Monty Hall Problem and the Principle of Equal Probability in Measurement Theory

Home , Monty Hall problem

Applied Mathematics, 2012, 3, 788-794 http://dx.doi.org/10.4236/am.2012.37117 Published Online July 2012 (http://www.SciRP.org/journal/am)

Shiro Ishikawa Department of Mathematics, Faculty of Science and Technology, Keio University, Yokohama, Japan Email: [email protected]

Received April 28, 2012; revised May 28, 2012; accepted June 5, 2012

ABSTRACT In this paper, we study the principle of equal probability (i.e., unless we have sufficient reason to regard one possible case as more probable than another, we treat them as equally probable) in measurement theory (i.e., the theory of quantum mechanical world view), which is characterized as the linguistic turn of quantum mechanics with the Copenhagen interpretation. This turn from physics to language does not only realize the remarkable extension of quantum mechanics but also establish the method of science. Our study will be executed in the easy example of the Monty Hall problem. Although our argument is simple, we believe that it is worth pointing out the fact that the principle of equal probability can be, for the first time, clarified in measurement theory (based on the dualism) and not the conventional statistics (based on Kolmogorov’s probability theory).

Keywords: Linguistic Interpretation; Quantum and Classical Measurement Theory; Philosophy of Statistics; Fisher Maximum Likelihood Method; Bayes’ Theorem

1. Introduction (cf. Figure 1: ③ later), constructed as the scientific theory formulated in a certain C*-algebra A (i.e., a norm 1.1. Monty Hall Problem closed subalgebra in the operator algebra BH  com- The Monty Hall problem is well-known and elementary. posed of all bounded operators on a Hilbert space H, cf. Also it is famous as the problem in which even great [14,15]). MT is composed of two theories (i.e., pure mathematician P. Erdös made a mistake (cf. [1]). The measurement theory (or, in short, PMT] and statistical Monty Hall problem is as follows: measurement theory (or, in short, SMT). That is, it has Problem 1 [Monty Hall problem 1]. You are on a the following structure: game show and you are given the choice of three doors. (A) MT (measurement theory) Behind one door is a car, and behind the other two are A:PMT1    goats. You choose, say, door 1, and the host, who knows  where the car is, opens another door, behind which is a pure measurement causality goat. For example, the host says that  AxiomP 1 Axiom2 (♭) the door 3 has a goat.    And further, He now gives you the choice of sticking  with door 1 or switching to door 2? What should you do? A:SMT In the framework of measurement theory [2-12], we  2  statistical measurement + causality shall present two answers of this problem in Sections 3.1    and 4.2. Although this problem seems elementary, we S  Axiom 1 Axiom2 assert that the complete understanding of the Monty Hall   problem can not be acquired within Kolmogorov’s prob- where Axiom 2 is common in PMT and SMT. For com- ability theory [13] but measurement theory (based on the pleteness, note that measurement theory (A) (i.e., (A ) dualism). 1 and (A2)) is not physics but a kind of language based on “the (quantum) mechanical world view”. As seen in [9], 1.2. Overview: Measurement Theory note that MT gives a foundation to statistics. That is, As emphasized in refs. [7,8], measurement theory (or in roughly speaking, short, MT) is, by a linguistic turn of quantum mechanics (B) it may be understandabl e to consider that PMT and

Figure 1. The development of the world views from our standing point. For the explanation of (①-⑧), see [8,10].

