Consider an operator algebra B ( H ) (i.e., an operator algebra composed of all bounded linear operators on a Hilbert space H with the norm ‖ F ‖ B ( H ) = sup ‖ u ‖ H = 1 ‖ F u ‖ H , [22] [23] [24] ), and consider the pair [ A , N ] B ( H ) (or, the triplet [ A ⊆ N ⊆ B ( H ) ] ), called a basic structure. Here, A ( ⊆ B ( H ) ) is a C^*-algebra, and N ( A ⊆ N ⊆ B ( H ) ) is a particular C^*-algebra (called a W^*-algebra) such that N is the weak closure of A in B ( H ) .

QL (= quantum language = measurement theory) is classified as follows.

(A) QL = { ( A 1 ) :   quantum   QL         ( when   A = C ( H ) ) ( A 2 ) :   classical   QL         ( when   A = C 0 ( Ω ) )

That is, when A = C ( H ) , the C^*-algebra composed of all compact operators on a Hilbert space H, the quantum QL (A₁) is also called quantum measurement theory (or, quantum system theory), which can be regarded as the linguistic aspect of quantum mechanics. Also, when A is commutative (that is, when A is characterized by C 0 ( Ω ) , the C^*-algebra composed of all continuous complex-valued functions vanishing at infinity on a locally compact Hausdorff space Ω (cf. refs. [23] [24] )), the classical QL (A₂) is also called classical measurement theory.

Also, note that, when A = C ( H ) , i.e., quantum cases,

(i) A * = T r ( H ) (=trace class), N = B ( H ) , N * = T r ( H ) (i.e., pre-dual space), thus, ( ρ , T ) T r ( H ) B ( H ) = T r H ( ρ T ) ( ρ ∈ T r ( H ) , T ∈ B ( H ) ) .

Also, when A = C 0 ( Ω ) , i.e., classical cases,

(ii) A * = M ( Ω ) (=“the space of all signed measures on Ω )”, N = L ∞ ( Ω , ν ) ( ⊆ B ( L 2 ( Ω , ν ) ) ) , N * = L 1 ( Ω , ν ) , where ν is some measure on Ω , thus, ( ρ , T ) L 1 ( Ω , ν ) L ∞ ( Ω , ν ) = ∫ Ω     ρ ( ω ) T ( ω ) ν ( d ω ) ( ρ ∈ L 1 ( Ω , ν ) , T ∈ L ∞ ( Ω , ν ) ) (cf. ref. [23] ).

Let A ( ⊆ N ⊆ B ( H ) ) be a C^*-algebra, and let A * be the dual Banach space of A . That is, A * = { ρ | ρ   isacontinuouslinearfunctionalon   A } , and the norm ‖ ρ ‖ A * is defined by sup { | ρ ( F ) | | F ∈ A   such   that   ‖ F ‖ A ( = ‖ F ‖ B ( H ) ) ≤ 1 } . Define the mixed state ρ ( ∈ A * ) such that ‖ ρ ‖ A * = 1 and ρ ( F ) ≥ 0 for all F ∈ A such that F ≥ 0 . And define the mixed state space S m ( A * ) such that

S m ( A * ) = { ρ ∈ A * | ρ   is   a   mixed   state } .

A mixed state ρ ( ∈ S m ( A * ) ) is called a pure state if it satisfies that ρ = θ ρ 1 + ( 1 − θ ) ρ 2 for some ρ 1 , ρ 2 ∈ S m ( A * ) and 0 < θ < 1 implies ρ = ρ 1 = ρ 2 . Put

S p ( A * ) = { ρ ∈ S m ( A * ) | ρ   is   a   pure   state } ,

which is called a state space. It is well known (cf. ref. [23] ) that S p ( C ( H ) * ) = { | u 〉 〈 u | ( i . e . , the Dirac notation ) | ‖ u ‖ H = 1 } , and S p ( C 0 ( Ω ) * ) = { δ ω 0 | δ ω 0   is a point measure at   ω 0 ∈ Ω } , where ∫ Ω f ( ω ) δ ω 0 ( d ω ) = f ( ω 0 ) ( ∀ f ∈ C 0 ( Ω ) ) . The latter implies that S p ( C 0 ( Ω ) * ) can be also identified with Ω (called a spectrum space or simply spectrum) such as

S p ( C 0 ( Ω ) * ) ( statespace ) ∋ δ ω ↔ ω ∈ Ω ( spectrum )

Thus, ω and Ω is also called a state and state space respectively.

For instance, in the above (ii) we must clarify the meaning of the value” of f ( ω 0 ) for f ∈ L ∞ ( Ω , ν ) and ω 0 ∈ Ω . An element f ( ∈ L ∞ ( Ω , ν ) ) is said to be essentially continuous at ω 0 ( ∈ Ω ) , if there uniquely exists a complex number α such that

· if ρ ( ∈ L 1 ( Ω , ν ) , ‖ ρ ‖ L 1 ( Ω , ν ) = 1 , ρ ≥ 0 ) converges to δ ω 0 ( ∈ M ( Ω ) ) in the sense of weak^* topology of M ( Ω ) , that is,

ρ ( G ) → G ( ω 0 )     ( ∀ G ∈ C 0 ( Ω ) ( ⊆ L ∞ ( Ω , ν ) ) ) ,

then ρ ( f ) ( = ∫ Ω   f ( ω ) ρ ( ω ) ν ( d ω ) ) converges to α .

And the value of f ( ω 0 ) is defined by the α .

Definition 1. [Observable] According to the noted idea (cf. ref. [25] [26] ), an observable O : = ( X , F , F ) in N is defined as follows:

(i) [σ-field] X is a set, F ( ⊆ 2 X , the power set of X) is a σ-field of X, that is, “ Ξ 1 , Ξ 2 , ⋯ ∈ F ⇒ ∪ n = 1 ∞ Ξ n ∈ F ”, “ Ξ ∈ F ⇒ X \ Ξ ∈ F ”.

