Philosophy of Science for Scientists: The Probabilistic Interpretation of Science

Recently we proposed “quantum language” (or, “the linguistic Copenhagen interpretation of quantum mechanics”, “measurement theory”) as the language of science. This theory asserts the probabilistic interpretation of science (=the linguistic quantum mechanical worldview), which is a kind of mathematical generalization of Born’s probabilistic interpretation of quantum mechanics. In this paper, we consider the most fundamental problems in philosophy of science such as Hempel’s raven paradox, Hume’s problem of induction, Goodman’s grue paradox, Peirce’s abduction, flagpole problem, which are closely related to measurement. We believe that these problems can never be solved without the basic theory of science with axioms. Since our worldview (=quantum language) has the axiom concerning measurement, these problems can be solved easily. Thus we believe that quantum language is the central theory in philosophy of science. Hence there is a reason to assert that quantum language gives the mathematical foundations to science.


Philosophy of Science Is about as Useful to Scientists as Ornithology Is to Birds
We think that philosophy of science is classified as follows.
(A 1 ) Criticism about science, The relation between society and science, History of science, etc.
(Non-scientists may be interested in these mainly, thus it is called "philosophy 2. Review: Quantum Language (=Measurement Theory (=MT))

Quantum Language Is the Language to Describe Science
Recently, in refs. [3]- [18], we proposed quantum language (or, "the linguistic Copenhagen interpretation of quantum mechanics", "measurement theory (=MT)"), which is a kind of the language of science. This is not only characterized as the metaphysical and linguistic turn of quantum mechanics but also the quantitative turn of Descartes = Kant epistemology and the dualistic turn of statistics. Thus, the location of this theory in the history of scientific worldviews is as follows (cf. refs. [8] [14] [18]): Figure 1 says that quantum language (=the linguistic Copenhagen interpretation) has the following three aspects (B 1 ) the linguistic Copenhagen interpretation is the true figure of so-called Copenhagen interpretation (⑦ in Figure 1), cf. refs. [3] [6] [7] [13] [16], particularly, Heisenberg's uncertainty relation was proved in [3], and the von Neumann-Lüders projection postulate (i.e., the meaning of the wave function collapse) was clarified in [13], (B 2 ) the scientific final goal of dualistic idealism (i.e., Descartes = Kant philosophy) (⑧ in Figure 1), cf. refs. [8] [14] [15] [17], particularly, the mind-body problem was clarified in [15], also, the paradox of brain in bat (cf. [19]) was  [17], (B 3 ) statistics (=dynamical system theory) with the concept measurement (⑨ in Figure 1), cf. refs. [4] [5] [6] [10] [11]. Thus, quantum language gives the answer of "Why statistics is used in science?" Hence, it is natural to assume that (B 4 ) quantum language proposes the probabilistic interpretation of science (=the linguistic quantum mechanical worldview), and thus, it is just the language to describe science. Thus, we assert that science is built on dualism (≈["observer" + "matter"] ≈ measurement) and idealism (≈metaphysics ≈ language).
which is the most important assertion of quantum language. Also, we assume that to make the language to describe science is one of main purposes of philosophy of science.
As criticism of philosophy of science, there is criticism that scientific philosophy is not very useful for scientists (as mentioned in Section 1). We agree to this criticism. However, as mentioned in the above (B 1 ) and (B 3 ), we say that quantum language is one of the most useful theories in science, and thus it should be regarded as a kind of unified theory of science.
Remark 1. Since space and time are independent in quantum language, the theory of relativity (and further, the theory of everything: ⑤ in Figure 1) cannot be described by quantum language. We think that the theory of relativity is too special, an exception. It is too optimistic to expect that all scientific propositions can be written in quantum language. However, we want to assert the (B 4 ), that is, quantum language is the most fundamental language for almost all familiar science. We believe that arguments without a worldview do not bring about the success of philosophy of science.

