Identifying Noise Variables in Singular Decisions using Counterfactual Reasoning Meghna Bhadra1,† , Steffen Hölldobler1,2,† 1 Technische Universität Dresden, Faculty of Computer Science, Germany 2 North Caucasus Federal University, Stavropol, Russian Federation Abstract In their book, Noise: A Flaw in Human Judgment, the authors Daniel Kanheman, Olivier Sibony and Cass R. Sunstein highlight the importance of minimizing bias, i.e. systematic deviation, and noise, i.e. variability, in judgments in order to reduce error. Bias has long been the subject of many discussions but noise is yet to gain the attention it deserves. In this paper, we discuss noise variables in decision-making, particularly in unique, non-recurrent or singular decisions. For this purpose we introduce and utilize the framework of the Weak Completion Semantics to discuss how noise variables may be identified using counterfactual reasoning. Keywords Weak Completion Semantics, Noise, Singular Decisions, Counterfactual Reasoning 1. Noise in Singular Decisions Bias in decision-making is an idea familiar to many. Bias may be comprehensively defined as a systematic deviation to or from a certain decision or choice mainly owing to a person’s psychological mechanism. Bias may be considered as an average of error in judgment. Although biases may be favourable in certain contexts, they may be quite unfavourable in some. Deciding on job applicants based on their racial profile, for example, is a social bias that is better avoided. Another example is overconfidence, which is an instance of cognitive bias. There are plenty of articles, talks, and books which have been dedicated to critical discussions on different kinds of biases and how judgement suffers (or not) from it. A deviation from this norm, the book Noise: A Flaw in Human Judgment [1], authored by Daniel Kanheman, Olivier Sibony and Cass R. Sunstein considers another parameter which although often overlooked, plays an important role in errors - noise. According to the authors, ”to understand error in judgment, we must understand both bias and noise”. Noise, unlike bias, is variability in error. It is not a psychological phenomenon but more of a statistical one, which arises from variability or differences across individuals. Put simply, noise can be characterized by different people making different judgments given the same situation. Thus noise has the general property of being recognizable without knowing the intended outcome of a situation or the bias involved. Noise results in variable judgments which are not systematic. For example, the IJCAI-ECAI’22, Workshop on Cognitive Aspects of Knowledge Representation, 2022, July 23, Vienna, Austria † Authors are listed alphabetically. $ meghna.bhadra@tu-dresden.de (M. Bhadra); sh43@posteo.de (S. Hölldobler) © 2022 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). CEUR Workshop Proceedings http://ceur-ws.org ISSN 1613-0073 CEUR Workshop Proceedings (CEUR-WS.org) same person being interviewed by different interviewers may be rated and assessed differently. This indicates variability and noise. People seeking asylum in the United States are admitted on a system resembling a lottery [2]. This indicates variability and noise. Decisions to grant patents are again variable and thus noisy [3]. When speaking of noise in decision-making, one needs to recognize two kinds of decisions: recurrent and singular. Recurrent decisions are those which we take on a repetitive basis. For example, deciding on the grades or performance of students appearing for an examination. Singular decisions are on the other hand unique and not repeated. For example, deciding on marriage with a particular person, or deciding whether to design a certain kind of mobile application can be considered as singular decisions. Obama’s decision to send three thousand healthcare workers and soldiers to West Africa when the Ebola epidemic broke out in 2014 is yet another example of a historically prominent singular decision [1]. Obama had never encountered such a situation before and never had to take such a decision prior to that experience. Thus, he did not have a previously formulated and pre-packaged template of actions or choices he could draw from. Moreover, he did not have the scope to repeat this decision; it was a unique situation which was not recurrent. Recurrent and singular decisions both suffer from noise and variability. It is however not very easy to identify noise in singular decisions, given their nature. While identifying the presence of noise in the exemplary situation above, one may imagine the following, in lines with [1]: would Obama have taken the same decision had he been surrounded by a different set of advisors? Or if his advisors had a different mindset? What if the situation and its gravity had been presented differently to the president? If we can imagine different alternative outcomes on tweaking these different parameters, we can sense the noise in the system. These parameters are therefore the noise variables. The authors of [1] suggest that noise in singular decisions could be identified using counterfactual reasoning. However, this point was not very elaborated upon. Thus, it is the goal of this paper to discuss an example of a singular decision, and using it as a prototypical model illustrate how counterfactual reasoning may be employed to recognize noise variables in the system. We use the Weak Completion Semantics (WCS) framework for our purposes. The WCS is a formal three-valued, computational, non-monotonic cognitive theory. Till date it has been used to adequately model the suppression task by [4] as shown by [5], disjunctive reasoning as shown by [6], human syllogistic reasoning as shown by [7], and its belief bias as discussed in [8]. It has also been used to model the four inference tasks associated with reasoning about a conditional sentence, namely affirmation of the antecedent, affirmation of the consequent in [9], denial of the antecedent in [10] and denial of the consequent in [11]. Counterfactual reasoning, as the name suggest, is reasoning with imagined alternatives which are contrary to facts. That is, how people reflect on past events which have occurred, and imagine possibilities like, ”What if . . . had happened? Would the outcome have been any different? What if . . . had not happened?”