1. Introduction
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] );
Figure 1. The history of the world-descriptions.
(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.
2. Elementary Review of Classical QL
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)
QL is classified as follows
(C)
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
be a state space, i.e., compact space. An element
in
is called a state. And let
be the commutative
-algebra, i.e., the space of all complex-valued continuous functions on
.
Definition 1. [Observable, Image observable] According to the noted idea (cf. refs. [29] [30] ), an observable
in
is defined as follows:
1) X is a finite set,
(
, the power set of X).
2) G is a mapping from
to
satisfying: a): for every
,
is a non-negative element in
such that
, b):
and
, where 0 and I are the 0-element and the identity in
respectively. c): [additivity]
(1)
for all
such that
.
If
, then
in
is a projective observable (or, crisp observable). Also,
in
is also called an X-valued observable. We will devote ourselves to binary (i.e.,
-valued) observables in most of the cases in this paper. Let Y be a finite set, and let
be a map. Then,
in
is also observable in
(which is called an image observable).
With any classical system S, a commutative
-algebra
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
and an observable is represented by an observable
in
. Also, the measurement of the observable O for the system S with the state
is denoted by
. An observer can obtain a measured value
by the measurement
.
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
obtained by the measurement
is given by
.
Next, we explain Axiom 2 in (B).
Let
be a (finite) tree-like semi-ordered set with the root
. Let
is the parent map, that is,
is defined such that
(2)
This
is also written by
.
For example see Figure 2 below, in which we see the root
, the parent map:
,
,
.
Let
be a tree-like semi-ordered set with the root
. For each
, define the homomorphism.
(see, for example, Figure 3), which is characterized as a continuous map
such that
(3)
Figure 2. Tree:
.
Figure 3. Homomorphism
.
Now we can propose Axiom 2 (i.e., causality). (For details, see ref. [6].)
Axiom 2. [Causality]; Let
be a (finite) tree-like semi-ordered set with the root
. Causality is represented by
.
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
in Axiom 2.
(D3) The Heisenberg picture
is used and the Schrödinger picture
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
be commutative observables in
. Define the quasi-product observable
such that
(4)
Also,
is called the quasi-product measurement of
. [(ii): Tensor
-algebra, tensor product observable, tensor quasi-product measurement]: Let
be observables in
,
. Define a tensor product observable
in the tensor
-algebra
. such that
(5)
Also,
is called a tensor product measurement of
.
3. Quantum Fuzzy Logic in Classical QL
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
be a binary observable (or, (TF)-observable,
-valued observable) in a commutative
-algebra
. A measurement
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
is given by
, we say that
(E1) a (TF)-measurement
is true with probability
.
Or,
(E2)
Also,
is called the membership function of O.
Definition 5. [Quantum fuzzy logic symbols (
,
,
,
))] Let
be binary observables (or,
-valued observable) in a commutative
-algebra
, (
). Fix the quasi-product observable
. Consider (TF)-measurement
(which is abbreviated as
) in a
-algebra
. Put
, and
.
(i; Negation): Put
. Remembering the image observable in Definition 1, let’s define
such that
(6)
where the map
is defined by
,
. Clearly, it holds that
.
(ii; And): Define
such that
(7)
where
is defined by
,
.
It holds that
.
(iii; Or): Define
such that
(8)
where
is defined by
,
.
It holds that
.
(iv; Implication): Define
such that
(9)
where
is defined by
,
.
It holds that
.
Example 6. As with mathematical logic, truth tables are useful in quantum fuzzy logic. Concerning
and
, we get the following QL version of the truth table (i.e., Table 1).
For example, we see that
(10)
4. Time in Quantum Fuzzy Logic
4.1. Parallel Times Series; John Is Always Hungry
Let
be a compact state space, each element of which is assumed
Table 1. Probabilistic truth table (Elementary propositions
,
).
to represent the state of the human mind. Consider the membership function
such that
which is usually interpreted as follows. That is,
is defined by the following:
(F) When asked, “Does the person with state
feel hungry?”,
percent said “yes”.