No Scientific Argument without Scientific Worldview
It is well known that the following problems are the most fundamental in philosophy of science: In Sections 3-6, we clarify these problems under the linguistic quantum mechanical worldview (B 4 ), since we believe that there is no scientific argument without scientific worldview (or, without scientific language). That is, the above are not problems in mathematics and logic. And we conclude that the reason that these problems are not yet clarified depends on lack of the concept of measurement in philosophy of science.

Mathematical Preparations
Following refs. [6] [7] [8] [18] (all our results until present are written in ref. [18]), we shall review quantum language, which has the following form: which asserts that "measurement" and "causality" are the most important concepts in science.
Consider an operator algebra ( )  The Hilbert space method for the mathematical foundations of quantum mechanics is essentially due to von Neumann (cf. ref. [20]). He devoted himself to quantum (D 1 ). On the other hand, in most cases, we devote ourselves to classical (D 2 ), and not (D 1 ). However, the quantum (D 1 ) is convenient for us, in the sense that the idea in (D 1 ) is often introduced into classical (D 2 ).
, the C * -algebra composed of all compact operators on a Hilbert space H, the (D 1 ) is called quantum measurement theory (or, quantum system theory), which can be regarded as the linguistic aspect of quantum mechanics. Also, when  is commutative (that is, when  is characterized by ( ) 0 C Ω , the C * -algebra composed of all continuous complex-valued functions vanishing at infinity on a locally compact Hausdorff space Ω (cf. refs. [21] [22]  ∈ Ω ∈ Ω ) (cf. ref. [22]).
In Sections 3-6 later, we devote ourselves to a compact space Ω with a probability measure ν (i.e., ( ) 1 ν Ω =) and thus, can be also identified with Ω (called a spectrum space or simply spectrum) such as In this paper, Ω and ( ) ω ∈ Ω is respectively called a state space and a state.
In Axiom 1 later, we need the value of ( ) The following definition is due to E.B. Davies (cf. ref. [24]).
in  is defined as follows: 2) [Countable additivity] F is a mapping from  to  satisfying: a): for where 0 and I is the 0-element and the identity in  respectively. c): for any countable decomposition in the sense of weak * topology in  .

Axiom 1 [Measurement] and Axiom 2 [Causality]
Quantum language (1) is composed of two axioms (i.e., Axioms 1 and 2) as follows. With any system S, a basic structure , which is usual in statistics. Thus, roughly speaking, statistics starts from "sample probability space", on the other hand, quantum language starts from "measurement".
2): Axiom 1 is the quantitative realization of the spirit: "there is no science without measurements". And, we think that Axiom 1 means the probabilistic interpretation of science since it is a kind of mathematical generalization of Born's probability interpretation of quantum mechanics.
, the closed interval in the real line  ) is the state space, and the ν is the Lebesgue measure on the Borel σ-field Ω  , that is, the smallest σ-field that contains all open sets in Ω .
Define the exact observable is not always a probability space. However, note that there exists a probability space ( ) In addition to the above 1) and 2), we assume that ( )

Axiom 2 [Causality]
; For each t ( T ∈ = "tree like semi-ordered set"), consider the basic structure: Then, the chain of causalities is represented by a sequential causal operator When parameters 1 t , 2 t ( 1 2 t t < ) are regarded as time, we usually consider that a causal operator t t t t Φ →   represents "causality". Thus, this axiom gives an answer to "What is causality?". That is, we consider that, if Axiom 2 is used in the quantum linguistic representation of a phenomenon, causality exists in the phenomenon.

