Quantum contextuality implies a logic that does not obey the principle ...

Quantum contextuality implies a logic that does not obey the principle of bivalence

arXiv:1802.07390v1 [quant-ph] 21 Feb 2018

Arkady Bolotin∗ Ben-Gurion University of the Negev, Beersheba (Israel) February 22, 2018

Abstract In the paper, a value assignment for projection operators relating to a quantum system is equated with assignment of truth-values to the propositions associated with these operators. In consequence, the Kochen-Specker theorem (its localized variant, to be exact) can be treated as the statement that a logic of those projection operators does not obey the principle of bivalence. This implies that such a logic has a gappy (partial) semantics or many-valued semantics. Keywords: Quantum mechanics; Kochen-Specker theorem; Contextuality; Truth values; Partial semantics; Many-valued semantics.

1

Introduction

Consider the triple (H, |Ψi, O) in which H is the Hilbert space of a quantum system, |Ψi ∈ H is the normalized vector describing the state of this system, and O = {Pˆ } is a finite set of projection operators Pˆ on H. Define an assignment function h as a function from the set O to the set of numerical values {0, 1}, namely, h : O → {0, 1}

,

(1)

such that

∗

h(ˆ0) = 0

,

(2)

h(ˆ1) = 1

,

(3)

Email : [email protected]

1

where ˆ0 and ˆ 1 are the zero-projection and identity-projection operators, respectively. On the word of [1], the assignment function h expresses the notion of a hidden variable, namely, h(Pˆ ) specifies in advance the result obtained from the measurement of an observable corresponding to the particular projection operator Pˆ . Also, define a subset C ⊂ O as a context if |C| ≥ 2 and any two projection operators in C, say, Pˆα and Pˆβ (where Pˆα 6= Pˆβ ), are orthogonal Pˆα Pˆβ = Pˆβ Pˆα = ˆ0

.

(4)

The context C is maximal (or complete) if the projection operators in C resolve to the identityprojection operator: X

Pˆ = ˆ1

.

(5)

Pˆ ∈ C

Now consider the following sentences: (a) The set O is value definite under h; in other words, h is a total function. (b) The value h(Pˆ ) depends only on Pˆ and not the context C containing Pˆ . (c) For a maximal C, the values of its projection operators h(Pˆ ) add up to 1; otherwise stated, the next entailment holds: 

h

X

Pˆ ∈ C



Pˆ= 1 =⇒

X

Pˆ ∈ C

  h Pˆ = 1

.

(6)

Provided that the sentences (b) and (c) are true, the sentence (a) must be denied in accordance with the Kochen-Specker theorem [2, 3]. That is, the assignment function h cannot be total and hence at least one projection operator Pˆ ∈ O must be value indefinite under h (i.e., must have the value neither 0 nor 1). What is more, according to the variant of the Kochen-Specker theorem localizing value indefiniteness [4], there is a set O containing projection operators Pˆα and Pˆβ such that if the system is prepared in the pure state |Ψα i in which h(Pˆα ) = 1, then both h(Pˆβ ) = 1 and h(Pˆβ ) = 0 lead to contradictions. On the other hand, consider a truth-value assignment function vC that denotes a truth valuation in a circumstance C, that is, a mapping from some subset of propositions P ⊆ {⋄} related to the quantum system (where the symbol ⋄ stands for any proposition, compound or simple) to the set of truth-values {0, 1} (where the value 0 represents “false” and the value 1 represents “true”) relative to a circumstance of valuation indicated by C (such a circumstance can be, for example, the state |Ψi in which the system is prepared or found): vC : P → {0, 1} 2

.

(7)

Commonly, it is written using the double-bracket notation, namely, vC (⋄) = [[⋄]]C . The truth-value assignment function vC expresses the notion of not-yet-verified truth values: It specifies in advance the truth-value obtained from the verification of a proposition ⋄. Let the following valuational axiom hold true vC (Pˆ⋄ ) = [[⋄]]C

,

(8)

where Pˆ⋄ is the projection operator uniquely (i.e., one-to-one) associated with the proposition ⋄. Assume that the function h coincides with the function vC . Then, the localized variant of the Kochen-Specker theorem is equivalent to the statement that a logic defined as the relations between projection operators Pˆ⋄ on H does not obey the principle of bivalence (according to which a proposition must be either true or false [5]). In other words, a logic of the projection operators Pˆ⋄ has a non-bivalent semantics, e.g., a gappy one (in which the function vC is partial and thus some propositions may have absolutely no truth-value) or many-valued one (in which there are more than two truth-values). Let us demonstrate this equivalence in the presented paper.