SMT is related to Fisher’s statistics and Bayesian statis- f  1 . The bi-linear functional   f  is also de- tics respectively. CΩ  Also, for the position of MT in science, see Figure 1, noted by , f , or in short , f . CΩ * CΩ which was precisely explained in [8,10]. When A  BH, the C*-algebra composed of all * c  Define the mixed state  C Ω  such that compact operators on a Hilbert space H, the (A) is called  *  1 and  f  0 for all fC Ω such that quantum measurement theory (or, quantum system the- CΩ    ory), which can be regarded as the linguistic aspect of f  0 . And put quantum mechanics. Also, when A is commutative (that * is, when A is characterized by C Ω , the C -algebra m ** 0  SCΩΩ C is a mixed state . composed of all continuous complex-valued functions     vanishing at infinity on a locally compact Hausdorff Also, for each   Ω , define the pure state space Ω (cf. [16])), the (A) is called classical meas-   SCm Ω * such that urement theory. Thus, we have the following classifica-     * , fffC Ω. And put tion: CΩ  CΩ   quantum MT when A  BH  c  p ** (C) MT  SC ΩΩ  C is a pure state , classical MT when ACΩ     0  which is called a state space. Note, by the Riesz theorem The purpose of this paper is to clarify the Monty Hall * problem in the classical PMT and classical SMT. (cf. [16]), that CM(Ω)(Ω)  is a signed measure on Ω and SCmmΩΩ*  M  is a meas- 2. Classical Measurement Theory (Axioms   1 and Interpretation) ure on Ω such that  Ω 1 . Also, it is clear that 2.1. Mathematical Preparations SCp Ω *    is a point measure at   Ω ,    00 0  Since our concern is the Monty Hall problem, we devote where ff d  fCΩ . This im-    0   0  ourselves to classical MT in (C). Throughout this paper, Ω we assume that Ω is a compact Hausdorff space. Thus, plies that the state space SCp Ω * can be also iden- we can put C0 Ω CΩ, which is defined by a Ba-  * nach space (or precisely, a commutative C -algebra) tified with Ω (called a spectrum space or simply, spec- composed of all continuous complex-valued functions on trum) such as a compact Hausdorff space Ω , where its norm f CΩ p * ∋ SC ΩΩ   is defined by max f  . Let C Ω * be the dual Ba- (1) Ω state space spectrum * nach space of C Ω . That is, C Ω    is a con- Also, note that C Ω is unital, i.e., it has the identity I (or precisely, I ), since we assume that Ω is com- tinuous linear functional on C Ω  , and the norm CΩ pact.

 * is defined by sup  ff: CΩ such that According to the noted idea (cf. [17]) in quantum me- CΩ   

chanics, an observable OXBF:,,  X  in C Ω is ciated. 2 defined as follows: (E2) For every tt,  T, a Markov operator X  12  (D1) [Field] X is a set, BX (2 , the power set of X) Φ : CΩ CΩ is defined (i.e., Φ  0, is a field of X, that is, “ 12, BBXX1  2 ”, tt12,  t2  t1 tt12, “ ΞΞBXXX  B”. Φ  I  I). And it satisfies that ΦΦ (D2) [Additivity] F is a mapping from B to C Ω tt12, CΩCΩ tt23,,tt13 X tt21 satisfying: 1): for every Ξ B , F Ξ is a non-nega- X holds for any tt, , tt, T2 . tive element in C Ω such that 0 F Ξ I , 2):  12  23  F 0 and F X I, where 0 and I is the 0-ele- The family of dual operators ment and the identity in C Ω respectively. 3): for any ** * mm Ξ , Ξ  B such that ΞΞ  , it holds that Φtt, : SCΩΩ t SC t 1 2 X 12  12  1  2  2 tt12,  T FFΞΞ12 Ξ 1FΞ 2. For the more precise argument (such as countably ad- is called a dual causal relation (due to the Schrödinger ditivity, etc.), see [7,9]. picture). When

** *  pp 2.2. Classical PMT in (A1) ΦΩtt, SC t SC Ω t 12 1  2 In this section we shall explain classical PMT in (A1). 2 * holds for any tt,   T, the causal relation is said to With any system S, a commutative C -algebra C Ω 12    be deterministic. can be associated in which the measurement theory (A) Here, Axiom 2 in the measurement theory (A) is pre- of that system can be formulated. A state of the system S sented as follows: is represented by an element   SCp Ω * and an Axiom 2 [Causality]. The causality is represented by a    causal relation Φ : CCΩΩ . observable is represented by an observable  tt12,  t 2t 1  2 tt12,  T OXBF:,, in C Ω . Also, the measurement of * X  For the further argument (i.e., the W -algebraic formu- the observable O for the system S with the state  is  lation) of measurement theory, see Appendix in [7]. denoted by MOS, or more precisely, CΩ   2.3. Classical SMT in (A2) MO:,,, XBFS . An observer can obtain CΩ X   It is usual to consider that we do not know the state  0 a measured value x X by the measurement when we take a measurement MOS, . That is C   MO, S. 0  CΩ  P because we usually take a measurement MO , S The Axiom 1 presented below is a kind of mathe- C       0   matical generalization of Born’s probabilistic interpreta- in order to know the state  . Thus, when we want to tion of quantum mechanics. And thus, it is a statement 0 without reality. emphasize that we do not know the the state  , 0 AxiomP 1 [Measurement]. The probability that a measured value x  X obtained by the measurement MOS,  is denoted by M OS, . Also,  C    C *   0  MOXBFS:,,, belongs to a set CΩ X  when we know the distribution 0 mm*  01MS  C of the unknown state  , Ξ BX is given by F Ξ 0  .    0 Next, we explain Axiom 2 in (A). Let T, be a  the MOS ,  is denoted by MOS,  . C     C * 0 tree, i.e., a partial ordered set such that “ t13 t and   0   S tt23 ” implies “ tt12 or tt21 ” In this paper, we The Axiom 1 presented below is a kind of mathe- assume that T is finite. Also, assume that there exists an matical generalization of AxiomP 1. S element tT0  , called the root of T, such that t0  t Axiom 1 [Statistical measurement] The probability 22 that a measured value x  X obtained by the meas- tT holds. Put Tt 12,tTtt 1 2. The fam-   urement MOXBFS:,,,  belongs to ily Φ : CCΩΩ is called a causal CΩ  X * 0  tt, t t 2  12 2 1 tt, T 12  a set  F  is given by relation (due to the Heisenberg picture), if it satisfies the following conditions (E1) and (E2). FF* ,   . * 00 CΩ CΩ  (E1) With each tT , a C -algebra C Ωt is asso-

