7 Hypothesis
7.1 Summary of this chapter
This chapter presents the development of a formal necessary condition hypothesis. It is part of a necessity theory and should be testable, non-trivial, plausible, and accompanied by a causal explanation. First, it is explained why NCA adopts a hypothesis formulation-and-testing approach to answer research and practice questions (Section 7.2). The next section (Section 7.3) clarifies why the hypothesis development process differs from usual practice (e.g., lack of past research on necessary conditions). This is followed in Section 7.4 by an overview of knowledge sources that can be used to develop necessity hypotheses (explicit and tacit knowledge from academia and practice). Subsequently, the three steps for developing a formal hypothesis are presented, including decision tools to support this process (Section 7.5). Step 1 establishes a preliminary necessity theory that consists of selecting a “best guess” necessity hypothesis from available sources of information that is embedded in theory. Step 2 uses thought experiments to challenge the hypothesis and, if needed, refine it. Step 3 deepens the causal explanation, and formulates the final formal hypothesis to be tested with empirical data.
7.2 Hypothesis formulation and testing
After the theoretical or practical goal is set, and it is decided to apply NCA, often the question of interest is: “Is \(X\) necessary for \(Y\)”? or “Which factors are necessary for \(Y\)”?
NCA advocates starting the study by providing a preliminary answer to the question before collecting empirical data. Such an answer can be based on existing sources of knowledge. NCA is an approach in which the hypothesis is first formulated and subsequently tested with empirical data. In this deductive approach, the hypothesis can have three states: ‘preliminary’, ‘formal’ and ‘tested’. A preliminary necessity hypothesis is a ‘best guess’ answer to the above questions based on existing sources of knowledge and, for example, abductive reasoning. A formal necessity hypothesis is a deepened preliminary hypothesis that is embedded in theory and that was challenged by thought experiments regarding testability, triviality, plausibility, and completeness. A tested necessity hypothesis is a formal hypothesis that was subjected to tests using empirical data, where the outcome of the test is that the hypothesis is rejected or not-rejected (supported) by the observed evidence. In this book, a hypothesis is considered supported or plausible when it is not rejected after being challenged empirically. Rejection may arise from the absence of a plausible causal explanation, a very small necessity effect size, or a large \(p\)-value. However, support for, or plausibility of, a necessary condition does not automatically establish its truth. Greater confidence requires, for example, support from replication studies. Even then, alternative causal explanations for the observed empty corner are possible (Section 2.7). Ultimately, any conclusion about a hypothesis involves human judgment, as discussed in Sections 2.7 and 6.5. Such a judgment may persist only briefly or endure for a long time until rejection.
As discussed below, a formal hypothesis specifies the expected necessity relationship, provides a causal explanation, and specifies the domain where it is expected to hold, among others. This is also helpful in different subsequent stages of an empirical study:
Study design: It helps to focus the study based on existing knowledge (Section 8.2).
Sampling or case selection: It specifies from which theoretical domain cases must be sampled or selected (Section 8.3).
Results: It satisfies NCA’s requirement for theoretical support for a decision about (non-)rejection of the hypothesis (Sections 6.2 and 9.10).
Discussion: It helps to interpret why the hypothesis is rejected or not, and what the theoretical and practical consequences are (Chapters 10 and 12).
In principle, NCA can also be used in a hypothesis-building (inductive) approach. In situations where the phenomenon of interest is novel or poorly understood and existing knowledge is not available, empirical data may be collected to extract a preliminary necessity hypothesis from data (exploratory or theory-building study). Such a situation is further discussed in Section 8.3.2. However, this book focuses on using empirical data only for hypothesis-testing and not for exploration/hypothesis-building for two reasons:
It is normally possible to formulate a preliminary necessity hypothesis from existing knowledge (Section 7.3) without collecting new empirical data.
To obtain an empirically tested hypothesis, it is more efficient to formulate a hypothesis first, even if existing knowledge is scarce, followed by one round of data collection for testing, compared to first having a round of data collection for empirically observing a preliminary hypothesis, followed by a second round of data collection with new data for empirically testing this hypothesis. Without testing the hypothesis with new data, the hypothesis remains untested, even if it is formalized and refined afterwards. According to a basic principle of the scientific methodology, using the same data for building and testing does not qualify for a proper empirical test of the hypothesis. This ‘predesignation’ principle was articulated in the Neyman–Pearson framework of hypothesis testing (Neyman & Pearson, 1933) and has an analogue in modern machine learning, where one dataset is used for training and another for testing. In other words, “It is never OK to snoop at the data before formulating a hypothesis—at least not if the same data are to be used in testing that hypothesis” (Mayo, 1996, p. 295).
As theories in the social sciences are usually semantic (expressed in words), a necessity hypothesis is formulated qualitatively as \(X\) is necessary for \(Y\) without specifying the exact levels of \(X\) and \(Y\), other than the possibility to specify presence/high level or absence/low level of \(X\) and \(Y\) (Chapter 3). Therefore, in this chapter only the in kind dichotomous version of the necessary condition (necessity-in-kind, NiK) is considered, where the presence/absence of \(X\) is necessary for the presence/absence of \(Y\) (Figure 3.6).
