Recently, in refs. [1] - [6], we proposed the linguistic Copenhagen interpretation (or, quantum language, measurement theory), which has a great linguistic power to describe both classical and quantum systems. Thus we think that quantum language can be viewed as the language of science. As seen in Figure 1 below, roughly speaking, QL has the following four aspects, that is,

(A₁) ⑦: the linguistic turn of quantum mechanics (cf. refs. [7] [8] [9] [10] );

(A₂) ⑧: the dualistic turn of statistics (cf. refs. [11] - [16] );

(A₃) ⑩: the scientific turn of Descartes = Kant philosophy (cf. refs. [17] [18] [19] );

(A₄) ⑬: the logical aspect (cf. refs. [20] [21] [22] ).

Thus the location of QL in the history of the world-descriptions is as in Figure 1 (cf. ref. [17] [22] [23] ):

The purpose of this paper is to study the logical aspect [⑬ + ⑭] of QL. Although it is generally said that logic and time do not go well together, many researchers have attempted to incorporate time into logic (e.g. ref. [24] ). In particular, Pnueli’s work (cf. [25] ) is highly regarded in the field of computing, and he was awarded the 1997 Turing Award for this achievement. It is natural to consider that mechanics and time series go hand in hand. Therefore, we can expect that quantum fuzzy logic (in [⑬ + ⑭]) and time series are also compatible. In fact, this paper shows that quantum fuzzy logic is as closely related to time as it is to quantum mechanics. However, “time” in everyday language has various aspects (e.g., tense, subjective time). Therefore, it is not possible to understand all of the “time” of everyday language by the “time” of quantum language.

In this section, we shall review quantum language (i.e., the linguistic Copenhagen interpretation of quantum mechanics, or measurement theory), which has the following form:

(B) Q u a n t u m   l a n g u a g e ( = m e a s u r e m e n t   t h e o r y ) ( q u a n t u m   m e c h a n i c a l   w o r l d v i e w ) = A x i o m   2 ( c a u s a l i t y )   +   A x i o m   1 ( m e a s u r e m e n t )   +   l i n g u i s t i c   ( C o p e n h a g e n )   interpretation ( h o w   t o   u s e   A x i o m s   1   a n d   2 )

QL is classified as follows

(C) ( ( C 1 ) : q u a n t u m   Q L   f o r   q u a n t u m   s y s t e m ,       ( w h i c h   i s   e s s e n t i a l l y   t h e   s a m e   a s   q u a n t u m   m e c h a n i c s ) ( C 2 ) : c l a s s i c a l   Q L   f o r   c l a s s i c a l   s y s t e m   ,     ( w h i c h   m a y   b e   v i e w e d   a s   s t a t i s t i c s   w i t h   a   m e a s u r e m e n t   c o n c e p t )

It is usual to discuss the above two simultaneously (cf. refs. [5] [6] [8] ). However, in this paper, we will devote ourselves to only classical QL. That is because we think that classical QL is easy to understand for readers who are not familiar with quantum theory. We do not want the difficulty of the mathematics (cf. refs. [26] [27] [28] ) to hinder the spread of QL.

Let Ω be a state space, i.e., compact space. An element ω in Ω is called a state. And let C ( Ω ) be the commutative C ∗ -algebra, i.e., the space of all complex-valued continuous functions on Ω .

Definition 1. [Observable, Image observable] According to the noted idea (cf. refs. [29] [30] ), an observable O = ( X , P ( X ) , G ) in C ( Ω ) is defined as follows:

1) X is a finite set, P ( X ) ( = 2 X , the power set of X).

2) G is a mapping from P ( X ) to C ( Ω ) satisfying: a): for every Ξ ∈ P ( X ) , G ( Ξ ) is a non-negative element in C ( Ω ) such that 0 ≤ G ( Ξ ) ≤ I , b): G ( ∅ ) = 0 and G ( X ) = I , where 0 and I are the 0-element and the identity in C ( Ω ) respectively. c): [additivity]

G ( Ξ 1 ) + G ( Ξ 2 ) = G ( Ξ 1 ∪ Ξ 2 ) (1)

for all Ξ 1 , Ξ 2 ∈ P ( X ) such that Ξ 1 ∩ Ξ 2 = ∅ .