This is essentially the same as the probabilistic interpretation of membership functions (cf. refs. [20] [23] ).
Define the observables
be a binary observable (or, (TF)-observable,
-valued observable) in a commutative
-algebra
such that
(11)
Let
be John’s mind state. We have the (TF)-measurement (i.e.,
, which can be identified with the following proposition:
(G) John is hungry.
That is, we see
(H1) a (TF)-measurement
is true with probability
.
Or,
(H2)
Next, we will study “John is always hungry”. For this, we must consider
.
Define the tree-like semi-ordered set
such that
. Assume that for each
, a
is associated (i.e.,
), and
. And, for any
, define a homomorphism
such that
satisfies that
(12)
(See, for example Figure 4)
According to (D2) in Remark 2 (The linguistic Copenhagen interpretation), we regard the above tree-like semi-ordered set
as time.
Let
be John’s mind state at time
. Thus, note that, for each
, the (TF)-measurement (i.e.,
can be identified with the following proposition:
(I) John is hungry at time
.
Figure 4. Homomorphism
.
Define the tensor product observables
in
and consider the measurement
. Define the map
such that
(13)
Thus, we get the (TF)-measurement
, which implies that
(J) John is always hungry.
That is, we see
(K1) a (TF)-measurement the
-measurement
Or,
(K2)
is true with probability
Remark 7 (i): For example, consider the following sentence:
(L) John was hungry yesterday and the day before yesterday.
This sentence (L) has to do with tense. Thus, as seen in Remark 2 (The linguistic Copenhagen interpretation), this is not a meaningful proposition within QL.
(ii): The above parallel time in Figure 4 does not match our daily senses, and the reader may feel uncomfortable. However, this time plays an essential role in the understanding of Hume’s problem of induction (cf. ref. [20] ). It is rather interesting that there is a gap between our everyday sensory understanding and our scientific understanding. This gap is the reason why Hume’s problem of induction has remained unresolved for so long.
4.2. If No One Is Scolded, No One Study
Here let us consider the following proposition:
(M) If no one is scolded, no one will study.
Our present purpose is to rewrite this proposition in quantum fuzzy logic. Let
be a compact state space, each element of which represents the mind of human being. Consider the membership function
such that
which is usually interpreted as follows.
(N) When asked, “Does the person feel scolded?”,
out of 100 respond “yes”.
This is essentially the same as the probabilistic interpretation of membership functions (cf. refs. [20] [23] ).
Further consider the membership function
such that
Also, consider the time evolution
, which is characterized by the continuous map
such that
(14)
which is illustrated in Figure 5 below.
Define the observables
be a binary observable (or, (TF)-observable,
-valued observable) in a commutative
-algebra
(
) such that
(15)
Let
be a state of the system S.
Here consider two (TF)-measurements (i.e., fuzzy propositions)
and
(which is respectively abbreviated as
). Here, it should be noted that the Heisenberg picture
is used and the Schrödinger picture
is not.
Fix the quasi-product observable
in
. And consider quasi-product measurement
. Put
,
, and
(see Table 2).
Figure 5. Homomorphism
.
Table 2. Probabilistic truth table (Elementary propositions
,
).
Therefore we see, by Table 2, that if “
”, then it holds that
Remark 8. As seen in Section 3, quantum fuzzy logic (e.g., implication) is produced by measurement (i.e., Axiom 1). On the other hand, causality (and time) arises from Axiom 2. For example, we can see both “implication” and “causality” in the above quasi-product measurement
. We think the difference between “implication” and “causality” is now clear in quantum language.
Remark 9. Bertrand Russell said in ref. [32] “The Analysis of Mind”, p. 223.
• We cannot deny the hypothesis that the world began five minutes ago.
which is true since this is a consequence of the linguistic Copenhagen interpretation (D4) (cf. ref. [19] ). Therefore, the sentence “the world began five minutes ago” is not a proposition in QL.
5. Conclusions
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.