Remark 1. Note that two statistical measurements Theorem 2 [Bayes’ method (cf. [9])]. Assume that a measured value obtained by a statistical measurement MOS,  and MOS,  can CΩ  0  CΩ  0  1 2  MOXBFSCX()  :,,, [*] 0 belongs to  BX . not be distinguished before measurements. In this sense, Then, there is a reason to infer that the posterior state we consider that, even if    , we can assume that 12 (i.e., the mixed state after the measurement) is equal to v , which is defined by MOS,, MOSpost CΩ  0(C )*  0  1  (2) F   0 d   D   D  MOS,. post  CΩ  0 F   d  2  0    

DB ;Borelfield. 2.4. Linguistic Interpretation The above two theorems are, of course, the most fun- Next, we have to answer how to use the above axioms as damental in statistics. Thus, if we believe in Figure 1, we follows. That is, we present the following linguistic in- can answer to the following problem (cf. [4,9]): terpretation (F) [= (F1) – (F3)], which is characterized as (G) What is statistics? Or, where is statistics in science? a kind of linguistic turn of so-called Copenhagen inter- which is certainly the most essential problem in the phi- pretation (cf. [7,8]). That is, we propose: losophy of statistics. (F1) Consider the dualism composed of “observer” and “system (= measuring object)”. And therefore, “ob- 3. The First Answer to Monty Hall Problem server” and “system” must be absolutely separated. 3.1. Fisher’s Method (The First Answer) (F2) Only one measurement is permitted. And thus, the state after a measurement is meaningless since it can not In this section, we present the first answer to Problem 1 be measured any longer. Also, the causality should be (Monty-Hall problem) in classical PMT. Put assumed only in the side of system, however, a state Ω  123,, with the discrete topology. Assume that never moves. Thus, the Heisenberg picture should be * each state   SCp Ω means adopted. m    (F3) Also, the observer does not have the space-time.   the state that the car is behind the door 1 Thus, the question: “When and where is a measured m (3) value obtained?” is out of measurement theory, and so on. m  1, 2, 3 This interpretation is, of course, common to both PMT 1,2,3 and SMT. Define the observable OF11 1, 2, 3 , 2 , in Remark 2. Note that quantum mechanics has many in- C Ω such that terpretations (i.e., several Copenhagen interpretation, FF10.0,2   0.5, many worlds interpretation, statistical interpretation, etc.). 11   1  1 On the other hand, we believe that the interpretation of FF30.5,1 0.0, 11   12  measurement theory (A) is uniquely determined as in the   above. This is our main reason to propose the linguistic FF1220.0,3  12 1.0, (4) interpretation of quantum mechanics. We believe that FF1310.0,2  1  31.0, this uniqueness is essential to the justification of Heisen-    berg’s uncertainty principle (cf. [10,18]).  F1330.0, 

