i=1 xi ⊓ ∼ ⊓2i=1 xi

(9)

,

in which the lattice elements xi are attributed to the propositions Xi such that v(xi ) = [[Xi ]]v . On the other hand, the truth values of disjunction, conjunction and negation can be decided through the truth degree functions F∨ , F∧ and F¬ of the corresponding logical connectives: [[Y ∨Z]]v = F∨ ([[Y ]]v , [[Z]]v )

,

(10)

[[Y ∧Z]]v = F∧ ([[Y ]]v , [[Z]]v )

,

(11)

[[¬Y ]]v = F¬ ([[Y ]]v )

,

(12)

where, as stated by [6], the most basic example for a truth degree function of a negation connective F¬ is 1 − [[Y ]]v (called Lukasiewicz negation). To meet this version of negation, we will assume v(∼y) = 1 − v(y). 4 Let us consider the case where the following inclusion relation holds: x1 ≤ x2 ; in this circumstance, the lattice join and the lattice meet must be ⊔2i=1 xi = x2 and ⊓2i=1 xi = x1 , which gives 4



In [8], the relation between the functions v(y ⊔ z) and F∨ [[Y ]]v , [[Z]]v as well as v(y ⊓ z) and F∧ [[Y ]]v , [[Z]]v is studied to examine whether Lukasiewicz operations can also be used to model conjunctions and disjunctions. As it is stated in the paper, Lukasiewicz disjunction and conjunction coincide with the truth-functions of joins and meets, namely, v(y ⊔ z) = min {v(y) + v(z), 1} and v(y ⊓ z) = max {v(y) + v(z) − 1, 0}, whenever these Lukasiewicz connectives can be defined.

4

v(x12 ) = v (x2 ⊓ (∼x1 )) =



x2 ≤∼ x1 ∼x1 ≤ x2

v(x2 ), 1 − v(x1 ),

.

(13)

As it has been already noted, were the truth-function v(x12 ) to possess pre-existing values (consistent with the occurrence of quantum interference and quantum collapse), it would have to obey the requirement {v(x12 )} = 6 {0, 1} which, according to (13), implies that the values of v(xi ) would have to meet the condition {v(xi )} = 6 {0, 1} before the verification. But together with that, after the verification, the values of v(xi ) would have to conform to bivalence, that is, {v(xi )} = {0, 1}. Consequently, the question is, how to combine in one theory those two mutually exclusive conditions on the values of the truth-functions v(xi )?

3

Many-valued quantum logic

A solution to this problem can be motivated by recalling that the total of all the individual probabilities equals 1, so, when one of the probabilities turns into 1, all the others become 0. Let us add some details to this idea 5 . Assume that there is a correspondence (homomorphism) between a lattice L = (L, ⊔, ⊓) and a family L(H) containing all closed, ordered by the subset relation subspaces of a (separable) Hilbert space H associated with a physical system under investigation. Explicitly, the inclusion relation between the lattice elements corresponds to the subset relation between the closed subspaces, the lattice meet ⊓ corresponds to the intersection ∩ of those subspaces and the lattice join ⊔ is the closed span of their union ∪, the least element of the lattice is the {0} subspace and the greatest element of the lattice is the identical subspace H. Consider a self-adjoint projection operator on H that represents a proposition declaring that the system possesses some experimentally verifiable property (such as a path that the particle takes getting through the barrier). Since each projection (denoted by the symbol ∗) leaves invariant any vector lying in its range, ran(∗), and annihilates any vector lying in its null space, ker(∗), giving in this way a decomposition of H into two complementary closed subspaces, namely, H = ran(∗)⊕ker(∗), there is a one-to-one correspondence between the closed subspaces and the projections. Therefore, one can consider the projections as the elements of L(H). In view of the homomorphism between L = (L, ⊔, ⊓) and L(H), the principles of calculus of truth values presented above are expected to survive the passage to the context of L(H). For this reason, one can put forward that v (ran(∗)) = v(∗) = [[⋄]]v 5

,

(14)

In fact, this idea – called many-valued quantum logic or fuzzy quantum logic – has already been developed in a series of papers [9, 10, 11, 12, 13, 14]; however, for the aim of this discussion, it is not necessary to follow those papers precisely. Also, for the discussion it is immaterial to present in its entirety the generally accepted interpretation of the elements of a quantum logic – an interested reader can be referred to any textbook on quantum logic: see, for example, [15] or [16].