(ii) [Countable additivity] F is a mapping from F to N satisfying: (a): for every Ξ ∈ F , F ( Ξ ) is a non-negative element in N such that 0 ≤ F ( Ξ ) ≤ I , (b): F ( ∅ ) = 0 and F ( X ) = I , where 0 and I is the 0-element and the identity in N respectively. (c): for any countable decomposition { Ξ 1 , Ξ 2 , ⋯ , Ξ n , ⋯ } of Ξ (i.e., Ξ , Ξ n ∈ F ( n = 1 , 2 , 3 , ⋯ ) , ∪ n = 1 ∞ Ξ n = Ξ , Ξ i ∩ Ξ j = ∅ ( i ≠ j )), it holds that F ( Ξ ) = ∑ n = 1 ∞   F ( Ξ n ) in the sense of weak^* topology in N .

Let ( Y , G ) be a measurable space, and let Θ : X → Y be a measurable map. Then, Θ ( O ) : = ( Y , G , F ( Θ − 1 ( ⋅ ) ) ) in N is also an observable in N (which is called an image observable). If F ( Ξ ) = F ( Ξ ) 2 ( ∀ Ξ ∈ F ) , then O : = ( X , F , F ) in N is a projective observable O : = ( X , F , F ) in N is also called anX-valued observable. I will devote myself to binary valued (i.e., { 1,0 } -valued) projective observables in most of the cases in this paper.

Measurement theory (A) is composed of two axioms (i.e., Axioms 1 and 2) as follows. With any system S, a basic structure [ A , N ] B ( H ) can be associated in which the measurement theory (A) of that system can be formulated. A state of the system S is represented by an element ρ ( ∈ S p ( A * ) ) and an observable is represented by an observable O : = ( X , F , F ) in N . Also, the measurement of the observable O for the system S with the state ρ is denoted by M ( O , S [ ρ ] ) (or more precisely, M N ( O , S [ ρ ] ) , M N ( O : = ( X , F , F ) , S [ ρ ] ) ). An observer can obtain a measured value x ( ∈ X ) by the measurement M ( O , S [ ρ ] ) .

The Axiom 1 presented below is a kind of mathematical generalization of Born’s probabilistic interpretation of quantum mechanics. And thus, it is a statement without reality.

Now we can present Axiom 1 in the W^*-algebraic formulation as follows.

Axiom 1 [Measurement]. Consider a basic structure [ A ⊆ N ⊆ B ( H ) ] . The probability that a measured value x ( ∈ X ) obtained by the measurement M N ( O : = ( X , F , F ) , S [ ρ ] ) belongs to a set Ξ ( ∈ F ) is given by ρ ( F ( Ξ ) ) if F ( Ξ ) is essentially continuous at ρ ( ∈ S p ( A * ) ) .

Remark 2. Recall that a statement whose truth or falsity is determined is called a proposition. Also, Axiom 1 says that a measured value is determined by a measurement M N ( O : = ( X , F , F ) , S [ ρ ] ) . Therefore, there is a point in thinking that “measured value” is a generalization of “truth value”. That is, we can regard “measurement” as a kind of generalizations of “proposition”. This idea is essential to Sec. 3.

Next, we explain Axiom 2 (which is not used in this paper). Let [ A 1 , N 1 ] B ( H 1 ) and [ A 2 , N 2 ] B ( H 2 ) be basic structures. A continuous linear operator Φ 1,2 : N 2 (with weak^* topology) → N 1 (with weak^* topology) is called a Markov operator, if it satisfies that (i): Φ 1,2 ( F 2 ) ≥ 0 for any non-negative element F 2 in N 2 , (ii): Φ 1,2 ( I 2 ) = I 1 , where I k is the identity in N k , ( k = 1 , 2 ) . In addition to the above (i) and (ii), we assume that Φ 1,2 ( A 2 ) ⊆ A 1 and sup { ‖ Φ 1,2 ( F 2 ) ‖ A 1 | F 2 ∈ A 2   such   that   ‖ F 2 ‖ A 2 ≤ 1 } = 1 .

It is clear that the dual operator Φ 1,2 * : A 1 * → A 2 * satisfies that Φ 1,2 * ( S m ( A 1 * ) ) ⊆ S m ( A 2 * ) . If it holds that Φ 1,2 * ( S p ( A 1 * ) ) ⊆ S p ( A 2 * ) , the Φ 1,2 is said to be deterministic. If it is not deterministic, it is said to be non-deterministic. Also note that, for any observable O 2 : = ( X , F , F 2 ) in N 2 , the ( X , F , Φ 1,2 F 2 ) is an observable in N 1 .

Definition 3. [Sequential causal operator; Heisenberg picture of causality] Let ( T , ≤ ) be a tree like semi-ordered set such that t 1 ≤ t 3 and t 2 ≤ t 3 implies t 1 ≤ t 2 or t 2 ≤ t 1 . The family { Φ t 1 , t 2 : N t 2 → N t 1 } ( t 1 , t 2 ) ∈ T ≦ 2 is called a sequential causal operator, if it satisfies that

(i) For each t ( ∈ T ) , a basic structure [ A t ⊆ N t ⊆ B ( H t ) ] is determined.

(ii) For each ( t 1 , t 2 ) ∈ T ≦ 2 , a causal operator Φ t 1 , t 2 : N t 2 → N t 1 is defined such as Φ t 1 , t 2 Φ t 2 , t 3 = Φ t 1 , t 3 ( ∀ ( t 1 , t 2 ) , ∀ ( t 2 , t 3 ) ∈ T ≦ 2 ) . Here, Φ t , t : N t → N t is the identity operator.

Now we can propose Axiom 2 (i.e., causality). (For details, see ref [9].)

Axiom 2 [Causality]; For each t ( ∈ T = “tree like semi-ordered set”), consider the basic structure:

[ A t ⊆ N t ⊆ B ( H t ) ]

Then, the chain of causalities is represented by a sequential causal operator { Φ t 1 , t 2 : N t 2 → N t 1 } ( t 1 , t 2 ) ∈ T ≦ 2 .

Since so-called Copenhagen interpretation is not firm (cf. ref. [27] ), we propose the linguistic Copenhagen interpretation in what follows. In the above, Axioms 1 and 2 are kinds of spells, (i.e., incantation, magic words, metaphysical statements), and thus, it is nonsense to verify them experimentally. Therefore, what we should do is not to understand” but to use”. After learning Axioms 1 and 2 by rote, we have to improve how to use them through trial and error.