2.5. Preliminary Fundamental Theorems where it is also possible to assume that F112     , F113101    . Thus we have a meas- We have the following two fundamental theorems in urement M OS, , which should be regarded as measurement theory: CΩ  1 *  Theorem 1 [Fisher’s maximum likelihood method (cf. the measurement theoretical representation of the meas- [9])]. Assume that a measured value obtained by a measurement that you say “door 1”. Here, we assume that urement M OXBFS:,,, belongs to 1) “measured value is obtained by the measurement CΩ  X *

M OS1, ”  The host says “Door 1 has a goat”; Ξ BX  . Then, there is a reason to infer that the un- CΩ  *  known state * is equal to  , where   Ω is   0 0   2) “measured v al ue is obtained by the measurement defined by M OS, ” The host says “Door 1 has a goat”; CΩ  1 *   FFΞ 0  max Ξ  . Ω  3) “measured v al ue is obtained by the measurement

M OS, ”  The host says “Door 1 has a goat”. MOS, CΩ 1 * urement CΩ  10*  ”  The host says “Door Recall that, in Pro blem 1, the host said “Door 3 has a 2 has a goat”; goat”. This implies that you get the measured value “3” 3) “measured value is obtained by the statistical meas- by the measurement M OS, . Therefore, Theo- urement MOS,  ”  The host says “Door CΩ 1 * CΩ  10*  rem 1 (Fisher’s maximum likelihood method) says that 1 has a goat”. you should pick door number 2. That is because we see Here, assume that, by the statistical measurement that MOS, C  10*   , you obtain a measured value 3,  F123  1.0 max  0.5,1.0,0.0  which corresponds to the fact that the host said “Door 3 has a goat”. Then, Theorem 2 (Bayes’ theorem) says   max3FFF111213  ,3  ,3 , m that the posterior state  post  M 1 Ω  is given by (5)

F103 and thus, there is a reason to infer that *   . Thus, 2  post  (7) you should switch to door 2. This is the first ans wer to  01,3F  Problem 1 (the Monty-Hall problem 1). That is, 3.2. Bayes’ Method (Answer to Modified Monty p Hall Problem 2) 1 p 2 ,,  2 In the sense mentioned in Remark 3 later, the following post 1pp post 2 11pp (8) modified Monty Hall problem (Problem 2) is completely 2222 different from Problem 1 (the Monty Hall problem 1). post 3  0. However, it is worth examining Problem 2 for the better understanding of Problem 3 later. Particularly, we see that Problem 2 [Modified Monty Hall problem 2]. Sup- (H) if ppp123 13, then it holds that pose you are on a game show, and you are given the  13,  23,  0 , choice of three doors (i.e., “number 1”, “number 2”, post 1 post 2 post 3 “number 3”). Behind one door is a car, behind the others, and thus, you should p ick Door 2. goats. You pick a door, say number 1. Then, the host, Remark 3. The difference between Problem 1 and who set a car behind a certain door, says Problem 2 should be remarked. Since the (#1) in Problem

(#1) the car was set behind the door decided by the 2 is the information from the host to you, Problem 1 and cast of the distorted dice. That is, the host set the car Problem 2 are completely different. Although the above behind the k-th door (i.e., “number k”) with probability (H) may be generally regarded as the proper answer of the Monty Hall problem, we do not admit that the (H) is pk (or, weight such that ppp1231 , 0,, ppp123  1 ). proper. That is, we consider that the (H) is not the second And further, the host says, for example, answer to the Monty Hall problem. (♭) the door 3 has a goat. He says to you, “Do you want to pick door number 2?” 4. The Second Answer to Monty Hall Is it to your advantage to switch your choice of doors? Problem In what follows we study this problem. Let  and In this section, we shall present the second answer. O be as in Section 3.1. Under the hypothesis (# ), de- 1 1 However, before it, we have to prepare the principle of fine the mixed state  M m   such that: 01 equal probability (i.e., unless we have sufficient reason to regard one possible case as more probable than another, 01pp 102,,  203  p3 (6) we treat them as equally probable). For completeness, Thus we have a statistical measurement note that measurement theory urges us to use only Axi- MOS,  . Note that oms 1 and 2. CΩ 10*  1) “measured value is obtained by the statistical meas- 4.1. The Principle of Equal Probability urement MOSCΩ 10, *  ”  The host says “Door  Put Ω  123,,,, n  with the discrete topology. 1 has a goat”; And consider any observable OXBF11 ,,X in C Ω. 2) “measured value is obtained by the statistical meas- Define the bijection 1 :   such that