7.3 Why developing necessity hypotheses is different
Developing a necessity hypothesis differs from developing a conventional probabilistic sufficiency hypothesis. Conventional hypothesis development draws on a wealth of existing knowledge about average effects. Reviewing established theories, causal explanations, and prior empirical studies is usually a central part of this process. In such cases, a literature review provides the main justification for the hypothesis, demonstrating that it is plausible, theoretically grounded, and worth (re)testing with empirical data.
In contrast, the current body of knowledge about necessity is still limited. Theoretical frameworks and empirical studies specifically addressing necessity are not yet widespread, and the traditional literature on probabilistic sufficiency offers little direct guidance. That literature hints at important factors that potentially could be necessary conditions, but usually provides no evidence. It may also contain indirect hints or isolated statements about necessity logic and causality. Although this can serve as inspiration or secondary support for developing necessity hypotheses, more is needed to develop well-grounded necessity hypotheses.
Because a cumulative body of theory and evidence on necessity causality is often lacking, developing necessity hypotheses requires an approach that relies more heavily on the analyst’s creativity, critical reasoning, and substantive expertise. By integrating solid knowledge of necessity logic and necessity causal reasoning with substantive domain knowledge, and by engaging in discussions with peers, analysts can formulate convincing and meaningful necessity hypotheses. It is often surprising how quickly a relevant preliminary necessity hypothesis can emerge once this reasoning process is clear.
Whereas developing an average-effect hypothesis generally involves dedicating most effort to the literature review, developing a necessity hypothesis centers on conceptual reasoning and logical argumentation. The goal is to construct a coherent narrative that justifies why a given condition must be present for the outcome to occur, and why the outcome will not occur if the condition is absent, and no compensation is possible.
Generally, such a narrative starts with identifying an important factor (known or presumed) that contributes to a desired outcome but is not the only factor that helps to produce the outcome. Conventional literature often establishes that such a factor is important on average.65 The next step is crucial for necessity reasoning: it formulates the expectation that the presence or a high level of the outcome will disappear if the condition is absent or is reduced, corresponding to necessity logic: if not \(X\), then not \(Y\). The narrative then explains why this disappearance occurs, for example, through what mechanisms the absence of \(X\) undermines \(Y\). This is followed by an argument that other factors cannot compensate for this absence: no substitute can replace the missing necessary condition; the outcome will not be possible without it. Finally, the narrative specifies the domain of applicability. This is the boundary within which the necessity claim holds. It clarifies whether it applies universally (e.g., oxygen is necessary for human life) or only within a specific group or context (e.g., being pregnant is necessary for having a child).
For example, it may be assumed that class attendance is necessary for school performance. A minimal narrative could be formulated as follows: Class attendance is widely recognized as an important determinant of school performance. Meta-analyses show that students who attend classes more regularly tend to achieve higher grades on average. Attendance is not the only factor: intelligence, motivation, prior preparation, and study habits also matter. However, class attendance is crucial, as it provides the behavioral foundation through which learning processes operate effectively. If class attendance is withdrawn, school performance deteriorates, a pattern consistent with necessity logic. This disappearance occurs for several reasons. Students who skip classes miss essential explanations, examples, and discussions on which exams and assignments depend. Absence removes opportunities for immediate correction and reinforcement, allowing misconceptions to persist. Reduced attendance may also weaken motivation and belonging, leading to further reductions in performance. Other factors cannot replace the absence of class attendance. Digital resources or peer notes can alleviate short-term gaps but do not replicate the real-time cognitive and social engagement that in-class participation provides. This necessity relationship applies primarily to students in traditional, face-to-face courses where class sessions deliver core content and interactive learning is integral to success. It may not apply universally, for example, in fully online, asynchronous courses where students can compensate for absence through other structured forms of engagement. Within the conventional classroom teaching format, however, attendance remains a necessary condition for school performance.66
This narrative defines not only the hypothesis (Class attendance is necessary for high school performance), but also gives a causal explanation (students miss essential learning elements, lack of interactions and motivation, no compensation possible by digital resources) and defines the theoretical domain (conventional classroom teaching).
While creativity is central for formulating the narrative, existing information sources can help generate, sharpen and justify hypotheses about necessary conditions. This chapter provides a detailed plan and practical tools for this process. While it may appear complex at first, it soon becomes almost automatic once one becomes familiar with the reasoning steps involved.
7.4 Knowledge sources for developing necessity hypotheses
The development of a formal necessity hypothesis begins with the selection of a preliminary hypothesis, based on existing knowledge. Various sources such as academic literature, practical insights, and expert judgment can inform this selection. To ensure the relevance of the necessity hypothesis for both theory and practice, knowledge from scholarly research, real-world experience, and the expertise of the analyst can be combined. Since formulating a necessity hypothesis is primarily a creative process it can rely both on explicit knowledge (documented knowledge from academia and practice) and on tacit knowledge (undocumented expert knowledge from scholars and practitioners). This process may be iterative and involve people with different knowledge backgrounds. The result is a defensible claim made by the analyst (and others), which requires a test with empirical data to be (non-)rejected. Or, according to Popper:
“[Great scientists] are men of bold ideas, but highly critical of their own ideas; they try to find whether their ideas are right by trying first to find whether they are not perhaps wrong” (K. R. Popper, 1985, p. 119)
This section elaborates on the two sources of information for selecting a preliminary necessity hypothesis: explicit knowledge and tacit knowledge.