If G ( Ξ ) = G ( Ξ ) 2 ( ∀ Ξ ∈ P ( X ) ) , then O = ( X , P ( X ) , G ) in C ( Ω ) is a projective observable (or, crisp observable). Also, O = ( X , P ( X ) , G ) in C ( Ω ) is also called an X-valued observable. We will devote ourselves to binary (i.e., { T , F } -valued) observables in most of the cases in this paper. Let Y be a finite set, and let Θ : X → Y be a map. Then, Θ ( O ) = ( Y , G , G ( Θ − 1 ( ⋅ ) ) ) in C ( Ω ) is also observable in C ( Ω ) (which is called an image observable).

With any classical system S, a commutative C * -algebra C ( Ω ) can be associated in which the measurement theory (B) of that system can be formulated. A state of the system S is represented by an element ω ( ∈ Ω ) and an observable is represented by an observable O = ( X , P ( X ) , G ) in C ( Ω ) . Also, the measurement of the observable O for the system S with the state ω 0 is denoted by M C ( Ω ) ( O = ( X , P ( X ) , G ) , S [ ω 0 ] ) . An observer can obtain a measured value x ( ∈ X ) by the measurement M ( O , S [ ω 0 ] ) .

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.

Axiom 1. [Measurement]. The probability that a measured value x ( ∈ X ) obtained by the measurement M C ( Ω ) ( O = ( X , P ( X ) , G ) , S [ ω 0 ] ) is given by [ G ( { x } ) ] ( ω 0 ) .

Next, we explain Axiom 2 in (B).

Let ( T ≡ { t 0 , t 1 , ⋯ , t n } , ≤ ) be a (finite) tree-like semi-ordered set with the root t 0 . Let Π : T \ { t 0 } → T is the parent map, that is, Π ( t ) = t ′ is defined such that

t ′ ≤ t ,   t ′ ≠ t ,   “ t ″ ≤ t ”     implies   that     t ″ ≤ t ′ (2)

This ( T ≡ { t 0 , t 1 , ⋯ , t n } , ≤ ) is also written by ( T ≡ { t 0 , t 1 , ⋯ , t n } , Π : T \ { t 0 } → T ) .

For example see Figure 2 below, in which we see the root t 0 , the parent map: Π ( t 3 ) = Π ( t 4 ) = t 2 , Π ( t 2 ) = Π ( t 5 ) = t 1 , Π ( t 1 ) = Π ( t 6 ) = Π ( t 7 ) = t 0 .

Let ( T ≡ { t 0 , t 1 , ⋯ , t n } , Π : T \ { t 0 } → T ) be a tree-like semi-ordered set with the root t 0 . For each t ∈ T \ { t 0 } , define the homomorphism. Φ Π ( t ) , t : C ( Ω t ) → C ( Ω Π ( t ) ) (see, for example, Figure 3), which is characterized as a continuous map Φ Π ( t ) , t : Ω Π ( t ) → Ω t such that

[ Φ Π ( t ) , t f 2 ] ( ω 1 ) = f 2 ( ϕ Π ( t ) , t ( ω 1 ) )   ( ∀ f 2 ∈ C ( Ω t ) , ∀ ω 1 ∈ Ω Π ( t ) ) (3)

Now we can propose Axiom 2 (i.e., causality). (For details, see ref. [6].)

Axiom 2. [Causality]; Let ( T ≡ { t 0 , t 1 , ⋯ , t n } , ≤ ) be a (finite) tree-like semi-ordered set with the root t 0 . Causality is represented by { homomorphism   Φ Π ( t ) , t : C ( Ω t ) → C ( Ω Π ( t ) ) : t ∈ T \ { t 0 } } .

Remark 2. [The linguistic Copenhagen interpretation] Since the linguistic Copenhagen interpretation is the manual to use Axioms 1 and 2, it consists of many rules (cf. refs. [6] [22] [23] ). However, for the purposes of this paper, it is sufficient to focus only on the following.