. People can imagine alternatives to something that has happened, by either deleting or undoing some aspect of it which had happened in the past reality, or by adding some new aspects to their simulation of reality [12]. To use an example from the field of explainable AI, discussed in [12], suppose a human tries to understand why an autonomous vehicle swerved into a wall to avoid hitting a pedestrian (thus injuring its passenger) and not hit the brakes instead. One might wonder, ”if the car detected the pedestrian earlier and braked, the passenger would not have been injured” or ”if the car had not swerved towards the wall then the passenger would not have been injured”. The former is called an additive counterfactual as it adds or considers new information for the reasoning process, and the latter is called subtractive because it deletes facts for the reasoning process. Summing up, during decision-making or judgment it is advisable that both noise and bias within a system be addressed and minimized as much as possible. In order to reduce noise, one needs to identify it first. Our aim is to propose one such modelling method using the WCS to identify noise variables within a given system. The paper is thus organized as follows: In Section 2 we formally introduce the WCS. A classification of conditional sentences relevant to the goal of this paper is given in Section 3. An example of a singular decision is presented in Section 4. Identification of noise variables in this particular system using the WCS is demonstrated in Section 5. Finally, in Section 6 we conclude and outline further possible research. 2. The Weak Completion Semantics We assume the reader to be familiar with logic and logic programming as presented in e.g. [13] and [14]. Let ⊤, ⊥, and U be truth constants denoting true, false, and unknown, respectively. A (logic) program is a finite set of clauses of the form 𝐵 ← body, where 𝐵 is an atom (also called a head ) and body is ⊤, or ⊥, or a finite, non-empty conjunction of literals. Clauses of the form 𝐵 ← ⊤, 𝐵 ← ⊥, and 𝐵 ← 𝐿1 , . . . , 𝐿𝑛 are called facts, assumptions, and rules, respectively, where 𝐿𝑖 , 1 ≤ 𝑖 ≤ 𝑛, are literals. We restrict our attention to propositional programs although the WCS extends to first-order programs as well [15]. Throughout this paper, 𝒫 will denote a program. An atom 𝐵 is defined in 𝒫 if and only if 𝒫 contains a clause of the form 𝐵 ← body. As an example consider the program 𝒫 = {𝐶 ← 𝐴 ∧ ¬ab, ab ← ⊥}, where 𝐴, 𝐶, and ab are atoms. 𝐶 and ab are defined, whereas 𝐴 is undefined. ab is an abnormality predicate which is assumed to be false. In the WCS, this program represents the conditional sentence if 𝐴 then 𝐶. In their everyday lives humans are often required to reason in situations where the information of all factors affecting the situation might not be complete. They still reason, unless new information which needs consideration comes to light. The abnormality predicate in the program serves the purpose of this (default) assumption, as was suggested in [16]. Let 𝑆 be a consistent subset of literals occurring in P, i.e. 𝑆 does not contain both an atom and its negation. Then, defs(𝒫, 𝑆) = {𝐵 ← body ∈ 𝒫 | 𝐵 ∈ 𝑆 𝑜𝑟 ¬𝐵 ∈ 𝑆}. E.g. let 𝒫0 = {𝑎 ← ⊤, 𝑐 ← 𝑏, 𝑑 ← ⊥} and 𝑆 = {𝑎, ¬𝑐}. Then, defs(𝒫0 , 𝑆) = {𝑎 ← ⊤, 𝑐 ← 𝑏}. Let us now consider the following transformation: (1) For all defined atoms 𝐵 occurring in a program 𝒫, replace all clauses of the form 𝐵 ← body 1 , 𝐵 ← body 2 , . . . by 𝐵 ← body 1 ∨ body 2 ∨ . . . . (2) Replace all occurrences of ← by ↔. The resulting set of equivalences is called the weak completion of 𝒫, denoted by wc(𝒫). It differs from the program completion defined in [17] in that undefined atoms in the weakly completed program are not mapped to false, but to unknown instead. Reconsidering the previous program 𝒫0 , the reader may note that while the undefined atom 𝑏 is mapped to false under the completion of 𝒫0 , it is mapped to unknown under its weak completion. Weak completion is necessary for the WCS framework to adequately model the suppression task (and other reasoning tasks) as demonstrated in [5]. As shown in [18], each weakly completed program admits a least model under the three- Table 1 The truth tables for the Łukasiewicz logic. As shown in the gray cells, U ← U = U ↔ U = ⊤. 𝐹 ¬𝐹 ∧ ⊤ U ⊥ ∨ ⊤ U ⊥ ← ⊤ U ⊥ ↔ ⊤ U ⊥ ⊤ ⊥ ⊤ ⊤ U ⊥ ⊤ ⊤ ⊤ ⊤ ⊤ ⊤ ⊤ ⊤ ⊤ ⊤ U ⊥ ⊥ ⊤ U U U ⊥ U ⊤ U U U U ⊤ ⊤ U U ⊤ U U U ⊥ ⊥ ⊥ ⊥ ⊥ ⊤ U ⊥ ⊥ ⊥ U ⊤ ⊥ ⊥ U ⊤ valued Łukasiewicz logic [19] (see Table 1). This model will be denoted by ℳ𝑤𝑐(𝒫) . It can be computed as the least fixed point of a semantic operator introduced in [20]. Let 𝒫 be a program and 𝐼 be a three-valued interpretation represented by the pair ⟨𝐼 ⊤ , 𝐼 ⊥ ⟩, where 𝐼 ⊤ and 𝐼 ⊥ are the sets of atoms mapped to true and false by 𝐼, respectively, and atoms which are not listed are mapped to unknown. We define Φ𝒫 𝐼 = ⟨𝐽 ⊤ , 𝐽 ⊥ ⟩,1 where 𝐽 ⊤ = {𝐵 | there exists 𝐵 ← body ∈ 𝒫 and 𝐼 body = ⊤}, 𝐽 ⊥ = {𝐵 | there exists 𝐵 ← body ∈ 𝒫 and for all 𝐵 ← body ∈ 𝒫 one finds 𝐼 body = ⊥}. Following [21] we consider an abductive framework ⟨𝒫, 𝒜𝒫 , ℐ𝒞, |=𝑤𝑐𝑠 ⟩, where 𝒫 is a program, 𝒜𝒫 = {𝐵 ← ⊤ | 𝐵 is undefined in 𝒫} ∪ {𝐵 ← ⊥ | 𝐵 is undefined in 𝒫} is the set of abducibles, ℐ𝒞 is a finite set of integrity constraints of the form ⊥ ← body or U ← body,2 and ℳ𝑤𝑐(𝒫) |=𝑤𝑐𝑠 𝐹 iff ℳ𝑤𝑐(𝒫) maps the formula 𝐹 to true. Let 𝒪 be an observation, i.e., a finite set of literals each of which does not follow from ℳ𝑤𝑐(𝒫) . We apply abduction to explain 𝒪, where 𝒪 is called explainable in the abductive framework ⟨𝒫, 𝒜𝒫 , ℐ𝒞, |=𝑤𝑐𝑠 ⟩ if and only if there exists a non-empty 𝒳 ⊆ 𝒜𝒫 called an explanation such that ℳ𝑤𝑐(𝒫∪𝒳 ) |=𝑤𝑐𝑠 𝐿 for all 𝐿 ∈ 𝒪 and ℳ𝑤𝑐(𝒫∪𝒳 ) satisfies ℐ𝒞. We have assumed that the set of explanations is non-empty as otherwise the observation already follows from the weak completion of the program. Formula 𝐹 follows credulously from 𝒫 and 𝒪 if and only if there exists an explanation 𝒳 for 𝒪 such that ℳ𝑤𝑐(𝒫∪𝒳 ) |=𝑤𝑐𝑠 𝐹 . 