The Linguistic Copenhagen Interpretation (=The Manual to Use Axioms 1 and 2)
It is well known (cf. ref. [25]) that the Copenhagen interpretation of quantum mechanics has not been established yet. For example, about the right or wrong of the wave function collapse, opinions are divided in the Copenhagen interpretation. Thus, the Copenhagen interpretation is often called "so-call Copenhagen interpretation". However, we believe that the linguistic Copenhagen interpretation of quantum language (B) (i.e., both quantum ( 1 B′ ) and classical ( 2 B′ )) is uniquely determined. For example, for the quantum linguistic opinion about the wave function collapse, see ref. [13] or §11.2 in ref. [18]. As mentioned in (B 1 ), we believe that the linguistic Copenhagen interpretation is the true figure of so called Copenhagen interpretation. Therefore, what we should do is not "to understand" but "to use". After learning Journal of Quantum Information Science Axioms 1 and 2 by rote, we have to improve how to use them through trial and error. We may do well even if we do not know the linguistic Copenhagen interpretation (=the manual to use Axioms 1 and 2). However, it is better to know the linguistic Copenhagen interpretation, if we would like to make progress quantum language early. We believe that the linguistic Copenhagen interpretation is the true Copenhagen interpretation, which does not belong to physics.
The essence of the manual is as follows: In Figure 2, we remark: (F 1 ) ○ x : it suffices to understand that interfere "is, for example, apply light".
○ y : perceive the reaction.
That is, "measurement" is characterized as the interaction between "observer" and "measuring object (= matter)". However, (F 2 ) in measurement theory (=quantum language), "interaction" must not be emphasized.
Therefore, in order to avoid confusion, it might better to omit the interaction "○ x and ○ y " in Figure 2.
After all, we think that: (F 3 ) it is clear that there is no measured value without observer (i.e., "I", "mind"). Thus, we consider that measurement theory is composed of three key-words: "measured value", "observable", "state" (cf. §3.1(p.63) in [18]). The linguistic Copenhagen interpretation says that (G 1 ) Only one measurement is permitted. And therefore, the state after a measurement is meaningless since it cannot be measured any longer. Thus, the collapse of the wavefunction is prohibited (cf. ref. [13]; projection postulate). We are not concerned with anything after measurement. Strictly speaking, the phrase after the measurement should not be used. Also, the causality should be assumed only in the side of system, however, a state never moves. Thus, the Heisenberg picture should be adopted, and thus, the Schrödinger picture should be prohibited. (G 2 ) "Observer" (="I") and "system" are completely separated in order not to make self-reference propositions appear. Hence, the measurement does not depend on the choice of observers. That is, any proposition (except Axiom 1) in quantum language is not related to "observer"(="I"), therefore, there is no "observer's space and time" in quantum language. And thus, it does not have tense (i.e., past, present, future).
2): We consider that the above (G 1 ) is closely related to Kolmogorov's extension theorem (cf. ref. [26]), which says that only one probability space is permitted. For details, see §4.1 in ref. [18].
3): The formula (1) says that scientific explanation is to explain phenomena in terms of "measurement" (Axiom 1) and "causality" (Axiom 2). If we are allowed to use the famous metaphor of Kant's Copernican revolution, to do familiar sciences is to see this world through colored glasses of measurement and causality (cf. [17]), or to use the metaphor of Wittgenstein's saying, the limits of quantum language are the limits of familiar science. Therefore, the explanation problem of scientific philosophy is automatically clarified in quantum language. 4): Violating the linguistic Copenhagen interpretation (G 2 ), we have many paradoxes of self-reference type such as "brain in a vat", "five-minute hypothesis", "I think, therefore I am", "McTaggart's paradox". Cf. ref. [17] or §10.8 in ref. [18].