(D₁) Only one measurement is permitted, and thus, the state after a measurement is non-sense. Thus, we proposed the new formulation of projection postulate (i.e., wavefunction collapse) (cf. ref. [9] ).

(D₂) Time should be represented by a tree-like semi-ordered set ( T ≡ { t 0 , t 1 , ⋯ , t n } , ≤ ) in Axiom 2.

(D₃) The Heisenberg picture M C ( Ω 0 ) ( Φ t 0 , t 1 O 1 , S [ ω t 0 ] ) is used and the Schrödinger picture M C ( Ω t 1 ) ( O 1 , S [ ϕ t 0 , t 1 ( ω 0 ) ] ) is not.

(D₄) The subjective time (i.e., observer’s time, tense) does not exist. (cf. Leibniz = Clarke correspondence in ref. [19].)

(D₅) The measurer cannot measure himself. Thus, the assertions: “I think”, “I am”, etc. are not scientific. Therefore, “I think, therefore I am” is not a scientific proposition.

We need the following definition for the above (D₁).

Definition 3. [(i): Quasi-product observable, quasi-product measurement]: Let O i = ( X i , P ( X i ) , G i ) ( i = 1,2, ⋯ , N ) be commutative observables in C ( Ω ) . Define the quasi-product observable × i = 1 , 2 , ⋯ , n q p O i = ( × i = 1 n X i , P ( × i = 1 n X i ) , × i = 1 , 2 , ⋯ , n q p G i ) such that

[ × i = 1,2, ⋯ , n q p G i ] ( X 1 × X 2 × ⋯ × X j − 1 × Ξ j × X j + 1 × ⋯ × X n ) = G j ( Ξ j ) ( ∀ Ξ j ∈ P ( X j ) , j = 1,2, ⋯ , n ) (4)

Also, M C ( Ω ) ( × i = 1 , 2 , ⋯ , n q p O i = ( × i = 1 n X i , P ( × i = 1 n X i ) , × i = 1 , 2 , ⋯ , n q p G i ) , S [ ω 0 ] ) is called the quasi-product measurement of M C ( Ω ) ( O i = ( X i , P ( X i ) , G i ) , S [ ω 0 ] ) ( i = 1,2, ⋯ , n ) . [(ii): Tensor C * -algebra, tensor product observable, tensor quasi-product measurement]: Let O i = ( X i , P ( X i ) , G i ) be observables in C ( Ω ) i , ( i = 1,2, ⋯ , n ) . Define a tensor product observable ⊗ i = 1 , 2 , ⋯ , n O i = ( × i = 1 n X i , P ( × i = 1 n X i ) , ⊗ i = 1 , 2 , ⋯ , n G i ) in the tensor C * -algebra ⊗ i = 1 n C ( Ω i ) ( = C ( × i = 1 n Ω i ) ) . such that

[ ⊗ i = 1 n G i ] ( × i = 1 n Ξ j ) = ⊗ i = 1 n G i ( Ξ i )   ( ∀ Ξ i ∈ P ( X i ) , i = 1,2, ⋯ , n ) (5)

Also, M ⊗ i = 1 n C ( Ω i ) ( ⊗ i = 1 , 2 , ⋯ , n O i = ( × i = 1 n X i , P ( × i = 1 n X i ) , ⊗ i = 1 , 2 , ⋯ , n G i ) , S [ ( ω 1 , ω 2 , ⋯ , ω n ) ] ) is called a tensor product measurement of M C ( Ω i ) ( O i = ( X i , P ( X i ) , G i ) , S [ ω i ] ) ( i = 1,2, ⋯ , n ) .

In the previous section, we introduced classical QL, which is the mathematical representation of the quantum mechanical worldview. In this section, we introduce the quantum fuzzy logic in the quantum mechanical worldview. We believe that Wittgenstein’s purpose of TLP (in ref. [31] ) is to propose the definition of “proposition” (i.e., to define “what we can speak about”). And thus, in ref. [22] (particularly Theorem 16 in ref. [22] is the fundamental theorem in quantum fuzzy logic), we asserted that “quantum fuzzy logic” realized Wittgenstein’s dream.

Let’s start with the following definition.