7.4.1 Explicit knowledge
Explicit knowledge is documented in the academic literature (e.g., books, journal articles) and practice literature (e.g., best practice documents, manuals). Since practice is more focused on ‘doing’ than on ‘documenting’, less explicit knowledge is available from practice than from academia. Practice documents could contain know-how and statements that refer to necessity using terms like ‘requirements’, ‘must have’, ‘critical success factors’, or ‘bottlenecks’, etc. This section focuses on academic literature since it contains much information that can be used to formulate a preliminary necessity hypothesis. Four approaches for identifying them can be distinguished:
Sometimes the academic literature includes explicit necessity theories, whose necessity hypotheses may need to be (re)tested.
The literature may also contain statements referring to necessity.
The literature contains causal models, which are formal models primarily developed for probabilistic sufficiency causality, but elements may be useful for necessity causality as well.
The literature contains causal narratives for describing probabilistic sufficiency relationships or more general causes that contribute to the outcome that can inspire necessity reasoning.
These sources of explicit scholarly knowledge can be used to derive a preliminary necessity hypothesis. Each source is discussed below in more detail.
7.4.1.1 Necessity theories
As discussed in Section 3.4, some theories explicitly state necessity relationships. Such theories can be established or emerging, and may be ‘pure’ necessity theories consisting only of necessity relationships, or ‘embedded’ necessity theories consisting of a combination of necessity relationship and other relationships. As the examples in Section 3.4 show, often, necessity theories have not been (extensively) tested empirically. (Re)testing selected necessity relationships from these theories can provide valuable insights into their empirical support and generalizability.
7.4.1.2 Necessity statements
Phrases using terms like ‘necessary’ or ‘necessary but not sufficient’ are very common in the academic literature (and in practice). Although several statements may use the word necessity loosely as a synonym for important, many refer explicitly to necessity logic and causality, in particular when it is stated ‘for what’ the condition is necessary. For example, in the field of supply chain management, Bokrantz & Dul (2023) systematically reviewed necessity statements in major journals and identified 157 statements that can be considered theoretical necessity claims. Most of these statements have not been tested at all, or not been tested properly for necessity (e.g., using regression analysis). A similar result was found by Richter & Hauff (2022) when analyzing necessity statements in the field of international business. The literature also uses phrases that are directly related to necessity, without using the words ‘necessity’ or ‘necessary’. Examples of such phrases are listed in Table 7.1. Several phrases reflect that the necessary condition must be present to have the outcome (enablers); others reflect that the absence of the necessary conditions guarantees the absence of the outcome (constraints). Existing necessity or equivalent statements can be reviewed, and selected statements can be candidates for a preliminary necessity hypothesis.
| Enablers (The presence of X …) | Constraints (The absence of X …) |
|---|---|
| X is necessary for Y | X constrains Y |
| X is needed for Y | X limits Y |
| X is critical for Y | X blocks Y |
| X is crucial for Y | X bounds Y |
| X is essential for Y | X restricts Y |
| X is indispensable for Y | X is a barrier for Y |
| X is a prerequisite for Y | X is a bottleneck for Y |
| X is a requirement for Y | Without X there cannot be Y |
| X is a conditio sine qua non for Y | |
| X is a pre-condition for Y | |
| X allows Y | |
| X enables Y | |
| There must be X to have Y | |
| Y requires X |
7.4.1.3 Causal models
The academic literature contains numerous causal models based on probabilistic sufficiency relationships between variables. Such models are generally evaluated through regression-based methods. All or some of the relationships in the models could be revisited with a necessity perspective on the relationship. Revisiting the relationship with a new causal perspective is not a replication or robustness test of the existing model. It yields a new necessity model or theory that provides new insights.
An example of a probabilistic sufficiency model is shown in Figure 7.1 about nine factors that contribute to academic achievement. While these factors have only been modeled as having an average effect on the outcome and were tested with regression-based studies, some studies in the literature refer to these factors as critical (e.g., X. Ma & Wang, 2001), in the meaning of ‘highly influential’. This model could be revisited from the perspective of necessity causality thereby re-interpreting the word critical as necessary.
Figure 7.1: Example of a causal model (originally for describing probabilistic sufficiency) for academic achievement with 9 factors that contribute to academic achievement. After: Walberg (1984).
It is also possible that two concepts that were not (directly) related in a probabilistic sufficiency-based model, are related in a necessity model. The original role of a concept (e.g., independent, dependent, mediator, moderator, control) is not relevant for formulating a new role for the concept in a different causal model (condition or outcome in a necessity model).
An additional possibility is to evaluate suggestions for future studies that are made in a study based on average effects. It is not uncommon that a publication about a model suggests to apply NCA in future studies to (also) investigate the phenomena from a necessity perspective. Likewise, literature reviews of studies with regression-based causal models and thus probabilistic sufficiency perspectives often suggest exploring the phenomenon with NCA and thus with a necessity perspective as well to further enrich the state of knowledge. Chapter 1 gives examples of publications that suggest using NCA in future research.
