MATEC Web of Conferences 336, 09032 (2021) https://doi.org/10.1051/matecconf/202133609032 CSCNS2020

The causal relationship k

Ilija Barukciˇ c´1,? 1Internist, Horandstrasse, DE-26441-Jever, Germany

Abstract. Aim: A detailed and sophisticated analysis of causal relationships and chains of causation in medicine, life and other sciences by logically consistent statistical meth- ods in the light of empirical data is still not a matter of daily routine for us. Methods: In this publication, the relationship between cause and e↵ect is characterized while using the tools of classical logic and probability theory. Results: Methods how to determine conditions are developed in detail. The causal rela- tionship k has been derived mathematically from the axiom +1 =+1. Conclusion: Non-experimental and experimental data can be analysed by the methods presented for causal relationships.

1 Introduction

Before we try to describe the relationship between a cause and an e↵ect mathematically in a logically consistent way, it is vital to consider whether it is possible to achieve such a goal in principle. In other words, why should we care about the nature of causation at all? Causation seemed painfully important to some[1–11], but not to others. The trial to establish a generally accepted mathemati- cal concept of causation is aggravated especially by the countless attacks [12] on the principle of causality[1, 7, 13–16] by many authors which even tried to get rid of this concept altogether and by the very long and rich history of the denialism of causality in Philosophy, Mathematics, Statistics, Physics and a number of other disciplines too. However, it is by no means a hopeless case to math- ematise the relationship between a cause and an e↵ect in accordance with the basic laws of classical logic, statistics and probability theory. In point of fact, George Boole (1815 - 1864) has been one of the first who successfully mathematised classical logic[17, 18]. Meanwhile, Boolean algebra is widely used and of highest value. However, logical connectives (also called logical operators) like conjunction (denoted as ), disjunction (denoted as ) or negation (denoted as ) et cetera which are ^ _ ¬ used to conjoin two statements Pt and Qt to form another statement can be used under conditions of probability theory too. Especially conditions like necessary and sufficient conditions et cetera can be expressed mathematically while using the tools of probability theory. In this context, notable propo- nents of conditionalism[19] like the German anatomist and pathologist David Paul von Hansemann [20] (1858 - 1920) and the biologist and physiologist Max Richard Constantin Verworn [21] (1863 - 1921) and of course other authors too favoured conditions one-sidedly with no objective reason. In his essay “Kausale und konditionale Weltanschauung”Verworn himself ignores cause and e↵ect rela- tionships completely. Verworn demands: “Das Ding ist also identisch mit der Gesamtheit seiner

?e-mail: [email protected]

© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). MATEC Web of Conferences 336, 09032 (2021) https://doi.org/10.1051/matecconf/202133609032 CSCNS2020

Bedingungen.”[21] In this publication, we will develop new mathematical methods in order to recog- nise necessary conditions, sufficient conditions, cause e↵ect relationships et cetera with the tools of statistics and probability theory.

2 Material and methods

2.2 Definitions Classical logic is a branch of philosophy but equally a branch of mathematics too. A Boolean variable, named after George Boole, represents mathematically (either +0 or +1) the two truth values of clas- sical logic and Boolean algebra [17]. However, it is very remarkable that Gottfried Wilhelm Leibniz (1646 - 1716) [22] published 1703 the first self-consistent binary number system [23, 24] representing all numeric values while using typically +0 (zero, false) and +1 (one, true).

2.2.1 The number +0

Definition 1 (The number +0). The number +0 is defined [7–11] as the expression

+ 0 (+1) (+0) (+0) (+1) +1 1 (1) ⌘ ⇥ ⌘ ⇥ ⌘ −

2.2.2 The number +1

Definition 2 (The number +1). The number +1 is defined [7–11] as the expression

+ 1 +1 + 0 +1 0 (2) ⌘ ⌘ −

2.2.3 The probability of a single event

Definition 3 (The probability of a single event). In consideration of the definitions before, let p(RXt) represent the probability of a single event RXt at Bernoulli trial t. Let (RXt) represent the wavefunc- tion, a probability amplitude [25] of an event or of finding an event inside a set at a given (period * of ) time / Bernoulli trial [26] t. Let (RXt) denote the complex conjugate of the wave-function. In general, it is

(RXt) p (RXt) (RXt) E(RXt) p (RXt) p (RXt) ⇥ ⌘ ⇥ (RXt) ⌘ (RXt) ⌘ RXt p ( X ) ( X X ) p ( X ) p ( X ) ( X X ) E ( X ) 2 p ( X ) R t ⇥ R t ⇥ R t R t ⇥ R t ⇥ R t ⇥ R t R t ⌘ R t ⇥ p ( X ) ( X X ) ⌘ p ( X ) ( X X ) ⌘ E X 2 R t ⇥ R t ⇥ R t R t ⇥ R t ⇥ R t R t ( X ) *( U ) (3) ⌘ R t ⇥ R t 

2.2.4 The n-th moment expectation value of X

Definition 4 (The n-th moment expectation value of X). Let RXt denote an event (at a certain (period of) time or Bernoulli trial t [26]. Let p(RXt) represent the probability of an event at a given n 1 Bernoulli trial t. Let E(RXt ) denote the n-th moment expectation value [27, 28] of RXt . Let E(RXt ) 2 denote the first moment expectation value of RXt . Let E(RXt ) denote the second moment expectation value of RXt . In general, the n-th moment expectation value of RXt is defined as

2 MATEC Web of Conferences 336, 09032 (2021) https://doi.org/10.1051/matecconf/202133609032 CSCNS2020

Bedingungen.”[21] In this publication, we will develop new mathematical methods in order to recog- nise necessary conditions, sufficient conditions, cause e↵ect relationships et cetera with the tools of n 1 1 1 statistics and probability theory. E (RXt ) RXt RXt RXt ... p (RXt) ⌘ 0 ⇥ ⇥ ⇥ 1 ⇥ B (n times) C B − C (B X n) p ( X ) C (4) 2 Material and methods ⌘ @BR| t ⇥ R{zt }AC

Furthermore, it is 2.2 Definitions

Classical logic is a branch of philosophy but equally a branch of mathematics too. A Boolean variable, n m 1 1 1 m m E (RXt ) RXt RXt RXt ... p (RXt) named after George Boole, represents mathematically (either +0 or +1) the two truth values of clas- ⌘ 0 ⇥ ⇥ ⇥ 1 ⇥ B (n times) C sical logic and Boolean algebra [17]. However, it is very remarkable that Gottfried Wilhelm Leibniz B − C (B X n) m p ( X ) m C (5) (1646 - 1716) [22] published 1703 the first self-consistent binary number system [23, 24] representing ⌘ @BR| t ⇥ {zR t }AC all numeric values while using typically +0 (zero, false) and +1 (one, true). The first moment expectation value of RXt follows as

2.2.1 The number +0 1 1 E RXt RXt p (RXt) ⌘ 0 1 ⇥ + + B(one times)C Definition 1 (The number 0). The number 0 is defined [7–11] as the expression ⇣ ⌘ B − C B 1 C BRX|{z}t pC(RXt) + 0 (+1) (+0) (+0) (+1) +1 1 (1) ⌘ @ ⇥ A ⌘ ⇥ ⌘ ⇥ ⌘ − (⇣ X )⌘ p ( X ) (6) ⌘ R t ⇥ R t 2.2.2 The number +1 The second moment expectation value of RXt follows as

Definition 2 (The number +1). The number +1 is defined [7–11] as the expression 2 1 1 E RXt RXt RXt p (RXt) ⌘ 0 ⇥ 1 ⇥ + 1 +1 + 0 +1 0 (2) B (two times) C ⌘ ⌘ − ⇣ ⌘ B − C B 2 C @BR|Xt {z p (}RACXt) (7) 2.2.3 The probability of a single event ⌘ ⇥ ⇣ ⌘ Definition 3 (The probability of a single event). In consideration of the definitions before, let p(RXt) 2.2.5 The n-th moment expectation value of anti X represent the probability of a single event X at Bernoulli trial t. Let ( X ) represent the wavefunc- R t R t Definition 5 (The n-th moment expectation value of anti X). Let p(RXt) represent the probability tion, a probability amplitude [25] of an event or of finding an event inside a set at a given (period of a single event RXt at a given Bernoulli trial t. Let (1-p(RXt)) represent the probability that a single of ) time / Bernoulli trial [26] t. Let * ( X ) denote the complex conjugate of the wave-function. In n R t event RXt will not occur, will not exist at a given Bernoulli trial t. Let E(RXt ) denote the n-th moment general, it is 1 expectation value [27, 28] of anti RXt . Let E(RXt ) denote the first moment expectation value of 2 anti RXt . Let E(RXt ) denote the second moment expectation value of anti RXt . In general, the n-th (RXt) p (RXt) (RXt) E(RXt) moment expectation value of anti RXt is defined as p (RXt) p (RXt) ⇥ ⌘ ⇥ (RXt) ⌘ (RXt) ⌘ RXt 2 n 1 1 1 p (RXt) (RXt RXt) p (RXt) p (RXt) (RXt RXt) E (RXt) E RXt RXt RXt RXt ... (1 p (RXt)) p (RXt) ⇥ ⇥ ⇥ ⇥ ⇥ 2 ⌘ 0 ⇥ ⇥ ⇥ 1 ⇥ − ⌘ ⇥ p (RXt) (RXt RXt) ⌘ p (RXt) (RXt RXt) ⌘ E RXt B (n times) C ⇥ ⇥ ⇥ ⇥ � � B − C * B C ( X ) ( U ) (3) (B X n) (1 p ( X )) C (8) ⌘ R t ⇥ R t  ⌘ @BR| t ⇥ −{z R t }AC

The first moment expectation value of anti RXt follows as 2.2.4 The n-th moment expectation value of X 1 1 E RXt RXt (1 p (RXt)) Definition 4 (The n-th moment expectation value of X). Let RXt denote an event (at a certain ⌘ 0 1 ⇥ − (period of) time or Bernoulli trial t [26]. Let p( X ) represent the probability of an event at a given B(one times)C R t ⇣ ⌘ B − C n 1 B 1 C Bernoulli trial t. Let E(RXt ) denote the n-th moment expectation value [27, 28] of RXt . Let E(RXt ) BRXt (C1 p (RXt)) 2 @ |{z} A denote the first moment expectation value of RXt . Let E(RXt ) denote the second moment expectation ⌘ ⇥ − ⇣(RXt )⌘ (1 p (RXt)) (9) value of RXt . In general, the n-th moment expectation value of RXt is defined as ⌘ ⇥ −

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The second moment expectation value of anti RXt follows as

2 1 1 E RXt RXt RXt (1 p (RXt)) ⌘ 0 ⇥ 1 ⇥ − B (two times) C ⇣ ⌘ B − C B 2 C @BR|Xt {z (1}ACp (RXt)) (10) ⌘ ⇥ − ⇣ ⌘ 2.2.6 The n-th moment expectation value of U and W

Definition 6 (The n-th moment expectation value of U and W). Let p(RUt, RWt) represent the joint probability of an occurring of the events RUt and RWt at the same (period of time or) Bernoulli trial t. n n 1 Let E(RUt , RWt ) denote the n-th moment expectation value of RUt and RWt . Let E(RUt ) denote the first moment expectation value of RUt . In general, the n-th moment expectation value of RUt and RWt is defined as