2

Truth-value assignment for projection operators

Consider the lattice L(H) formed by the column spaces (ranges) of the projection operators Pˆα , Pˆβ , . . . on H. Let the lattice ordering relation ≤ correspond to the subset relation ran(Pˆα ) ⊆ ran(Pˆβ ), the lattice operation meet ⊓ correspond to the interception ran(Pˆα ) ∩ ran(Pˆβ ), and the lattice operation join ⊔ correspond to the smallest closed subspace of H containing the union ran(Pˆα )∪ran(Pˆβ ). Let the lattice L(H) be bounded, i.e., let it have the greatest element ran(ˆ1) = H and the least element ran(ˆ 0) = {0}, which satisfy ran(ˆ0) ⊆ ran(Pˆ ) ⊆ ran(ˆ1)

(9)

for every ran(Pˆ ) in L(H). Let Pˆα ⊓ Pˆβ , Pˆα ⊔ Pˆβ and Pˆα + Pˆβ denote projections on the interception ran(Pˆα ) ∩ ran(Pˆβ ), the union ran(Pˆα ) ∪ ran(Pˆβ ) and the sum ran(Pˆα ) + ran(Pˆβ ), respectively. One can write then ran(Pˆα ⊓ Pˆβ ) = ran(Pˆα ) ∩ ran(Pˆβ )

,

(10)

ran(Pˆα ⊔ Pˆβ ) = ran(Pˆα ) ∪ ran(Pˆβ )

,

(11)

ran(Pˆα + Pˆβ ) = ran(Pˆα ) + ran(Pˆβ )

.

(12)

3

Clearly, if the column spaces ran(Pˆα ) and ran(Pˆβ ) are orthogonal to each other, their union coincides with their sum, i.e., ran(Pˆα ) ∩ ran(Pˆβ ) = ran(ˆ 0)

=⇒

ran(Pˆα ) ∪ ran(Pˆβ ) = ran(Pˆα ) + ran(Pˆβ )

.

(13)

Hence, for any two projection operators in the context C one must get Pˆα ⊓ Pˆβ = Pˆα Pˆβ = ˆ0 Pˆα ⊔ Pˆβ = Pˆα + Pˆβ

,

(14)

.

(15)

Next, consider the truth-value assignments of the projection operators in the lattice L(H). ˆ = ker(ˆ Given that ran(1) 0) = H, any arbitrary state of the system |Ψi ∈ H resides in the column space of the identity-projection operator, i.e., |Ψi ∈ ran(ˆ1). But then, being in ran(Pˆ ) means Pˆ |Ψi = 1 · |Ψi; so, in agreement with the eigenstate assumption [1], one can presume that the function v|Ψi assigns the truth value 1 to the projection operator ˆ1 in any admissible state of the system |Ψi ∈ H. At the same time, any arbitrary state of the system |Ψi ∈ H also resides in the null space of the null-projection operator, i.e., |Ψi ∈ ker(ˆ0). Since being in ker(Pˆ ) must mean Pˆ |Ψi = 0 · |Ψi, one can presume that the function v|Ψi assigns the truth value 0 to the projection operator ˆ0 in any admissible state of the system |Ψi ∈ H. This can be written down as |Ψi ∈



ran(ˆ1) = H ker(ˆ0) = H

⇐⇒



v|Ψi (ˆ1) = 1 v|Ψi (ˆ0) = 0

.

(16)

Let the system be prepared in a pure state |Ψα i lying in the column space of the projection operator Pˆα in the context C. Since being in ran(Pˆα ) means Pˆα |Ψα i = 1 · |Ψα i, one can assume that the function v|Ψα i assigns the truth value 1 to the projection operator Pˆα in the state |Ψα i. Contrariwise, if the truth value of the projection operator Pˆα is 1 in the state |Ψα i, one can deduce that the state |Ψα i is in the column space of the projection operator Pˆα . These two presumptions can be recorded together as the following logical biconditional: |Ψα i ∈ ran(Pˆα )

⇐⇒

v|Ψα i (Pˆα ) = 1

.