5

where v(∗) is a truth-function value at projection operator ∗ which corresponds to a truth value of proposition ⋄. Suppose that the state of the system is characterized by the unit vector |Ψi. It is not difficult to see that the truth values of the proposition Y will coincide with eigenvalues of the projection operator yˆ, i.e., {v(ˆ y )} = {0, 1}, if and only if the given vector |Ψi lies in either ran(ˆ y ) or ker(ˆ y ). However, if |Ψi is such a unit vector in H that |Ψi 6∈ ran(ˆ y ) as well as |Ψi 6∈ ker(ˆ y ), one can only get {hΨ|ˆ y |Ψi} = {¯ y ∈ R | 0 < y¯ < 1}, where y¯ is the expected value of the observable y corresponding to the projection yˆ. Consistent with the orthodox quantum theory, the value hΨ|ˆ y |Ψi can be interpreted as the probability that the measurement of the observable y will produce the answer 1 in the state of the system given by |Ψi (meaning that the proposition Y will be proved to be true), i.e., P[Y ] ≡ hΨ|ˆ y |Ψi

.

(15)

But, as stated by the idea being discussed here, the value hΨ|ˆ y |Ψi should be also regarded as the pre-existing truth value of the proposition Y , namely, v(ˆ y ) ≡ hΨ|ˆ y |Ψi

.

(16)

Accordingly, in general, i.e., if |Ψi is any unit vector, {v(ˆ y )} = {v ∈ R | 0 ≤ v ≤ 1} which can be interpreted as a generalization of the Boolean domain V2 = {0, 1}. Along these lines, one should assume that the value hΨ|ˆ y |Ψi represents the degree to which the proposition Y is true prior to the verification (i.e., before the experiment designed to verify the fact of possessing the property declared by the proposition Y could be completed) 6 . When the experiment is completed, the probability that the system possesses the said property becomes either 1 or zero. Consequently, the truth-function v(ˆ y ) will be then contained in the twopoint set V2 . Though it seems convincing from the mathematical point of view, treating the probability P[Y ] as the pre-existing truth value of the proposition Y presents at least two conceptual difficulties. Let us analyze those difficulties. 6

At the same time, v(∼ yˆ) ≡ 1 − hΨ|ˆ y|Ψi represents the degree to which the pre-existing truth value of the proposition Y is not true (that is, the degree to which the system does not possess the mentioned property prior to the verification).

6

4

Propositions vs. probabilities

First, the hypothesis (16) is parallel to the requirement that a truth assignment be indistinguishable from a probability assignment, namely, [[Y ]]v ≡ P[Y ]

.

(17)

Suppose, for example, that a proposition Y is assigned the true or the false; then, in accordance with the equality (17), the probability assignment P[Y ] is supposed to be likewise true or false. However, according to the epistemic interpretation of probability (favorable in probabilistic semantics, see, e.g., [17]), P[Y ] is treated as the degree to which, given the state |Ψi as an assumption, one is warranted in believing Y to be true. As much as a belief or a confidence is neither true nor false, a probability assignment cannot be true or false. 7 Furthermore, the right-hand side of the equality (17) is subjective since it depends upon someone (called the agent), at the same time as the left-hand side of this equality is objective. Indeed, gathering data allows the agents to update their probability assignments (by using Bayes’ theorem); so, the probability P[Y ] always depends on the agents’ prior probabilities as well as on the data and thus can be different (e.g. not equal to 1) for agents in possession of the same data. 8 The same relates to the quantum state |Ψi which is every bit as subjective as any Bayesian probability. In contrast, the truth assignment [[Y ]]v = 1 is the statement that y = 1 occurs, and for this reason it is a fact for any agent. Finally, the equality (17) cannot be right because it represents as equivalent two different states of affair, namely, the vague (fuzzy) valuation [[Y ]]v ∈ V∞ , which uses infinite-valued truth values (degrees of truth) as a mathematical model of vagueness (to be exact, as a tool for modeling reasoning with vague predicates and propositions), and the probability P[Y ] which is a mathematical model of ignorance. Thus, P[Y ] means that the variable y is either equal to 1 or not equal to 1: One does not know which and so describes one’s ignorance with the probability. On the other hand, the fuzzy valuation [[Y ]]v measures the degree to which the variable y is equal to 1, not whether this variable happens to be 1. Even if at some point P[Y ] and [[Y ]]v can be superficially equivalent in terms of their numerical uncertainty (say, both are 21 ), one cannot assume that they will always stay the same: After all, they are ontologically distinct as one is “random” while the other is “fuzzy”. 9 7