E ( U n, W n) U 1 W 1 U 1 W 1 ... p ( U , W ) R t R t ⌘ 0 R t ⇥ R t ⇥ R t ⇥ R t ⇥ 1 ⇥ R t R t B C B⇣ ⌘ (n ⇣times) ⌘ C B − C B( U n W n) p ( U , W ) C (11) ⌘ @BR| t ⇥ R t ⇥ {zR t R t }AC

The first moment expectation value of RUt and RWt follows as

1 1 1 1 E RUt , RWt RUt RWt p (RUt , RWt ) ⌘ 0 ⇥ 1 ⇥ B (one times) C ⇣ ⌘ B − C B 1 1 C @BR|Ut {zRWt}AC p (RUt , RWt ) ⌘ ⇥ ⇥ ⇣( U W ) ⌘ p ( U , W ) (12) ⌘ R t ⇥ R t ⇥ R t R t

2.2.7 The variance

Definition 7 (The variance). Sir Ronald Aylmer Fisher (1890 – 1962), an English statistician, “the single most important figure in 20th century statistics”[29] coined the term variance as follows: “It is therefore desirable in analysing the causes of variability to deal with the square of the standard deviation as the measure of variability. We shall term this quantity the Variance ... ”[see 30, p. 399] Again, let p(RXt) represent the probability of a single event RXt at a given point in space-time or Bernoulli trial t. Let E(RXt) denote again the expectation value of RXt. The expectation value of RXt is defined as E ( X ) p ( X ) ( X ) ( X ) X *( X ) (13) R t ⌘ R t ⇥ R t ⌘ R t ⇥ R t ⇥ R t The expectation value of the other of RXt, of the complementary [7, 10] of RXt, of the opposite of RXt, of the anti RXt, denoted by RXt, is defined as E X (1 p ( X )) ( X ) (14) R t ⌘ − R t ⇥ R t 2 In this context, E(RXt ) is the expectation� � value of the second moment of RXt . The expectation value 2 of RXt is defined as

E X 2 p ( X ) X 2 p ( X ) ( X X ) (15) R t ⌘ R t ⇥ R t ⌘ R t ⇥ R t ⇥ R t ⇣ ⌘ ⇣ ⌘

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The second moment expectation value of anti X follows as R t 2 Let σ(RXt) denote the standard deviation of RXt. Let σ(RXt) denote the variance of RXt. In general, the variance [see 31, p. 42] is defined[7, 10] as 2 1 1 E RXt RXt RXt (1 p (RXt)) 2 ⌘ 0 ⇥ 1 ⇥ − σ(RXt) σ (RXt) σ (RXt) B (two times) C ⌘ ⇥ ⇣ ⌘ B − C 2 B 2 C E (RXt E (RXt)) @BR|Xt {z (1}ACp (RXt)) (10) ⌘ − ⌘ ⇥ − E X 2 (E ( X )) 2 ⇣ ⌘ ⌘ R t − R t 2.2.6 The n-th moment expectation value of U and W ⇣X 2 ⌘p ( X ) (p ( X ) X ) 2 ⌘ R t ⇥ R t − R t ⇥ R t 2 2 Definition 6 (The n-th moment expectation value of U and W). Let p(RUt, RWt) represent the joint ⇣RXt p (RX⌘t) p (RXt) ⌘ ⇥ − probability of an occurring of the events RUt and RWt at the same (period of time or) Bernoulli trial t. 2 n n 1 ⇣RXt ⌘ (⇣p (RXt) (1 p (R⌘Xt))) Let E(RUt , RWt ) denote the n-th moment expectation value of RUt and RWt . Let E(RUt ) denote the ⌘ ⇥ ⇥ − first moment expectation value of U . In general, the n-th moment expectation value of U and R⇣ Xt ⌘(p (RXt) RXt (1 p (RXt))) R t R t ⌘ ⇥ ⇥ ⇥ − RWt is defined as E ( X ) X (1 p ( X )) ⌘ R t ⇥ R t ⇥ − R t E ( X ) E X (16) ⌘ R t ⇥ R t E ( U n, W n) U 1 W 1 U 1 W 1 ... p ( U , W ) From equation 16 follows that R t R t ⌘ 0 R t ⇥ R t ⇥ R t ⇥ R t ⇥ 1 ⇥ R t R t � � B C B⇣ ⌘ (n ⇣times) ⌘ C σ(RUt) B − C RUt (17) B n n C 2 B( U W ) p ( U , W ) C (11) ⌘ p ( U ) (1 p ( U )) ⌘ @BR| t ⇥ R t ⇥ {zR t R t }AC R t ⇥ − R t

and that p The first moment expectation value of RUt and RWt follows as σ( W ) W R t (18) R t 2 ⌘ p (RWt) (1 p (RWt)) 1 1 1 1 ⇥ − E RUt , RWt RUt RWt p (RUt , RWt ) ⌘ 0 ⇥ 1 ⇥ p B (one times) C ⇣ ⌘ B − C 2.2.8 The n-th moment co-variance B 1 1 C @BR|Ut {zRWt}AC p (RUt , RWt ) ⌘ ⇥ ⇥ Definition 8 (The n-th moment co-variance). Let p(RUt, RWt) represent the joint probability of RUt n n ⇣(RUt RWt ) ⌘ p (RUt , RWt ) (12) and RWt at the same (period of time) Bernoulli trial t. Let E(RUt , RWt ) denote the n-th moment ⌘ ⇥ ⇥ n expectation value of RUt and RWt . Let E(RUt ) denote the n-th moment expectation value of RUt . Let n 2.2.7 The variance E(RWt ) denote the n-th moment expectation value of RWt . Let σ(RUt, RWt) denote the co-variance between RUt and RWt. In general, the n-th moment co-variance between RUt and RWt is defined[7] Definition 7 (The variance). Sir Ronald Aylmer Fisher (1890 – 1962), an English statistician, “the as single most important figure in 20th century statistics”[29] coined the term variance as follows: “It σ ( U n, W n) ( U n W n) (p ( U , W ) (p ( U ) p ( W ))) is therefore desirable in analysing the causes of variability to deal with the square of the standard R t R t R t R t R t R t R t R t ⌘ n⇥ n ⇥ − n ⇥ n deviation as the measure of variability. We shall term this quantity the Variance ... ”[see 30, p. ((RUt RWt p (RUt , RWt )) ((RUt RWt ) p (RUt ) p (RWt ))) ⌘ n⇥ n ⇥ n − ⇥n ⇥ ⇥ 399] Again, let p(RXt) represent the probability of a single event RXt at a given point in space-time or E ( U , W ) ( U p ( U )) ( W p ( W )) ⌘ R t R t − R t ⇥ R t ⇥ R t ⇥ R t Bernoulli trial t. Let E(RXt) denote again the expectation value of RXt. The expectation value of RXt is n n n n E (RUt , RWt ) (E (RUt ) E (RWt )) (19) defined as ⌘ − ⇥ E ( X ) p ( X ) ( X ) ( X ) X *( X ) (13) From equation 19 follows equally that R t ⌘ R t ⇥ R t ⌘ R t ⇥ R t ⇥ R t The expectation value of the other of X , of the complementary [7, 10] of X , of the opposite of X , σ (RUt , RWt ) (RUt RWt ) (p (RUt , RWt ) (p (RUt ) p (RWt ))) R t R t R t ⌘ ⇥ ⇥ − ⇥ of the anti RXt, denoted by RXt, is defined as (( U W p ( U , W )) (( U W ) p ( U ) p ( W ))) ⌘ R t ⇥ R t ⇥ R t R t − R t ⇥ R t ⇥ R t ⇥ R t E X (1 p ( X )) ( X ) (14) E (RUt , RWt ) (RUt p (RUt )) (RWt p (RWt )) R t ⌘ − R t ⇥ R t ⌘ − ⇥ ⇥ ⇥ E (RUt , RWt ) (E (RUt ) E (RWt )) (20) 2 ⌘ − ⇥ In this context, E(RXt ) is the expectation� � value of the second moment of RXt . The expectation value 2 Equation 20 demands too that of RXt is defined as σ ( U , W ) 2 2 R t R t E RXt p (RXt) RXt p (RXt) (RXt RXt) (15) RUt RWt (21) ⌘ ⇥ ⌘ ⇥ ⇥ ⇥ ⌘ (p (RUt , RWt ) (p (RUt ) p (RWt ))) − ⇥ ⇣ ⌘ ⇣ ⌘

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2.2.9 Two by two table of Bernoulli random variables Definition 9 (Two by two table of Bernoulli random variables). The two by two or contingency table as introduced by Karl Pearson[32] in 1904 harbours a large variety of topics and debates. Central to these is the problem to apply the laws of classical logic on data sets, which concerns the justification of inferences that extrapolate from sample data to general facts. However, a contingency table is still an appropriate theoretical model too for studying the relationships between two Bernoulli[33] (i. e. +0/+1) distributed random variables existing or occurring at the same Bernoulli trial [26] (period of time) t. In this context, let a Bernoulli distributed random variable At denote a risk factor, a condition or a cause et cetera and occur or exist with the probability p(At) at the Bernoulli trial [26] (period of time) t. Let E( At) denote the expectation value of At. In the case of +0/+1 distributed Bernoulli random variables it is E (A ) A p (A ) t ⌘ t ⇥ t p (a ) + p (b ) ⌘ t t (+0 + 1) p (A ) ⌘ ⇥ t p (A ) (22) ⌘ t Let a Bernoulli distributed random variable Bt denote an outcome, a conditioned event or an e↵ect and occur or exist et cetera with the probability p(Bt) at the Bernoulli trial (period of time) t. Let E( Bt) denote the expectation value of Bt. It is E (B ) B p (B ) t ⌘ t ⇥ t p (a ) + p (c ) ⌘ t t (+0 + 1) p (B ) ⌘ ⇥ t p (B ) (23) ⌘ t Let p(a )= p(A B ) denote the joint probability distribution of A and B at the same Bernoulli t t ^ t t t trial (period of time) t. In general it is E (a ) E (A B ) t ⌘ t ^ t (A B ) p(A B ) ⌘ t ⇥ t ⇥ t ^ t p(A B ) ⌘ t ^ t p (a ) (24) ⌘ t Let p(b )= p(A B ) denote the joint probability distribution of A and not B at the same t t ^ ¬ t t t Bernoulli trial (period of time) t. In general it is E (b ) E (A B ) t ⌘ t ^ ¬ t (A B ) p(A B ) ⌘ t ⇥ ¬ t ⇥ t ^ ¬ t p(A B ) ⌘ t ^ ¬ t p (b ) (25) ⌘ t Let p(c )= p( A B ) denote the joint probability distribution of not A and B at the same t ¬ t ^ t t t Bernoulli trial (period of time) t. In general it is E (c ) E ( A B ) t ⌘ ¬ t ^ t ( A B ) p( A B ) ⌘ ¬ t ⇥ t ⇥ ¬ t ^ t p( A B ) ⌘ ¬ t ^ t p (c ) (26) ⌘ t