5):
We want to understand that Zeno's paradox is not a problem concerning geometric series or spatial division, but the problem concerning the worldview.
That is, "Propose a certain scientific worldview, in which Zeno's paradox should be studied!" That is because we think that there is no scientific argument without scientific language (≈scientific worldview). And our answer (cf. § 14.4 in ref. [18]) is "If Zeno's paradox is a problem in science, it should be studied in quantum language". That is because our assertion is "Quantum language is the language of science". Also, Monty Hall problem, two envelope problem, three pris-Journal of Quantum Information Science oners problem etc. are not only mathematical puzzles but also profound problems in quantum language (cf. refs. [11] [18]).
As the further explanation of parallel measurement in the linguistic Copenhagen interpretation (G 5 ), we have to add the following definition. Definition 8. [Parallel measurement (cf. [18])] Though the parallel measurement can be defined in both classical and quantum systems, we, for simplicity, devote ourselves to classical systems as follows. Let  be a classical basic structure, where we assume, for simplicity, that Ω is compact space and ν is a measure such that . However, the linguistic Copenhagen interpretation (G 1 ) says "Only one measurement is permitted". Therefore, instead of the family of measurements, we consider the parallel is the finite product σ-field, i.e., the smallest σ-field that includes Then, Axiom 1 [measurement] says that (H) the probability that a measured value obtained by the parallel measure- Remark 9. The above finite parallel measurement can be generalized to the case that the index set Λ is infinite. That is, The existence of the parallel measurement is guaranteed in both classical and quantum systems. Cf. §4.2 in ref. [18]. It is not so difficult to extend the above finite parallel measurements to infinite parallel measurements for mathemati-Journal of Quantum Information Science cians. However, in this paper, we are not concerned with the infinite parallel measurement. That is because our concern is not mathematics but foundations of philosophy of science.
Here we add the following definition, which will be used in be an observable such that ( ) ( ) That is because the probability that a measured value ( ) Remark 11.

Inference; Fisher's Maximum Likelihood Method
We begin with the following notation: Here, note that (I 1 ) in most cases that the measurement That is because It is an easy consequence of Axiom 1 (cf. §5.2 in ref. [18]).
[Inference and Control cf. §5.2 in ref. [18]] The inference problem is characterized as the reverse problem of measurements. That is, we consider that On the other hand

What Is Hempel's Raven Paradox?
Although all results mentioned in this paper hold in both classical and quantum systems, we, for simplicity, devote ourselves to classical systems.
In this section we discuss Hempel's raven paradox (cf. ref. [ Its contraposition is denoted by (K 2 ) "Every non-black bird is a nonraven": Of course, the two (K 1 ) and (K 2 ) are equivalent. However, If (K 1 ) and (K 2 ) are equivalent, then we have the following questions (i.e., raven problem): (K 3 ) Why is the actual verification of (K 2 ) much more difficult than the actual verification of (K 1 )?
(K 4 ) Why can the truth of (K 1 ): "any raven is black" be known by (K 2 ), i.e., without seeing a raven also at once?
Some may think that these are nonsense questions. However, in this section, we clarify the true meaning of (K 3 ) and (K 4 ) in the linguistic quantum mechanical worldview. And further, we conclude that Hempel's raven paradox may suggest the importance of measurement in science, that is, the language of science is not logic but quantum language.
. This is wrong. That is, as shown in the next section, the questions arise even in the case that . In order to avoid misunderstanding, we assume that U is a finite set.