(17)

For the reason that all the projection operators Pˆ⋄ ∈ C are commutative and for each pair of them, namely, Pˆα and Pˆ⋄6=α , the equality holds Pˆα Pˆ⋄6=α = ˆ0, one gets Pˆ⋄6=α |Ψα i = 0 · |Ψα i. This means that the vector |Ψα i must also reside in the null space of any other projection operator Pˆ⋄6=α in the context C, and therefore all other truth values v|Ψα i (Pˆ⋄6=α ) relating to the context C are zero: |Ψα i ∈ ker(Pˆ⋄6=α )

⇐⇒ 4

v|Ψα i (Pˆ⋄6=α ) = 0

.

(18)

This obviously gives X v|Ψα i (Pˆα ) + v|Ψ i (Pˆ⋄ ) = 1 | {z } ⋄6=α α =1 {z } |

.

(19)

=0

But let’s say that the state |Ψα i resides in ker(Pˆα ) and so v|Ψα i (Pˆα ) = 0. As ker(Pˆα ) = ran(ˆ1 − Pˆα ), one gets that in the maximal context C the entailment holds |Ψα i ∈ ran(ˆ 1 − Pˆα )

⇐⇒



v|Ψα i

X

⋄6=α



Pˆ⋄ = 1

,

(20)

where the value v|Ψα i (Pˆ⋄6=α ) can be either 0 or 1 since 0 · Pˆ⋄6=α |Ψα i = 0 · |Ψα i. Take the projection operator Pˆβ6=α in C and let v|Ψα i (Pˆβ ) = 1. In such a case, |Ψα i is in the null spaces of the rest of the projection operators in C and, in accordance with (18), X v|Ψα i (Pˆα ) + v|Ψα i (Pˆβ ) + v|Ψα i (Pˆ⋄ ) = 1 | {z } | {z } ⋄6=α, β =0 =1 {z } |

.

(21)

=0

If v|Ψα i (Pˆβ )= 0, then, according to (20),



X



v|Ψα i (Pˆα ) + v|Ψα i (Pˆβ ) + v|Ψα i Pˆ⋄  = 1 | {z } | {z } ⋄6=α, β =0 =0 | {z }

.

(22)

=1

Repeating the arguments, one ultimately finds

P

ˆ

⋄ v|Ψα i (P⋄ )

= 1.

Thus, if the system is prepared (found) in the state lying in the column or null space of any projection operator in the maximal context C, then among all the propositions {⋄}C associated with C exactly one would be true while the others would be false. Suppose that there is a different context C ′ ⊂ O, whose members Pˆ⋄′ do not commute with Pˆ⋄ . Let the vector |Φi be arranged in the column space of the projection operator Pˆ⋄′ ∈ C ′ , |Φi ∈ ran(Pˆ⋄′ ), and so v|Φi (Pˆ⋄′ ) = 1. Because of the non-commutativity, i.e., Pˆ⋄ Pˆ⋄′ 6= Pˆ⋄′ Pˆ⋄ , the vector |Φi cannot reside in the column or null space of any projection operator in C, i.e., |Φi ∈ / ran(Pˆ⋄ ) and |Φi ∈ / ker(Pˆ⋄ ). Therefore, under the valuations (17) and (18), the truth-value function v|Φi must assign neither 1 nor 0 to any projection operator Pˆ⋄ ∈ C, that is, v|Φi (Pˆ⋄ ) 6= 1 and v|Φi (Pˆ⋄ ) 6= 0. Consequently, the propositions {⋄}C cannot be bivalent under v|Φi , that is, 5

|Φi ∈ /



ran(Pˆ⋄ ) ker(Pˆ⋄ )

⇐⇒ v|Φi(Pˆ⋄ ) = [[ ⋄ ∈ {⋄}C ]]|Φi ∈ / {0, 1}

.

(23)

As a result, for any set O containing non-commuting contexts, the bivaluation relation vC : {Pˆ } → {0, 1} cannot be a total function.