As it is put in [18], “The truth of statements concerning probabilities cannot be decided, not even approximately, and not even in principle. The usual method of “verifying” probabilities, through the outcomes of repeated trials, yields outcome frequencies, which belong to the category of events and propositions and are not probabilities. Probability theory allows one to assign a probability (e.g., p = 0.99) to the proposition “the outcome frequency is in the interval 0.29 ≤ f ≤ 0.31”, and the truth of this proposition can be unambiguously decided, but this is a proposition about an outcome frequency, not about probability.” 8

As stated by Bayesian approach to probability theory, probabilities are degrees of belief, not facts. Probabilities cannot be derived from facts alone. Two agents who agree on the facts can legitimately assign different prior probabilities. In this sense, probabilities are not objective, but subjective (see, e.g., [19, 20, 21, 22]). 9 More thorough discussion of the conceptual and theoretical differences between ignorance and fuzziness can be found in many papers: see, e.g., [23, 24, 25]. But, in a nutshell, it can be caught by a slogan coined in [26], “Fuzziness does not require that God play dice.”

7

5

Two-valued conversion of many-valued logic

Second, the hypothesis (16) implies that there is an assignment of pre-existing truth values, in particular, truth-values of infinite-valued logics, to all the elements of the family L(H) which can yield empirically observed outcomes. But then again, if there is a theory able to assign truth values of infinite-valued logics to the experimentally provable propositions before their validation, then such a theory is expected to explain how an infinite-valued semantics can be associated with a two-valued semantics emerging during the verification. Otherwise, why should one even consider a theory that uses logics with more than two values if only a bivalent logic can be observed? 10 That is to say, in order to be predictive, such a theory must provide the function composition: {V (v (∗))} = V2

(18)

,

where v is the truth-function corresponding to the pre-existing logic with the set of truth values VN , whilst V denotes the function V : VN → V2

(19)

that during the verification maps the objects of the set VN = {v} into the objects of the 2-point set V2 with respect to the order, i.e., if

v′ ≤ v′′

V (v′ ) ≤ V (v′′ )

then

.

(20)

Let us assume that the said composite function exists. A particular form of this function may depend on the grading of multivalued truth values VN . Thus, suppose that every truth-value of the set VN = DN ∪ UN is graded in terms of its membership to the sets of designated DN and undesignated UN truth-values such that 1 ∈ DN ⊆ VN

,

(21)

0 ∈ UN ⊆ V N

,

(22)

together with DN ∩ UN = ∅. Then, as it is follows from, e.g., [28], the residual bivalence embodied in the distinction between designated and undesignated values can allow to effectively reformulate an original N -valued semantics using at most two truth-values so that 10 To borrow an idea from [27], even if one uses a many-valued logic as a certain way of speaking about situations in everyday life, one must use two-valued logic in the metalanguage (e.g., to make statements about statements in a many-valued language).

8

V (v (∗)) = 1

if

v (∗) ∈ DN ,

otherwise

V (v (∗)) = 0

(23)

.