Definition 4. [(TF)-measurement (=Fuzzy proposition), Fuzzy set (= Membership function)] Let O = ( { T , F } , P ( { T , F } ) , G ) be a binary observable (or, (TF)-observable, { T , F } -valued observable) in a commutative C * -algebra C ( Ω ) . A measurement M C ( Ω ) ( O , S [ ω 0 ] ) is called a (TF)-measurement, which is also called a fuzzy proposition. Since Axiom 1 says that the probability that a measured value T is obtained by (TF)-measurement M C ( Ω ) ( O , S [ ω 0 ] ) is given by [ G ( { T } ) ] ( ω 0 ) , we say that

(E₁) a (TF)-measurement M C ( Ω ) ( O , S [ ω 0 ] ) is true with probability [ G ( { T } ) ] ( ω 0 ) .

Or,

(E₂) Prob [ M C ( Ω ) ( O , S [ ω 0 ] ) ; { T } ] = [ G ( { T } ) ] ( ω 0 )

Also, G ( { T } ) ( ∈ C ( Ω ) ) is called the membership function of O.

Definition 5. [Quantum fuzzy logic symbols ( ¬ , ∧ , ∨ , → ))] Let

O i = ( { T , F } , P ( { T , F } ) , G i ) be binary observables (or, { T , F } -valued observable) in a commutative C * -algebra C ( Ω ) , ( i = 1 , 2 ). Fix the quasi-product observable O 1 × q p O 2 = ( { T , F } 2 , P ( { T , F } 2 ) , G 1 × q p G 2 ) . Consider (TF)-measurement M C ( Ω ) ( O i = ( { T , F } , P ( { T , F } ) , G i ) , S [ ω 0 ] ) (which is abbreviated as P i ) in a C * -algebra C ( Ω ) . Put μ i ( Ξ ) = [ G ( { T } ) ] ( ω 0 ) ( Ξ ∈ { T , F } , i = 0,1,2 ) , and ( × i = 1,2 q p μ i ) ( Ξ 1 × Ξ 2 ) = [ ( G 1 × q p G 2 ) ( Ξ 1 × Ξ 2 ) ] ( ω 0 ) ( Ξ 1 , Ξ 2 ∈ 2 { T , F } ) .

(i; Negation): Put i = 1 , 2 . Remembering the image observable in Definition 1, let’s define ¬ M C ( Ω ) ( O i , S [ ω 0 ] ) such that

¬ M C ( Ω ) ( O i , S [ ω 0 ] ) = M C ( Ω ) ( π ¬ O i , S [ ω 0 ] ) (6)

where the map π ¬ { T , F } → { T , F } is defined by π ¬ ( T ) = F , π ¬ ( F ) = T . Clearly, it holds that Prob [ ¬ M C ( Ω ) ( O i , S [ ω 0 ] ) ; { T } ] = [ G i ( { F } ) ] ( ω 0 ) = μ i ( { F } ) .

(ii; And): Define M C ( Ω ) ( O 1 , S [ ω 0 ] ) ∧ M C ( Ω ) ( O 2 , S [ ω 0 ] ) such that

M C ( Ω ) ( O 1 , S [ ω 0 ] ) ∧ M C ( Ω ) ( O 2 , S [ ω 0 ] ) = M C ( Ω ) ( π ∧ ( O 1 × q p O 2 ) , S [ ω 0 ] ) (7)

where π ∧ : { T , F } 2 → { T , F } is defined by π ∧ ( T , T ) = T , π ∧ ( T , F ) = π ∧ ( F , T ) = π ∧ ( T , F ) = F .

It holds that

Prob [ M C ( Ω ) ( O 1 , S [ ω 0 ] ) ∧ M C ( Ω ) ( O 2 , S [ ω 0 ] ) ; { T } ] = [ ( G 1 × q p G 2 ) ( ( π ∧ ) − 1 ( { T } ) ) ] ( ω 0 ) = ( μ 1 × q p μ 2 ) ( { ( T , T ) } ) .