The most important statement in the linguistic Copenhagen interpretation is as follows.

(B) Only one measurement is permitted. Thus, Axiom 1 can be used only once. And therefore, the state after a measurement is meaningless since it cannot be measured any longer. That is, the state is only one.

Referring ref. [9], we have the following definition:

Definition 4. [(i; a)] Simultaneous observable]: Let O i : = ( X i , F i , F i ) ( i = 1,2, ⋯ , N ) be commutative projective observables in N . Let

( × i = 1 N X i , ⊠ i = 1 N F i ) is a product measurable space of { ( X i , F i ) } i = 1 N . Then, there uniquely exists a projective observable × i = 1 N O i = ( × i = 1 N X i , ⊠ i = 1 N F i , × i = 1 N F i ) such that

[ × i = 1 N F i ] ( X 1 × X 2 × ⋯ × X j − 1 × Ξ j × X j + 1 × ⋯ × X N ) = F j ( Ξ j )   ( ∀ Ξ j ∈ F i , j = 1,2, ⋯ , N )

This × i = 1 N O i is called a simultaneous observable (or, product observable) of { O i | i = 1,2, ⋯ , N } . Note that the existence and uniqueness is guaranteed (cf., ref. [9] ).

[(i; b)] Simultaneous measurement]: A measurement M N ( × i = 1 N O i = ( × i = 1 N X i , ⊠ i = 1 N F i , × i = 1 N F i ) , S [ ρ ] ) is called a simultaneous measurement concerning commutative O i ( i = 1,2, ⋯ , N ) in N .

[(ii; a)] Parallel observable]: Let O i : = ( X i , F i , F i ) be a projective observable in N i ( i = 1,2, ⋯ , N ) . Let ( × i = 1 N X i , ⊠ i = 1 N F i ) is a product measurable space of { ( X i , F i ) } i = 1 N . Then, there uniquely exists an observable ⊗ i = 1 N O i = ( × i = 1 N X i , ⊠ i = 1 N F i , ⊗ i = 1 N F i ) in a tensor algebra ⊗ i = 1 N N i such that

[ ⊗ i = 1 N F i ] ( X 1 × X 2 × ⋯ × X j − 1 × Ξ j × X j + 1 × ⋯ × X N ) = I 1 ⊗ I 2 ⊗ ⋯ ⊗ I j − 1 ⊗ F i ( Ξ j ) ⊗ I j + 1 ⊗ ⋯ ⊗ I N     ( ∀ Ξ j ∈ F j , j = 1,2, ⋯ , N )

This ⊗ i = 1 N O i is called a parallel observable (or, tensor observable) of { O i | i = 1,2, ⋯ , N } in a tensor algebra ⊗ i = 1 N N i

[(ii; b)] Parallel measurement]: A measurement M ⊗ i = 1 N N i ( ⊗ i = 1 N O i , S [ ⊗ i = 1 N ρ i ] ) is called a parallel measurement concerning { M N i ( O i : = ( X i , F i , F i ) , S [ ρ i ] ) } i = 1 N .

We have the following theorem:

Theorem 5. In a class of binary projective measurements in classical QL, measurement has properties like logic:

In this section we will devote ourselves to the above proof as follows.

Consider a classical basic structure

[ C 0 ( Ω ) ⊆ L ∞ ( Ω , ν ) ⊆ B ( L 2 ( Ω , ν ) ) ]

Here, assume that Ω is a locally compact space with a Borel measure ν on Ω such that ν ( D ) > 0 (for any open set D ( ⊆ Ω , D ≠ ∅ ). Also, without loss of generality, we assume that ν ( Ω ) = 1 . Consider many tomatoes, that is, roughly speaking, consider T as the set of all tomatoes. Assume that any tomato t ( ∈ T ) is represented by a state ω , which is an element of the state space Ω . Thus, we have the map ω ^ : T → Ω . That is, the quantitative property of a tomato t is represented by ω ^ ( t ) . For example, it suffices to consider Ω such that Ω ⊆ ℝ N (=N-dimensional real space), where N is sufficiently large natural number (or, N = ∞ ). That is,

Ω ∋ ω = ( ω ( 1 ) ( = weight ) , ω ( 2 ) ( = diameter ) , ω ( 3 ) ( = diameter ) ,                           ω ( 4 ) ( = color   value ) , ω ( 5 ) ( = calorie ) ,                           ω ( 6 ) ( = sugar   content ) , ⋯ , ω ( N ) ( = ⋯ ) ) ∈ ℝ N

Consider a binary projective observable (i.e., { 1,0 } -valued projective observable, or { x 1 , x 0 } -valued projective observable) O ≡ ( X ,2 X , F ) in L ∞ ( Ω , ν ) , where X = { 1,0 } (or, X = { x 1 , x 0 } ) and F ( Ξ ) = [ F ( Ξ ) ] 2 ( ∀ Ξ ∈ F ) .

Further, as shown in Figure 1 below, we, for the sake of simplicity, assume that { ω ∈ Ω | [ F ( { 1 } ) ( ω ) = 1, a . e . } ( ≡ Γ ) is an open set such that

Γ = [ Γ ¯ ] ∘ ,   ν ( Ω \ ( Γ ∪ [ Γ c ] ∘ ) ) = 0

where D c is the complement of D, i.e., Ω \ D , D ¯ = “the closure of D”, D ∘ = “the interior of D”. Note that we can assume that Γ c = [ Γ c ] ∘ in the sense of “almost everywhere concerning ν ” (i.e., “ ν ( Γ c \ [ Γ c ] ∘ ) = 0 ”), which will be

frequently used without refusal in this paper. The { 1,0 } -valued projective observable O ≡ ( X ,2 X , F ) in L ∞ ( Ω , ν ) is also denoted by

O Γ ≡ ( X ,2 X , F Γ ) (2)

Remark 6. (i): Someone might say that the term “the set of all tomatoes” is as ambiguous as “the set of all dinosaurs”. However, for the sake of convenience, here we use the term “the set of all tomatoes”. This problem is the same as that of the Hempel’ raven paradox. For further discussion about this, see ref. [20] [21].