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2.2.9 Two by two table of Bernoulli random variables Let p(d )= p( A B ) denote the joint probability distribution of not A and not B at the same t ¬ t ^ ¬ t t t Bernoulli trial (period of time) t. In general it is Definition 9 (Two by two table of Bernoulli random variables). The two by two or contingency table as introduced by Karl Pearson[32] in 1904 harbours a large E (d ) E ( A B ) t ⌘ ¬ t ^ ¬ t variety of topics and debates. Central to these is the problem to apply the laws of classical logic on data ( At Bt) p( At Bt) sets, which concerns the justification of inferences that extrapolate from sample data to general facts. ⌘ ¬ ⇥ ¬ ⇥ ¬ ^ ¬ p( A B ) However, a contingency table is still an appropriate theoretical model too for studying the relationships ⌘ ¬ t ^ ¬ t p (d ) (27) between two Bernoulli[33] (i. e. +0/+1) distributed random variables existing or occurring at the ⌘ t same Bernoulli trial [26] (period of time) t. In this context, let a Bernoulli distributed random variable In general, it is At denote a risk factor, a condition or a cause et cetera and occur or exist with the probability p(At) at p (at) + p (bt) + p (ct) + p (dt) +1 (28) the Bernoulli trial [26] (period of time) t. Let E( At) denote the expectation value of At. In the case ⌘ of +0/+1 distributed Bernoulli random variables it is Table 1 provide us with an overview of the definitions above. E (A ) A p (A ) t ⌘ t ⇥ t p (at) + p (bt) Table 1: The two by two table of Bernoulli random variables ⌘ (+0 + 1) p (A ) ⌘ ⇥ t p (A ) (22) ⌘ t Conditioned Bt Let a Bernoulli distributed random variable Bt denote an outcome, a conditioned event or an e↵ect TRUE FALSE and occur or exist et cetera with the probability p(Bt) at the Bernoulli trial (period of time) t. Let E( Condition TRUE p(at) p(bt) p(At) Bt) denote the expectation value of Bt. It is At FALSE p(ct) p(dt) p(At) p(B ) p(B ) +1 E (B ) B p (B ) t t t ⌘ t ⇥ t p (a ) + p (c ) ⌘ t t (+0 + 1) p (B ) ⌘ ⇥ t p (B ) (23) ⌘ t 2.2.10 Two by two table of Binomial random variables Let p(a )= p(A B ) denote the joint probability distribution of A and B at the same Bernoulli t t ^ t t t Definition 10 (Two by two table of Binomial random variables). trial (period of time) t. In general it is Under conditions where the probability of an event, an outcome, a success et cetera is constant E (at) E (At Bt) ⌘ ^ from Bernoulli trial to Bernoulli trial t, it is (A B ) p(A B ) ⌘ t ⇥ t ⇥ t ^ t p(A B ) ⌘ t ^ t A = N E (At) p (at) (24) ⇥ ⌘ N (At p (At)) Let p(bt)= p(At Bt) denote the joint probability distribution of At and not Bt at the same ⌘ ⇥ ⇥ ^ ¬ N (p (A ) + p (B )) Bernoulli trial (period of time) t. In general it is ⌘ ⇥ t t N p (A ) (29) E (b ) E (A B ) ⌘ ⇥ t t ⌘ t ^ ¬ t (A B ) p(A B ) and ⌘ t ⇥ ¬ t ⇥ t ^ ¬ t p(A B ) ⌘ t ^ ¬ t p (bt) (25) B = N E (B ) ⌘ ⇥ t Let p(c )= p( A B ) denote the joint probability distribution of not A and B at the same N (B p (B )) t ¬ t ^ t t t ⌘ ⇥ t ⇥ t Bernoulli trial (period of time) t. In general it is N (p (A ) + p (c )) ⌘ ⇥ t t E (ct) E ( At Bt) N p (B ) (30) ⌘ ¬ ^ ⌘ ⇥ t ( A B ) p( A B ) ⌘ ¬ t ⇥ t ⇥ ¬ t ^ t where N denotes the population size. Furthermore, it is p( A B ) ⌘ ¬ t ^ t p (c ) (26) a N (E (At)) N (p (At)) (31) ⌘ t ⌘ ⇥ ⌘ ⇥

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and b N (E (B )) N (p (B )) (32) ⌘ ⇥ t ⌘ ⇥ t and c N (E (c )) N (p (c )) (33) ⌘ ⇥ t ⌘ ⇥ t and d N (E (d )) N (p (d )) (34) ⌘ ⇥ t ⌘ ⇥ t and a + b + c + d A + A B + B N (35) ⌘ ⌘ ⌘ Table 2 provide us again an overview of a two by two table of Binomial random variables.

Table 2: The two by two table of Binomial random variables

Conditioned Bt TRUE FALSE Condition TRUE a b A At FALSE c d A BBN

2.2.11 Independence

Definition 11 (Independence).

In general, an event At at the Bernoulli trial t need not but can be independent of the existence or of the occurrence of another event Bt at the same Bernoulli trial t. Mathematically, independence [34, 35] in terms of probability theory is defined at the same (period of) time t (i. e. Bernoulli trial t) as p (A B ) p (A ) p (B ) (36) t ^ t ⌘ t ⇥ t

2.2.12 Dependence

Definition 12 (Dependence).

The dependence of events [see 1, p. 57-61] is defined as

p At Bt Ct ... p (At) p (Bt) p (Ct) ... (37) 0 ^ ^ ^ 1 ⌘ n ⇥ ⇥ ⇥ n n B C s B C @B| {z }AC | {z }

8 MATEC Web of Conferences 336, 09032 (2021) https://doi.org/10.1051/matecconf/202133609032 CSCNS2020 and 2.2.13 Exclusion relationship b N (E (B )) N (p (B )) (32) ⌘ ⇥ t ⌘ ⇥ t Definition 13 (Exclusion relationship [EXCL]). and Mathematically, the exclusion (EXCL) relationship, denoted by p(A B ) in terms of statistics and t | t c N (E (c )) N (p (c )) (33) probability theory, is defined[see 1, p. 68-70] as ⌘ ⇥ t ⌘ ⇥ t p (A B ) p (b ) + p (c ) + p (d ) and t | t ⌘ t t t N (p (bt) + p (ct) + p (dt)) d N (E (dt)) N (p (dt)) (34) ⇥ ⌘ ⇥ ⌘ ⇥ ⌘ N N and A B t t + + a + b + c + d A + A B + B N (35) t=1 _ b c d ⌘ ⌘ ⌘ ⌘ P � N � ⌘ N Table 2 provide us again an overview of a two by two table of Binomial random variables. +1 (38) ⌘ Based on the Henry M. She↵er, 1913 relationship, it is p (A B ) 1 p (A B ) (see table 3). t ^ t ⌘ − t | t Table 2: The two by two table of Binomial random variables

Table 3: At excludes Bt and vice versa.

Conditioned Bt TRUE FALSE Conditioned (COVID-19) Bt Condition TRUE a b A TRUE FALSE At FALSE c d A Condition (Vaccine) TRUE +0 p(bt) p(At) BBN At FALSE p(ct) p(dt) p(At) p(Bt) p(Bt) +1

2.2.11 Independence Remark 1. Pfizer Inc. and BioNTech SE announced on Monday, November 09, 2020 - 06:45am results from a Phase 3 COVID-19 vaccine trial with 43.538 participants which provides evidence that their vaccine (BNT162b2) is preventing COVID-19 in participants without evidence of prior SARS- Definition 11 (Independence). CoV-2 infection. In toto, 170 confirmed cases of COVID-19 were evaluated, with 8 in the vaccine group versus 162 in the placebo group. The exclusion relationship can be calculated as follows. In general, an event At at the Bernoulli trial t need not but can be independent of the existence or of the occurrence of another event Bt at the same Bernoulli trial t. Mathematically, independence p (Vaccine : BNT162b2 COVID 19(in f ection)) p (bt) + p (ct) + p (dt) [34, 35] in terms of probability theory is defined at the same (period of) time t (i. e. Bernoulli trial t) | − ⌘ 1 p (a ) as ⌘ − t 8 p (At Bt) p (At) p (Bt) (36) 1 ^ ⌘ ⇥ ⌘ − 43538 ! +0, 99981625 (39) ⌘ 2.2.12 Dependence with a P Value = 0,000184. Definition 12 (Dependence). 2.2.14 The goodness of fit test of an exclusion relationship The dependence of events [see 1, p. 57-61] is defined as Definition 14 (The χ˜ 2 goodness of fit test of an exclusion relationship).

Under some well known circumstances, testing hypothesis about an exclusion relationship p(At 2 2 | p At Bt Ct ... p (At) p (Bt) p (Ct) ... (37) Bt) is possible by the chi-square distribution (also chi-squared orχ ˜ -distribution) too. Theχ ˜ goodness 0 ^ ^ ^ 1 ⌘ n ⇥ ⇥ ⇥ n n of fit test of an exclusion relationship with degree of freedom (d. f.) of d. f. = 1 is calculated as B C s B C @B| {z }AC | {z }

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(b (a + b)) 2 χ˜ 2 ((A B ) A) − + Calculated t | t | ⌘ A (c + d) A 2 − A  a2 + 0 ⌘ A a2 (40) ⌘ A or equally as

(c (a + c)) 2 χ˜ 2 ((A B ) B) − + Calculated t | t | ⌘ B (b + d) B 2 − B  a2 + 0 ⌘ B a2 (41) ⌘ B and can be compared with a theoretical chi-square value at a certain level of significance ↵. The χ˜ 2-distribution equals zero when the observed values are equal to the expected/theoretical values of an exclusion relationship/distribution p(A B ), in which case the null hypothesis to be accepted. Yate’s t | t [36] continuity correction has not been used under these circumstances.

2.2.15 The left-tailed p Value of an exclusion relationship

Definition 15 (The left-tailed p Value of an exclusion relationship).

It is known that as a sample size, N, increases, a sampling distribution of a special test statistic approaches the normal distribution (central limit theorem). Under these circumstances, the left-tailed (lt) p Value [37] of an exclusion relationship can be calculated as follows.

1 p A B pValue (A B ) 1 e−( − ( t| t)) lt t | t ⌘ − (a/N) 1 e− (42) ⌘ − A low p-value may provide some evidence of statistical significance.

2.2.16 Neither nor conditions

Definition 16 (Neither At nor Bt conditions [NOR]).