𝐹 follows skeptically from 𝒫 and 𝒪 if and only if 𝒪 can be explained and for all explanations 𝒳 for 𝒪 we find ℳ𝑤𝑐(𝒫∪𝒳 ) |=𝑤𝑐𝑠 𝐹 . The latter is an application of the so-called Gricean implicature [22]: humans normally do not quantify over things which do not exist. Meaning, (unlike classical logic) all explanations for an observation 𝒪 may only be taken into account to skeptically decide on a formula 𝐹 , when 𝒪 is explainable and these so-called explanations exist in the first place. If a formula 𝐹 does not follow skeptically from 𝒫 and 𝒪, we conclude that nothing follows. Furthermore, one should also observe that if an observation 𝒪 cannot be explained, then nothing follows credulously as well as skeptically. In all examples discussed in this paper the set of integrity constraints is empty; they are not relevant to the goal of this paper. However they are needed in other applications of the WCS like human disjunctive reasoning [6]. For the purposes of our current goal we utilize a so-called revision operator rev , defined in [23]. Let us consider a consistent set of literals 𝑆, whose elements in the general case may be mapped to false under the least model of wc(𝒫). We then define the revision of 𝒫 with respect to 𝑆 as, 𝑟𝑒𝑣(𝒫, 𝑆) = (𝒫 ∖ defs(𝒫, 𝑆)) ∪ {𝐴 ← ⊤ | 𝐴 ∈ 𝑆} ∪ {𝐴 ← ⊥ | ¬𝐴 ∈ 𝑆}. 1 Whenever we apply a unary operator like Φ𝒫 to an argument like 𝐼, we omit the parenthesis and write Φ𝒫 𝐼 instead. Likewise, we write 𝐼 body instead of 𝐼(body). 2 Please note that assumptions like 𝐴 ← ⊥ are weakly completed. Their truth value is propagated via Φ. Moreover, they can be overridden by another clause of the form 𝐴 ← body. On the other hand, the constraint ⊥ ← 𝐴 cannot be part of the program. It’s only purpose is to eliminate models, where 𝐴 is true. Overall, given premises, general knowledge, and observations, reasoning in the WCS is modelled in six steps: 1. Reasoning towards a logic program 𝒫 following [20]. 2. Weakly completing the program, leading to 𝑤𝑐(𝒫). 3. Computing the least model ℳ𝑤𝑐(𝒫) of 𝑤𝑐(𝒫) under the three-valued Łukasiewicz logic. 4. Reasoning with respect to ℳ𝑤𝑐(𝒫) . 5. If observations cannot be explained, then applying skeptical abduction. 6. Searching for counterexamples [10]. 3. A Classification of Conditional Sentences In this section we introduce a classification of conditionals and their antecedents which we consider relevant for the identification of noise variables in a system. Pragmatics, life experiences and cultural differences play a deciding role in how different individuals comprehend or classify both. Following [24] we identify two types of conditional sentences: obligational and factual. We call a conditional sentence obligational if the truth of the consequent appears to be obligatory when given that the antecedent is true. For each obligational conditional there are two initial possibilities humans comprehending the conditional think about. The first possibility is the conjunction of the antecedent and the consequent, which is permitted in the viewpoint of the individual. The second possibility is the conjunction of the antecedent and the negation of the consequent, which is forbidden. Exceptions are possible but unlikely. Let us consider for example, if Maria drinks alcoholic beverages in pubs then she must be over 19 years of age. In many countries the law demands that a person may only drink alcohol publicly when they are above a certain age group (e.g. 19 years). This implies that for an individual from such a background, Maria is drinking alcoholic beverages in a pub and she is older than 19 years is a permitted possibility, whereas Maria is drinking alcoholic beverages in a pub and she is not older than 19 years is a forbidden one. Hence, this particular conditional is an obligational one. As another example consider, if plants get water then they grow. Here, plants getting water and plants growing is a permitted possibility. But as many plant enthusiasts would know, plants getting water and plants not growing is also possible. There are many factors, such as lack of light, overwatering, pest infestation, etc. which may hinder a plant’s growth. Conditionals such as these, where the (truth of the) consequent is not obligatory given the antecedent, we call factual conditionals. In particular, in such a case the truth of the antecedent is deemed inconsequential to that of the consequent by the individual in question. After the above discussion a question that may naturally arise is, what happens when the antecedent of a conditional sentence is not satisfied? To that end, the antecedent 𝐴 of a conditional sentence if 𝐴 then 𝐶 can be classified as necessary with respect to the consequent 𝐶, if and only if 𝐶 cannot be true unless 𝐴 is true. This implies that if 𝐴 does not hold, 𝐶 cannot either. For example, in case of the conditional if plants get water then they grow, the antecedent plants get water is a necessary one for the consequent plants will grow. If a plant is not watered at all, it will very likely die. This does not imply however, that the antecedent need always be a precondition for the consequent, per se. On the other hand, the antecedent 𝐴 of a conditional sentence if 𝐴 then 𝐶 is said to be non-necessary with respect to the consequent 𝐶, if 𝐶 can be true despite the falsity of 𝐴. This implies, if 𝐴 does not hold, 𝐶 may or may not hold. In the conditional, if Maria drinks alcoholic beverages in pubs then she must be over 19 years of age, the falsity of drinking alcoholic beverages in a pub is inconsequential to the truth of the consequent older than 19 years. There are plenty of adults (over 19 years) who do not drink alcohol. The antecedent of the conditional sentence is therefore called non-necessary. In order to account for the aforementioned classifications in the WCS, given a clause {𝐶 ← 𝐴 ∧ ¬ab} in a program 𝒫, the previous definition of the set of abducibles 𝒜𝒫 (see Section 2) can be extended to include 𝒜𝑛𝑛 𝒫 = {𝐶 ← ⊤} when the antecedent is deemed 𝑓 non-necessary, and to include 𝒜𝒫 = {ab ← ⊤} when the conditional is deemed factual. An interested reader may find this discussed in a lot more detail in [10]. 