(ii): If we want to both tomato’s world Ω 1 and apple’s world Ω 2 , it suffices to start from the product space Ω 1 × Ω 2 . Thus, in general we consider the world also is represented by a large state space Ω ^ .

Definition 7. [Measured value] Consider a measurement M L ∞ ( Ω , ν ) ( O Γ , S [ ω ] ) , where O Γ ≡ ( X ( = { 1,0 } ) ,2 X , F Γ ) is a binary projective observable in L ∞ ( Ω , ν ) . Denote the measured value of M L ∞ ( Ω , ν ) ( O Γ , S [ ω ] ) by [ M V ] ( M L ∞ ( Ω , ν ) ( O Γ , S [ ω ] ) ) . Then, we see that

[ M V ] ( M L ∞ ( Ω , ν ) ( O Γ , S [ ω ] ) ) = ( 1   ( ω ∈ Γ ) 0   ( ω ∉ Γ )

(with probability 1).

Let X = { 1,0 } , Ω and ω ^ ( t ) be as before. Put Γ = RD , or Γ = SW in the formula (2) (see Figure 2 below). Consider a binary projective observables O RD ≡ ( X ( = { 1,0 } ) ,2 X , F RD ) and O SW ≡ ( X ,2 X , F SW ) in L ∞ ( Ω , ν ) . Consider a measurement M L ∞ ( Ω , ν ) ( O RD , S [ ω ^ ( t ) ] ) . That is, we consider that the following three are equivalent (i.e., Axiom 1 (measurement) says that (C₁) Û (C₂). Also, (C₃) is the expression of (C₁) in ordinary language):

(C₁) A measured value 1 is obtained by the measurement M L ∞ ( Ω , ν ) ( O RD , S [ ω ^ ( t ) ] ) (i.e., [ M V ] ( M L ∞ ( Ω , ν ) ( O RD , S [ ω ^ ( t ) ] ) ) = 1 ) (strictly speaking, the probability that a measured value 1 is obtained by the measurement M L ∞ ( Ω , ν ) ( O RD , S [ ω ^ ( t ) ] ) is equal to 1.)

(C₂) ω ^ ( t ) ∈ RD ( ≡ { ω ∈ Ω | [ F RD ( { 1 } ) ( ω ) = 1 } )

(C₃) A tomato t is “red”.

Similarly, as shown in Figure 2 below, consider a measurement M L ∞ ( Ω , ν ) ( O SW , S [ ω ^ ( t ) ] ) . That is, we consider that the following three are equivalent:

( C ′ 1 ) A measured value 1 is obtained by the measurement M L ∞ ( Ω , ν ) ( O SW , S [ ω ^ ( t ) ] ) (i.e., [ M V ] ( M L ∞ ( Ω , ν ) ( O SW , S [ ω ^ ( t ) ] ) ) = 1 ).

( C ′ 2 ) ω ^ ( t ) ∈ SW ( ≡ { ω ∈ Ω | [ F SW ( { 1 } ) ( ω ) = 1 } )

( C ′ 3 ) A tomato t is “sweet”.

[Not]

It is clear that the following four are equivalent:

(D₀) [ M V ] ( M L ∞ ( Ω , ν ) ( O SW , S [ ω ^ ( t ) ] ) ) = 0 .

(D₁) A measured value 1 is obtained by the measurement M L ∞ ( Ω , ν ) ( Θ ¬ ( O SW ) , S [ ω ^ ( t ) ] ) (which is also denoted by ¬ M L ∞ ( Ω , ν ) ( O SW , S [ ω ^ ( t ) ] ) ), where Θ ¬ : { 1,0 } → { 1,0 } is defined by Θ ¬ ( 1 ) = 0 , Θ ¬ ( 0 ) = 1 (cf. Definition 1; image observable). Thus, [ M V ] ( ¬ M L ∞ ( Ω , ν ) ( O SW , S [ ω ^ ( t ) ] ) ) = 1 .

(D₂) ω ^ ( t ) ∈ [ SW ¯ ] c ( ≡ { ω ∈ Ω | [ F SW ( { 0 } ) ] ( ω ) = 0 } )

(D₃)A tomato t is not “sweet”.

[And]

We see that the following four are equivalent:

(E₀) A measured value ( 1,1 ) is obtained by the measurement M L ∞ ( Ω , ν ) ( O SW × O RD , S [ ω ^ ( t ) ] ) .

(E₁) A measured value 1 is obtained by the measurement M L ∞ ( Ω , ν ) ( Θ ∧ ( O SW × O RD ) , S [ ω ^ ( t ) ] ) (which is also denoted by M L ∞ ( Ω , ν ) ( O SW , S [ ω ^ ( t ) ] ) ∧ M L ∞ ( Ω , ν ) ( O RD , S [ ω ^ ( t ) ] ) ), where Θ ∧ : { 1,0 } 2 → { 1,0 } is defined by Θ ∧ ( 1,1 ) = 1 , Θ ∧ ( 1,0 ) = Θ ∧ ( 0,1 ) = Θ ∧ ( 0,0 ) = 0 (cf. Definition 1; image observable). Thus, [ M V ] ( M L ∞ ( Ω , ν ) ( O SW , S [ ω ^ ( t ) ] ) ∧ M L ∞ ( Ω , ν ) ( O RD , S [ ω ^ ( t ) ] ) ) = 1 .