Mathematically, a neither At nor Bt condition (rejection, Jean Nicod’s statement 1924) relationship (NOR), denoted by p(A B ) in terms of statistics and probability theory, is defined [see 1, p. 68-70] t # t

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2 as 2 (b (a + b)) χ˜ Calculated ((At Bt) A) − + | | ⌘ A p (At Bt) p (dt) 2 # ⌘ (c + d) A N N − N (At Bt) At Bt A − = _ = ^ N (p (d ))  t 1 t 1 ⇥ t a2 ⌘ P N ⌘ P � N � ⌘ N + 0 ⌘ A d a2 ⌘ N (40) +1 (43) ⌘ A ⌘ or equally as 2.2.17 The Chi square goodness of fit test of a neither nor condition relationship (c (a + c)) 2 χ˜ 2 ((A B ) B) − + Definition 17 (The χ˜ 2 goodness of fit test of a neither A nor B condition relationship). Calculated t | t | ⌘ B t t 2 (b + d) B A neither At nor Bt condition relationship p(At Bt) can be tested by the chi-square distribution − 2 2 # B (also chi-squared orχ ˜ -distribution). Theχ ˜ goodness of fit test of a neither At nor Bt condition  a2 relationship with degree of freedom (d. f.) of d. f. = 1 may be calculated as + 0 ⌘ B a2 (d (c + d)) 2 (41) χ˜ 2 ((A B ) A) − + ⌘ B Calculated t # t | ⌘ A ((a + b) A) 2 and can be compared with a theoretical chi-square value at a certain level of significance ↵. The − χ˜ 2-distribution equals zero when the observed values are equal to the expected/theoretical values of an A exclusion relationship/distribution p(A B ), in which case the null hypothesis to be accepted. Yate’s c2 t | t + 0 (44) [36] continuity correction has not been used under these circumstances. ⌘ A

or equally as 2.2.15 The left-tailed p Value of an exclusion relationship (d (b + d)) 2 χ˜ 2 ((A B ) B) − + Definition 15 (The left-tailed p Value of an exclusion relationship). Calculated t # t | ⌘ B ((a + c) B) 2 It is known that as a sample size, N, increases, a sampling distribution of a special test statistic − approaches the normal distribution (central limit theorem). Under these circumstances, the left-tailed B b2 (lt) p Value [37] of an exclusion relationship can be calculated as follows. + 0 (45) ⌘ B 1 p A B pValue (A B ) 1 e−( − ( t| t)) lt t | t ⌘ − Yate’s [36] continuity correction has not been used in this context. (a/N) 1 e− (42) ⌘ − A low p-value may provide some evidence of statistical significance. 2.2.18 The left-tailed p Value of a neither nor B condition relationship

Definition 18 (The left-tailed p Value of a neither At nor Bt condition relationship). 2.2.16 Neither nor conditions The left-tailed (lt) p Value [37] of a neither At nor Bt condition relationship can be calculated as follows. Definition 16 (Neither At nor Bt conditions [NOR]). (1 p(At Bt)) pValuelt (At Bt) 1 e− − # Mathematically, a neither At nor Bt condition (rejection, Jean Nicod’s statement 1924) relationship # ⌘ − p(At Bt) (NOR), denoted by p(A B ) in terms of statistics and probability theory, is defined [see 1, p. 68-70] 1 e− _ t # t ⌘ − ((a+b+c)/N) 1 e− (46) ⌘ −

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where may denote disjunction or logical inclusive or. In this context, a low p-value indicates again _ a statistical significance. In general, it is p (A B ) 1 p (A B ) (see table 4). t _ t ⌘ − t # t

Table 4: Neither At nor Bt relationship.

Conditioned Bt YES NO Condition At YES 0 0 0 NO 0 1 1 0 11

2.2.19 Necessary condition

Definition 19 (Necessary condition [Conditio sine qua non]). Mathematically, the necessary condition (SINE) relationship, denoted by p(A B ) in terms of t t statistics and probability theory, is defined [see 1, p. 15-28] as

N At Bt = _ p (A B ) p A B t 1 t t ⌘ t _ t ⌘ P � N � � � p (a ) + p (b ) + p (d ) ⌘ t t t N (p (a ) + p (b ) + p (d )) ⇥ t t t ⌘ N a + b + d ⌘ N +1 (47) ⌘ It is p (A < B ) 1 p (A B ) (see Table 5). t− t ⌘ − t t

Table 5: Necessary condition.

Conditioned Bt TRUE FALSE Condition TRUE p(at) p(bt) p(At) At FALSE +0 p(dt) p(At) p(Bt) p(Bt) +1

Remark 2. A necessary condition At is characterized itself by the property that another event Bt will not occur if At is not given, if At did not occur [1–11]. Example. A human being cannot live without water. A human being cannot live without gaseous oxygen et cetera. Water itself is a necessary

12 MATEC Web of Conferences 336, 09032 (2021) https://doi.org/10.1051/matecconf/202133609032 CSCNS2020 where may denote disjunction or logical inclusive or. In this context, a low p-value indicates again condition of human life. However, gaseous oxygen is a necessary condition of human life too. Thus _ a statistical significance. In general, it is p (A B ) 1 p (A B ) (see table 4). far, even if water is given and even if water is a necessary condition of human life, without gaseous t _ t ⌘ − t # t oxygen there will be no human life. In general, if a conditioned or an outcome Bt depends on the necessary condition At and equally on numerous other necessary conditions, an event Bt will not Table 4: Neither At nor Bt relationship. occur if At itself is not given independently of the occurrence of other necessary conditions.

Conditioned Bt YES NO 2.2.20 The Chi-square goodness of fit test of a necessary condition relationship Condition At YES 0 0 0 Definition 20 (The χ˜ 2 goodness of fit test of a necessary condition relationship). NO 0 1 1 0 11 Under some well known circumstances, hypothesis about the conditio sine qua non relationship p(A B ) can be tested by the chi-square distribution (also chi-squared or χ2-distribution), first t t described by the German statistician Friedrich Robert Helmert [38] and later rediscovered by Karl Pearson [39] in the context of a goodness of fit test. Theχ ˜ 2 goodness of fit test of a conditio sine qua non relationship with degree of freedom (d. f.) of d. f. = 1 is calculated as 2.2.19 Necessary condition

Definition 19 (Necessary condition [Conditio sine qua non]). (a (a + c)) 2 χ˜ 2 (A B B) − + Calculated t t | ⌘ B Mathematically, the necessary condition (SINE) relationship, denoted by p(A B ) in terms of t t (b + d) B 2 statistics and probability theory, is defined [see 1, p. 15-28] as − B  N c2 At Bt + 0 t=1 _ ⌘ B p (At Bt) p At Bt ⌘ _ ⌘ P � N � c2 (48) � � p (a ) + p (b ) + p (d ) ⌘ t t t ⌘ B N (p (a ) + p (b ) + p (d )) ⇥ t t t or equally as ⌘ N 2 a + b + d 2 (d (c + d)) χ˜ Calculated At Bt A − + ⌘ N | ⌘ A +1 (47)  ((a + b) A) 2 ⌘ − A It is p (A < B ) 1 p (A B ) (see Table 5). t− t ⌘ − t t c2 + 0 ⌘ A c2 Table 5: Necessary condition. (49) ⌘ A and can be compared with a theoretical chi-square value at a certain level of significance ↵. It has Conditioned B t not yet been finally clarified whether the use of Yate’s [36] continuity correction is necessary at all. TRUE FALSE Condition TRUE p(at) p(bt) p(At) 2.2.21 The left-tailed p Value of the conditio sine qua non relationship At FALSE +0 p(dt) p(At) + p(Bt) p(Bt) 1 Definition 21 (The left-tailed p Value of the conditio sine qua non relationship). The left-tailed (lt) p Value [37] of the conditio sine qua non relationship can be calculated as follows. A necessary condition A is characterized itself by the property that another event B will Remark 2. t t 1 p A B pValue (A B ) 1 e−( − ( t t)) not occur if At is not given, if At did not occur [1–11]. Example. A human being cannot live without lt t t ⌘ − (c/N) water. A human being cannot live without gaseous oxygen et cetera. Water itself is a necessary 1 e− (50) ⌘ −

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2.2.22 Sufficient condition

Definition 22 (Sucient condition [Conditio per quam]).

Mathematically, the sufficient condition (IMP) relationship, denoted by p(A B ) in terms of t ! t statistics and probability theory, is defined[see 1, p. 68-70] as

N At Bt = _ p (A B ) p A B t 1 t ! t ⌘ t _ t ⌘ P � N � � � p (a ) + p (c ) + p (d ) ⌘ t t t N (p (a ) + p (c ) + p (d )) ⇥ t t t N a + c + d ⌘ N +1 (51) ⌘ It is p (A > B ) 1 p (A B ) (see Table 6). t − t ⌘ − t ! t

Table 6: Sufficient condition.

Conditioned Bt TRUE FALSE Condition TRUE p(at) +0 p(At) At FALSE p(ct) p(dt) p(At) p(Bt) p(Bt) +1

Remark 3. A sufficient condition At is characterized by the property that another event Bt will occur if At is given, if At itself occured [1–11]. Example. The ground, the streets, the trees, human beings and many other objects too will become wet during a heavy rain. Especially, if it is raining (event At), then human beings will be wet (event Bt). However, even if this is a common human wisdom, a human being equipped with an appropriate umbrella (denoted by Rt) need not to become wet even during a heavy rain. An appropriate umbrella (Rt) is similar to an event which can counteract the occurrence of another event (Bt) and can be understood something as an anti-dot of another event. In other words, an appropriate umbrella is an antidote of the e↵ect of rain on human body, an appropriate umbrella has the potential to protect humans from the e↵ect of rain on their body. It is a good rule of thumb that the following relationship

p (A B ) + p (R B ) +1 (52) t ! t t ^ t ⌘

indicates that Rt is an antidote of At. However, taking a shower, swimming in a lake et cetera may make human hair wet too. More than anything else, however, these events does not a↵ect the final outcome, the e↵ect of raining on human body.

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2.2.22 Sufficient condition 2.2.23 The Chi square goodness of fit test of a sufficient condition relationship

Definition 22 (Sucient condition [Conditio per quam]). Definition 23 (The χ˜ 2 goodness of fit test of a sucient condition relationship).

Mathematically, the sufficient condition (IMP) relationship, denoted by p(At Bt) in terms of Under some well known circumstances, testing hypothesis about the conditio per quam relation- ! 2 statistics and probability theory, is defined[see 1, p. 68-70] as ship p(At Bt) is possible by the chi-square distribution (also chi-squared orχ ˜ -distribution) too. ! Theχ ˜ 2 goodness of fit test of a conditio per quam relationship with degree of freedom (d. f.) of d. f. N = 1 is calculated as At Bt t=1 _ p (At Bt) p At Bt ! ⌘ _ ⌘ P � N � 2 2 (a (a + b)) � � p (at) + p (ct) + p (dt) χ˜ (A B A) − + ⌘ Calculated t ! t | ⌘ A N (p (a ) + p (c ) + p (d )) ⇥ t t t (c + d) A 2 N − A a + c + d  b2 ⌘ N + 0 +1 (51) ⌘ A ⌘ b2 (53) It is p (At > Bt) 1 p (At Bt) (see Table 6). ⌘ A − ⌘ − ! or equally as

Table 6: Sufficient condition. (d (b + d)) 2 χ˜ 2 A B B − + Calculated t ! t | ⌘ B  ((a + c) B) 2 Conditioned Bt − TRUE FALSE B b2 Condition TRUE p(at) +0 p(At) + 0 At FALSE p(ct) p(dt) p(At) ⌘ B p(Bt) p(Bt) +1 b2 (54) ⌘ B

and can be compared with a theoretical chi-square value at a certain level of significance ↵. The 2 Remark 3. A sufficient condition At is characterized by the property that another event Bt will occur χ˜ -distribution equals zero when the observed values are equal to the expected/theoretical values of if A is given, if A itself occured [1–11]. Example. The ground, the streets, the trees, human beings the conditio per quam relationship/distribution p(A B ), in which case the null hypothesis accepted. t t t ! t and many other objects too will become wet during a heavy rain. Especially, if it is raining (event At), Yate’s [36] continuity correction has not been used in this context. then human beings will be wet (event Bt). However, even if this is a common human wisdom, a human being equipped with an appropriate umbrella (denoted by Rt) need not to become wet even during a 2.2.24 The left-tailed p Value of the conditio per quam relationship heavy rain. An appropriate umbrella (Rt) is similar to an event which can counteract the occurrence of another event (Bt) and can be understood something as an anti-dot of another event. In other words, an appropriate umbrella is an antidote of the e↵ect of rain on human body, an appropriate Definition 24 (The left-tailed p Value of the conditio per quam relationship). umbrella has the potential to protect humans from the e↵ect of rain on their body. It is a good rule of thumb that the following relationship The left-tailed (lt) p Value [37] of the conditio per quam relationship can be calculated as follows.