Be that as it may, let us consider a situation where the pre-existing truth values v(ˆ y ) and v(∼ˆ y ) of the proposition Y and its negation ¬Y do not satisfy the standard condition, explicitly, v (ˆ y ) 6∈ DN

and v (∼ yˆ) = (1 − v(ˆ y )) 6∈ DN

(24)

,

which can be if the hypothesis (16) holds true: e.g., v (ˆ y) = hΨ|ˆ y |Ψi = 21 and 21 6∈ DN . Accordingly, the truth values v(ˆ y ), v(∼ˆ y ) ∈ VN must be converted to ones of the set V2 such that V (v(ˆ y )) = [[Y ]]V = 0 V (v(∼ˆ y )) = [[¬Y ]]V = 0

(25)

,

(26)

.

The problem, however, is not quite that both Y and ¬Y come out false in this case 11 but rather that the expression for [[¬Y ]]V appears to be not well-defined. To be sure, this expression gives contradictory results when its form (the way in which it is presented) is changed but the value of its input, i.e., v(ˆ y ), is not changed, namely, V (v(∼ˆ y)) = [[¬Y ]]V = V (1 − v(ˆ y )) = 0

(27)

,

V (v(∼ˆ y )) = [[¬Y ]]V = F¬ ([[Y ]]v ) = 1 − [[Y ]]V = 1 − V (v(ˆ y )) = 1

.

(28)

Explained differently, since negation can be regarded as a logical connective, i.e., an operation on the truth value of a proposition (namely, subtraction from the numerical representation of true) and – at the same time – as a function defined on the set of truth values, the expression (26) cannot be unambiguous. So, unless one believes that in a quantum domain, a truth value is not an object that can be converted or(and) a negation connective is not Lukasiewicz negation (i.e., subtraction from the true), there cannot be a well-defined conversion [[⋄]]v → [[⋄]]V and, as a result, the use of logical manyvaluedness in quantum theory cannot have predictive power even in principle. Otherwise stated, regardless of what underlying system of pre-existing logical many-valuedness is presupposed, it may have no effect on the truth values of experimental propositions decided by the way things stand with respect to a verification macrophysical domain. 11

There is a bivalent semantics where such a valuation is possible: In this semantic the first half of the principle of bivalence, namely, “a proposition cannot be neither true nor false”, holds even though the principle of excluded middle, i.e., “a proposition and its negation cannot be false together ”, does not.

9

6

Introducing logical zero-valuedness in quantum mechanics

A way out of the difficulties posed in Sections 4 and 5 might lie in the acknowledgment that in orthodox quantum mechanics, truth values of the future non-certain events simply do not exist. Such could be if any lattice element different from the least and the greatest elements carries no truth values, that is, {v (∗) | ∗ = 6 0L and ∗ = 6 1L } = ∅ whereas

v(0L ) = 0 and

v(1L ) = 1

.

(29)

Along the lines of this assumption, prior to the verification, an object S = {⋄} of a formal language to which an element of L(H) other than the {0}-subspace and the identical subspace H is attributed, should not be called a proposition – i.e., a primary bearer of truth-value – but a propositional formula or a sentence or anything that carries no intrinsic meaning of truth or falsehood. Only subsequent to the act of verification, i.e., when the image of ⋄ under the valuation comes to be either 1 or 0, one can call the aforesaid object a proposition. 12 Let’s take, for example, a system whose state before the verification is given by a quantum superposition of the type |Ψi = c1 |Ψ1 i + c2 |Ψ2 i such that |Ψ1 i ∈ ran(ˆ x1 ) and |Ψ2 i ∈ ran(ˆ x2 ) where ran(ˆ x1 ) and ran(ˆ x2 ) are closed subspaces of H that have no element in common except {0}, and c1 , c2 are the superposition coefficients. The subspaces ran(ˆ x1 ) and ran(ˆ x2 ) are identified with disjoint (i.e., mutually exclusive) properties of the system, possession of which are declared by the propositional formulas X1 and X2 associated with the orthogonal projection operators x ˆ1 and x ˆ2 . It is straightforward that the superposition c1 |Ψ1 i + c2 |Ψ2 i would correspond to the direct sum of ran(ˆ x1 ) and ran(ˆ x2 ) bringing in a decomposition of H, namely, H = ran(ˆ x1 ) ⊕ ran(ˆ x2 ) = {c1 |Ψ1 i + c2 |Ψ2 i}

(30)

.