(E₂) ω ^ ( t ) ∈ SW ( ≡ { ω ∈ Ω | [ F SW ( { 1 } ) ] ( ω ) = 1 } )                   ∩ RD ( ≡ { ω ∈ Ω | [ F RD ( { 1 } ) ] ( ω ) = 1 } )

(E₃) A tomato t is “sweet” and “red”

Remark 8. When ω 1 ≠ ω 2 , it should be noted that the simbol “ M L ∞ ( Ω , ν ) ( O SW , S [ ω 1 ] ) ∧ M L ∞ ( Ω , ν ) ( O RD , S [ ω 2 ] ) ” is not yet defined. Recall the linguistic Copenhagen interpretation “Only one measurement is permitted”. Thus,

“there is only one state”. Therefore, this should be defined by the parallel M L ∞ ( Ω , ν ) ⊗ L ∞ ( Ω , ν ) ( Θ ∧ ( O SW ⊗ O RD ) , S [ ( ω 1 , ω 2 ) ] ) . More generally, the simbol “ ∧ λ ∈ Λ M L ∞ ( Ω λ , ν λ ) ( O Γ λ , S [ ω λ ] ) ” is defined by

M ⊗ λ ∈ Λ L ∞ ( Ω λ , ν λ ) ( ⊗ λ ∈ Λ O Γ λ , S [ ( ω λ ) λ ∈ Λ ] )

[Or]

We see that the following four are equivalent:

(F₀) A measured value ( x 1 , x 2 ) obtained by the measurement M L ∞ ( Ω , ν ) ( O SW × O RD , S [ ω ^ ( t ) ] ) belongs to { ( 1,1 ) , ( 1,0 ) , ( 0,1 ) }

(F₁) A measured value 1 is obtained by the measurement M L ∞ ( Ω , ν ) ( Θ ∨ ( O SW × O RD ) , S [ ω ^ ( t ) ] ) (which is also denoted by M L ∞ ( Ω , ν ) ( O SW , S [ ω ^ ( t ) ] ) ∨ M L ∞ ( Ω , ν ) ( O RD , S [ ω ^ ( t ) ] ) ), where Θ ∨ : { 1,0 } 2 → { 1,0 } is defined by Θ ∨ ( 1 , 1 ) = Θ ∨ ( 1 , 0 ) = Θ ∨ ( 0 , 1 ) = 1 , Θ ∨ ( 0 , 0 ) = 0 .

Thus, [ M V ] ( M L ∞ ( Ω , ν ) ( O SW , S [ ω ^ ( t ) ] ) ∨ M L ∞ ( Ω , ν ) ( O RD , S [ ω ^ ( t ) ] ) ) = 1 .

(F₂) ω ^ ( t ) ∈ SW ( ≡ { ω ∈ Ω | [ F SW ( { 1 } ) ] ( ω ) = 1 } )                   ∪ RD ( ≡ { ω ∈ Ω | [ F RD ( { 1 } ) ] ( ω ) = 1 } )

(F₃) A tomato t is “sweet” or “red”

[Implication]

We see that the following four are equivalent:

(G₀) A measured value ( x 1 , x 2 ) obtained by the measurement M L ∞ ( Ω , ν ) ( O SW × O RD , S [ ω ^ ( t ) ] ) belongs to { ( 1,1 ) , ( 0,1 ) , ( 0,0 ) }

(G₁) A measured value 1 is obtained by the measurement M L ∞ ( Ω , ν ) ( Θ → ( O SW × O RD ) , S [ ω ^ ( t ) ] ) (which is also denoted by M L ∞ ( Ω , ν ) ( O SW , S [ ω ^ ( t ) ] ) → M L ∞ ( Ω , ν ) ( O RP , S [ ω ^ ( t ) ] ) ), where Θ → : { 1,0 } 2 → { 1,0 } is defined by Θ → ( 1 , 1 ) = Θ → ( 0 , 1 ) = Θ → ( 0 , 0 ) = 1 , Θ → ( 1,0 ) = 0 . Thus, [ M V ] ( M L ∞ ( Ω , ν ) ( O SW , S [ ω ^ ( t ) ] ) → M L ∞ ( Ω , ν ) ( O RD , S [ ω ^ ( t ) ] ) ) = 1 .

(G₂) ω ^ ( t ) ∈ SW c ( ≡ { ω ∈ Ω | [ F SW ( { 1 } ) ] ( ω ) = 0 } )                   ∪ RD ( ≡ { ω ∈ Ω | [ F RD ( { 1 } ) ] ( ω ) = 1 } )

(G₃) A tomato t is not “sweet”, or it is “red”

Summing up the above, we have the following theorem:

Theorem 9. We see that for each ω ∈ Ω ,

(a) [ M V ] ( ¬ M L ∞ ( Ω , ν ) ( O Γ , S [ ω ^ ( t ) ] ) ) = 1 − [ M V ] ( M L ∞ ( Ω , ν ) ( O Γ , S [ ω ^ ( t ) ] ) )

(b) [ M V ] ( M L ∞ ( Ω , ν ) ( O Γ 1 , S [ ω ^ ( t ) ] ) ∧ M L ∞ ( Ω , ν ) ( O Γ 2 , S [ ω ^ ( t ) ] ) ) = min { [ M V ] ( M L ∞ ( Ω , ν ) ( O Γ 1 , S [ ω ^ ( t ) ] ) ) , [ M V ] ( M L ∞ ( Ω , ν ) ( O Γ 2 , S [ ω ^ ( t ) ] ) ) }

(c) [ M V ] ( M L ∞ ( Ω , ν ) ( O Γ 1 , S [ ω ^ ( t ) ] ) ∨ M L ∞ ( Ω , ν ) ( O Γ 2 , S [ ω ^ ( t ) ] ) ) = max { [ M V ] ( M L ∞ ( Ω , ν ) ( O Γ 1 , S [ ω ^ ( t ) ] ) ) , [ M V ] ( M L ∞ ( Ω , ν ) ( O Γ 2 , S [ ω ^ ( t ) ] ) ) }

(d) [ M V ] ( M L ∞ ( Ω , ν ) ( O Γ 1 , S [ ω ^ ( t ) ] ) → M L ∞ ( Ω , ν ) ( O Γ 2 , S [ ω ^ ( t ) ] ) ) = max { 1 − [ M V ] ( M L ∞ ( Ω , ν ) ( O Γ 1 , S [ ω ^ ( t ) ] ) ) , [ M V ] ( M L ∞ ( Ω , ν ) ( O Γ 2 , S [ ω ^ ( t ) ] ) ) }

Therefore, we can expect that a class of classical binary projective measurements has a logical structure. In this sense, this class can be regarded as a class of “propositions”.