7.4.1.4 Causal narratives
When proposing or justifying hypotheses, academic publications often use causal narratives: stories with explanations that connect causes to effects. For example, in describing probabilistic sufficiency relations, the narrative outlines the mechanism through which a cause contributes to an effect. These causal mechanisms can be critically evaluated by assessing whether the absence of a particular factor disrupts the mechanism. By systematically analyzing causal terminology (Table 7.1), and identifying factors that could break the mechanism (potential necessary conditions) it may be possible to formulate a preliminary necessity hypothesis. For example, Hausknecht et al. (2008) argue that organizational commitment has a negative effect on absenteeism as follows:
“[…] [O]rganizational commitment is defined as a collective sense of affective or emotional attachment to an organization […]. Employees are thought to develop organizational attachments via experiences in the unit where they work […]. For example, employees of high-commitment units may engage in more community maintenance behaviors, including regular attendance at work. Thus, high-commitment work units are likely to be associated with stricter attendance norms […]. At the individual level, meta-analytic evidence has supported the negative relationship between commitment and absenteeism […]. In view of these findings, and the premise that high commitment reflects a strong collective attachment to organizational values and goals, including attendance at work, we expect that at the unit level of analysis, organizational commitment is negatively related to absenteeism” (Hausknecht et al., 2008, p. 1225)
Although in this quote the relationship between organizational commitment and absenteeism is clearly described as a probabilistic sufficiency relationship, such a causal narrative could be approached from the perspective of necessity as well. One might wonder if the absence of organizational commitment would stop low absenteeism. In this example, the authors themselves explicitly suggest this by stating:
“When the level of analysis is the work unit, organizational commitment may be a necessary but insufficient condition for low absenteeism” (Hausknecht et al., 2008, p. 1225)
and after conducting a regression analysis they conclude (although this cannot be supported by a regression-based analysis):
“Taken together, the findings […] show that high organizational commitment is necessary to avoid high levels of absenteeism” (Hausknecht et al., 2008, p. 1239)
Explicit statements about necessity on causal reasoning in the literature may support the necessity argument, even though necessity was not tested in the original study (Section 7.4.1.2).
7.4.2 Tacit knowledge
Undocumented expert knowledge can be used to formulate a preliminary necessity hypothesis. Experts often possess theoretical or practical insights that can be interpreted in terms of necessity, in particular when these insights are described with phrases from Table 7.1. This approach is especially useful in fields where explicit knowledge is scarce, while experts believe necessity relationships exist.
Two types of experts can be distinguished: academic experts (scholars) and practitioner experts (practitioners). Scholars are persons with deep theoretical or empirical knowledge developed through research. Drawing upon their expertise, comprehensive understanding of the academic literature, and possibly practical experience, they may identify potential necessary conditions either directly by recognizing explicit hypotheses, or indirectly, by recognizing important factors for an outcome that could also be interpreted as necessary conditions. Practitioners are professionals with hands-on experience in a field outside academia (e.g., consultants, managers, employees, engineers, policy-makers, educators, designers). While they may not articulate their knowledge as hypotheses, they often operate based on an implicit “theory in use” (Dul & Hak, 2008), a solid belief grounded in practice and experience that explains why a certain factor causes a certain effect. These informal theories may include necessity logic. This is not unlikely as practitioners commonly focus on factors that are required for success (‘critical success factors’), or on fail factors that prevent success by identifying and addressing ‘bottlenecks’. Their practical insights can be systematically translated into preliminary necessity hypotheses.
An array of qualitative methods exists to capture experts’ tacit knowledge, ranging from informal conversations, in-depth interviews, and creative group sessions to more structured approaches such as think-aloud protocols, and Delphi studies. However, tacit knowledge has a complex status in academia. It is often valued implicitly (many scholars rely on it), but devalued explicitly when used as evidence or justification, because it does not fit the norms of transparency, replicability, and formal theorizing that define “scientific” knowledge. If a hypothesis is (partly) developed based on tacit practitioner knowledge, the approach used for obtaining this knowledge can be reported and it can be justified that such knowledge is not anecdotal, but accumulated through extensive experience with the practice, system, or process, thus empirically grounded.
7.5 Developing a formal necessity hypothesis
A formal necessity hypothesis has five characteristics:
Theory-grounded: the hypothesis is embedded in a necessity theory. According to Chapter 3, a theory is defined when four elements are defined: focal unit, concepts (condition and outcome), (direction of) the hypothesis67, and theoretical domain.
Testable. A hypothesis is empirically testable if the concepts \(X\) and \(Y\) can be operationalized into measurable variables.
Non-trivial. A hypothesis is non-trivial when both the absence of the condition and presence of the outcome are possible, otherwise it is trivial.
Plausible. A hypothesis is plausible when virtually no cases are reasonably expected in the empty corner.
Complete. A hypothesis is complete when a plausible explanation for why condition \(X\) is necessary for outcome \(Y\) is available.
The development process of a formal necessity hypothesis consists of three steps that are shown in Figures 7.2, 7.4, and 7.6, respectively.
In the first step (Figure 7.2), a preliminary necessity hypothesis is selected based on the available sources of information (Section 7.4), and is embedded in a preliminary necessity theory by defining focal unit, concepts, (direction of) the hypothesis, and theoretical domain. Concepts are defined such that they are measurable and the hypothesis becomes testable. After this step the hypothesis is theory-grounded and testable.