3

Projection operators of a spin-half particle

Let us illustrate this inference with an example of the set O comprising the projection operators of a spin-half particle. This set O includes three non-commuting maximal contexts CQ corresponding to the spin projection onto the Q = {x, y, z} axis, namely, CQ = {Pˆ↑Q , Pˆ↓Q }

,

Pˆ↑Q Pˆ↓Q = Pˆ↓Q Pˆ↑Q = ˆ0 Pˆ↑Q + Pˆ↓Q = ˆ1

(24) ,

(25)

,

(26)

where the projection operators Pˆ↑Q = |↑Q ih↑Q | and Pˆ↓Q = |↓Q ih↓Q | are associated with the propositions ↑Q and ↓Q asserting that a spin- 12 particle exists in the spin state “Q-up”, i.e., |↑Q i, and the spin state “Q-down”, i.e., |↓Q i, respectively. Consider, for example, the contexts Cz and Cx :      10 00 ˆ ˆ Cz = P↑z = , P↓z = , 00 01      1 1 1 1 1 −1 , Pˆ↓x = Cx = Pˆ↑x = 2 11 2 −1 1

(27) .

(28)

Their column and null spaces are        a 0  ˆ ˆ  z z ran( P ) = : a ∈ R , ker( P ) = : a ∈ R  ↑ ↑  0   a    Cz : 0 a   : a ∈ R , ker(Pˆ↓z ) = : a∈R  ran(Pˆ↓z ) = a 0        a a  ˆ ˆ  : a ∈ R , ker(P↑x ) = : a∈R  ran(P↑x ) =   a  −a   Cx : a a   ˆ ˆ : a∈R  ran(P↓x ) = : a ∈ R , ker(P↓x ) = a −a 6

,

(29)

.

(30)

For the propositions ↑z and ↓z to have the bivalent values {0, 1}, the spin-half particle must be prepared in the spin-state lying in the column or null space of the context Cz . Likewise, the propositions ↑x and ↓x are bivalent if the particle is prepared in the state lying in the column or null space of the context Cx . For example,   ( ran(Pˆ↑z ) ⇐⇒ v|↑z i (Pˆ↑z ) = 1 1 |↑ i = ∈ 0 ker(Pˆ↓z ) ⇐⇒ v|↑z i (Pˆ↓z ) = 0 z

  ( 1 ran(Pˆ↑x ) ⇐⇒ v|↑x i (Pˆ↑x ) = 1 1 |↑x i = √ ∈ ker(Pˆ↓x ) ⇐⇒ v|↑x i (Pˆ↓x ) = 0 2 1

⇐⇒ v|↑z i (Pˆ↑z ) + v|↑z i (Pˆ↓z ) = 1

⇐⇒ v|↑x i (Pˆ↑x ) + v|↑x i (Pˆ↓x ) = 1

,

(31)

.

(32)

As it is readily seen, neither the column spaces nor the null spaces of the context Cz are subsets or supersets of the column or null spaces of the context Cx :    a : a∈R  0   0 : a∈R a

    a   : a ∈ R  6⊂  a   a  6⊃    : a∈R   −a    

.

(33)

Under the valuations (17) and (18), this implies that if the particle is prepared in, say, the state |↑z i in which v|↑z i (Pˆ↑z ) = 1, then both v|↑z i (Pˆ↑x ) = 1 and v|↑z i (Pˆ↑x ) = 0 lead to the contradiction 0 = 1:     a     : a ∈ R ⇐⇒ v|↑z i (Pˆ↑x ) = 1 =⇒ 0 = 1  1 a z    |↑ i = ∈ 0 a   : a ∈ R ⇐⇒ v|↑z i (Pˆ↑x ) = 0 =⇒ 0 = 1  −a

.

(34)

Equally, if the particle is prepared in the state |↑x i in which v|↑x i (Pˆ↑x ) = 1, then both v|↑x i (Pˆ↑z ) = 1 and v|↑x i (Pˆ↑z ) = 0 will lead to the same contradiction 0 = 1:     a     : a ∈ R ⇐⇒ v|↑x i (Pˆ↑z ) = 1 =⇒ 0 = 1  1 1 0 x   ∈ |↑ i = √ 0  2 1  : a ∈ R ⇐⇒ v|↑x i (Pˆ↑z ) = 0 =⇒ 0 = 1  a

.