(1 p(At Bt)) pValuelt (At Bt) 1 e− − ! p (At Bt) + p (Rt Bt) +1 (52) ! ⌘ − ! ^ ⌘ (b/N) 1 e− (55) ⌘ − indicates that Rt is an antidote of At. However, taking a shower, swimming in a lake et cetera may make human hair wet too. More than anything else, however, these events does not a↵ect the final Again, a low p-value indicates a statistical significance. outcome, the e↵ect of raining on human body.

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2.2.25 Necessary and sufficient conditions

Definition 25 (Necessary and sucient conditions [EQV]).

The necessary and sufficient condition (EQV) relationship, denoted by p(A B ) in terms of t $ t statistics and probability theory, is defined[see 1, p. 68-70] as

N At Bt At Bt = _ ^ _ p (A B ) t 1 t $ t ⌘ P �� N� � �� p (a ) + p (d ) ⌘ t t N (p (a ) + p (d )) ⇥ t t ⌘ N a + d ⌘ N +1 (56) ⌘

2.2.26 The Chi square goodness of fit test of a necessary and sufficient condition relationship

Definition 26 (The χ˜ 2 goodness of fit test of a necessary and sucient condition relationship).

Even the necessary and sufficient condition relationship p(A B ) can be tested by the chi-square t $ t distribution (also chi-squared orχ ˜ 2-distribution) too. Theχ ˜ 2 goodness of fit test of a necessary and sufficient condition relationship with degree of freedom (d. f.) of d. f. = 1 is calculated as

(a (a + b)) 2 χ˜ 2 (A B A) − + Calculated t $ t | ⌘ A d ((c + d)) 2 − A b2 c2 + (57) ⌘ A A

or equally as

(a (a + c)) 2 χ˜ 2 (A B B) − + Calculated t $ t | ⌘ B d ((b + d)) 2 − B c2 b2 + (58) ⌘ B B

The calculatedχ ˜ 2 goodness of fit test of a necessary and sufficient condition relationship can be compared with a theoretical chi-square value at a certain level of significance ↵. Under conditions where the observed values are equal to the expected/theoretical values of a necessary and sufficient condition relationship/distribution p(A B ), theχ ˜ 2-distribution equals zero. It is to be cleared t $ t whether Yate’s [36] continuity correction should be used at all.

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2.2.25 Necessary and sufficient conditions 2.2.27 The left-tailed p Value of a necessary and sufficient condition relationship

Definition 25 (Necessary and sucient conditions [EQV]). Definition 27 (The left-tailed p Value of a necessary and sucient condition relationship). The necessary and sufficient condition (EQV) relationship, denoted by p(A B ) in terms of t $ t statistics and probability theory, is defined[see 1, p. 68-70] as The left-tailed (lt) p Value [37] of a necessary and sufficient condition relationship can be calcu- N lated as follows. At Bt At Bt t=1 _ ^ _ 1 p A B p (At Bt) pValue (A B ) 1 e−( − ( t$ t)) $ ⌘ P �� N� � �� lt t $ t ⌘ − ((b+c)/N) p (a ) + p (d ) 1 e− (59) ⌘ t t ⌘ − N (p (a ) + p (d )) ⇥ t t ⌘ N In this context, a low p-value indicates again a statistical significance. Table 7 may provide an a + d overview of the theoretical distribution of a necessary and sufficient condition. ⌘ N +1 (56) ⌘ Table 7: Necessary and sufficient condition. 2.2.26 The Chi square goodness of fit test of a necessary and sufficient condition relationship Conditioned Bt YES NO Definition 26 (The χ˜ 2 goodness of fit test of a necessary and sucient condition relationship). Condition At YES 1 0 1 Even the necessary and sufficient condition relationship p(A B ) can be tested by the chi-square NO 0 1 1 t $ t distribution (also chi-squared orχ ˜ 2-distribution) too. Theχ ˜ 2 goodness of fit test of a necessary and 1 12 sufficient condition relationship with degree of freedom (d. f.) of d. f. = 1 is calculated as

2 2 (a (a + b)) χ˜ Calculated (At Bt A) − + 2.2.28 Either or conditions $ | ⌘ A d ((c + d)) 2 − A Definition 28 (Either At or Bt conditions [NEQV]). b2 c2 + (57) ⌘ A A Mathematically, an either At or Bt condition relationship (NEQV), denoted by p(At >-< Bt) in terms of statistics and probability theory, is defined[see 1, p. 68-70] as or equally as

N (a (a + c)) 2 χ˜ 2 (A B B) − + At Bt At Bt Calculated t $ t | ⌘ B t=1 ^ _ ^ p (At > < Bt) d ((b + d)) 2 − ⌘ P �� N� � �� − p (b ) + p (c ) B ⌘ t t + c2 b2 N (p (bt) p (ct)) + (58) ⇥ ⌘ B B ⌘ N b + c The calculatedχ ˜ 2 goodness of fit test of a necessary and sufficient condition relationship can be ⌘ N +1 (60) compared with a theoretical chi-square value at a certain level of significance ↵. Under conditions ⌘ where the observed values are equal to the expected/theoretical values of a necessary and sufficient condition relationship/distribution p(A B ), theχ ˜ 2-distribution equals zero. It is to be cleared It is p (A > < B ) 1 p (A < > B ) (see Table 8). t $ t t − t ⌘ − t − t whether Yate’s [36] continuity correction should be used at all.

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Table 8: Either At or Bt relationship.

Conditioned Bt YES NO Condition At YES 0 1 1 NO 1 0 1 1 12

2.2.29 The Chi-square goodness of fit test of an either or condition relationship

Definition 29 (The χ˜ 2 goodness of fit test of an either or condition relationship).

An either or condition relationship p(At >-< Bt) can be tested by the chi-square distribution (also chi-squared orχ ˜ 2-distribution) too. Theχ ˜ 2 goodness of fit test of an either or condition relationship with degree of freedom (d. f.) of d. f. = 1 is calculated as

(b (a + b)) 2 χ˜ 2 ((A > < B ) A) − + Calculated t − t | ⌘ A c ((c + d)) 2 − A a2 d2 + (61) ⌘ A A

or equally as

(c (a + c)) 2 χ˜ 2 ((A > < B ) B) − + Calculated t − t | ⌘ B b ((b + d)) 2 − B a2 d2 + (62) ⌘ B B

Yate’s [36] continuity correction has not been used in this context.

2.2.30 The left-tailed p Value of an either or condition relationship

Definition 30 (The left-tailed p Value of an either or condition relationship). The left-tailed (lt) p Value [37] of an either or condition relationship can be calculated as follows.

1 p A > < B ) 1 e−( − ( t − t)) lt t − t ⌘ − ((a+d)/N) 1 e− (63) ⌘ − In this context, a low p-value indicates again a statistical significance.

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2.2.31 Causal relationship k Table 8: Either At or Bt relationship. Definition 31 (Causal relationship k). Conditioned B t Nonetheless, mathematically, the causal relationship [1–5] between a cause U (German: Ursache) YES NO t and an e↵ect W (German: Wirkung), denoted by k(U ,W), is defined at each single Bernoulli trial t Condition A YES 0 1 1 t t t t in terms of statistics and probability theory as NO 1 0 1 1 12 σ (Ut, Wt) k (U , W ) t t ⌘σ (U ) σ (W ) t ⇥ t p (U W ) p (U ) p (W ) t ^ t − t ⇥ t (64) ⌘ 2 (p (U ) (1 p (U ))) (p (W ) (1 p (W ))) t ⇥ − t ⇥ t ⇥ − t 2.2.29 The Chi-square goodness of fit test of an either or condition relationship p 2 Definition 29 (The χ˜ goodness of fit test of an either or condition relationship). where σ (Ut ,Wt) denotes the co-variance between a cause Ut and an e↵ect Wt at every single Bernoulli trial t, σ (Ut) denotes the standard deviation of a cause Ut at the same single Bernoulli An either or condition relationship p(At >-< Bt) can be tested by the chi-square distribution (also trial t, σ (W ) denotes the standard deviation of an e↵ect W at same single Bernoulli trial t. Table 9 2 2 t t chi-squared orχ ˜ -distribution) too. Theχ ˜ goodness of fit test of an either or condition relationship illustrates the theoretically possible relationships between a cause and an e↵ect. with degree of freedom (d. f.) of d. f. = 1 is calculated as

Table 9: Sample space and the causal relationship k (b (a + b)) 2 χ˜ 2 ((A > < B ) A) − + Calculated t − t | ⌘ A c ((c + d)) 2 E↵ect Bt − A TRUE FALSE Cause TRUE p(a ) p(b ) p(U ) a2 d2 t t t + (61) At FALSE p(ct) p(dt) p(Ut) ⌘ A A p(Wt) p(Wt) +1 or equally as

(c (a + c)) 2 χ˜ 2 ((A > < B ) B) − + Calculated t − t | ⌘ B 2.3 Axioms b ((b + d)) 2 − 2.2.1 Axiom I. Lex identitatis B In this context, we define[7] axiom I as a2 d2 + (62) + 1 =+1 (65) ⌘ B B

Yate’s [36] continuity correction has not been used in this context. 2.2.2 Axiom II. Lex contradictionis In this context[7], axiom II or lex contradictionis, the negative of lex identitatis, or 2.2.30 The left-tailed p Value of an either or condition relationship + 0 =+1 (66) Definition 30 (The left-tailed p Value of an either or condition relationship). is of no minor importance too. The left-tailed (lt) p Value [37] of an either or condition relationship can be calculated as follows. 2.2.3 Axiom III. Lex negationis 1 p A > < B ) 1 e−( − ( t − t)) lt t − t ⌘ − ((a+d)/N) (0) 0 = 1 (67) 1 e− (63) ¬ ⇥ ⌘ − where denotes (logical [18] or natural) negation[7]. In this context, there is some evidence that ¬ In this context, a low p-value indicates again a statistical significance. (1) 1 = 0. In other words, it is ( (1) 1) ( (0) 0) = 1. ¬ ⇥ ¬ ⇥ ⇥ ¬ ⇥

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3 Results

3.1 Anti Chebyshev - The exact probability of a single event

Theorem 1 (Anti Chebyshev - The exact probability of a single event). The Pafnuty Lvovich Cheby- shev’s (1821 – 1894) inequality (also called the Irénée-Jules Bienaymé [40] (1796 – 1878) – Cheby- shev inequality) enables us to obtain bounds on the probability of an event when both the mean and variance of a random variable are known. According to Kolmogoro↵[see 31, p. 42], Chebyshev’s inequality can be changed to

σ 2 2 2 (RXt) p RXt E (RXt) E RXt 2 (68) | − |≥  E RXt q ! � � However, it is necessary to emphasize that Chebyshev’s inequality� as proved� by Chebyshev [41] him- self in 1867 and later by his student Andrey Markov (1856–1922) in his 1884 Ph.D. thesis provides in this form only an approximate value of the exact probability of a single event. Thus far and in contrast to Chebyshev’s inequality [see 31, p. 42], the exact value of the probability of a single event (Chebyshev’s equality) is given by the relationship