In the second step (Figure 7.4) a thought experiment is conducted in which the preliminary necessity theory is challenged and possibly revised. After this step the hypothesis is also non-trivial and plausible.
In the third step (Figure 7.6) a detailed causal explanation for the hypothesis is given (Chapter 2). This final step makes the hypothesis complete. The formal hypothesis is now ready to be tested with empirical data.
Below, each step is discussed in more detail and illustrated with an example of the preliminary hypothesis that ‘An internet connection is necessary for attending an online meeting’.
7.5.1 Step 1: Define a preliminary necessity theory
The first step transforms the preliminary hypothesis into a preliminary necessity theory. This is a theory that has measurable concepts, a hypothesis describing the expected necessity relationship between the concepts, and a description of the theory’s focal unit and theoretical domain (Chapter 3). However, this theory is not yet challenged in a thought experiment, which is the topic of Step 2. Figure 7.2 shows the hypothesis development tool for Step 1 of the development process.
Figure 7.2: Hypothesis development tool for Step 1 of the development process of a formal necessity hypothesis: Defining a preliminary necessity theory.
To begin, a preliminary hypothesis is formulated based on existing knowledge (Section 7.4). This hypothesis is then elaborated into a preliminary necessity theory by precisely defining the four constituent elements of a necessity theory:
- Precise definition of focal unit. The focal unit is the entity of the theory to which the theory applies (e.g., person, team, country, project, etc.).
Sometimes, defining the focal unit is simple. For example, the Theory of Planned Behavior (TPB, Ajzen, 1991) is a theory about individual behavior. The focal unit is evident: ‘individual’ or ‘person’. Other times, the focal unit may not be immediately clear, for example, when theorizing the relationship between parent height and child height (Galton, 1886).68 The focal unit is neither parent nor child but ‘parent-child pair’. The focal unit is expressed as a singular noun, not a plural noun. Specification is important because cases to be sampled or selected for testing the hypothesis must be specific units/examples of the focal unit (e.g., a person, a parent-child pair). Additionally, the concepts of the theory are characteristics of the focal unit.
- Precise definition of the concepts. The concepts are the varying characteristics that are relevant for the theory. For a necessity theory, the concepts are condition \(X\) and outcome \(Y\). The outcome often represents something either desirable or undesirable. Desirable outcomes include, health, well-being, performance, innovation, success, etc. Undesirable outcomes may be sickness, risk, disease, failure, etc. The condition is the concept that must be present to have the outcome.
In the TPB example the condition is Intention for the behavior (of a person), and the outcome is Performance of the behavior (of a person). In the body height example the condition is Parent body height (of a parent-child pair) and the outcome is Child body height (of a parent-child pair).
- Precise specification of the hypothesis. Once the concepts are defined, the hypothesis can be formulated as \(X\) is necessary for \(Y\). If the condition is desirable, the hypothesis could be formulated using one of the enabling terms of Table 7.1 to indicate that the necessary condition supports the outcome. If the condition is undesirable, the hypothesis could be formulated using one of the constraining terms of Table 7.1 to indicate that the absence of the necessary condition prevents the outcome.
Usually, theories in the social sciences are semantic, such that levels of its concepts are only expressed in binary terms and qualitatively (e.g., in terms of absence/presence, or low/high), without giving exact levels. For example, the presence of \(X\) is necessary for the presence of \(Y\), or high \(X\) is necessary for high \(Y\). This is the implicit meaning of \(X\) is necessary for \(Y\). The conceptual model of this hypothesis is shown in Figure 3.6-top-left. This figure shows a specification of the direction of necessity as or high-high (Section 3.5): presence/high \(X\) is necessary for presence/high \(Y\). Other possible directions are absence/low \(X\) is necessary for presence/high \(Y\) (; low-high), presence/high \(X\) is necessary for absence/low \(Y\) (; high-low), and absence/low \(X\) is necessary for absence/low \(Y\) (; low-low) (see the other conceptual models in Figure 3.6). The specified direction of the hypothesis defines the expected empty corner: which corner of an \(XY\)-plot is expected to be empty if the hypothesis holds (Figures 3.7 and 3.8).
For the TPB example, the preliminary hypothesis is: the presence of intention for the behavior is necessary for the presence of performance of the specific behavior. For the body height example, the preliminary hypothesis is: a high parent body height is necessary for a high child body height. In both hypotheses, the direction is high-high.
- Precise definition of the theoretical domain. The theoretical domain specifies the universe of units (cases) of the focal unit to which the hypothesis is expected to apply. This universe can be defined in various ways, such as temporally (e.g., a specific time period), geographically (e.g., a region, country, or global context), demographically (e.g., age group, profession), or conceptually (e.g., type of organization or system). The specified theoretical domain determines which cases must be sampled (from a population in the theoretical domain) or selected (directly from the theoretical domain).
For the TPB example, the theoretical domain is ‘All adults in the world’. For the body height example, the theoretical domain is ‘All adult parent-child pairs in the world, when children have reached adulthood’.
After the preliminary hypothesis is embedded in theory, the first question (Q1) of Figure 7.2 checks if precise definitions are available for four elements of the theory: focal unit, concepts, hypothesis and theoretical domain.