Remark 10. Note that propositional logic (i.e., ¬ , ∧ , ∨ , → ) and predict logic (i.e., ¬ , ∧ , ∨ , → , ∀ , ∃ ) are essentially the same since P 1 ∧ P 2 ∧ P 3 ∧ ⋯ = ( ∀ n ) [ P n ] and P 1 ∨ P 2 ∨ P 3 ∨ ⋯ = ( ∃ n ) [ P n ] . Thus, this paper does not distinguish propositional logic and predict logic.

Further, consider a measurement M L ∞ ( Ω , ν ) ( O RP , S [ ω ^ ( t ) ] ) . That is, we consider that the following three are equivalent:

(H₁) A measured value 1 is obtained by the measurement M L ∞ ( Ω , ν ) ( O RP , S [ ω ^ ( t ) ] ) .

(H₂) ω ^ ( t ) ∈ RP ( ≡ { ω ∈ Ω | [ F RP ( { 1 } ) ] ( ω ) = 1 } )

(H₃) A tomato t is “ripe”.

Theorem 11. [Syllogism]:

Let t be a tomato, and let ω ^ ( t ) ( ∈ Ω ) be the state of t. Assume the followings:

(I₀) A measured value ( x 1 , x 2 ) obtained by the measurement M L ∞ ( Ω , ν ) ( O SW × O RP , S [ ω ^ ( t ) ] ) belongs to { ( 1,1 ) , ( 0,1 ) , ( 0 , 0 ) }

which is equivalent to

(I₁) [ M V ] ( M L ∞ ( Ω , ν ) ( O SW , S [ ω ^ ( t ) ] ) → M L ∞ ( Ω , ν ) ( O RP , S [ ω ^ ( t ) ] ) ) = 1

(I₂) ω ^ ( t ) ∉ SW ( ≡ { ω ∈ Ω | [ F SW ( { 1 } ) ] ( ω ) = 1 } )   ∨ ω ^ ( t ) ∈ RP ( ≡ { ω ∈ Ω | [ F RP ( { 1 } ) ] ( ω ) = 1 } )

(I₃) A tomato t is not “sweet”, or it is “ripe”.

and

( I ′ 0 ) A measured value ( x 2 , x 3 ) obtained by the measurement M L ∞ ( Ω , ν ) ( O RP × O RD , S [ ω ^ ( t ) ] ) belongs to { ( 1,1 ) , ( 0,1 ) , ( 0 , 0 ) }

which is equivalent to

( I 1 ) [ M V ] ( M L ∞ ( Ω , ν ) ( O RP , S [ ω ^ ( t ) ] ) → M L ∞ ( Ω , ν ) ( O RD , S [ ω ^ ( t ) ] ) ) = 1

which is equivalent to

( I 2 ) ω ^ ( t ) ∉ RP ( ≡ { ω ∈ Ω | [ F RP ( { 1 } ) ] ( ω ) = 1 } )   ∨ ω ^ ( t ) ∈ RD ( ≡ { ω ∈ Ω | [ F RD ( { 1 } ) ] ( ω ) = 1 } )

( I 3 ) A tomato t is not “ripe”, or it is “red”.

Then the following holds:

(J₀) A measured value ( x 1 , x 3 ) obtained by the measurement M L ∞ ( Ω , ν ) ( O SW × O RD , S [ ω ^ ( t ) ] ) belongs to { ( 1,1 ) , ( 0,1 ) , ( 0 , 0 ) }

(J₁) [ M V ] ( M L ∞ ( Ω , ν ) ( O SW , S [ ω ^ ( t ) ] ) → M L ∞ ( Ω , ν ) ( O RD , S [ ω ^ ( t ) ] ) ) = 1

(J₂) ω ^ ( t ) ∉ SW ( ≡ { ω ∈ Ω | [ F RD ( { 1 } ) ] ( ω ) = 1 } )   ∨ ω ^ ( t ) ∈ RD ( ≡ { ω ∈ Ω | [ F RP ( { 1 } ) ] ( ω ) = 1 } )

(J₃) A tomato t is not “sweet”, or it is “red”.

[Proof]: Recalling the linguistic Copenhagen interpretation “Only one measurement is permitted” (in Sec. 2.3), we have enough to see the simultaneous observable O SW × O RP × O RD , which uniquely exists (cf. Definition 4 (i)). Thus, we have the measurement M L ∞ ( Ω , ν ) ( O SW × O RP × O RD , S [ ω ^ ( t ) ] ) . Let ( x 1 , x 2 , x 3 ) be the measured value. We easily see that ( x 1 , x 3 ) belongs to { ( 1,1 ) , ( 0,1 ) , ( 0 , 0 ) } . Thus, (J) holds.

However, we should add the following. This proof is not self-evident since the existence and uniqueness of the simultaneous observable O SW × O RP × O RD is not trivial (cf. Definition 4 (i)). Also, see Sec. 5 (i.e., Syllogizm does not always hold in quantum sistems).

Consider the state space Ω , which is finite (or, countable) with a metric d (i.e., d ( ω 1 , ω 2 ) = 1   ( ω 1 ≠ ω 2 ) , = 0   ( ω 1 = ω 2 ) .

Definition 12. Let λ be any element of Ω . Putting Γ = { λ } in the formula (2), define the elementary binary projective observable O { λ } = ( X ( = { 1,0 } ) ,2 X , F { λ } ) in L ∞ ( Ω , ν ) such that

[ F { λ } ( { 1 } ) ] ( ω ) = ( 1 ( if     ω = λ ) 0 ( if     ω ≠ λ ) , [ F { λ } ( { 0 } ) ] ( ω ) = 1 − [ F { λ } ( { 1 } ) ] ( ω )             ( ∀ ω ∈ Ω )

The measurement M L ∞ ( Ω , ν ) ( O { λ } , S [ ω ] ) ( λ , ω ∈ Ω ) is called an elementary measurement.