2 σ(RXt) p (RXt) 1 2 (69) ⌘ − E RXt Proof by modus ponens. If the premise � � +1 =+1 (70) (Premise) is true, then the conclusion | {z } 2 σ (RXt) p (RXt) 1 2 (71) ⌘ − E RXt is also true, the absence of any technical errors presupposed.� � The premise + 1 +1 (72) ⌘ 2 is true. Multiplying this premise (i. e. axiom) by the variance σ(RXt)

σ ( X ) 2 σ ( X ) 2 (73) R t ⌘ R t Equation 73 can be rearranged (see definition 7, equation 16) as

E X 2 (E ( X )) 2 σ ( X ) 2 (74) R t − R t ⌘ R t or as ⇣ ⌘ E X 2 (E ( X )) 2 + σ ( X ) 2 (75) R t ⌘ R t R t The normalised form of the variance⇣ follows⌘ as (E ( X )) 2 σ ( X ) 2 R t + R t + 2 2 1 (76) E RXt E RXt ⌘ Rearranging equation 76, it is � � � � 2 2 (E (RXt)) σ (RXt) 2 1 2 (77) E RXt ⌘ − E RXt � � � �

20 MATEC Web of Conferences 336, 09032 (2021) https://doi.org/10.1051/matecconf/202133609032 CSCNS2020

3 Results Equation 77 simplifies (see definition 3, equation 3) as

3.1 Anti Chebyshev - The exact probability of a single event σ ( X ) 2 p ( X ) 1 R t (78) R t ⌘ − E X 2 Theorem 1 (Anti Chebyshev - The exact probability of a single event). The Pafnuty Lvovich Cheby- R t shev’s (1821 – 1894) inequality (also called the Irénée-Jules Bienaymé [40] (1796 – 1878) – Cheby- � � Quod erat demonstrandum. shev inequality) enables us to obtain bounds on the probability of an event when both the mean and variance of a random variable are known. According to Kolmogoro↵[see 31, p. 42], Chebyshev’s inequality can be changed to Remark 4. Example. Let RXt = At Bt . The exact probability of this relationship follows according _ 2 σ At Bt 2 to theorem 1, equation 78 as 1 � �_ The exact probability[7] of other events, including 2 σ(RXt) 2 p X E ( X ) E X 2 − E At Bt R t R t R t 2 (68) � _ � | − |≥  E RXt the numerous relationships developed in this publication can be calculated by the same method too. q ! � � �� � � However, it is necessary to emphasize that Chebyshev’s inequality� as proved� by Chebyshev [41] him- self in 1867 and later by his student Andrey Markov (1856–1922) in his 1884 Ph.D. thesis provides 3.2 The causal relationship k in this form only an approximate value of the exact probability of a single event. Thus far and in contrast to Chebyshev’s inequality [see 31, p. 42], the exact value of the probability of a single event Theorem 2 (Causal relationship k). Thus far, let p(RUt) represent the probability from the point of (Chebyshev’s equality) is given by the relationship view of a stationary observer R of a certain cause RUt (in German: U like Ursache), i. e. a random variable or a quantum mechanical observable or a cluster inside a set, at a certain Bernoulli trial t. 2 2 2 σ(RXt) Let E(RUt ) denote the expectation value of the cause RUt . Let E(RUt) denote the expectation value p (RXt) 1 2 (69) 2 ⌘ − E RXt of the cause RUt. Let σ(RUt) denote the standard deviation of the cause RUt. Let σ(RUt) denote the variance of the cause RCt. Let p(RWt) represent the probability from the point of view of a stationary Proof by modus ponens. If the premise � � observer R of its own e↵ect RWt (in German: W like Wirkung), i. e. a random variable or a quantum + =+ 2 1 1 (70) mechanical observable or a cluster inside a set, at a certain Bernoulli trial t. Let E(RWt ) denote the 2 (Premise) expectation value of the e↵ect RWt . Let E(RWt) denote the expectation value of the e↵ect RWt. Let ↵ 2 ↵ is true, then the conclusion | {z } σ(RWt) denote the standard deviation of the e ect RWt. Let σ(RWt) denote the variance of the e ect 2 σ ( X ) RWt. Let σ(RUt, RWt) denote the co-variance of cause RUt and e↵ect RWt. The causal relationship, p ( X ) 1 R t (71) R t 2 , ⌘ − E RXt denoted as k (RUt RWt), inside a sets can be calculated as is also true, the absence of any technical errors presupposed.� � The premise σ ( U , W ) , R t R t + 1 +1 (72) k (RUt RWt) 2 ⌘ ⌘ σ ( U ) 2 σ ( W ) 2 R t ⇥ R t 2 σ ( U , W ) is true. Multiplying this premise (i. e. axiom) by the variance σ(RXt) p R t R t ⌘ σ (RUt) σ (RWt) σ ( X ) 2 σ ( X ) 2 (73) ⇥ R t ⌘ R t (RUt RWt) (p (RUt, RWt) (p (RUt) p (RWt))) 2 ⇥ ⇥ − ⇥ ⌘ U 2 (p ( U ) (1 p ( U ))) W 2 (p ( W ) (1 p ( W ))) Equation 73 can be rearranged (see definition 7, equation 16) as R t ⇥ R t ⇥ − R t ⇥ R t ⇥ R t ⇥ − R t , p (RUt RWt) (p (RUt RWt) (p (RUt) p (RWt))) 2 2 2 �� � ⇥2 ⇥ −� � ⇥ � E RXt (E (RXt)) σ (RXt) (74) ⌘ ( U W ) ((p ( U ) (1 p ( U ))) (p ( W ) (1 p ( W )))) − ⌘ R t ⇥ R t ⇥ R t ⇥ − R t ⇥ R t ⇥ − R t ⇣ ⌘ (p ( U W ) (p ( U ) p ( W ))) or as R pt ^ R t − R t ⇥ R t (79) 2 2 2 2 E X (E ( X )) + σ ( X ) (75) ⌘ ((p (RUt) (1 p (RUt))) (p (RWt) (1 p (RWt)))) R t ⌘ R t R t ⇥ − ⇥ ⇥ − The normalised form of the variance⇣ follows⌘ as Proof by modus ponens.p If the premise (E ( X )) 2 σ ( X ) 2 +1 =+1 (80) R t + R t + 2 2 1 (76) (Premise) E RXt E RXt ⌘ is true, then the conclusion | {z } Rearranging equation 76, it is � � � � 2 2 (E (RXt)) σ (RXt) σ ( U , W ) 1 (77) k ( U , W ) R t R t (81) E X 2 ⌘ − E X 2 R t R t ⌘ σ( U ) σ( W ) R t R t R t ⇥ R t � � � �

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is also true, the absence of any technical errors presupposed. The premise or respectively axiom I

+ 1 +1 (82) ⌘

is true. Multiplying this premise (i. e. axiom I) by the a single cause (RUt) at a certain Bernoulli trial t, it is

( U ) ( U ) (83) R t ⌘ R t

Multiplying the cause (equation 83) by its own e↵ect (RWt) it is

( U W ) ( U W ) (84) R t ⇥ R t ⌘ R t ⇥ R t This basic relationship between a cause and an e↵ect as described by equation 84 may be of use especially under conditions of classical logic (either +0 or +1 values) and is illustrated in detail by table 10.

Table 10: Causal relationship under conditions of classical logic

Bernoulli trial t U W ( U W ) ( U W ) R tRt R t ^ R t ⌘ R t ⇥ R t

1 +1 +1 +1 2 +1 +0 +0 3 +0 +1 +0 4 +0 +0 +0 ......

It should be noted at this early stage that we will not further discuss at this point the relationship 11 11 12 or the relationship 01 01 02 [9]. As a side e↵ect, table 10 provide an evidence of the ⇥ ⌘ ⇥ ⌘ identity of the logical operation, denoted by , and the algebraic operation multiplication. However, ^ as soon as RUt or RWt take on other values than either +0 or +1, the previous relationship (equation 84) is not appropriate enough to describe the causal relationship completely and may lead to a cum hoc ergo propter hoc logical fallacy [42]. In general, it is necessary to deal with such circumstances too. According to equation 17 it is

σ(RUt) RUt (85) ⌘ 2 p ( U ) (1 p ( U )) R t ⇥ − R t with the consequence that equation 84 changesp to

σ(RUt) (RUt RWt) RWt (86) ⇥ ⌘ 2 p ( U ) (1 p ( U )) ⇥ 0 R t ⇥ − R t 1 B C According to equation 18 it is @B p AC

σ(RWt) RWt (87) ⌘ 2 p ( W ) (1 p ( W )) R t ⇥ − R t p

22 MATEC Web of Conferences 336, 09032 (2021) https://doi.org/10.1051/matecconf/202133609032 CSCNS2020 is also true, the absence of any technical errors presupposed. The premise or respectively axiom I Therefore, equation 86 changes to

+ 1 +1 (82) σ(RUt) σ(RWt) ⌘ (RUt RWt) (88) ⇥ ⌘ 2 ⇥ 2 0 p (RUt) (1 p (RUt))1 0 p (RWt) (1 p (RWt))1 is true. Multiplying this premise (i. e. axiom I) by the a single cause (RUt) at a certain Bernoulli trial ⇥ − ⇥ − B C B C t, it is According to definition 8, equation@B p 21, it is AC @B p AC