Q1: Are all four elements of the necessity theory precisely defined? When all elements of a necessity theory are specified, the hypothesis is considered to be embedded in theory.
Q2: Are the \(X\) and \(Y\) concepts measurable? This question checks whether the concepts can be transformed into measurable variables. When concepts cannot be measured directly, substitute measurements (proxies) can be considered, such as measuring an informant’s perception or belief about an otherwise unmeasurable behavior or event, although this approach may compromise measurement validity. NCA accepts any meaningful, valid, and reliable scores of \(X\) and \(Y\) as input (Chapter 8). When the \(X\) and \(Y\) concepts of the necessity theory can be operationalized into measurable variables, the hypothesis is considered testable.
7.5.1.1 Example Step 1
The hypothesis development tool for Step 1 (Figure 7.2) can be applied to the example about the necessity of an Internet connection for attending an online meeting. First the preliminary necessity hypothesis that is formulated based on expert knowledge is selected.
Select preliminary hypothesis: Internet connection is necessary for Online meeting attendance.
Next, the hypothesis is embedded in theory.
Elaborate hypothesis into theory:
Q1: Are all four elements of the necessity theory precisely defined? - Focal unit: ‘Person’. A person may or may not have an Internet connection and the person may or may not attend an online meeting.
Condition concept \(X\): Internet connection, defined as ‘the communication pathway between a device such as a computer or smartphone and the global network known as the Internet’. This concept can be absent (no Internet connection) or present (Internet connection).
Outcome concept: \(Y\): Online meeting attendance. This concept is defined as ‘being part of a real-time, multimedia interaction among multiple participants’. This concept can also be absent (no Online meeting attendance) or present (Online meeting attendance).
Direction of the hypothesis: the presence of an Internet connection is necessary for the presence of an Online meeting attendance (high-high).
Theoretical domain: All persons in the world
Answer: yes.
Q2: Are the concepts Internet connection (\(X\)) and Online meeting attendance (\(Y\)) measurable? The presence or absence of a person’s Internet connection, and the presence or absence of a person’s online meeting attendance can be measured in different ways such as by analyzing the platform and network information, user activity logs, observations, questionnaires, and other data sources. This makes the hypothesis testable \(\rightarrow\) yes.
Figure 7.3 shows the conceptual model and the expected empty corner for this preliminary theory. The preliminary hypothesis predicts that all persons that do not have an Internet connection also do not attend an online meeting, and that all persons that attend an online meeting have an Internet connection.
Figure 7.3: Conceptual model and \(XY\)-table with expected empty corner when the necessity proposition: having an Internet connection is necessary for Online meeting attendance.
7.5.2 Step 2: Conduct thought experiments
The second step of the development process of a formal necessity hypothesis consists of conducting thought experiments with the preliminary theory including its hypothesis, which is now elaborated into theory and is testable. Figure 7.4 shows Step 2 of the development process.
Figure 7.4: Step 2 of the development process of a formal necessity theory: using thought experiments to challenge the preliminary theory.
A thought experiment in NCA is a mental evaluation to check the validity of the necessity theory and its hypothesis. Since a necessity hypothesis has only one \(X\) and one \(Y\), the prediction of the necessity theory is straightforward: the outcome is absent if the condition is absent, and the condition is present if the outcome is present. Therefore it is usually easy to evaluate the hypothesis mentally. The basic idea is to evaluate if the expected empty corner is indeed empty. Assuming that presence/high \(X\) is necessary for presence/high \(Y\), the expected empty corner is the upper-left corner.
A thought experiment can help to check if the preliminary theory may be trivial or implausible. Testing trivial or implausible hypotheses with empirical data is not interesting and a waste of resources. Furthermore, if the thought experiment shows that many counterexamples exist (the expected empty space is not empty) the hypothesis should be rejected. With a few counterexamples, further evaluation is needed and the theory may be refined by redefining the concepts or the theoretical domain, such that the (revised) hypothesis of the revised theory is plausible. Three thought experiments can be done (references to the questions in Figure 7.4 are given in brackets):
Thought experiment for non-triviality (Q3, Q4).
Thought experiment for plausibility (Q5, Q6).
Thought experiment for counterexamples and exceptions (Q7, Q8, Q9).
7.5.2.1 Non-triviality
A trivial necessity hypothesis is a hypothesis that is expected to hold, but is not informative because the condition or the outcome is nearly (constant) in the situation of interest (i.e., in the theoretical domain where the hypothesis is supposed to hold). Triviality applies when the condition is virtually always present, or the outcome is virtually always absent. The first type of triviality where the condition is constant (always present) and the outcome varies, is depicted in Figure 3.11-bottom-left. An example is the hypothesis that oxygen (\(X\)) is necessary for attending an online meeting (\(Y\)). This hypothesis holds because when an online meeting is held (\(Y\) is present), oxygen (\(X\)) is present and needed for the participants to have the meeting. The hypothesis is trivial because oxygen is virtually never absent. The second type of triviality where the outcome is constant (always absent) and the condition varies, is depicted in Figure 3.11-top-left. An example is the hypothesis that having good food during lifetime (\(X\)) is necessary for a human to reach age 125. The necessity claim of this hypothesis is reasonable, but it is trivial because reaching age 125 is rare.