(35)

/ {0, 1}, the function / {0, 1} as well as v|↑x i (Pˆ↑z ) ∈ As v|↑z i (Pˆ↑x ) ∈

vC : {Pˆ↑z , Pˆ↓z , Pˆ↑x , Pˆ↓x , Pˆ↑y , Pˆ↓y } → {0, 1} is not total.

7

(36)

4

Cabello’s set of 4×4 matrices

In the paper [6] Cabello et al. presented proof of the Bell-Kochen-Specker theorem using the set of 18 four-dimensional matrices (i.e., projection operators Pˆ on the Hilbert space H = C4 ) which (Q) involved 9 maximal contexts CQ = {Pˆi }4i=1 where Q = {1, . . . , 9}. Consider the projection operators in the contexts C1 and C6 :        1010 1 0 ¯1 0 0000 0000    0 0 0 0    (1)  ˆ (1)  0 1 0 0  ˆ (1) 1  0 0 0 0  ˆ (1) 1  0 0 0 0  Pˆ1 =   0 0 0 0  , P2 =  0 0 0 0  , P3 = 2  1 0 1 0  , P4 = 2  ¯1 0 1 0  , 0000 0000 0001 0000         1¯ 1¯ 11 1111 1 0 0 ¯1 0000    ¯    1¯ 111¯ 1 (6)  , Pˆ (6) = 1  1 1 1 1  , Pˆ (6) = 1  0 0 0 0  , Pˆ (6) = 1  0 1 1 0  , Pˆ1 =  4 3 2 111¯ 1 4¯ 41 1 1 1 20 0 0 0 2  0 ¯1 1 0  ¯ ¯ ¯ 1001 0000 1111 1111 

(37)

(38)

where ¯1 stands for −1. Their column and null spaces are:       0 a               0 b (1) (1)  : a ∈ R , ker(Pˆ ) =   : a, b, c ∈ R ran(Pˆ1 ) =  , 1  0  c             a 0       0 a               a 0 (1) (1) ˆ   ˆ   ran(P2 ) =   : a ∈ R , ker(P2 ) =   : a, b, c ∈ R , 0 b             0 c       0 −b               0 a (1) (1) ˆ ˆ     ran(P3 ) =   : a ∈ R , ker(P3 ) =  : a, b, c ∈ R , a b             0 c       a b               0 a (1) (1)  : a ∈ R , ker(Pˆ ) =   : a, b, c ∈ R ran(Pˆ4 ) =  ; 4  −a   b             0 c       a a+b−c               −a a (6) (6)  : a ∈ R , ker(Pˆ ) =   : a, b, c ∈ R ran(Pˆ1 ) =  1  −a   b             a c

8

(39)

(40)

(41)

(42)

,

(43)

      −a − b − c a               a a (6) (6)  ˆ   ˆ  : a, b, c ∈ R ran(P2 ) =   : a ∈ R , ker(P2 ) =  b a             a c       a c               0 a (6) (6)  : a ∈ R , ker(Pˆ ) =   : a, b, c ∈ R ran(Pˆ3 ) =  , 3  0  b             −a c       0 a               a b (6) (6) ˆ   ˆ   ran(P4 ) =  : a ∈ R , ker( P ) = : a, b, c ∈ R . 4  b −a              0 c

,

(44)

(45)

(46)

Suppose that the system is prepared in the state |1(1) i lying in the column space of the projection (1) operator Pˆ1 in the context C1 , namely,   0  0  ˆ (1) |1(1) i =   0  ∈ ran(P1 ) 1

.

(47)

As it can be seen, the ket |1(1) i resides in the null spaces of the rest of the projections in the context (6) C1 and, in addition, in the null space of Pˆ4 : (1) |1(1) i ∈ ker(Pˆi6=1 )

,

(48)

(6) |1(1) i ∈ ker(Pˆ4 )

.

(49)

But together with that, the vector |1(1) i does not reside in the column or null space of any other projection operator in C6 (1)

|1 i ∈ /

(

(6)

ran(Pˆi ) (6) ker(Pˆ )

.