4. A Singular Decision: Dr. Snow and the Broad Street Pump As an example of a singular decision let us consider the following excerpt from the life of the father of epidemiology, Dr. John Snow, synthesized from [25, 26]. The human race has been threatened by a multitude of diseases, among which Cholera alone has claimed the lives of many over the years. Its patients suffer from mild to severe symptoms, the latter including intense diarrhoea and dehydration which may cause the person’s skin to turn bluish-grey and even be life-threatening. This symptom gives Cholera its colloquial nickname, "the blue death". Although Cholera still poses a problem in certain parts of the world, sewage and water treatment systems have helped curb it in many parts of the globe. This was not however always the case. England for example suffered high Cholera outbreaks in 1831, 1848 and 1854. No effective treatment had yet been in sight, and people did not yet have the knowledge that bacteria transmitted the disease. It was a common (mis)conception that Cholera spread through "miasmas" or toxic gases from open graves, garbage dumps, sewers etc. During the Cholera outbreak in 1848, Dr. John Snow, who had received his medical degree in London was a practising physician there. The outbreak led Snow to examine many Cholera patients and become heavily involved in the study of the disease. It is also noteworthy that during the 1831 outbreak, young Snow who aimed to be a physician had been working as an apprentice to a physician who treated Cholera patients. These experiences ultimately led Snow to publish a book on specific conclusions he had reached about the nature of the disease which was unfortunately rejected by most of his colleagues in the medical community. This did not discourage Snow however, and in 1854 when Cholera broke out once again Snow conducted his own investigation. His investigations showed that most death cases were approximately concentrated around a hand pump in Broad Street. Furthermore, numerous (fruitful) interviews with the families of the deceased indicated that the victims had consumed water from the aforementioned hand pump. Convinced that the hand pump’s contaminated water played an important role in the epidemic, Snow appeared for an interview with the Board of Guardians of the St. James parish. The board paid heed to Snow’s advice and had the hand pump removed the following day. This helped control the epidemic to a large extent, the repercussions of which echoed through time. The decision taken by the Board of Guardians to remove the Broad Street pump was a singular one because it was a unique decision with unique circumstances that was never repeated. This singular decision, however, was not entirely free from noise. In other words, had certain Table 2 Background conditional statements and their corresponding program clauses. Conditional Statements Program Clauses (1) Because Dr. Snow had an interview with the Board of Guardians, the board removed the Broad 𝑟𝑒𝑚 ← 𝑖𝑛𝑡 ∧ ¬ab 𝑖𝑛𝑡 , Street hand pump. ab 𝑖𝑛𝑡 ← ⊥ (2) Because Dr. Snow was convinced of the Broad Street pump playing an important role in the 𝑖𝑛𝑡 ← 𝑐𝑜𝑛𝑣 ∧ ¬ab 𝑐𝑜𝑛𝑣 , epidemic, he had an interview with the Board of Guardians. ab 𝑐𝑜𝑛𝑣 ← ⊥ (3) Because Dr. Snow had previous experience with Cholera patients, he was convinced of the Broad 𝑐𝑜𝑛𝑣 ← 𝑒𝑥𝑝 ∧ ¬ab 𝑒𝑥𝑝 , Street pump playing an important role in the epidemic. ab 𝑒𝑥𝑝 ← ⊥ (4) Because Dr. Snow’s investigations were fruitful, he was convinced of the Broad Street pump playing 𝑐𝑜𝑛𝑣 ← 𝑖𝑛𝑣 ∧ ¬ab 𝑖𝑛𝑣 , an important role in the epidemic. ab 𝑖𝑛𝑣 ← ⊥ (5) Because Dr. Snow was practising in London during the 1848 outbreak, he had previous experience 𝑒𝑥𝑝 ← 𝑝𝑟𝑎𝑐 ∧ ¬ab 𝑝𝑟𝑎𝑐 , with Cholera patients. ab 𝑝𝑟𝑎𝑐 ← ⊥ (6) Because Dr. Snow had been an apprentice during the 1831 outbreak, he had previous experience 𝑒𝑥𝑝 ← 𝑎𝑝𝑝 ∧ ¬ab 𝑎𝑝𝑝 , with Cholera patients. ab 𝑎𝑝𝑝 ← ⊥ (7) Because John received his medical degree in London, he was practising there during the 1848 𝑝𝑟𝑎𝑐 ← 𝑑𝑒𝑔 ∧ ¬ab 𝑑𝑒𝑔 , outbreak. ab 𝑑𝑒𝑔 ← ⊥ parameters in the background story been changed, the singular decision might have had a different outcome. Identification of the noise, or more specifically the noise variables can be brought about by reasoning counterfactually about the singular decision. How this can be modelled using the WCS framework will be discussed in the next section. 5. Identifying Noise Variables using Counterfactual Reasoning 5.1. Representing Background Knowledge using Causal Conditionals Table 2 lists the various conditional statements which may be used to comprehensively summa- rize the background knowledge presented in Section 4. For the convenience of the reader it also includes the corresponding clauses in a logic program representing the said conditionals, which shall be further used during the modelling of noise variables in the next sub-section. The reader may observe that the abnormality predicate in each clause has been assumed to be false. This may be overridden in the subsequent discussion when we take the classification of the conditionals and their antecedents into account while searching for noise variables. 