The situation of a constant condition that is always absent and an outcome that varies (Figure 3.11-bottom-right) and the situation of a constant outcome that is always present and a condition that varies (Figure 3.11-top-right) both result in a clear rejection because the outcome can be present when the condition is absent. In these situations the necessary condition hypothesis is immediately rejected, not because it is trivial but because it is not necessary.
Therefore, there are two questions to be raised and answered in a thought experiment for checking triviality of the theory and its hypothesis (Figure 7.4):
Q3. Do cases exist with the outcome? This question checks if the outcome is virtually always absent. If not, the theory is trivial.
Q4. Do cases exist without the condition? This question checks if the condition is virtually always present. If not, the theory is trivial.
If either of these questions is answered negatively, the theory and its hypothesis can be considered trivial and not worth testing with empirical data.
7.5.2.2 Plausibility
A plausible hypothesis is a non-trivial hypothesis (both the condition and outcome can vary) that cannot be rejected in a thought experiment. The goal of this thought experiment is to consider condition and outcome together to find supportive cases for the hypothesis, and in particular possible counterexamples that challenge the hypothesis. This thought experiment for plausibility resembles an empirical test of a necessity hypothesis using a small number of cases (Section 8.3.2).
To find supportive cases, two questions can be asked:
Q5. Do all imaginable cases with the outcome have the condition? If the answer yes, this is an indication that the hypothesis may hold.
Q6. Do all imaginable cases without the condition lack the outcome? If the answer yes, this is another indication that the hypothesis may hold.
If the answer to both questions is affirmative, all cases from the theoretical domain that can be imagined are supportive cases for the hypothesis. When only supportive cases and no counterexamples can be imagined, the hypothesis is plausible. If the answer to one or both questions is negative, counterexamples are identified. These counterexamples need further evaluation before a decision can be made about the plausibility of the hypothesis.
7.5.2.3 Counterexamples and exceptions
When counterexamples outnumber supportive cases, it is clear that the hypothesis needs to be rejected. However, when there is a limited number of counterexamples compared to supportive cases an additional evaluation is needed. Therefore, when counterexamples exist, the following question is answered first:
Q7: Are counterexamples uncommon? If counterexamples are common, the theory should be rejected. If counterexamples are uncommon, they need further evaluation before the analyst makes a decision to reject or keep the hypothesis, possibly after refinement of the theory or adoption of the typicality perspective (see below).
The further analysis consists of considering the possibility of incorporating or excluding counterexamples in the theory by redefining the condition \(X\), the outcome \(Y\), or the theoretical domain. Broadening the condition \(X\) or narrowing the outcome \(Y\) can make counterexamples become supportive cases within the redefined theory. Narrowing the theoretical domain excludes the counterexamples from the redefined theory.69
Therefore, the analyst has three options to handle counterexamples: (1) broadening the condition, (2) narrowing the outcome, and (3) narrowing the theoretical domain. It is the analyst’s choice which option to adopt. Option 1 broadens the ways in which the condition can allow the outcome. Often, the broader condition is a higher-order concept that integrates the original concept and the counterexample concept. By broadening the condition, counterexamples that previously displayed an absence of the condition now display a presence of the condition, and become supportive cases under the refined theory. However, this risks making the condition too broad and weakening its explanatory power: almost any facilitating mechanism could then be considered part of the condition. Option 2 narrows the outcome to a more specific form, reducing the scope of what is being explained. With the redefined outcome concept, the exception becomes a supportive case that does not violate the hypothesis. However, this risks excluding cases that are theoretically relevant. Option 3 restricts the theoretical domain by focusing only on a subset of cases where necessity holds. With a new definition of the theoretical domain, the exception should not have been selected for testing the hypothesis because the claim of the hypothesis is only valid in a defined theoretical domain. While this approach preserves the original concepts and hypothesis, it reduces the generalizability of the theory. Nevertheless, it seems reasonable to adopt option 3 since a theoretical domain is seldom homogeneous and universal. In option 3, the original hypothesis is retained but its theoretical domain is narrowed. This strategy preserves the preliminary hypothesis while specifying the scope of its application more precisely, provided that no known counterexamples exist.
If any of the options for handling counterexamples is selected, it is assumed that the redefined theory holds for all imaginable cases selected from the theoretical domain.70
If counterexamples exist the following question needs to be answered:
Q8. Can counterexamples be handled by redefining concepts/domain? If the answer is yes, the theory is adjusted, so that there are no more counterexamples. After the counterexamples are handled by redefining the necessity theory (concepts or domain) it is assumed that the hypothesis holds for all imaginable cases of the theoretical domain, such that the testing of the hypothesis can start with a deterministic view on necessity (Section 2.4.1).
If the answer to Q8 is no, it may be that (some) counterexamples remain. The analyst now has two options to proceed:
Adopting the deterministic perspective of necessity, single counterexamples reject the hypothesis (Section 2.4.1).
Adopting the typicality perspective of necessity. The counterexamples are rare and cannot be explained or handled. They are considered exceptions that are not driven by necessity and do not reject the necessity hypothesis (Section 2.4.3). If counterexamples are not rare, the hypothesis should be rejected. Note the difference between exceptions and noise (Section 4.5.4).