(iii; Or): Define M C ( Ω ) ( O 1 , S [ ω 0 ] ) ∨ M C ( Ω ) ( O 2 , S [ ω 0 ] ) such that

M C ( Ω ) ( O 1 , S [ ω 0 ] ) ∨ M C ( Ω ) ( O 2 , S [ ω 0 ] ) = M C ( Ω ) ( π ∨ ( O 1 × q p O 2 ) , S [ ω 0 ] ) (8)

where π ∨ : { T , F } 2 → { T , F } is defined by π ∨ ( T , T ) = π ∨ ( T , F ) = π ∨ ( F , T ) = T , π ∨ ( F , F ) = F .

It holds that Prob [ M C ( Ω ) ( O 1 , S [ ω 0 ] ) ∨ M C ( Ω ) ( O 2 , S [ ω 0 ] ) ; { T } ] = [ ( G 1 × q p G 2 ) ( ( π ∨ ) − 1 ( { T } ) ) ] ( ω 0 ) = ( μ 1 × q p μ 2 ) ( { ( T , T ) , ( T , F ) , ( F , T ) } ) .

(iv; Implication): Define M C ( Ω ) ( O 1 , S [ ω 0 ] ) → M C ( Ω ) ( O 2 , S [ ω 0 ] ) such that

M C ( Ω ) ( O 1 , S [ ω 0 ] ) → M C ( Ω ) ( O 2 , S [ ω 0 ] ) = M C ( Ω ) ( π → ( O 1 × q p O 2 ) , S [ ω 0 ] ) (9)

where π → : { T , F } 2 → { T , F } is defined by π → ( T , T ) = π → ( F , T ) = π → ( F , F ) = 1 , π → ( T , F ) = F .

It holds that Prob [ M C ( Ω ) ( O 1 , S [ ω 0 ] ) → M C ( Ω ) ( O 2 , S [ ω 0 ] ) ; { T } ] = [ ( G 1 × q p G 2 ) ( ( π → ) − 1 ( { T } ) ) ] ( ω 0 ) = ( μ 1 × q p μ 2 ) ( { ( T , T ) , ( F , F ) , ( F , T ) } ) .

Example 6. As with mathematical logic, truth tables are useful in quantum fuzzy logic. Concerning M C ( Ω ) ( O 1 , S [ ω 0 ] ) and M C ( Ω ) ( O 2 , S [ ω 0 ] ) , we get the following QL version of the truth table (i.e., Table 1).

For example, we see that

Prob [ [ P 1 → P 2 ] ; { T } ] = p 12 + p 1 ¯ 2 + p 1 ¯   2 ¯ (10)

It is usual that “logic” is constructed in mathematics (e.g., see ref. [33] ). On the other hand, our logic (i.e., quantum fuzzy logic in ref. [22] ) is constructed in the quantum language (which has been proposed as the “language of science”). And thus, we can expect that quantum fuzzy logic plays an important role in science. This is the motivation for us to write this paper.

It is generally said that mathematical logic and time are not compatible. We believe this is certain. However, our logic is not mathematical logic, but quantum fuzzy logic and quantum fuzzy logic is a logic born from quantum mechanics. It is natural to think that mechanics and time are compatible, so there is a reason to expect quantum fuzzy logic and time to be compatible. As this paper shows, it can be said that this expectation was correct to some extent. In fact, in Section 4, we showed the denial of the proposition of “John is always hungry” and the contraposition of the proposition of “If no one is scolded, no one study” in quantum fuzzy logic.

In everyday language, we are free to use tense and subjective time, but in the quantum language (or science), the use of tense and subjective time is forbidden. Quantum language is a scientific language, so it is natural that we cannot use the word “time” unscientifically (cf. Remark 9: Russell’s “the world began five minutes ago”). On the other hand, although the parallel time is scientific time, it is interesting that this time has a gap with the nuances of everyday language (cf. Remark 7: Related to Hume’s problem of induction).

As a discussion about logic and time within the framework of science, we believe that our proposal is the best. We hope that our proposal will be examined from various points of view.

The authors declare no conflicts of interest regarding the publication of this paper.

Ishikawa, S. and Kikuchi, K. (2021) Quantum Fuzzy Logic and Time. Journal of Applied Mathematics and Physics, 9, 2609-2622. https://doi.org/10.4236/jamp.2021.911168