(50)

i6=4

According to (17) and (18), this implies 4 X

(1) v|1(1) i (Pˆi ) = 1

.

(51)

i=1

(6) v|1(1) i (Pˆ4 ) = 0

9

,

(52)

(6)

/ {0, 1} v|1(1) i (Pˆi6=4 ) ∈

,

(53)

which demonstrates that Cabello’s set O = {CQ }9Q=1 is value indefinite under the bivaluation.

5

Interpretation of the Kochen-Specker theorem

For the valuation relation vC : {Pˆ } → {0, 1} the failure of being a total function can be described by way of the truth-value gaps, namely, |Φi ∈ /



ran(Pˆ⋄ ) ker(Pˆ⋄ )

⇐⇒

n

o v|Φi(Pˆ⋄ ) = ∅

.

(54)

This expression means that in a state |Φi not residing in the column or null space of a projection operator Pˆ⋄ , a proposition ⋄ associated with Pˆ⋄ has no truth-value at all, i.e., {[[⋄]]|Φi } = ∅. A semantics defined by set of these truth-value gaps in conjunction with the valuations (17) and (18) is gappy and yet two-valued. Accordingly, it can be called a supervaluationist semantics (for details of such semantics see [7, 8] and also [9]). This semantics is, in general, not truth-functional: Thus, according to (17), (18) and (54), in any admissible state |Ψi of the system, the function v|Ψi assigns the value of the truth to the sum of the projection operators Pˆ⋄ in a maximal context C, even though there is a state |Ψi = |Φi where at least one of these projection operators has no truth-value:

|Φi ∈ /



ran(Pˆ⋄ ) ker(Pˆ⋄ )



=⇒ v|Φi

X

Pˆ⋄ ∈C



Pˆ⋄= 1 =⇒

X

Pˆ⋄ ∈C

v|Φi (Pˆ⋄ ) 6= 1

.

(55)

For example, the values v|↑x i (Pˆ↑x ) and v|↑x i (Pˆ↓x ) are bivalent in supervaluationist semantics but the values v|↑z i (Pˆ↑x ) and v|↑z i (Pˆ↓x ) do not exist before their verification, that is |↑z i ∈ /



ran(Pˆ↑x ) ker(Pˆ↓x )

⇐⇒

o v|↑z i (Pˆ↑x ) = ∅ o n v|↑z i (Pˆ↓x ) = ∅ n

.

(56)

(6) In the same manner, the values v|1(1) i (Pˆi6=4 ) of the projection operators in the context C6 of Cabello’s set are nonexistent within supervaluationist semantics:

(1)

|1 i ∈ /

(

(6)

ran(Pˆi ) (6) ker(Pˆ ) i6=4

⇐⇒

n

o (6) v|1(1) i (Pˆi6=4 ) = ∅

.

(57)

Alternatively, the failure of the principle of bivalence for projection operators Pˆ on H can be described using multivalued semantics in which projection operators Pˆ may have more than two values, specifically, 10

vC : {Pˆ } → VN

,

(58)

where VN denotes a set of truth-values whose cardinality is N > 2 and whose upper and lower bounds are 1 (that represents “true” or “absolutely true”) and 0 (that represents “false” or “absolutely false”), respectively. To accomplish that, instead of the truth-value gaps (54) one can introduce the following valuation |Φi ∈ /



ran(Pˆ⋄ ) ker(Pˆ⋄ )

⇐⇒ v|Φi(Pˆ⋄ ) = hΦ|Pˆ⋄ |Φi

,

(59)

where the function v|Φi is determined by the probability P[[[⋄]]|Φi= 1] = hΦ|Pˆ⋄ |Φi. As said by [10, 11] the value v|Φi (Pˆ⋄ ) represents the degree to which the projection operator Pˆ⋄ ∈ C has the value 1 in the state |Φi. Because hΦ|Pˆ⋄ |Φi ∈ {x ∈ R | 0 < x < 1}, a semantics defined by the set of the valuations (17), (18) and (59) is infinite-valued. For example, even though the values v|↑x i (Pˆ↑x ) and v|↑x i (Pˆ↓x ) are respectively 1 and 0 in the said semantics, the values v|↑z i (Pˆ↑x ) and v|↑z i (Pˆ↓x ) are neither 1 nor 0, that is, z

|↑ i ∈ /



ran(Pˆ↑x ) ker(Pˆ↓x )