The Measurement Theoretical Answer to the Raven Paradox
Although most results mentioned in Sections 3~6 hold in both classical and quantum systems, we, for simplicity, devote ourselves to classical systems: Let U be the universal set of all birds. Let ( ) B U ⊆ be a set of all black birds.
Let R be the set of all ravens. Assume that U is finite. Thus, "Any raven is black" is logically denoted by ∈ → ∈ ; "any raven is black" This is logically equivalent to the following (L 2 ) and (L 3 ): (i.e., (L 1 ) ⇔ (L 2 ) ⇔ (L 3 )) x U R ∀ ∈ → ∈ ; "every non-black bird is a nonraven"  ; "a non-black raven does not exist" In what follows we try to explain the measurement theoretical (i.e., dualistic) representations of the logical (or set theoretical) statements (L 1 ), (L 2 ) and (L 3 ): Let Ω be the state space. Without loss of generality (and, for simplicity), the state space Ω is assumed to be a compact space. Let ν be a measure on Ω such that ( ) 1 ν Ω = and ( ) 0 . And further assume that where D  is the interior of ( ) , , However, we devote ourselves to the above general situation.
[Step (I)]: The measurement theoretical representation of (L 1 ); Any raven is black" Now let us study the measurement theoretical representation of (L 1 ): "Any raven is black". Define the observable where "b" and " b " means "black" and "non-black" respectively.
This is one of the measurement theoretical representations of the statement: "Any raven is black". However, the above term "for any raven" is too logical and not concrete. Thus which is a formal (i.e., measurement theoretical) expression of the (L 1 ). Also, this may mean "All ravens are black" rather than "Any raven is black". For completeness, note that "any (=arbitrary)" and "all" are distinguished in quantum language.
[Step (II)]: The measurement theoretical representation of (L 2 ); "Every Journal of Quantum Information Science non-black bird is a nonraven" Here let us study the measurement theoretical representation of (L 2 ): "Every non-black bird is a nonraven". Define the observable where "r" and " r " means "raven" and "nonraven" respectively. Now, for any This is one of the measurement theoretical representations of the statement: "Every non-black bird is a nonraven". However, the above term "for any non-black bird" is too logical and not concrete. Thus, we may, by using the parallel measurement (cf. (H) in Definition 8), rewrite the (N 1 ) to the following: which is a formal (i.e., measurement theoretical) expression of the (L 2 ). Also, this may mean "All the birds non-black are nonravens" rather than "Any non-black bird is a nonraven".
However, the above term "Let t be any bird" is too logical and not concrete.
Thus, we may, by using the parallel measurement (cf. (H) in Definition 8), rewrite the (O 1 ) to the following: [measurement] says that the probability that ( , which includes more measurements than the parallel measurement Thus, some people think that the actual verification of (L 1 ) (or, (L 2 )) may be easier than that of (L 3 ). However, this is not true. That is because if the (M 2 ) asserts "all ravens are black" ( R B ⊆ ), we have to prove that R is the set of all ravens. That is, (P 1 ) Before the measurement (M 2 ), we have to prove that R is the set of all ravens, namely, we have to obtain the measured value ( )  is the set of all birds. However, it is usually difficult to prove that U is the set of all birds as there may be some birds in unexplored land. Therefore, in most case, it is impossible to be convinced of "All ravens are black". Similarly, to assert "every non-black bird is a nonraven" or "a non-black raven does not exist", we have to examine all the birds ( t U ∀ ∈ ). This is impossible in most cases.
Therefore, we can completely understand the questions (K 3 ) and (K 4 ). As the answer to the (K 3 ) and (K 4 ), some may directly find the (P 2 ) without quantum language (and thus, without the arguments [Step(I)~Step(IV)]). If so, they may be somewhat excellent. However, it is not worth so much. That is because we think that the reason why Hempel's problem is famous is that many researchers know the (P 2 ) (i.e., the difference between definition and proof) unconsciously.
Thus, we think that to solve Hempel's raven problems (K 3 ) and (K 4 ) is to answer the following: (P 3 ) Propose a worldview! And further derive the assertion (P 2 ) from the axioms of its worldview!
Remark 17. 1): As seen in the above, the logical implication "→" has various interpretation in quantum language. In this paper we are not concerned with Axiom 2 [Causality], which is also related to "implication". That is because Axiom 2 causality t t <  → "state at time 2 t "] can be regarded as a kind of implication. But it's a little unreasonable to regard causality as an implication. For example, consider the following famous puzzle: (P 4 ) Describe the contraposition of "If he is not scolded, he does not study"! This is not so difficult as puzzle. Also, some may associate temporal logic. But, we think, from the quantum linguistic point of view, that this puzzle is unnatural.
That is because we consider that the language of science is not logic but quantum language.
2): For the sake of completeness, we sum up and add the following correspondence:

Falsification Test
As seen in the above (P 2 ), in most case, it is impossible to be convinced of "All ravens are black". Thus, our next problem is to answer the problem "How do we believe it?". For example, assume the following fact: (Q 1 ) It was found that one hundred ravens were black continuously. What we can do is to reject the null hypothesis of the (Q 1 ). For instance, it is usual to assume the following null hypothesis: (Q 2 ) Non-black ravens can be observed at 3 time of a rate to 100 times. A simple calculation shows that this null hypothesis (Q 2 ) is represented in quantum language as the measurement  )). Thus, we may reject the null hypothesis (Q 2 ) since probability 0.048 is quite rare.
Remark 18. Note that the above argument is popular as statistical hypothesis testing in statistics, though statistics does not have the concept of "measurement". In our worldview (i.e., linguistic quantum mechanical worldview), we consider that Popper's falsificationism (cf, ref. [27]) and statistical hypothesis testing are almost the same. However his theory was supported by philosophers rather than scientists since his proposal was not proposed under a certain scientific worldview. That is, Popper's falsificationism belongs to (A 1 ): "philosophy of science for the general public".