 j1 jn  MOXBFS: ,,,  . CΩ  11Xe*  1 j   1 jn  It should be noted that the principle of equal p robabil- and define the observable OXBFkX ,,k in C Ω ity is not “principle” but “theorem” in measurement the- such that ory. Remark 4. This theorem was also discussed in [5,6], FFkk 11    where we missed the formula (2) in Remark 1. Thus, the  ,1,2,,kn argument in [5,6] was too abstract. And thus, it might be regarded as ambiguous and vague. In fact, we must admit where 0     Ω and that the explanation in [5,6] is not yet accepted generally. Therefore, we recommend readers to read [5,6] after the    ,1,2,,kn . kk11 understanding of the concrete explanation (I) in the linguistic interpretation (F). Also, note that Theorem 3 is Let pkk  1, , n be a non-negative real number n independent of Axiom 2. And further, for the principle of such that p  1. k 1 k equal (a priori) probabilities in equilibrium statistical (I) For example, fix a state  mn 1, 2, , . And, m mechanics, see [11], in which how to use measurement by the cast of the distorted dice, you choose an observ- theory (and thus statistics) in statistical mechanics is ex- able OXBF,, with probability p . And further, kXk  k plained. you take a measurement 4.2. The Second Answer to Monty Hall Problem MO: XBFS,,, . C kXk  m (i.e., Modified Monty Hall Problem 3) Here, we can easily see that the probability that a As an application of Theorem 3, we consider the follow- measured value obtained by the measurement (I) belongs ing modified Monty-Hall problem: to Ξ BX is given by Problem 3 [Modified Monty Hall problem 3]. Sup- pose you are on a game show, and you are given the nn pFΞ ,pFΞ  (9) choice of three doors (i.e., “number 1”, “number 2”, kk11kk m  kk m “number 3”). Behind one door is a car, behind the others, which is equal to FpΞ , n  . This implies goats. 1 k 1 k km1 that the measurement (I) is equivalent to a statistical (#2) You choose a door by the cast of the fair dice, i.e., measurement: with probability 1/3. According to the rule (#2), you pick a door, say num- n MO(: XBFS , , , p . ber 1, and the host, who knows where the car is, opens CΩ 11Xk k 1 km1   m another door, behind which is a goat. For example, the

Note that the (9) depends on the state m . Thus, we host says that can not calculate the (9) such as the (8). (♭) the door 3 has a goat. He says to you, “Do you want to pick door number 2?” However, if it holds that pnkk 11,, n, we Is it to your advantage to switch your choice of doors? 1 n [Answer]. Consider Ω and O1 as in Section 3.1. see that k 1 is independent of the choice of n km1 Then, Theorem 3 says that the answer of Problem 3 is the 1 n same as the (H). Thus, you should pick the door 2. the state  . Thus, putting   , we m k 1  e n km1 Remark 5. The difference between the (#1) in Problem see that the me asurement (I) is equivalent to the statisti- 2 and the (#2) in Problem 3 is clear in the dualism (F). The former is host’s selection, but the latter is your se- cal measurement MOS,  , which is also CΩ 1  e lection (i.e., observer’s selection). That is, in Problem 3, m the information from host to you is only the (♭). This equivalent to MOS, (from the formula CΩ 1 *  e situation is the same as that of Problem 1. In this sense, (2) in Remark 1). we think that Problems 1 and 3 are similar. That is, we Thus, under the above notation, we have the following can conclude that Problem 1 [resp. Problem 3] is the Monty Hall problem in PMT [resp. SMT]. Also, our re- theorem. cent report [19] will promote a better understanding of Theorem 3 [The principle of equal probability (i.e., measurement theory. the equal probability of selection)]. If pnk  1  kn 1, , , the measurement (I) is independ ent of the 5. Conclusions choice of the state m . Hence, the (I) is equivalent to a statistical measurement In the conventional statistics based on Kolmogorov’s