⇐⇒

v|↑z i (Pˆ↑x ) = h↑z |Pˆ↑x |↑z i = v|↑z i (Pˆ↓x ) = h↑z |Pˆ↓x |↑z i =

1 2 1 2

,

(60)

which demonstrates that this semantics is truth-functional, namely, v|↑z i (Pˆ↑x + Pˆ↓x ) = 1 =⇒ v|↑z i (Pˆ↑x ) + v|↑z i (Pˆ↓x ) = 1. (6)

Analogously, in the infinite-valued semantics, the values v|1(1) i (Pˆi6=4 ) of Cabello’s set are defined by 1 4

,

(61)

1 (6) (6) v|1(1) i (Pˆ2 ) = h1(1) |Pˆ2 |1(1) i = 4

,

(62)

1 (6) (6) v|1(1) i (Pˆ3 ) = h1(1) |Pˆ3 |1(1) i = 2

;

(63)

(6)

(6)

v|1(1) i (Pˆ1 ) = h1(1) |Pˆ1 |1(1) i =

P (6) thus, 4i=1 v|1(1) i (Pˆi ) = 1. This shows that despite its value indefiniteness under the bivaluation, Cabello’s set O = {CQ }9Q=1 is value definite under the infinite-valued assignment (59). Hence, only within supervaluationist semantics the negation of the sentence (a) (i.e., the principle of bivalence) can be interpreted as a sign that the verification of the “gappy” (i.e., having no truthvalues) propositions must result in the ex nihilo creation of the bivalent values of these propositions.

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While on the contrary, in many-valued semantics, the failure of bivalence implies that non-classical (i.e., different from 1 and 0) truth-values must exist before the verification and they become bivalent as a result of the verification (thus, a bivalent semantic merely emerges at the end of the verification process). In such a sense, one may say that the measurements (verifications) produce the output that yield pre-existing elements of physical reality. For that reason, the Kochen-Specker theorem (along with its localized variant) cannot justify the belief that quantum mechanics is indeterministic, that is, that there are no hidden variables (or not-yet-verified truth values of the propositions) determining somehow the outcome of a measurement (verification) in advance. This theorem only shows that if those hidden variables were to exist, they would have to comply with a logic which does not obey the principle of bivalence.

References [1] A. Abbott, C. Calude, J. Conder, and K. Svozil. Strong Kochen-Specker theorem and incomputability of quantum randomness. Phys. Rev. A, 86(062109):1–11, 2012. [2] S. Kochen and E. Specker. The problem of hidden variables in quantum mechanics. J. Math. Mech., 17(1):59–87, 1967. [3] A. Peres. Two simple proofs of the Kochen-Specker theorem. Phys. A: Math. Gen., 24:L175– L178, 1991. [4] A. Abbott, C. Calude, and K. Svozil. A variant of the Kochen-Specker theorem localizing value indefiniteness. J. Math. Phys., 56(102201):1–17, 2015. [5] J.-Y. B`eziau. Bivalence, Excluded Middle and Non Contradiction. In L. Behounek, editor, The Logica Yearbook 2003, pages 73–84. Academy of Sciences, Prague, 2003. [6] A. Cabello, J. Estebaranz, and G. Garcia-Alcaine. Bell-Kochen-Specker theorem: A proof with 18 vectors. Phys. Letters A, 212:183–187, 1996. [7] A. Varzi. Supervaluationism and Its Logics. Mind, 116:633–676, 2007. [8] R. Keefe. Theories of Vagueness. Cambridge University Press, Cambridge, 2000. [9] A. Bolotin. Quantum supervaluationism. J. Math. Phys., 58(12):122106–1–7, 2017. doi: 10.1063/1.5008374. [10] J. Pykacz. Quantum Logic as Partial Infinite-Valued Lukasiewicz Logic. Int. J. Theor. Phys., 34(8):1697–1710, 1995. [11] J. Pykacz. Quantum Physics, Fuzzy Sets and Logic. Steps Towards a Many-Valued Interpretation of Quantum Mechanics. Springer, 2015.

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