Problem of Induction in the Linguistic Quantum Mechanical World View
Although is said to satisfy the uniformity principle of nature (concerning µ ), if there exists a proba- Under this definition, we assert the following theorem, which should be re-  says that the probability that a measured value ( ) Thus, the sequence { } N i i n x = − can be regarded as independent random variables with the identical distribution µ . Hence, using the law of large numbers, we can immediately get the formula (7). Also, this theorem is a direct consequence of the law of large numbers for parallel measurements (cf. refs. [5], or §4.2 in ref. [18]).
Remark 21. 1): Recall that the law of large numbers (which is almost equivalent to Theorem 20) says that "frequency probability" = "the probability in Axiom 1" (cf. ref. [5]) though the probability in Axiom 1 has the several aspects. Journal of Quantum Information Science Also, note that the law of large numbers in statistics (cf. ref. [26]) has already been accepted as the fundamental theorem in science. Therefore, even if Theorem 20 ([Inductive reasoning] + (7)) is called the fundamental theorem in philosophy of science, we don't think it's exaggerated. We believe that our proposal (i.e., Theorem 20) is completely true in our worldview. Thus, we think that the solution of Hume's problem of induction was practically already found as the law of large numbers. In the framework of our worldview, we are convinced that the above is the definitive solution to Hume's problem. However, there may be another idea if some start from another worldview. Hence, as described at the end of this paper, we hope that many philosophers propose various mathemati- 3): It may be understandable to consider two measurements: The reason that we do not consider two measurements is due to the linguistic Copenhagen interpretation (G 1 ), i.e., only one measurement is permitted.
That is, a family of measurements Clearly, a family of measurements

Grue Paradox Cannot Be Represented in Quantum Language
If our understanding of inductive reasoning (mentioned in the above) is true, we can solve the grue paradox (cf. ref. [28]). Let us mention it as follows.
(R 1 ) Define that Y has a grue property iff Y is green at time i such that 0 i ≤ and Y is blue at time i such that 0 i < . Suppose that we have examined the emeralds at , 1, , 1, 0 n n − − + −  , and found them to all be green (and hence also grue). Then, "so-called inductive reasoning" says that emeralds at 1, 2, , N  have the grue property (and hence blue) as well as green. Thus, a contradiction is gotten.
However, we think that this (R 1 ) cannot be described in quantum language. If we try to describe the (R 1 ), we may consider as follows.
Hence Theorem 20 [Inductive reasoning] cannot be applied. Therefore Goodman's grue paradox (R 1 ) cannot be described in quantum language.
Remark 25. We believe that there is no scientific argument without scientific worldview. Thus, we can immediately conclude that Goodman's discussion (R 1 ) is doubtful since his argument is not based on any scientific worldview. In this sense, the above arguments (R 2 ) and (R 3 ) may not be needed. That is, the confusion of grue paradox is due to lack of the understanding of Hume's problem of induction in the linguistic quantum mechanical worldview, and not lack of the term "grue" is non-projectible (cf. ref. [28]). Thus, we think that to solve Goodman's grue paradox is to answer the following: (R 4 ) Propose a worldview! And further formulate Hume's induction as the fundamental theorem in the worldview! In this formulation, confirm that Goodman's paradox is eliminated naturally.
What we did is this.

Deduction and Abduction in "Logic"
A typical example of deduction is as follows: (In the following, (

The Measurement Theoretical Representation of Deduction and Abduction
In this section, we show that the abduction [(T 1 )-(T 3 where "w" and "b" means "white" and "black" respectively.
Thus, we have the measurement This implies (T 3 ). Therefore, the above (U 2 ) is the measurement theoretical representation of abduction (i.e., (T 1 )-(T 3 )). For the sake of completeness, note that (U 1 ) and (U 2 ) are in reverse problem (cf. Remark 14). That is, we have the following correspondence: Thus, the scientific meaning of abduction can be completely clarified in the translation from logic to quantum language.