5.2. Modelling Noise Variables using the Weak Completion Semantics The idea behind the approach that we demonstrate in this paper is to pick up the thread of the singular decision or the first statement, counterfactually gauge which antecedent could have altered the consequent, which in turn hints at the former being a noise variable. Then, go further back in the chain of events to explore the rules of inference which led to the aforementioned antecedent, thus repeating the exercise. Beginning with statement (1) in Table 2, upon considering the nature of the antecedent as discussed in Section 3, one may deem the antecedent to have been necessary for the consequent. In other words, one may not easily imagine a possibility where Snow had no interview with the board yet the board decided to remove the hand pump. Had Snow not approached the board for an interview, the very idea of removing the hand pump may have slipped their attention. The logic program 𝒫1 which may be constructed from statement (1) consists of the following: {𝑟𝑒𝑚 ← 𝑖𝑛𝑡 ∧ ¬ab 𝑖𝑛𝑡 , ab 𝑖𝑛𝑡 ← ⊥, 𝑖𝑛𝑡 ← ⊤}, where rem denotes the board removed the hand pump and int denotes Snow had an interview with the board. ab int is an abnormality predicate denoting anything that went wrong with regard to the interview, e.g. one of the board members fell sick on the spot and the meeting was adjourned. As there was no such case it is assumed to be false. The last clause represents a fact. The weak completion of 𝒫1 , 𝑤𝑐(𝒫1 ), is {𝑟𝑒𝑚 ↔ 𝑖𝑛𝑡 ∧ ¬ab 𝑖𝑛𝑡 , ab 𝑖𝑛𝑡 ↔ ⊥, 𝑖𝑛𝑡 ↔ ⊤}, which has the least model ℳ𝑤𝑐(𝒫1 ) = ⟨{𝑖𝑛𝑡, 𝑟𝑒𝑚}, {ab 𝑖𝑛𝑡 }⟩.3 This signifies 𝑖𝑛𝑡 and 𝑟𝑒𝑚 are true, while ab 𝑖𝑛𝑡 is false. When reasoning with the situation counterfactually, one may use the conditional if Dr. Snow did not have an interview with the board, then the board would not have removed the hand pump or, if ¬𝑖𝑛𝑡 then ¬𝑟𝑒𝑚. Upon evaluating ¬𝑖𝑛𝑡 under the aforementioned least model, the reader may find that it is false. Hence we call the condi- tional a subtractive counterfactual conditional, in lines with [12]. Revising 𝒫1 with respect to {¬𝑖𝑛𝑡} using the revision operator rev (see Section 2) leads us to the revised program rev (𝒫1 , {¬int}) = (𝒫1 ∖ {𝑖𝑛𝑡 ← ⊤}) ∪ {𝑖𝑛𝑡 ← ⊥}. Its weak completion, {𝑟𝑒𝑚 ↔ 𝑖𝑛𝑡 ∧ ¬ab 𝑖𝑛𝑡 , ab 𝑖𝑛𝑡 ↔ ⊥, 𝑖𝑛𝑡 ↔ ⊥} admits the least model ⟨∅, {𝑖𝑛𝑡, 𝑟𝑒𝑚, ab 𝑖𝑛𝑡 }⟩. In particular, 𝑟𝑒𝑚 which was true is now false. Thus, the variability in the truth of the consequent leads to variability in the outcome, which indicates noise. Consequently, int is a noise variable. Now we further explore the atom int and hence statement (2). Considering that the antecedent may be deemed necessary for the consequent, any possibility that Snow himself was not convinced that the hand pump should be removed yet had an interview with the board about the same may be discounted. The logic program 𝒫2 constructed from the statements (1) and (2) is: {𝑟𝑒𝑚 ← 𝑖𝑛𝑡 ∧ ¬ab 𝑖𝑛𝑡 , ab 𝑖𝑛𝑡 ← ⊥, 𝑖𝑛𝑡 ← 𝑐𝑜𝑛𝑣 ∧ ¬ab 𝑐𝑜𝑛𝑣 , ab 𝑐𝑜𝑛𝑣 ← ⊥, 𝑐𝑜𝑛𝑣 ← ⊤}, where conv denotes Snow was convinced about the hand pump, and ab conv is an abnormality predicate assumed to be false. The last clause represents a fact. 𝑤𝑐(𝒫2 ) admits the least model ℳ𝑤𝑐(𝒫2 ) = ⟨{𝑐𝑜𝑛𝑣, 𝑖𝑛𝑡, 𝑟𝑒𝑚}, {ab 𝑐𝑜𝑛𝑣 , ab 𝑖𝑛𝑡 }⟩. Reasoning counterfactually, one may imagine the conditional if John had not been convinced then he would not have asked for an interview viz. if ¬𝑐𝑜𝑛𝑣 then ¬𝑖𝑛𝑡. As ¬𝑐𝑜𝑛𝑣 evaluates to false under the aforementioned least model, this is a subtractive counterfactual conditional. Then, rev (𝒫2 , {¬conv }) leads to (𝒫2 ∖ {𝑐𝑜𝑛𝑣 ← ⊤}) ∪ {𝑐𝑜𝑛𝑣 ← ⊥}. Its weak completion admits the least model ⟨∅, {𝑐𝑜𝑛𝑣, 𝑖𝑛𝑡, 𝑟𝑒𝑚, ab 𝑐𝑜𝑛𝑣 , ab 𝑖𝑛𝑡 }⟩. This indicates that 𝑐𝑜𝑛𝑣 is a noise variable. Going further back on the chain of events, we now consider the statements (1) to (4). In case of statements (3) and (4), both antecedents of the conditionals may be considered necessary for the consequent. Hence, in such a case the logic program 𝒫3 is: 3 This model can be computed as follows. Starting with the interpretation ⟨∅, ∅⟩, we obtain Φ𝒫1 ⟨∅, ∅⟩ = ⟨{𝑖𝑛𝑡}, {ab 𝑖𝑛𝑡 }⟩ and Φ𝒫1 ⟨{𝑖𝑛𝑡}, {ab 𝑖𝑛𝑡 }⟩ = ⟨{𝑖𝑛𝑡, 𝑟𝑒𝑚}, {ab 𝑖𝑛𝑡 }⟩ = Φ𝒫1 ⟨{𝑖𝑛𝑡, 𝑟𝑒𝑚}, {ab 𝑖𝑛𝑡 }⟩. {𝑟𝑒𝑚 ← 𝑖𝑛𝑡 ∧ ¬ab 𝑖𝑛𝑡 , ab 𝑖𝑛𝑡 ← ⊥, 𝑖𝑛𝑡 ← 𝑐𝑜𝑛𝑣 ∧ ¬ab 𝑐𝑜𝑛𝑣 , ab 𝑐𝑜𝑛𝑣 ← ⊥, 𝑐𝑜𝑛𝑣 ← 𝑒𝑥𝑝 ∧ ¬ab 𝑒𝑥𝑝 , 𝑐𝑜𝑛𝑣 ← 𝑖𝑛𝑣 ∧ ¬ab 𝑖𝑛𝑣 , ab 𝑒𝑥𝑝 ← ⊥, ab 𝑒𝑥𝑝 ← ¬𝑖𝑛𝑣, ab 𝑖𝑛𝑣 ← ⊥, ab 𝑖𝑛𝑣 ← ¬𝑒𝑥𝑝, 𝑒𝑥𝑝 ← ⊤, 𝑖𝑛𝑣 ← ⊤}, where exp denotes Snow had a lot of experience with Cholera patients and inv denotes Snow’s investigations proved fruitful. ab exp and ab inv are abnormality predicates. The last two clauses represent facts. One may observe that we have now added the clauses ab 𝑒𝑥𝑝 ← ¬𝑖𝑛𝑣 and ab 𝑖𝑛𝑣 ← ¬𝑒𝑥𝑝 and we attempt to clarify this in what follows. As the antecedents of both the conditionals if 𝑒𝑥𝑝 then 𝑐𝑜𝑛𝑣 and if 𝑖𝑛𝑣 then 𝑐𝑜𝑛𝑣 have been deemed necessary for the consequent, the possibility that one of the antecedents is false but the consequent is true is discounted. Not having enough experience with Cholera patients could have prevented Snow from being convinced about the removal of the hand pump even if his investigations had proved fruitful. Likewise, an unproductive investigation could have prevented Snow from being convinced about the pump, despite him having enough experience. Thus given the two conditionals if 𝑒𝑥𝑝 then 𝑐𝑜𝑛𝑣 and if 𝑖𝑛𝑣 then 𝑐𝑜𝑛𝑣, where both the antecedents are deemed to be necessary, we characterize ¬𝑖𝑛𝑣 as an abnormality with respect to the former conditional and ¬𝑒𝑥𝑝 as an abnormality with respect to the latter. Hence the additional clauses ab 𝑒𝑥𝑝 ← ¬𝑖𝑛𝑣 and ab 𝑖𝑛𝑣 ← ¬𝑒𝑥𝑝. Such a characterization is along the lines of so-called enabling relations as discussed in [27]. Using this we have also modelled experiments involving additional arguments in the suppression task [4, 5]. The reader may note that in 𝑤𝑐(𝒫3 ), the assumptions ab 𝑖𝑛𝑣 ← ⊥ and ab 𝑒𝑥𝑝 ← ⊥ are overridden by ab 𝑖𝑛𝑣 ← ¬𝑒𝑥𝑝 and ab 𝑒𝑥𝑝 ← ¬𝑖𝑛𝑣, respectively. 