On the other hand, given that the meet of the orthogonal projections x ˆi is ⊓2i=1 x ˆi = x ˆ1 x ˆ2 = x ˆ2 x ˆ1 = 0, the direct sum (30) would correspond to the projection x ˆ1 + x ˆ2 and, hence, the join ⊔2i=1 x ˆi . In accordance with the definition (14), one gets then
v (ran(ˆ x1 )∩ran(ˆ x2 )) = v ⊓2i=1 x ˆi = [[X1 ∧X2 ]]v

,

(31)


v ({c1 |Ψ1 i + c2 |Ψ2 i}) = v ⊔2i=1 x ˆi = [[X1 ∨X2 ]]v

.

(32)

Consistent with the assumption (29), {v(ran(ˆ xi ))} = ∅ at the same time as v(ran(ˆ x1 ) ∩ ran(ˆ x2 )) = v({0}) = 0 and v({c1 |Ψ1 i + c2 |Ψ2 i}) = v(H) = 1, which would give 12

This is reminiscent of the logical system of intuitionistic logic that lacks a complete set of truth values because its semantics is specified in terms of provability conditions.

10

{[[Xi ]]v } = ∅

(33)

,

[[X1 ∧X2 ]]v = 0

,

(34)

[[X1 ∨X2 ]]v = 1

.

(35)

Accordingly, ahead of the verification, the sentence “Out of two contradictory properties, the system possesses one or the other, but not both” would be true (and thus it would be a proposition) despite the fact that the sentence “Out of two contradictory properties, the system possesses a particular one” would have no truth value at all (implying that before the verification the truth degree functions of the logical connectives F∧ ([[X1 ]]v , [[X2 ]]v ), F∨ ([[X1 ]]v , [[X2 ]]v ) and F¬ ([[Xi ]]v ) would not be defined on [[X1 ]]v and [[X2 ]]v ). 13 But even so, provided a probability assignment for the latter sentence is possible in a way that 0 < P[Xi ] < 1 if

{[[Xi ]]v } = ∅

,

(36)

the probability that this sentence will be proved to be true given the particular property i ∈ {1, 2} can be estimated by P[Xi ] = hΨ|ˆ xi |Ψi = |ci |2 where |c1 |2 + |c2 |2 = 1. As a rule, H 6= ({c1 |Ψ1 i + c2 |Ψ2 i} and so in general {[[X1 ∨X2 ]]v } = ∅ along with {[[X1 ∧X2 ]]v } = ∅. This implies that prior to the verification the law of alternatives, i.e., P[X1 ∨ X2 ] = P[X1 ] + P[X2 ], would not be valid on the whole. So, essentially, the assumption (29) suggests that in the scope of orthodox quantum mechanics and a related quantum logic, one should – to paraphrase the remark made in the paper [30] – focus on maps from the family L(H) to the unit interval [0, 1] that generalize the classical idea of probability, rather than that of truth.

References [1] D. Bohm. A Suggested Interpretation of the Quantum Theory in Terms of “Hidden” Variables. I. Physical Review, 85:166–179, 1952. [2] D. Bohm. A Suggested Interpretation of the Quantum Theory in Terms of “Hidden” Variables. II. Physical Review, 85:180–193, 1952. [3] D. Bohm and J. Bub. A Proposed Solution to the Measurement Problem in Quantum Mechanics by a Hidden Variable Theory. Rev. of Mod. Phys., 38(3):453–469, 1966. 13 This inference concurs with the conclusion drawn in the paper [29] arguing that the major transformation from classical to quantum physics lies in the shift from intrinsic to extrinsic properties. In consequence, a compound property such as X ∨Y may have a truth value, even though neither X nor Y has one.