Flagpole Problem
Let us explain the flagpole problem as follows. Suppose that the sun is at an elevation angle α  in the sky. Assume that tan There is a flagpole which is 0 0 ω meters tall. The flagpole casts a shadow 0 1 ω meters long. Suppose that we want to explain the length of the flagpole's shadow. On Hempel's model, the following explanation is sufficient.
The sun is at an elevation angle α  in the sky.
3 Similarly, we may consider as follows.
(V 2 ) 1) The sun is at an elevation angle α  in the sky.
3) The length of the shadow is 0 That is, it satisfies that 1 1 , X Ω = Ω =   (i.e., the Borel field in Ω ), Thus, we have the measurement 2ω is given by 1.
which is the measurement theoretical representation of (V 1 ). That is, we consider that the (V 1 ) is the simplified form (or, the rough representation) of (W 1 which is the measurement theoretical representation of (V 2 ). That is, we consider that the (V 2 ) is the simplified form (or, the rough representation) of (W 2 ).
Thus, we conclude that "scientific explanation" is to describe by quantum language. Also, we have to add that the flagpole problem is not trivial but significant, since this is never solved without Axiom 1 (measurement) and Axiom 2 (causality) (i.e., the answers to the problems "What is measurement?" and "What is causality?"). In this paper, we asserted that (# 3 ), rather than (# 1 ) and (# 2 )], more essential.

Conclusion
In what follow, again let us examine this: [(# 1 ): Logic]: Some may say "Science is to describe phenomena by logic", which may be due to the logical positivism (or, the tradition of Aristotle's syllogism). However, as seen in Sections 3-6, Hempel's raven paradox, Hume's problem of induction, Goodman's grue paradox, Peirce's abduction and flagpole problem are related to the concept of measurement (=inference), and thus, these problems cannot be adequately handled by logic alone. Thus, we think that logic is the language of mathematics, and not the language of science. Mathematical logic (i.e., the language of mathematics) should not be confused with usual logic.
As seen throughout this paper, we believe that the representation using "logic" is rough in most cases. So-called logic plays an essential role in everyday conversation (e.g., trial, business negotiations, politics, romance, etc. Therefore some may say "Science is to describe phenomena in the classical mechanical worldview (≈statistics ≈ dynamical system theory)". This answer may be somewhat better as follows.
(X 1 ) economics is to describe economical phenomena by statistics (it is usual to regard economics as the application of dynamical system theory (≈statistics)) (X 2 ) psychology is to describe psychological phenomena by statistics (X 3 ) biology is to describe biological phenomena by statistics (X 4 ) medicine is to describe medical phenomena by statistics (i.e., medical sta-Journal of Quantum Information Science