𝑤𝑐(𝒫3 ) has the least model ℳ𝑤𝑐(𝒫3 ) = ⟨{𝑒𝑥𝑝, 𝑖𝑛𝑣, 𝑐𝑜𝑛𝑣, 𝑖𝑛𝑡, 𝑟𝑒𝑚}, {ab 𝑒𝑥𝑝 , ab 𝑖𝑛𝑣 , ab 𝑐𝑜𝑛𝑣 , ab 𝑖𝑛𝑡 }⟩ where inv and exp are true, and so is rem. In particular no least models of 𝑤𝑐(𝒫3 ) where inv , exp or both are false but conv is true are constructed. Reasoning with the situation counterfactually now gives rise to the conditionals, if Snow had no experience with Cholera patients then he would not have been convinced, viz. if ¬𝑒𝑥𝑝 then ¬𝑐𝑜𝑛𝑣, and if his investigations were unsuccessful then he would not have been convinced viz. if ¬𝑖𝑛𝑣 then ¬𝑐𝑜𝑛𝑣. Upon evaluation of ¬𝑒𝑥𝑝 and ¬𝑖𝑛𝑣 with respect to ℳ𝑤𝑐(𝒫3 ) the reader may find that they are both false. Hence these are subtractive counterfactual conditionals. So, rev (𝒫3 , {¬exp, ¬inv }) results in the program (𝒫3 ∖ {𝑒𝑥𝑝 ← ⊤, 𝑖𝑛𝑣 ← ⊤}) ∪ {𝑒𝑥𝑝 ← ⊥, 𝑖𝑛𝑣 ← ⊥}. Its weak completion now admits a least model where inv and exp are false, and (now) so is rem. Thus the variability in the truth of rem indicates that inv and exp are noise variables. Let us now consider statements (1) to (6), and in particular statements (5) and (6). While practising in London during 1848 may be deemed necessary for the level of experience Snow had with Cholera patients, his apprenticeship during 1831 may be deemed non-necessary by some individuals. That is while one may not readily imagine a possibility where Snow did not practise in London, but had experience with Cholera patients, one may imagine Snow not doing the apprenticeship yet having experience. The reader is pointed out that the opposite may also hold for some individuals where they may consider apprenticeship to be necessary but the London practice to be non-necessary. Or some individuals may even consider both antecedents to be non-necessary for the consequent. For the sake of modelling and demonstration purposes, we assume the first case. So, we consider the conditional if 𝑎𝑝𝑝 then 𝑒𝑥𝑝, where 𝑎𝑝𝑝 is deemed non-necessary for 𝑒𝑥𝑝. And consider if 𝑝𝑟𝑎𝑐 then 𝑒𝑥𝑝, where 𝑝𝑟𝑎𝑐 is deemed necessary for 𝑒𝑥𝑝, owing to which we characterize ¬𝑝𝑟𝑎𝑐 as an abnormality with respect to the former conditional.4 Thus, the program 𝒫4 has the clauses: {𝑟𝑒𝑚 ← 𝑖𝑛𝑡 ∧ ¬ab 𝑖𝑛𝑡 , ab 𝑖𝑛𝑡 ← ⊥, 𝑖𝑛𝑡 ← 𝑐𝑜𝑛𝑣 ∧ ¬ab 𝑐𝑜𝑛𝑣 , ab 𝑐𝑜𝑛𝑣 ← ⊥, 𝑐𝑜𝑛𝑣 ← 𝑒𝑥𝑝 ∧ ¬ab 𝑒𝑥𝑝 , ab 𝑒𝑥𝑝 ← ⊥, ab 𝑒𝑥𝑝 ← ¬𝑖𝑛𝑣, 𝑐𝑜𝑛𝑣 ← 𝑖𝑛𝑣 ∧ ¬ab 𝑖𝑛𝑣 , ab 𝑖𝑛𝑣 ← ⊥, ab 𝑖𝑛𝑣 ← ¬𝑒𝑥𝑝, 𝑒𝑥𝑝 ← 𝑝𝑟𝑎𝑐 ∧ ¬ab 𝑝𝑟𝑎𝑐 , ab 𝑝𝑟𝑎𝑐 ← ⊥, 𝑒𝑥𝑝 ← 𝑎𝑝𝑝 ∧ ¬ab 𝑎𝑝𝑝 , ab 𝑎𝑝𝑝 ← ⊥, ab 𝑎𝑝𝑝 ← ¬𝑝𝑟𝑎𝑐, 𝑖𝑛𝑣 ← ⊤, 𝑝𝑟𝑎𝑐 ← ⊤, 𝑎𝑝𝑝 ← ⊤}, where prac denotes Snow was practising in London during the 1848 outbreak and app denotes Snow was an apprentice for a physician during the 1831 outbreak. ab prac and ab app are abnormality predicates. The last three clauses represent facts. The reader may note that we now have the additional clause ab 𝑎𝑝𝑝 ← ¬𝑝𝑟𝑎𝑐, and that the assumption ab 𝑎𝑝𝑝 ← ⊥ is overridden by ab 𝑎𝑝𝑝 ← ¬𝑝𝑟𝑎𝑐 in 𝑤𝑐(𝒫4 ). The least model of 𝑤𝑐(𝒫4 ) is: ℳ𝑤𝑐(𝒫4 ) = ⟨{𝑖𝑛𝑣, 𝑝𝑟𝑎𝑐, 𝑎𝑝𝑝, 𝑒𝑥𝑝, 𝑐𝑜𝑛𝑣, 𝑖𝑛𝑡, 𝑟𝑒𝑚}, {ab 𝑝𝑟𝑎𝑐 , ab 𝑎𝑝𝑝 , ab 𝑖𝑛𝑣 , ab 𝑒𝑥𝑝 , ab 𝑐𝑜𝑛𝑣 , ab 𝑖𝑛𝑡 }⟩. Here, app and prac are both true and so is rem. Because prac has been deemed necessary for exp, irrespective of whether he worked as an apprentice or not, as long as Snow did not practice in London in 1848 he would not have the level of experience. Reasoning counterfactually about the situation we may thus imagine, if Snow would not have been practising in London then he would not have the experience viz. if ¬𝑝𝑟𝑎𝑐 then ¬𝑒𝑥𝑝. Upon evaluating ¬𝑝𝑟𝑎𝑐 with respect to ℳ𝑤𝑐(𝒫4 ) , we find that it is false. Hence, this is a subtractive counterfactual conditional. Therefore revision using rev (𝒫4 , {¬prac}) leads us to the program (𝒫4 ∖ {𝑝𝑟𝑎𝑐 ← ⊤}) ∪ {𝑝𝑟𝑎𝑐 ← ⊥}. Its weak completion admits a least model where prac is false, and so is rem. The variability in the truth of rem indicates that prac is a noise variable. Now, reconsidering statements (5) and (6) from a different angle, we might ask ourselves the following - ”if Snow was a physician’s apprentice in 1831 does it necessarily mean that he would have a high level of experience with Cholera patients?” or that ”if Snow practised in London in 1848, does it necessarily that he would have a high level of experience with Cholera patients?”. In other words, we may imagine a possibility where Snow did practise in London in 1848 but due to some (additional) reasons he could not have a high level of experience with Cholera patients. Or that, he was an apprentice but certain reasons hindered his opportunity to have the needed experience. As discussed in Section 3, imagining such (alternative) possibilities count for statements (5) and (6) being comprehended as factual. Meaning in such a case, given if 𝐴 then 𝐶 and affirming 𝐴, both 𝐶 and ¬𝐶 are deemed possible. For the current moment, we are particularly interested in this latter possibility. Within the WCS it can be modelled by considering 𝐴 as an observation and applying abduction in order to explain it using the set of abducibles for factual conditionals, 𝒜𝑓𝒫 , mentioned in Section 3. We attempt to clarify the process in the current context in what follows. We consider the program 𝒫5 = 𝒫4 ∖ {𝑝𝑟𝑎𝑐 ← ⊤, 𝑎𝑝𝑝 ← ⊤}, and consider 𝑝𝑟𝑎𝑐 and 𝑎𝑝𝑝 to be observations instead, meaning 𝒪 = {𝑝𝑟𝑎𝑐, 𝑎𝑝𝑝}. Now we apply abduction in order to explain 𝒪. Following the definition of abducibles as described in Section 2, since both 𝑝𝑟𝑎𝑐 and 4 This is in lines with the prior discussion regarding statement (3) and (4). 𝑎𝑝𝑝 are undefined in 𝒫5 , 𝒜𝒫5 = {𝑝𝑟𝑎𝑐 ← ⊤, 𝑝𝑟𝑎𝑐 ← ⊥, 𝑎𝑝𝑝 ← ⊤, 𝑎𝑝𝑝 ← ⊥}. Moreover, comprehending statements (5) and (6) as factual entails extending the set of abducibles that can be derived from 𝒫5 to 𝒜𝑒𝒫5 = 𝒜𝒫5 ∪ 𝒜𝑓𝒫5 , where 𝒜𝑓𝒫5 includes {ab 𝑎𝑝𝑝 ← ⊤, ab 𝑝𝑟𝑎𝑐 ← ⊤}. While there is a minimal explanation for 𝒪, viz. {𝑝𝑟𝑎𝑐 ← ⊤, 𝑎𝑝𝑝 ← ⊤}, there is also a non-minimal explanation viz. {𝑝𝑟𝑎𝑐 ← ⊤, ab 𝑝𝑟𝑎𝑐 ← ⊤, 𝑎𝑝𝑝 ← ⊤, ab 𝑎𝑝𝑝 ← ⊤}. Adding the former to 𝒫5 would again result in the program 𝒫4 . However, the latter results in a program 𝒫5 ′ : {𝑟𝑒𝑚 ← 𝑖𝑛𝑡 ∧ ¬ab 𝑖𝑛𝑡 , ab 𝑖𝑛𝑡 ← ⊥, 𝑖𝑛𝑡 ← 𝑐𝑜𝑛𝑣 ∧ ¬ab 𝑐𝑜𝑛𝑣 , ab 𝑐𝑜𝑛𝑣 ← ⊥, 𝑐𝑜𝑛𝑣 ← 𝑒𝑥𝑝 ∧ ¬ab 𝑒𝑥𝑝 , ab 𝑒𝑥𝑝 ← ⊥, ab 𝑒𝑥𝑝 ← ¬𝑖𝑛𝑣, 𝑐𝑜𝑛𝑣 ← 𝑖𝑛𝑣 ∧ ¬ab 𝑖𝑛𝑣 , ab 𝑖𝑛𝑣 ← ⊥, ab 𝑖𝑛𝑣 ← ¬𝑒𝑥𝑝, 𝑒𝑥𝑝 ← 𝑝𝑟𝑎𝑐 ∧ ¬ab 𝑝𝑟𝑎𝑐 , ab 𝑝𝑟𝑎𝑐 ← ⊥, ab 𝑝𝑟𝑎𝑐 ← ⊤, 𝑒𝑥𝑝 ← 𝑎𝑝𝑝 ∧ ¬ab 𝑎𝑝𝑝 , ab 𝑎𝑝𝑝 ← ⊥, ab 𝑎𝑝𝑝 ← ¬𝑝𝑟𝑎𝑐, ab 𝑎𝑝𝑝 ← ⊤, 𝑖𝑛𝑣 ← ⊤, 𝑝𝑟𝑎𝑐 ← ⊤, 𝑎𝑝𝑝 ← ⊤}. Its weak completion, 𝑤𝑐(𝒫5 ′ ), admits the least model: ℳ𝑤𝑐(𝒫5 ′ ) = ⟨{𝑎𝑝𝑝, 𝑝𝑟𝑎𝑐, 𝑖𝑛𝑣, ab 𝑎𝑝𝑝 , ab 𝑝𝑟𝑎𝑐 , ab 𝑖𝑛𝑣 }, {𝑒𝑥𝑝, ab 𝑒𝑥𝑝 , 𝑐𝑜𝑛𝑣, ab 𝑐𝑜𝑛𝑣 , 𝑖𝑛𝑡, ab 𝑖𝑛𝑡 , 𝑟𝑒𝑚}⟩, where 𝑎𝑝𝑝, 𝑝𝑟𝑎𝑐, ab 𝑎𝑝𝑝 and ab 𝑝𝑟𝑎𝑐 are true, but exp is false. The reader is pointed out that the additional clauses {ab 𝑝𝑟𝑎𝑐 ← ⊤} and {ab 𝑎𝑝𝑝 ← ⊤} signify that there could be (other additional) reasons which could have hindered Snow’s experience with Cholera patients. This is in line with additive counterfactuals as discussed in [12]. In other words, reasoning counterfactually one may thus use statements such as if something abnormal had happened with respect to his apprenticeship, then John would not have the experience, i.e. if ab 𝑎𝑝𝑝 then ¬𝑒𝑥𝑝 or, if something abnormal had happened with respect to his practice, then John would not have the experience, i.e. if ab 𝑝𝑟𝑎𝑐 then ¬𝑒𝑥𝑝. Information here is not taken away like in case of (the previous) subtractive counterfactual statements, but rather added to the simulation of reality. As rem is false in ℳ𝑤𝑐(𝒫5 ′ ) , it signifies variability in the outcome. Thus ab 𝑎𝑝𝑝 and ab 𝑝𝑟𝑎𝑐 are noise variables which could be explored further. The case of non-necessary antecedents begs a lengthier discussion than the current spatial constraints of the paper would allow. Given if 𝐴 then 𝐶, comprehending 𝐴 to be non-necessary for 𝐶 signifies that in case of ¬𝐴, both ¬𝐶 and particularly 𝐶 are deemed possible. This implies two least models, one in which 𝐶 is false, and the other where 𝐶 is true. Both can be computed by employing the abductive framework using the extended set of abducibles 𝒜𝑛𝑛 𝒫 mentioned in Section 3, as discussed in detail in [10]. However, the implications of dealing with these multiple least models when identifying noise variables need further analysis. For now, as a small example let us consider statement (6) and explore the atom 𝑑𝑒𝑔. Thus, we consider the program 𝒫6 = (𝒫4 ∖ {𝑝𝑟𝑎𝑐 ← ⊤}) ∪ {𝑝𝑟𝑎𝑐 ← 𝑑𝑒𝑔 ∧ ¬ab 𝑑𝑒𝑔 , ab 𝑑𝑒𝑔 ← ⊥}. In statement (6), the antecedent may be deemed non-necessary for the consequent. Meaning, considering the question, ”if John had not received his medical degree in London, could he (yet) be practising there during the 1848 outbreak?”, one may deem it possible for John to not have received his medical de- gree in London but to have been practising there. This can be modelled within the WCS using the aforementioned abductive framework as follows. Supposing 𝒪 = {¬𝑑𝑒𝑔} to be an observation for which we look for an explanation, leads us to apply abduction. Since 𝑑𝑒𝑔 is undefined in 𝒫6 , 𝒜𝒫6 includes {𝑑𝑒𝑔 ← ⊤, 𝑑𝑒𝑔 ← ⊥}. Furthermore as 𝑑𝑒𝑔 here is a non-necessary antecedent, 𝒜𝑒𝒫6 = 𝒜𝒫6 ∪ 𝒜𝑛𝑛 𝒫6 where 𝒜𝒫6 includes { 𝑝𝑟𝑎𝑐 ← ⊤}. While there is a minimal explanation 𝑛𝑛 to 𝒪 viz. {𝑑𝑒𝑔 ← ⊥}, there is also a non-minimal explanation, viz. {𝑑𝑒𝑔 ← ⊥, 𝑝𝑟𝑎𝑐 ← ⊤}. The latter results in the revised program 𝒫6 ∪ {𝑑𝑒𝑔 ← ⊥, 𝑝𝑟𝑎𝑐 ← ⊤}. Its weak completion admits a least model where we find that 𝑝𝑟𝑎𝑐 is true, although 𝑑𝑒𝑔 is false. Summing up in simpler words, there could be some reason due to which John practised in London (i.e. 𝑝𝑟𝑎𝑐 could be true) even if he had not received his medical degree there (i.e. 𝑑𝑒𝑔 was false). Such an exercise in turn motivates further questions along the lines - ”what if this particular reason had not occurred?”. This so-called reason hints at being a noise variable and could be explored further. In all that has been demonstrated so far, we have attempted to illustrate the identification of some of the noise variables in our system such as, 𝑖𝑛𝑡, 𝑒𝑥𝑝, 𝑐𝑜𝑛𝑣, 𝑖𝑛𝑣, 𝑝𝑟𝑎𝑐, ab 𝑎𝑝𝑝 , ab 𝑝𝑟𝑎𝑐 etc. Thus we have identified the system noise. At this point it must also be acknowledged that while there are variables which may contribute to noise, there may also be those which do not. One possible means to identify the latter could be to guage the relevance of the antecedent of a conditional with respect to the consequent, in line with [28]. For example, consider 𝒫7 = 𝒫6 ∪ {𝑑𝑒𝑔 ← ⊤, 𝑤𝑖𝑛𝑡 ← ⊤}, where 𝑤𝑖𝑛𝑡 ← ⊤ represents the fact that the winter of 1848 was particularly harsh, and a conditional, if Snow received his medical degree from London then the winter of 1848 was particularly harsh viz. if 𝑑𝑒𝑔 then 𝑤𝑖𝑛𝑡. In the common knowledge of an individual, 𝑑𝑒𝑔 is very likely not relevant to 𝑤𝑖𝑛𝑡. This may be modelled following [29], using the notion of the so-called strong relevance, the core idea of which is to check whether 𝑤𝑖𝑛𝑡 loses support as soon as the support of 𝑑𝑒𝑔 is withdrawn.5 The reader may observe that both 𝑑𝑒𝑔 and 𝑤𝑖𝑛𝑡 are true in ℳ𝒫7 . Now, removing the support of 𝑑𝑒𝑔 from 𝒫7 which leads to 𝒫7 ′ = 𝒫 ∖ {𝑑𝑒𝑔 ← ⊤}, we still find that 𝑤𝑖𝑛𝑡 is true in ℳ𝒫7 ′ . Hence, in this case we may conclude that 𝑑𝑒𝑔 is not strongly relevant to 𝑤𝑖𝑛𝑡. The said individual may thus ask - ”if Snow had not received his medical degree from London, would the winter of 1848 still be harsh?”. And the answer may well be yes. With this remark we cease the discussion about variables that do not add to noise, as it requires further research and is best reserved for another occasion. 6. Conclusion Noise just like bias, may be undesirable in much of our decision-making and judgment. As the human race journeys further into the age of digitalization and artificial intelligence becomes more and more involved in our daily lives, creating systems which minimize bias and noise, both of which contribute to errors in judgement seems important. In order to create any kind of AI system with decision-making capabilities with minimal noise and bias, it is essential to discuss how they can be identified in our own, humane judgments, whether it be in economy, judiciary, education, healthcare, or even personal. Regardless of whether the decisions are recurrent or singular the aim is to identify and minimize the noise in both. In this paper, we have particularly looked into singular decisions because in comparison with recurrent decisions the noise in these systems may be less apparent. Considering the singular decision to remove the hand pump in Broad Street which not only helped save the lives of many during the 1854 Cholera outbreak 5 Unfortunately the current spatial constraints of the paper disallows us from being more detailed about strong relevance, but an interested reader is encouraged to read [29]. in London, but