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).