It is clear that it holds that

(K₁) A measured value 1 is obtained by the elementary measurement M L ∞ ( Ω , ν ) ( O { λ } , S [ ω ] ) ⇔ λ = ω

(K₂) A measured value 1 is obtained by the elementary measurement M L ∞ ( Ω , ν ) ( O { λ } , S [ ω ] ) ⇔ λ ≠ ω

Under the above preparation, the following theorem is clear

Theorem 13. Let Γ be a subset of Ω . And let ω ∈ Ω . Then we see that

M L ∞ ( Ω , ν ) ( O Γ , S [ ω ] ) = ∨ λ ∈ Γ M L ∞ ( Ω , ν ) ( O { λ } , S [ ω ] )

Remark 14. Many readers may not consider the above theorem to be particularly important. I have the same view. However, this theorem was prepared in preparation for Sec.4 (Theorem 23). Of course, the spirit of expressing complex observables in simple observables is quite important. In quantum language, this spirit is realized by von Neumann’s spectral decomposition theorem (see the formula (3), or [22] ) and Holevo’s theorem (cf. ref. [26] ), that is, “Any observable can be represented by the product of commutative binary projective observables in QL (i.e., both classical and quantum QL)”,

Remark 15. In this section, I devoted myself to the classical cases. Our arguments in this section are invalid in quantum cases. Consider a binary projective observable O = ( { 1,0 } ,2 { 1,0 } , F ) in B ( H ) , where F ( { 1 } ) = P ( = projection ) , F ( { 0 } ) = I − P . Define the state ρ by | e 〉 〈 e | (where e ∈ H such that ‖ e ‖ H = 1 ). Axiom 1 say that

· The probability that a measured value 1 ( ∈ X = { 1 , 0 } ) is obtained by the measurement M ( O : = ( X ,2 { 1,0 } , F ) , S [ ρ ] ) is given by T r [ ρ F ( { 1 } ) ] ( = ‖ P e ‖ H )

Thus, if e ∉ P H ∪ ( 1 − P ) H (i.e., in most case) the “probability” belongs to the open interval ( 0,1 ) . That is, in quantum cases, the probability concept cannot be covered up. Thus, the arguments in this section are invalid in quantum cases.

The question “What is a proposition?” is easy in mathematics. However, outside of mathematics, this question is not easy. Wittgenstein’s purpose is to clarify “proposition” in science.

Let Ω , Γ , ω , ω ^ ( t ) , SW , RP , RD , ⋯ be the same as in the previous section. Let us start from the following definition.

Definition 17. [Proposition, Truth value] A pair ( Γ , ω ) ( Γ ⊆ Ω , ω ∈ Ω ) is called a proposition in Ω (i.e., “a system with a state ω has Γ -property”), which is denoted by P Ω ( Γ , S [ ω ] ) . Define the truth value of the proposition P Ω ( Γ , S [ ω ] ) (which is denoted by [ T V ] ( P Ω ( Γ , S [ ω ] ) ) ) by

[ T V ] ( P Ω ( Γ , S [ ω ] ) ) = ( T   ( true )       ( if   ω ∈ Γ ) F   ( false )     ( if   ω ∉ Γ )  

That is, truth values { T , F } ( = X ) correspond to measured values { 1,0 } ( = X ) (or, X = { x 1 , x 0 } ). Thus, I think that the above is essentially the same as naivie set theory (cf. ref. [28] ).

If [ T V ] ( P Ω ( Γ , S [ ω ^ ( t ) ] ) ) = “ T ” , we may say that

(M) a tomato t has Γ -property.

As shown in Figure 3 below, consider a proposition P Ω ( RD , S [ ω ^ ( t ) ] ) . Thus, the following three are equivalent (i.e., (N₁) Û (N₂). Also, (N₃) is the expression of (N₁) in ordinary language):

(N₁) [ T V ] ( P Ω ( RD , S [ ω ^ ( t ) ] ) ) = “ T ” .

(N₂) ω ^ ( t ) ∈ RD

(N₃) A tomato t is “red”.

Similarly, consider a proposition P Ω ( SW , S [ ω ^ ( t ) ] ) . Thus, we see that the following three are equivalent:

(O₁) [ T V ] ( P Ω ( SW , S [ ω ^ ( t ) ] ) ) = “ T ” .

(O₂) ω ^ ( t ) ∈ SW

(O₃) A tomatot is “sweet”.

[Not]

It is clear that the following three are equivalent:

(R₁) [ T V ] ( P Ω ( SW c , S [ ω ^ ( t ) ] ) ) = “ T ” . (Here, P Ω ( SW c , S [ ω ^ ( t ) ] ) is also denoted by ¬ P Ω ( SW , S [ ω ^ ( t ) ] ) ), Thus, [ T V ] ( ¬ P Ω ( SW , S [ ω ^ ( t ) ] ) ) = “ T ” .

(R₂) ω ^ ( t ) ∈ [ SW ] c

(R₃) A tomato t is not “sweet”.

[And]

We see that the following three are equivalent:

(S₁) A truth value of the proposition P Ω ( SW ∩ RD , S [ ω ^ ( t ) ] ) is equal to “T” (which is also denoted by P Ω ( SW , S [ ω ^ ( t ) ] ) ∧ P Ω ( RD , S [ ω ^ ( t ) ] ) ).

Thus, [ T V ] ( P Ω ( SW , S [ ω ^ ( t ) ] ) ∧ P Ω ( RD , S [ ω ^ ( t ) ] ) ) = “ T ” .

(S₂) ω ^ ( t ) ∈ SW ∩ RD

(S₃) A tomato t is “sweet” and “red”

[Or]

We see that the following three are equivalent:

(T₁) A truth value of the proposition P Ω ( Θ ∨ ( SW ∪ RD ) , S [ ω ^ ( t ) ] ) (which is also denoted by P Ω ( SW , S [ ω ^ ( t ) ] ) ∨ P Ω ( RD , S [ ω ^ ( t ) ] ) ) is equal to “T”) That is, [ T V ] ( P Ω ( SW , S [ ω ^ ( t ) ] ) ∨ P Ω ( RD , S [ ω ^ ( t ) ] ) ) = “ T ” .