also influenced healthcare for the better around the world, we have attempted to demonstrate how the historical outcome was not entirely free from noise. Had some of the influencing causal factors been tweaked, the outcome may have been different. The outcome’s variability thus becomes apparent. To that end, we have used the WCS framework to model the identification of some of these so-called noise variables using counterfactual reasoning. The protoypical modelling is not limited to the discussion in this paper however and there is scope for future development and general formalization. Some avenues that present themselves for furture exploration through the current exercise are modelling non-necessary antecedents as noise variables, handling obligational conditionals when reasoning counterfactually and eventually minimizing noise in a system. References [1] C. R. S. Daniel Kanheman, Olivier Sibony, Noise: A flaw in Human Judgment, Little, Brown Spark, 2021. [2] J. Ramji-Nogales, A. I. Schoenholtz, P. G. Schrag, Refugee roulette: Disparities in asylum adjudication, Stan. L. Rev. 60 (2007) 295. [3] M. A. Lemley, B. Sampat, Examiner characteristics and patent office outcomes, Review of economics and statistics 94 (2012) 817–827. [4] R. M. J. Byrne, Suppressing valid inferences with conditionals, Cognition 31 (1989) 61–83. [5] E.-A. Dietz, S. Hölldobler, M. Ragni, A computational logic approach to the suppression task, Proceedings of the Annual Conference of the Cognitive Science Society, 34 (2012) 1500–1505. Retrieved from https://escholarship.org/uc/item/2sd6d61q. [6] I. Hamada, S. Hölldobler, On disjunctions and the weak completion semantics, https: //tu-dresden.de/ing/informatik/ki/krr/ressourcen/dateien/hh2021.pdf/view, 2021. Accepted at MathPsych/ICCM2021. [7] A. Oliviera da Costa, E.-A. Dietz Saldanha, S. Hölldobler, M. Ragni, A computational logic approach to human syllogistic reasoning, Proceedings of the Annual Conference of the Cognitive Science Society, 39 (2017) 883–888. [8] E.-A. Dietz, A computational logic approach to the belief bias in human syllogistic reasoning, in: Proceedings of the 10th International and Interdisciplinary Conference on Modeling and Using Context (CONTEXT), volume 10257 of Lecture Notes in Computer Science, 2017, pp. 691–707. [9] M. Cramer, S. Hölldobler, M. Ragni, Modeling human reasoning about conditionals, in: L. Amgoud, R. Booth (Eds.), Proceedings of the 19th International Workshop on Non- Monotonic Reasoning, 2021, pp. 223–232. [10] M. Bhadra, S. Hölldobler, The weak completion semantics and counter examples, in: C. Beierle, M. Ragni, F. Stolzenburg, M. Thimm (Eds.), Proceedings of the 7th Workshop on Formal and Cognitive Reasoning, volume 2961 of CEUR Workshop Proceedings, 2021, pp. 60–73. [11] M. Cramer, S. Hölldobler, M. Ragni, When are humans reasoning with modus tollens?, Proceedings of the Annual Conference of the Cognitive Science Society, 43 (2021) 2337– 2343. Retrieved from https://escholarship.org/uc/item/9x33q50g. [12] R. M. Byrne, Counterfactuals in explainable artificial intelligence (xai): Evidence from human reasoning., in: IJCAI, 2019, pp. 6276–6282. [13] M. Fitting, First–Order Logic and Automated Theorem Proving, 2nd ed., Springer-Verlag, Berlin, 1996. [14] J. W. Lloyd, Foundations of Logic Programming, Springer-Verlag, 1984. [15] S. Hölldobler, Weak completion semantics and its applications in human reasoning, in: U. Furbach, C. Schon (Eds.), Bridging 2015 – Bridging the Gap between Human and Automated Reasoning, volume 1412 of CEUR Workshop Proceedings, CEUR-WS.org, 2015, pp. 2–16. http://ceur-ws.org/Vol-1412/. [16] K. Stenning, M. van Lambalgen, Semantic interpretation as computation in nonmonotonic logic: The real meaning of the suppression task, Cognitive Science 29 (2005). [17] K. L. Clark, Negation as failure, in: H. Gallaire, J. Minker (Eds.), Logic and Databases, Plenum, New York, 1978, pp. 293–322. [18] S. Hölldobler, C. D. P. Kencana Ramli, Logic programs under three-valued Łukasiewicz’s semantics, in: P. M. Hill, D. S. Warren (Eds.), Logic Programming, volume 5649 of Lecture Notes in Computer Science, Springer-Verlag Berlin Heidelberg, 2009, pp. 464–478. [19] J. Łukasiewicz, O logice trójwartościowej, Ruch Filozoficzny 5 (1920) 169–171. English translation: On Three-Valued Logic. In: Jan Łukasiewicz Selected Works. (L. Borkowski, ed.), North Holland, 87-88, 1990. [20] K. Stenning, M. van Lambalgen, Human Reasoning and Cognitive Science, MIT Press, 2008. [21] A. C. Kakas, R. A. Kowalski, F. Toni, Abductive Logic Programming, Journal of Logic and Computation 2(6) (1992) 719–770. [22] H. P. Grice, Logic and conversation, in: P. Cole, J. L. Morgan (Eds.), Syntax and Semantics, volume 3, Academic Press, New York, 1975, pp. 41–58. [23] E.-A. Dietz, S. Hölldobler, A new computational logic approach to reason with conditionals, in: F. Calimeri, G. Ianni, M. Truszczynski (Eds.), Logic Programming and Nonmonotonic Reasoning, 13th International Conference, LPNMR, volume 9345 of Lecture Notes in Artifi- cial Intelligence, Springer, 2015, pp. 265–278. [24] R. M. J. Byrne, The Rational Imagination: How People Create Alternatives to Reality, MIT Press, Cambridge, MA, USA, 2005. [25] Wikipedia contributors, John snow — Wikipedia, the free encyclopedia, 2022. URL: https: //en.wikipedia.org/w/index.php?title=John_Snow&oldid=1086361182, [Online; accessed 13-May-2022]. [26] SMU Department of Physics, Scientific thinking in medicine: dr. john snow and cholera, 2022. URL: https://www.physics.smu.edu/pseudo/ThinkingMed/, [Online; accessed 13- May-2022]. [27] G. A. Miller, P. N. Johnson-Laird, Language and Perception, Belknap Press, 1976. [28] P. N. Johnson-Laird, R. M. J. Byrne, Conditionals: A theory of meaning, pragmatics, and inference, Psychological Review 109 (2002) 646–678. [29] S. Hölldobler, Conditional reasoning and relevance, in: U. Schmid, F. Klügl, D. Wolter (Eds.), Proceedings of the KI 2020: Advances in Artificial Intelligence, Springer, 2020, pp. 73–87.