Copyright © 2012 SciRes. AM 794 S. ISHIKAWA probability theory, Problem 3 may be unconsciously pp. 277-306. doi:10.1016/S0165-0114(96)00114-5 confused with Problem 2. On the other hand, as men- [4] S. Ishikawa, “Statistics in Measurements,” Fuzzy Sets and tioned in Remark 5, the difference between Problems 2 Systems, Vol. 116, No. 2, 2000, pp. 141-154. and 3 can be clearly described in measurement theory doi:10.1016/S0165-0114(98)00280-2 (based on the dualism (F)). This is the merit of measure- [5] S. Ishikawa, “Mathematical Foundations of Measurement ment theory. Theory,” Keio University Press Inc., 2006. What we executed in this paper may be merely the http://www.keio-up.co.jp/kup/mfomt/ translation from “ordinary language” to “scientific lan- [6] S. Ishikawa, “Monty Hall Problem in Unintentional Ran- guage”, that is, dom Measurements,” Far East Journal of Dynamical Sys- tems, Vol. 3, No. 2, 2009, pp. 165-181. Monty Hall problem Monty Hall Problem [7] S. Ishikawa, “A New Interpretation of Quantum Mechan- Problem 1 MT  Section 3.1, 4.2 ics,” Journal of Quantum Information Science, Vol. 1, No. translation 2, 2011, pp. 35-42. doi:10.4236/jqis.2011.12005 ordinary language scientific language[8] S. Ishikawa, “Quantum Mechanics and the Philosophy of We believe that this translation is just “the mechanical Language: Reconsideration of Traditional Philosophies,” Journal of Quantum Information Science, Vol. 2, No. 1, world view” or “the method of science” (at least, science 2012, pp. 2-9. doi:10.4236/jqis.2012.21002 in the series ○L of Figure 1). That is, ordinary science (at [9] S. Ishikawa, “A Measurement Theoretical Foundation of least, its basic statements) should be described in terms Statistics,” Applied Mathematics, Vol. 3, No. 3, 2012, pp. of measurement theory. For example, for the translation 183-192. of equilibrium statistical mechanics and the Zeno’s [10] S. Ishikawa, “The Linguistic Interpretation of Quantum paradoxes, see [11] and [12] respectively. Probably, we Mechanics,” 2012. http://arxiv.org/pdf/1204.3892.pdf refrained from the publication of [12], if we were not [11] S. Ishikawa, “Ergodic Hypothesis and Equilibrium Statis- sure of “MT = the method of science (or the form of sci- tical Mechanics in the Quantum Mechanical World entific thinking)”. View,” World Journal of Mechanics, Vol. 2, No. 2, 2012, In this paper (as well as [9]), we showed one of ad- pp. 125-130. doi:10.4236/wjm.2012.22014 vantages of the measurement theoretical foundation of [12] S. Ishikawa, “Zeno’s Paradoxes in the Mechanical World statistics through the examination of the Monty Hall View,” 2012. http://arxiv.org/pdf/1205.1290.pdf problem. Also, recall that measurement theory possesses [13] A. Kolmogorov, “Foundations of the Theory of Probabil- a great power to answer to the problem (G). However, ity (Translation),” Chelsea Pub Co. Second Edition, New our methodology should be tested from various points of York, 1960. view, because the classic statistics methodology (based [14] G. J. Murphy, “C*-Algebras and Operator Theory,” Aca- on Kolmogorov’s probability theory) can be good ap- demic Press, Boston, 1990. plied in many fields. We hope that our approach will be [15] J. von Neumann, “Mathematical Foundations of Quantum examined from various view points. Mechanics,” Springer Verlag, Berlin, 1932. [16] K. Yosida, “Functional Analysis,” 6th Edition, Springer- REFERENCES Verlag, Berlin, 1980. [17] E. B. Davies, “Quantum Theory of Open Systems,” Aca- [1] P. Hoffman, “The Man Who Loved Only Numbers, the demic Press, London, 1976. Story of Paul Erdös and the Search for Mathematical [18] S. Ishikawa, “Uncertainty Relation in Simultaneous Mea- Truth,” Hyperion, New York, 1998, surements for Arbitrary Observables,” Reports on Mathe- [2] S. Ishikawa, “Fuzzy Inferences by Algebraic Method,” matical Physics, Vol. 29, No. 3, 1991, pp. 257-273. Fuzzy Sets and Systems, Vol. 87, No. 2, 1997, pp. 181-200. doi:10.1016/0034-4877(91)90046-P doi:10.1016/S0165-0114(96)00035-8 [19] S. Ishikawa, “What Is Statistics? The Answer by Quan- [3] S. Ishikawa, “A Quantum Mechanical Approach to Fuzzy tum Language,” arXiv:1207.0407v1 [physics.data-an], 2012. Theory,” Fuzzy Sets and Systems, Vol. 90, No. 3, 1997, http://arxiv.org/abs/1207.0407v1