(RUt) (RUt) (83) σ (RUt , RWt ) ⌘ RUt RWt (89) ⇥ ⌘ (p (RUt , RWt ) (p (RUt ) p (RWt ))) Multiplying the cause (equation 83) by its own e↵ect (RWt) it is − ⇥ Simplifying equation 88, it is ( U W ) ( U W ) (84) R t ⇥ R t ⌘ R t ⇥ R t σ (RUt , RWt ) ↵ This basic relationship between a cause and an e ect as described by equation 84 may be of use (p (RUt , RWt ) (p (RUt ) p (RWt ))) ⌘ especially under conditions of classical logic (either +0 or +1 values) and is illustrated in detail by − ⇥ ! σ( U ) table 10. R t 2 p ( U ) (1 p ( U )) 0 R t ⇥ − R t 1 B C B C σ(RWt) Table 10: Causal relationship under conditions of classical logic @ p A (90) ⇥ 2 0 p (RWt) (1 p (RWt))1 B ⇥ − C B C Bernoulli trial t U W ( U W ) ( U W ) Further rearrangement of equation 90 yields the causal relationship@ p between the cause RUt Aand the R tRt R t ^ R t ⌘ R t ⇥ R t e↵ect RWt, denoted as k (RUt , RWt ), as 1 +1 +1 +1 σ (RUt , RWt ) 2 +1 +0 +0 k (RUt , RWt ) ⌘ σ(RUt) σ(RWt) 3 +0 +1 +0 ⇥ (p (RUt , RWt ) (p (RUt ) p (RWt ))) 4 +0 +0 +0 − ⇥ (91) ⌘ 2 p ( U ) (1 p ( U )) p ( W ) (1 p ( W )) ...... R t ⇥ − R t ⇥ R t ⇥ − R t p Quod erat demonstrandum. It should be noted at this early stage that we will not further discuss at this point the relationship 11 11 12 or the relationship 01 01 02 [9]. As a side e↵ect, table 10 provide an evidence of the ⇥ ⌘ ⇥ ⌘ identity of the logical operation, denoted by , and the algebraic operation multiplication. However, 3.3 The law of nature relationship g ^ as soon as U or W take on other values than either +0 or +1, the previous relationship (equation R t R t Theorem 3 (The law of nature relationship g). In this theorem, we are specifying the probability 84) is not appropriate enough to describe the causal relationship completely and may lead to a cum measure on the sample space of an experiment as being equal to p = 1. In other words, it is for hoc ergo propter hoc logical fallacy [42]. In general, it is necessary to deal with such circumstances sure that an event occurred. Thus far, let the sample space Y denote the set of all possible out- too. According to equation 17 it is t comes of an experiment at a certain Bernoulli trial t. Let yt denote a random variable, a real-valued σ(RUt) function defined on a single element of the sample space Yt at a Bernoulli trial t. In general, it is RUt (85) 2 Yt y1t,y2t,...,ynt, . Let E(yt) denote the expectation value of yt. Let the sample space Xt denote ⌘ p (RUt) (1 p (RUt)) ⌘ { } ⇥ − the set of all possible outcomes of X at a certain Bernoulli trial t. Let xt denote a random variable, with the consequence that equation 84 changesp to a real-valued function defined on a single element of the sample space Xt at a Bernoulli trial t. In general, it is X x , x ,...,x , . Let E(y ) denote the expectation value of y . Let f(x ) denote t ⌘ { 1t 2t nt } t t t σ(RUt) a mathematical function which describes the behaviour of each element of a set Xt, let E(f( xt)) de- (RUt RWt) RWt (86) ⇥ ⌘ 2 ⇥ note the expectation value of f(xt). The law of nature relationship is based on a quantity dominated, 0 p (RUt) (1 p (RUt))1 ⇥ − mechanical understanding of the relationship between two factors like y and f( x ). Let g(y , f(x x )) B C t t t t According to equation 18 it is @B p AC denote the law of nature relationship, ‘der gesetzmäßige Zusammenhang’. The law of nature relation- ship is defined as σ W (R t) σ (yt, f (xt)) RWt (87) 2 g (yt, f (xt)) (92) ⌘ p (RWt) (1 p (RWt)) ⌘ σ (y ) σ ( f (x )) ⇥ − t ⇥ t p

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Proof by modus ponens. If the premise +1 =+1 (93) (Premise) is true, then the conclusion | {z }

E ((y E (y )) ( f (x ) E ( f (x ) ))) g (y , f (x )) t − t ⇥ t − t t t ⌘ E (y E (y )) E ( f (x ) E ( f (x ) )) t − t ⇥ t − t σ (y , f (x )) t t (94) ⌘ σ (y ) σ ( f (x )) t ⇥ t is also true, the absence of any technical errors presupposed. The premise

+ 1 +1 (95) ⌘

is true. Multiplying this premise (i. e. axiom) by yt, it is

y y (96) t ⌘ t

The law of nature relationship g(yt, f(xt)) is based on the demand that an outcome, denoted as yt is determined exactly by f(xt) at every run of an experiment, at every Bernoulli trial t. In other words, it is yt = f (xt). Based on this fundamental assumption, equation 94 can be rearranged as

y f (x ) (97) t ⌘ t Equation 97 leads to E (y ) E ( f (x )) (98) t ⌘ t Equation 97 demands too that y 2 f (x ) 2 (99) t ⌘ t Equation 99 demands that E y 2 E f (x ) 2 (100) t ⌘ t Equation 97 can be rearranged as ⇣ ⌘ ⇣ ⌘

y E (y ) f (x ) E (y ) (101) t − t ⌘ t − t According to equation 98, equation 101 changes to

y E (y ) f (x ) E ( f (x )) (102) t − t ⌘ t − t In other words, we must accept the equality of

E (y E (y )) E ( f (x ) E ( f (x ) )) (103) t − t ⌘ t − t By squaring equation 103, it is

E (y E (y )) 2 E ( f (x ) E ( f (x ) )) 2 (104) t − t ⌘ t − t or E (y E (y )) 2 E ( f (x ) E ( f (x ) )) E ( f (x ) E ( f (x ) )) (105) t − t ⌘ t − t ⇥ t − t

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Proof by modus ponens. If the premise or +1 =+1 (93) E (y E (y )) E (y E (y )) E ( f (x ) E ( f (x ) )) E ( f (x ) E ( f (x ) )) (106) t − t ⇥ t − t ⌘ t − t ⇥ t − t (Premise) Based on equation 103, equation 106 can be rearranged as is true, then the conclusion | {z } E (yt E (yt )) E ( f (xt) E ( f (xt) )) E (( f (xt) E ( f (xt) )) ( f (xt) E ( f (xt) ))) (107) E ((y E (y )) ( f (x ) E ( f (x ) ))) − ⇥ − ⌘ − ⇥ − g (y , f (x )) t − t ⇥ t − t t t ⌘ E (y E (y )) E ( f (x ) E ( f (x ) )) Based on equation 97 and equation 98, equation 107 can be rearranged as t − t ⇥ t − t σ (yt, f (xt)) (94) E (yt E (yt )) E ( f (xt) E ( f (xt) )) E ((yt E (yt )) ( f (xt) E ( f (xt) ))) (108) ⌘ σ (y ) σ ( f (x )) − ⇥ − ⌘ − ⇥ − t ⇥ t Rearranging equation 108, the law of nature relationship g(yt, f(x xt)) follows as is also true, the absence of any technical errors presupposed. The premise E ((y E (y )) ( f (x ) E ( f (x ) ))) g (y , f (x )) t − t ⇥ t − t + 1 +1 (95) t t ⌘ E (y E (y )) E ( f (x ) E ( f (x ) )) ⌘ t − t ⇥ t − t σ (yt, f (xt)) is true. Multiplying this premise (i. e. axiom) by yt, it is (109) ⌘ σ (y ) σ ( f (x )) t ⇥ t y y (96) t ⌘ t Quod erat demonstrandum.

The law of nature relationship g(yt, f(xt)) is based on the demand that an outcome, denoted as yt is Moving away from confusing and logically inconsistent theories of causation seems in- determined exactly by f(xt) at every run of an experiment, at every Bernoulli trial t. In other words, it Remark 5. evitable due to proofs provided in this publication. Especially structural equation modelling (SEM) or is yt = f (xt). Based on this fundamental assumption, equation 94 can be rearranged as counterfactual claims et cetera failed to provide a coherent mathematical foundation for the analysis yt f (xt) (97) of cause and e↵ect relationships. In the 1920’s by Sewall Wright [see 43, p. 557] was one of the ⌘ first to derive a kind of a structural equation modelling from the coefficient of correlation. Wright Equation 97 leads to points out: “The present paper is an attempt to present a method of measuring the direct influence E (y ) E ( f (x )) (98) along each separate path in such a system and thus of finding the degree to which variation of a given t ⌘ t e↵ect is determined by each particular cause. The method depends on the combination of knowledge Equation 97 demands too that of the degrees of correlation among the variables in a system with such knowledge as may be pos- y 2 f (x ) 2 (99) t ⌘ t sessed of the causal relations ”[see 43, p. 557]. Wright himself is writing that “The method depends Equation 99 demands that on the ... correlation among the variables ... ”[see 43, p. 557]. The law of nature relationship 2 2 g (y , f (x )), in contrast to Pearl’s do(X=x) operator [see 44, p. 204], provide a logically consistent E yt E f (xt) (100) t t ⌘ mathematical alternative to the structure equation modelling proposed analysis of potential causal Equation 97 can be rearranged as ⇣ ⌘ ⇣ ⌘ dependencies between exogenous and endogenous variables. Even if multiplied by a sample size N, the law of nature relationship need not to change. We obtain yt E (yt ) f (xt) E (yt ) (101) − ⌘ − N N σ (y , f (x )) g (y , f (x )) ⇥ ⇥ t t (110) According to equation 98, equation 101 changes to t t ⌘ N σ (y ) N σ ( f (x )) ⇥ t ⇥ ⇥ t yt E (yt ) f (xt) E ( f (xt)) (102) − ⌘ − 4 Discussion In other words, we must accept the equality of In general, combining classical logic and probability theory into one single mathematical framework might appear somewhat difficult and strange at first sight. It is particularly noteworthy that classical E (yt E (yt )) E ( f (xt) E ( f (xt) )) (103) − ⌘ − logic as such is concerned more or less with absolutely certain truths and inferences. In contrast to By squaring equation 103, it is classical logic, probability theory deals primarily with uncertainties. In particular, these and similar difficulties should not prevent us form the possibility to recognise the relationships between events of 2 2 E (yt E (yt )) E ( f (xt) E ( f (xt) )) (104) objective reality as the same are while relying on logically consistent methods. In this context, espe- − ⌘ − cially causation is a live topic across a number of scientific disciplines. Thus far, what is the causal or relationship k? The causal relationship k (see theorem 2)[1–5] enable us to evaluate cause-e↵ect E (y E (y )) 2 E ( f (x ) E ( f (x ) )) E ( f (x ) E ( f (x ) )) (105) relationships including causal chains[see 1, p. 139-160] hypotheses in the light of empirical facts. t − t ⌘ t − t ⇥ t − t

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However, the causal relationships k should not be confused neither with Bravais [45] (1811-1863) - Pearson’s coecient of correlation [46, 47] nor with Pearson’s phi coecient [32]. Karl Pearson (1857-1936) himself “rejected causal thinking”[see 48, p. 39] as such. Following Pearson, causa- tion is without any scientific significance, Pearson is demanding unconditionally that “. . . correlation . . . have to replace . . . causation ”[see 49, p. 157]. Pearson is an advocate of anti-causality in the ex- treme and much more than that. “Pearson categorically denies the need for an independent concept of causal relation beyond correlation ... he exterminated causation from statistics before it had a chance to take root ”[see 44, p. 340]. However, it is appropriate to consider that every single event RWt has its own cause RUt. Therefore and completely in contrast to Pearson’s demonstrably anti-causal statistical methods[32, 47], the causal relationship k is defined, derived and valid at every single run of an experiment, at every single Bernoulli trial t.

5 Conclusion

Experimental and non-experimental data can be analysed for di↵erent conditions, causal relationships and for laws of nature relationships.