tistics)
Also, since dynamical system theory is considered as a kind of mathematical generalization of Newtonian mechanics, we may be allowed to say: (X 5 ) Newtonian mechanics is to describe classical mechanical phenomena by statistics (=dynamical system theory). Also, it is clear that dynamical system theory plays a central role in engineering.
though Newtonian mechanics is physics, and thus, it belongs to the realistic worldview in Figure 1.
However, statistics (≈dynamical system theory (cf. Remark 14)) is too mathematical. Hence, "Science is to describe phenomena in the classical mechanical worldview (≈statistics ≈ dynamical system theory)" is almost the same as "Science is to describe phenomena using the mathematical theories of probability and differential equation". And thus, the framework of the classical mechanical worldview is ambiguous. Since statistics (≈dynamical system theory) does not have clear axioms, we think that it is a little unreasonable to say that statistics is the language of science.
For example, we don't know how to attack Hempel's raven problem (K 3 ) and (K 4 ) from the statistical point of view, since statistics does not have the concept of measurement. As seen below, the relationship between science and statistics is revealed by quantum language (cf. ⑨ in Figure 1).
[(# 3 ): Quantum language; the linguistic quantum mechanical worldview]: We choose quantum language (i.e., the linguistic quantum mechanical worldview, or the probabilistic interpretation of science), and we assert "Science is to describe phenomena by quantum language". That is, in a similar sense of (X 1 )-(X 5 ), we say that (Y 1 ) economics is to describe economical phenomena by quantum language (Y 2 ) psychology is to describe psychological phenomena by quantum language (cf. Chapter 18 in ref [18]) (Y 3 ) biology is to describe biological phenomena by quantum language (Y 4 ) medicine is to describe medical phenomena by quantum language (Y 5 ) Newtonian mechanics is to describe classical mechanical phenomena by quantum language (in the same meaning as the (X 5 ), also recall the history: ② → ⑦ → ⑩ in Figure 1). Also, it is clear that classical system theory (=dynamical system theory, cf. ( 2 B′ )) plays a central role in engineering.
(Y 6 ) statistical mechanics is to describe statistical mechanical phenomena by quantum language (cf. ref. [12]) (Y 7 ) quantum mechanics (i.e., quantum information theory) is to describe quantum mechanical phenomena by quantum language (cf. (B 1 )). worldview. Therefore, quantum language guarantees that these problems are scientific. On the other hand, Goodman's grue paradox (R 1 ) cannot be described by quantum language. Thus, it is not scientific. Also, it should be noted that these results are consequences of Axiom 1 [measurement].
etc. Quantum language has the advantage of having the concept of "measurement". And thus, as seen in this paper, "logic" can be paraphrased in detail in terms of measurement, and thus, precise expression is obtained.
[Can logic and statistics be regarded as kinds of worldviews?] Logic and statistics has respectively various aspects. However, when we say roughly, logic is the language of mathematics, and statistics is a quite useful mathematical theory. If so, how can we regard logic and statistics as kinds of worldviews? In this paper, we see that logic and statistics respectively has aspects such as simplified forms of quantum language. That is,  (6), (11)). If so, logic and statistics are scientific as simplified forms of quantum language. Thus, we conclude that so-called logic (i.e., non-mathematical logic) is essential in usual conversation (e.g., argument in a trial, etc.) and not in science.

Summing Up; Quantum Language Is the Language of Science
In this paper, we clarify the following unsolved problems in the linguistic quantum mechanical worldview: (Z 4 ) [Peirce's abduction in Section 5]: As shown in the formula (11), Peirce's abduction is characterized as the simplified form of Fisher's maximum likelihood method in quantum language.
(Z 5 ) [The flagpole problem in Section 6]: The confusion concerning the flagpole problem is due to relying only on "logic", and not using quantum language.
Hence we conclude that the reason that these problems are not yet clarified depends on lack of the worldview with the concept of measurement in philosophy of science. Again we emphasize the importance of worldview in science. That is because, if we do not have the worldview, we do not know what to rely on to proceed with the discussion. Therefore, it is no exaggeration to say that there is no science without a scientific worldview.
As mentioned in Remark 1, quantum language does not cover all sciences. However, we consider that familiar sciences are described by quantum language. And further, we believe that quantum language (i.e., the probabilistic interpretation of science) plays a central role in almost familiar sciences. That is, we believe that quantum language gives the mathematical foundations to science.
However, the answer to "What is science?" may not necessarily be determined uniquely. Although there are several aspects of philosophy of science, we believe that it is the central theme of the philosophy of science to find a mathematical structure that is common to almost all sciences. If so, we feel like knowing other interpretations (i.e., other scientific worldview) besides ours (i.e., the probabilistic interpretation of science). Thus, we hope that various mathematical foundations (e.g., category theoretical approach, modal logic approach, etc.) of scientific philosophy will be proposed. And we expect that fundamental problems such as raven problem, problem of induction, grue paradox, etc. will be investigated in these interpretations. And we hope that philosophy of science will progress with such competition.
We hope that our proposal will be examined from various points of view 1 .