(T₂) ω ^ ( t ) ∈ SW ∪ RD

(T₃) A tomato t is “sweet” or “red”

[Implication]

We see that the following three are equivalent:

(U₁) A truth value of the proposition P Ω ( SW c ∪ RD , S [ ω ^ ( t ) ] ) (which is also denoted by P Ω ( SW , S [ ω ^ ( t ) ] ) → P Ω ( RP , S [ ω ^ ( t ) ] ) ) is equal to “T”, That is, [ T V ] ( P Ω ( SW , S [ ω ^ ( t ) ] ) → P Ω ( RD , S [ ω ^ ( t ) ] ) ) = “ T ” .

(U₂) ω ^ ( t ) ∈ SW c ∪ RD

(U₃) a tomato t is not “sweet”, or it is “red”

Remark 18. Note that the above is essentially the same as logical operation (Boolean algebra; [ ¬ , ∧ , ∨ , → ]). However, it should be noted that Definition 17 is essential to the above argument.

Further, consider a proposition P Ω ( RP , S [ ω ^ ( t ) ] ) . That is, we consider that the following three are equivalent:

(V₁) [ T V ] ( P Ω ( RP , S [ ω ^ ( t ) ] ) ) = “ T ” .

(V₂) ω ^ ( t ) ∈ RP

(V₃) A tomato t is “ripe”.

Theorem 19. [Syllogism]:

Let t be a tomato, and let ω ^ ( t ) ( ∈ Ω ) be the state of t. Assume the followings:

(W₁) [ T V ] ( P Ω ( SW , S [ ω ^ ( t ) ] ) → P Ω ( RP , S [ ω ^ ( t ) ] ) ) = “ T ” .

which is equivalent to

(W₂) [ ω ^ ( t ) ∉ SW ] ∨ [ ω ^ ( t ) ∈ RP ]

(W₃) A tomato t is not “sweet”, or it is “ripe”.

And further, assume

( W ′ 1 ) [ T V ] ( P Ω ( RP , S [ ω ^ ( t ) ] ) → P Ω ( RD , S [ ω ^ ( t ) ] ) ) = “ T ” .

which is equivalent to

( W ′ 2 ) [ ω ^ ( t ) ∉ RP ] ∨ [ ω ^ ( t ) ∈ RD ]

( W ′ 3 ) A tomato t is not “ripe”, or it is “red”.

Then the following holds:

(X₁) [ T V ] ( P Ω ( SW , S [ ω ^ ( t ) ] ) → P Ω ( RD , S [ ω ^ ( t ) ] ) ) = “ T ” .

(X₂) [ ω ^ ( t ) ∉ SW ] ∨ [ ω ^ ( t ) ∈ RD ]

(X₃) A tomato t is not “sweet”, or it is “red”.

[Proof using Definition 17]

A simple calculation shows that

[ S W c ∪ R P ] ∩ [ P R c ∪ R D ] = [ S W c ∩ P R c ] ∪ [ S W c ∩ R D ] ∪ [ R P ∩ P R c ] ∪ [ R P ∩ R D ]   = [ S W c ∩ P R c ] ∪ [ S W c ∩ R D ] ∪ [ R P ∩ R D ] ⊆ S W c ∪ R D

Recalling (W₂) and ( W ′ 2 ), we immediately see (X₂). and thus, (X₁), (X₃).

Remark 20. Note that the proof of this theorem (due to Definition 17) is simple compared to Theorem 11 (using Axiom 1 and the linguistic Copenhagen interpretation in Sec. 2.3). That is, in the proof of Theorem 19 we do not need to check the existence and uniqueness of the simultaneous observable O SW × O RP × O RD .

The following exercise will promote the reader’s understanding of “proposition”.

Exercise 21. Let Γ ⊆ Ω and Λ ⊆ Ω . Then, we have the following question.

· Is the statement “ Λ ⊆ Γ ” a proposition?

[Answer]:

Λ ⊆ Γ

⇔ ( ∀ λ ∈ Λ ) [ λ ∈ Γ ]

⇔ ( ω λ ) λ ∈ Λ ∈ × λ ∈ Λ Γ λ   ( where   ω λ : = λ   ( ∀ λ ∈ Λ ) ) ,   Γ λ : = Γ   ( ∀ λ ∈ Λ )

⇔ P Ω Λ [ × λ ∈ Λ Γ λ , S [ ( ω λ ) λ ∈ Λ ] ]     ( where   Ω Λ   is   Λ -dimensional   product   space )

Thus, the statement “ Λ ⊆ Γ ” is a proposition in Ω Λ .

Consider the state space Ω , which is finite (or, countable) with a metric d (i.e., d ( ω 1 , ω 2 ) = 1   ( ω 1 ≠ ω 2 ) , = 0   ( ω 1 = ω 2 ) . Further, assume that the Borel measure ν is defined by the point measure, i.e., ν ( { ω } ) = 1   ( ∀ ω ∈ Ω ) .

Definition 22. Let λ be any element of Ω . Putting Γ = { λ } , define the proposition P Ω ( { λ } , S [ ω ] ) , which is called an elementary proposition.

It is clear that it holds that

· A truth value of the elementary proposition P Ω ( { λ } , S [ ω ] ) is equal to “T” ⇔ λ = ω

· A truth value of the elementary proposition P Ω ( { λ } , S [ ω ] ) is equal to “F” ⇔ λ ≠ ω

Theorem 23. Let Γ be a subset of Ω . And let ω ∈ Ω . Then we see that

P Ω ( Γ , S [ ω ] ) = ∨ λ ∈ Γ P Ω ( { λ } , S [ ω ] )

That is, every proposition can be represented by the sum of elementary propositions. This is not trivial since Exercise 21 is not trivial.

[Proof].

We see that

The true value of P Ω ( Γ , S [ ω ] ) is equal to “T”

⇔ ω ∈ Γ

⇔ ∃ λ ( ∈ Γ ) [ ω ∈ { λ } ]

⇔ The true value of ∨ λ ∈ Γ P Ω ( { λ } , S [ ω ] ) is equal to “T”

Remark 24. I believe that the above is the main assertion in Wittgenstein’s picture theory. However many readers may not consider the above theorem to be particularly important. I have the same view. In fact, this theorem holds only in special cases such as Ω is a finite (or Ω has a discrete topology).