References

[1] I. Barukciˇ c,´ Die Kausalität, 1st edn. (Wiss.-Verl., Hamburg, 1989), ISBN 3-9802216-0-1 [2] I. Barukciˇ c,´ Die Kausalität, 2nd edn. (Scientia, Wilhelmshaven, 1997), ISBN 3-9802216-4-4 [3] I. Barukciˇ c,´ Causality: New statistical methods (Books on Demand GmbH, Norderstedt, Ger- many, 2005), ISBN 978-3-8334-3645-1 [4] I. Barukciˇ c,´ The Mathematical Formula of the Causal Relationship k, International Journal of Applied Physics and Mathematics 6, 45 (2016) [5] I. Barukciˇ c,´ Theoriae causalitatis principia mathematica (Books on Demand, Norderstedt, 2017), ISBN 978-3-7448-1593-2 [6] I. Barukciˇ c,´ Die Kausalität, reprint of first edition 1989. edn. (Books on Demand, Norderstedt, 2017), ISBN 978-3-7448-1595-6 [7] I. Barukciˇ c,´ Causal relationship k, International Journal of Mathematics Trends and Technology IJMTT 66, 76 (2020) [8] I. Barukciˇ c,´ O. Ufuoma, Analysis of Switching Resistive Circuits. A Method Based on the Uni- fication of Boolean and Ordinary Algebras, first edition edn. (Books on Demand, Norderstedt, 2020), ISBN 978-3-7519-8474-4 [9] I. Barukciˇ c,´ Zero and infinity. Mathematics without frontiers, 1st edn. (Books on Demand GmbH, Hamburg-Norderstedt, 2020), ISBN 978-3-7519-1873-2 [10] I. Barukciˇ c,´ Locality and Non locality, European Journal of Applied Physics 2 (2020) [11] I. Barukciˇ c,´ N-th index D-dimensional Einstein gravitational field equations. Geometry un- chained., Vol. 1, 1st edn. (Books on Demand GmbH, Hamburg-Norderstedt, 2020), ISBN 978- 3-7526-4490-6 [12] W. Heisenberg, Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Zeitschrift für Physik 43, 172 (1927) [13] I. Barukciˇ c,´ The Equivalence of Time and Gravitational Field, Physics Procedia 22, 56 (2011) [14] I. Barukciˇ c,´ Anti Heisenberg-Refutation Of Heisenberg’s Uncertainty Relation, in American In- stitute of Physics - Conference Proceedings (Växjö, (Sweden), 2011), Vol. 1327, pp. 322–325, https://aip.scitation.org/doi/abs/10.1063/1.3567453

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However, the causal relationships k should not be confused neither with Bravais [45] (1811-1863) - [15] I. Barukciˇ c,´ Anti Heisenberg – Refutation of Heisenberg’s Uncertainty Principle, International Pearson’s coecient of correlation [46, 47] nor with Pearson’s phi coecient [32]. Karl Pearson Journal of Applied Physics and Mathematics 4, 244 (2014) (1857-1936) himself “rejected causal thinking”[see 48, p. 39] as such. Following Pearson, causa- [16] I. Barukciˇ c,´ Anti Heisenberg—The End of Heisenberg’s Uncertainty Principle, Journal of Ap- tion is without any scientific significance, Pearson is demanding unconditionally that “. . . correlation plied Mathematics and Physics 04, 881 (2016) . . . have to replace . . . causation ”[see 49, p. 157]. Pearson is an advocate of anti-causality in the ex- [17] G. Boole, The Mathematical Analysis of Logic – Being an Essay Towards a Cal- treme and much more than that. “Pearson categorically denies the need for an independent concept of culus of Deductive Reasoning (Macmillan, Cambrdge, 1847), https://archive. causal relation beyond correlation ... he exterminated causation from statistics before it had a chance org/details/mathematicalanal00booluoft,https://archive.org/details/ to take root ”[see 44, p. 340]. However, it is appropriate to consider that every single event RWt mathematicalanal00bool has its own cause RUt. Therefore and completely in contrast to Pearson’s demonstrably anti-causal [18] G. Boole, An investigation of the laws of thought, on which are founded mathematical theories of statistical methods[32, 47], the causal relationship k is defined, derived and valid at every single logic and probabilities (New York, Dover, 1854), http://archive.org/details/bub_gb_ run of an experiment, at every single Bernoulli trial t. DqwAAAAAcAAJ [19] G. Kröber, Der Konditionalismus und seine Kritik in der sowjetischen Wissenschaft, Wis- 5 Conclusion senschaftliche Zeitschrift der Karl-Marx Universität Leipzig 10, 137 (1961) [20] D.P.v. Hansemann, Über das konditionale Denken in der Medizin und seine Bedeutung fur die Experimental and non-experimental data can be analysed for di↵erent conditions, causal relationships Praxis (Hirschwald, Berlin., 1912) and for laws of nature relationships. [21] M. Verworn, Kausale und konditionale Weltanschauung (Verlag von Gustav Fischer, Jena, 1912) [22] G.W.F.v. Leibniz, Explication de l’arithmétique binaire, qui se sert des seuls caractères O et I avec des remarques sur son utilité et sur ce qu’elle donne le sens des anciennes figures chinoises References de Fohy, Mémoires de mathématique et de physique de l’Académie royale des sciences (1703) [1] I. Barukciˇ c,´ Die Kausalität, 1st edn. (Wiss.-Verl., Hamburg, 1989), ISBN 3-9802216-0-1 [23] I. Barukciˇ c,´ The Interior Logic of Inequalities, International Journal of Mathematics Trends and [2] I. Barukciˇ c,´ Die Kausalität, 2nd edn. (Scientia, Wilhelmshaven, 1997), ISBN 3-9802216-4-4 Technology IJMTT 65, 146 (2019) [3] I. Barukciˇ c,´ Causality: New statistical methods (Books on Demand GmbH, Norderstedt, Ger- [24] I. Barukciˇ c,´ Classical Logic And The Division By Zero, International Journal of Mathematics many, 2005), ISBN 978-3-8334-3645-1 Trends and Technology IJMTT 65, 31 (2019) [4] I. Barukciˇ c,´ The Mathematical Formula of the Causal Relationship k, International Journal of [25] M. Born, Zur Quantenmechanik der Stoßvorgänge, Zeitschrift für Physik 37, 863 (1926) Applied Physics and Mathematics 6, 45 (2016) [26] J.v. Uspensky, Introduction To Mathematical Probability (McGraw-Hill Company, New York [5] I. Barukciˇ c,´ Theoriae causalitatis principia mathematica (Books on Demand, Norderstedt, (USA), 1937) 2017), ISBN 978-3-7448-1593-2 [27] C. Huygens, F. van Schooten, De ratiociniis in ludo alae: In: Exercitationum mathematicarum [6] I. Barukciˇ c,´ Die Kausalität, reprint of first edition 1989. edn. (Books on Demand, Norderstedt, liber primus [- quintus] (ex officina Johannis Elsevirii, Lugdunum Batavorum (Leiden, The 2017), ISBN 978-3-7448-1595-6 Netherlands), 1657), https://www.e-rara.ch/zut/doi/10.3931/e-rara-8813 [7] I. Barukciˇ c,´ Causal relationship k, International Journal of Mathematics Trends and Technology [28] W.A. Whitworth, Choice and Chance. With 1000 Exercises, fifth eidtion edn. (Deighton, Bell & https://archive.org/details/choicechancewith00whituoft IJMTT 66, 76 (2020) Co., Cambridge, 1901), [8] I. Barukciˇ c,´ O. Ufuoma, Analysis of Switching Resistive Circuits. A Method Based on the Uni- [29] B. Efron, R. A. Fisher in the 21st century (Invited paper presented at the 1996 R. A. Fisher fication of Boolean and Ordinary Algebras, first edition edn. (Books on Demand, Norderstedt, Lecture), Statistical Science 13, 95 (1998), publisher: Institute of Mathematical Statistics 2020), ISBN 978-3-7519-8474-4 [30] R.A. Fisher, XV.—The Correlation between Relatives on the Supposition of Mendelian Inheri- [9] I. Barukciˇ c,´ Zero and infinity. Mathematics without frontiers, 1st edn. (Books on Demand GmbH, tance., Transactions of The Royal Society of Edinburgh 52, 399 (1918), publisher: Royal Society Hamburg-Norderstedt, 2020), ISBN 978-3-7519-1873-2 of Edinburgh Scotland Foundation [10] I. Barukciˇ c,´ Locality and Non locality, European Journal of Applied Physics 2 (2020) [31] A.N. Kolmogorov, Foundations of the theory of probability, second english edition. translated by nathan morrison edn. (Chelsea Publishing Company, New York, 1956), ISBN 978-0-486-82159- [11] I. Barukciˇ c,´ N-th index D-dimensional Einstein gravitational field equations. Geometry un- 7 chained., Vol. 1, 1st edn. (Books on Demand GmbH, Hamburg-Norderstedt, 2020), ISBN 978- 3-7526-4490-6 [32] K. Pearson, Mathematical contributions to the theory of evolution. XIII. On the theory of contin- gency and its relation to association and normal correlation, Biometric Series I (Dulau and Co., [12] W. Heisenberg, Über den anschaulichen Inhalt der quantentheoretischen Kinematik und London, 1904) Mechanik, Zeitschrift für Physik 43, 172 (1927) [33] J. Bernoulli, Ars conjectandi, Opus posthumus: Accedit Tractatus de seriebus infinitis ; et [13] I. Barukciˇ c,´ The Equivalence of Time and Gravitational Field, Physics Procedia 22, 56 (2011) epistola Gallice scripta De Ludo Pilae Reticularis (Impensis Thurnisiorum [Tournes], fratrum, [14] I. Barukciˇ c,´ Anti Heisenberg-Refutation Of Heisenberg’s Uncertainty Relation, in American In- Basileae (Basel, Suisse), 1713) stitute of Physics - Conference Proceedings (Växjö, (Sweden), 2011), Vol. 1327, pp. 322–325, https://aip.scitation.org/doi/abs/10.1063/1.3567453

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[34] A.d. Moivre, The Doctrine of Chances or a Method of Calculating the Probability of Events in Play (printed by W. Pearson for the author, London, 1718) [35] A. Kolmogoro↵, Grundbegri↵e der Wahrscheinlichkeitsrechnung (Springer Berlin Heidelberg, Berlin, Heidelberg, 1933), ISBN 978-3-642-49596-0 [36] F. Yates, Contingency Tables Involving Small Numbers and the Chi square Test, Supplement to the Journal of the Royal Statistical Society 1, 217 (1934) [37] I. Barukciˇ c,´ The P Value of likely extreme events, International Journal of Current Science Re- search 5, 1841 (2019) [38] F.R. Helmert, Ueber die Wahrscheinlichkeit der Potenzsummen der Beobachtungsfehler und über einige damit im Zusammenhange stehende Fragen, Zeitschrift für Mathematik und Physik 21, 102 (1876) [39] K. Pearson, X. On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 50, 157 (1900) [40] I.J. Bienaymé, Considérations a l’appui de la découverte de Laplace sur la loi de probabilité dans la méthode des moindres carrés, Comptes rendus des séances de l’Académie des Sciences des Paris 37, 309 (1853) [41] P.L. Tschébychef, Des valeurs moyennes, Journal de Mathématiques Pures et Appliquées 12, 177 (1867) [42] G.S. Kienle, Cum hoc, ergo propter hoc (2020) [43] S. Wright, Correlation and causation, Journal of Agricultural Research 20 (1921) [44] J. Pearl, Causality: models, reasoning, and inference (Cambridge University Press, Cambridge, U.K. ; New York, 2000), ISBN 978-0-521-89560-6 [45] A. Bravais, Analyse mathématique sur les probabilités d es erreurs de situation d’un point, Mémoires Présentées Par Divers Savants À L’Académie Royale Des Sciences De L’Institut De France 9, 255 (1846) [46] F. Galton, Typical Laws of Heredity, Nature 15, 492 (1877) [47] K. Pearson, VII. Mathematical contributions to the theory of evolution.—III. Regression, hered- ity, and panmixia, Philosophical Transactions of the Royal Society of London. Series A, Con- taining Papers of a Mathematical or Physical Character 187, 253 (1896) [48] H.M. Blalock, Causal inferences in nonexperimental research, 6th edn. (Univ. of North Carolina Press, Chapel Hill, NC, 1972), ISBN 978-0-8078-0917-4 [49] K. Pearson, The Grammar of Science (Adam and Charles Black, London (GB), 1911), ISBN 978-1-108-07711-8

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