11

[4] D. Rainer. Advanced Quantum Mechanics: Materials and Photons. Springer-Verlag, New York, 2012. [5] K. Jung. Is the de Broglie-Bohm interpretation of quantum mechanics really plausible? Journal of Physics: Conference Series, 442(012060):1–9, 2013. [6] S. Gottwald. A Treatise on Many-Valued Logics. Research Studies Press Ltd., Baldock, Hertfordshire, England, 2001. [7] D. Miller and M. Thornton. Multiple valued logic: Concepts and representations. Morgan & Claypool Publishers, San Rafael, CA, 2008. [8] J. Pykacz and P. Fr¸ackiewicz. The Problem of Conjunction and Disjunction in Quantum Logics. Int. J. Theor. Phys., DOI 10.1007/s10773-017-3402-y, 2017. [9] J. Pykacz. Fuzzy quantum logics and infinite-valued Lukasiewicz logic. Int. J. Theor. Phys., 33:1403–1416, 1994. [10] J. Pykacz. Quantum Logic as Partial Infinite-Valued Lukasiewicz Logic. Int. J. Theor. Phys., 34(8):1697–1710, 1995. [11] J. Pykacz. Lukasiewicz operations in fuzzy set and many-valued representations of quantum logics. Found. Phys., 30:1503–1524, 2000. [12] J. Pykacz. Unification of two approaches to quantum logic: Every Birkhoff - von Neumann quantum logic is a partial infinite-valued Lukasiewicz logic. Stud. Logica, 95:5–20, 2010. [13] J. Pykacz. Towards many-valued/fuzzy interpretation of quantum mechanics. Int. J. Gen. Syst., 40:11–21, 2011. [14] J. Pykacz. Can Many-Valued Logic Help to Comprehend Quantum Phenomena? Theor. Phys., 54:4367–4375, 2015.

Int. J.

[15] E. Beltrametti and G. Cassinelli. The Logic of Quantum Mechanics. Addison-Wesley, Reading MA, 1981. [16] I. Pitowsky. Quantum Probability – Quantum Logic, Lecture Notes in Physics, Volume 321. Springer-Verlag, Berlin, 1989. [17] C. Morgan and H. Leblanc. Probability Theory, Intuitionism, Semantics, and the Dutch Book Argument. Notre Dame Journal of Formal Logic, 24(3):289–304, 1983. [18] C. Caves, C. Fuchs, and R. Schack. Subjective probability and quantum certainty. arXiv:quantph/0608190, 2007. [19] L. Savage. Foundations of Statistics, 2nd ed. Dover, New York, 1972. [20] B. de Finetti. Theory of Probability. Wiley, New York, 1990. [21] J. Bernardo and A. Smith. Bayesian Theory. Wiley, Chichester, 1994. [22] R. Jeffrey. Subjective Probability. The Real Thing. Cambridge University Press, Cambridge, 2004. 12

[23] R. Giles. A Non-Classical Logic for Physics. Studia Logica, 33(4):397–415, 1974. [24] H. Nguyen and A. Elbert. A First Course in Fuzzy Logic. Chapman and Hall, 2005. [25] N. Smith. Vagueness and Degrees of Truth. Oxford University Press, Oxford, 2005. [26] B. Kosko. Fuzziness vs. Probability. Int. J. General Systems, 17:211–240, 1990. [27] H. Putnam. Mathematics, Matter and Method: Philosophical Papers, Vol. 1. Cambridge University Press, Cambridge, 1985. [28] C. Caleiro, W. Carnielli, M. Coniglio, and J. Marcos. Two’s Company: “The Humbug of Many Logical Values”. In J. Beziau, editor, Logica Universalis, pages 169–189. Birkh¨auser Verlag, 2005. [29] S. Kochen. A Reconstruction of Quantum Mechanics. Found. Phys., 45(5):557–590, 2015. [30] R. Sorkin. An exercise in “anhomomorphic logic”. arXiv:quant-ph/0703276, 2007.

13