Adopting either perspective is a matter of analyst judgment. The deterministic perspective may be adopted when strict logical necessity is required without exceptions, when the theoretical domain contains few cases, or when logical rigor is prioritized. The typicality perspective may be adopted when it is recognized that rare exceptions can exist that cannot be explained with necessity logic.
Therefore, if counterexamples remain the following question needs to be answered:
Q9. Is the typicality perspective adopted? If the answer is no, the theory’s hypothesis is rejected; if yes, it is plausible and (rare) exceptions are allowed.
7.5.2.4 Example Step 2
The development tool for Step 2 (Figure 7.4) can be applied to the example about the necessity of an Internet connection for attending an online meeting. First, the preliminary necessity theory from Step 1 is selected. The first question for the triviality check is:
Q3. Do people exist who attend an online meeting? In 2021, about 500 million people daily had online meetings. Since then the number of daily users presumably has considerably increased \(\rightarrow\) yes.
The second question of the triviality check is:
Q4. Do people exist without an Internet connection? In 2025, about one-third of the world population had no access to the Internet \(\rightarrow\) yes.
Since both questions are answered affirmatively, the hypothesis is non-trivial for the selected theoretical domain (all people in the world). To check for the plausibility of the hypothesis, another two questions are raised:
Q5. Do all imaginable persons attending an online meeting have an Internet connection?
Q6. Do all imaginable persons without Internet connection not attend an online meeting? The answer to both questions is no. It is possible that persons attend an online meeting at home or in an organization in a Local Area Network (LAN). These persons attend an online meeting without being connected to the Internet. Both questions Q5 and Q6 are therefore answered negatively, indicating that counterexamples exist. Trying to find counterexamples, even if they seem uncommon, is an essential part of conducting a thought experiment to test plausibility of a necessity hypothesis \(\rightarrow\) no.
Since counterexamples exist, the next step is to evaluate if they are uncommon:
Q7: Are people with LAN connections uncommon? For the online meeting example, the number of persons attending an online meeting with a LAN connection compared to the number of persons attending an online meeting with an Internet connection is relatively small. Attending an online meeting with a LAN connection is relatively uncommon, suggesting further analysis of these cases is needed to refine the theory \(\rightarrow\) yes.
Q8: Can counterexamples be handled by redefining concepts/domain? The three options to handling counterexamples are:
Broadening the condition means that the original definition of Internet connection (the communication pathway of devices such as computer or smartphone to the global network known as Internet) is broadened such that it incorporates LAN networks. The concept Internet connection could be broadened to the concept Digital network connection defined as the communication pathway of devices such as computer or smartphone to a digital network. ‘Digital network’ now includes Internet and LAN.
Narrowing the outcome means that the original definition of the outcome \(Y\) (Online meeting attendance: being part of a real-time, multimedia interaction among multiple participants) is narrowed such that it incorporates the exception. For example, Online meeting attendance could be narrowed to Public online meeting attendance defined as being part of a real-time, multimedia interaction among multiple participants from the general public, excluding private online meetings (that use LAN) from the definition. For the cases that were previously counterexamples, the condition ‘Internet connection’ is absent (though LAN is present), and the redefined outcome concept ‘Public online meeting attendance’ is also absent (because the LAN meeting is considered a private meeting).
The third way to handle an exception is to exclude counterexamples from the theory by narrowing the theoretical domain. In the online meeting example, narrowing the theoretical domain means that the original definition, ‘all persons in the world’, would be revised such that it excludes people that use a LAN network in a private meeting. For example, the domain All persons in the world could be narrowed to All persons who are geographically distant from each other, which would exclude those in a local gathering (home, organization) from the theoretical domain.
In summary, for the example, the three possible ways to handle the exception by redefining concepts/domain are:
Option 1 (broadening condition):
Hypothesis: Digital network connection is necessary for Online meeting attendance. Focal unit: Person. Condition \(X\): Digital network connection: the communication pathway of devices such as computer or smartphone to a digital network. Outcome \(Y\): Online meeting attendance: real-time, multimedia interaction among multiple participants. Theoretical domain: All persons in the world.
Option 2 (narrowing the outcome):
Hypothesis: Internet connection is necessary for Public online meeting attendance. Focal unit: Person. Condition \(X\): Internet connection: the communication pathway of devices such as computer or smartphone to the global network known as Internet. Outcome \(Y\): Public online meeting attendance defined as a real-time, multimedia interaction among multiple participants from the general public. Theoretical domain: All persons in the world.
Option 3 (narrowing the theoretical domain):
Hypothesis: Internet connection is necessary for Online meeting attendance. Focal unit: Person. Condition \(X\): Internet connection: the communication pathway of devices such as computer or smartphone to the global network known as Internet. Outcome \(Y\): Online meeting attendance: real-time, multimedia interaction among multiple participants. Theoretical domain: All persons who are geographically distant from each other.
It is supposed that the second option is selected for further development in Step 3 (Section 7.5.3.1) \(\rightarrow\) yes.
7.5.3 Step 3: Provide causal explanation
In the previous step, the preliminary necessity theory including its hypothesis was either rejected or accepted during a thought experiment, possibly after revision of concepts or theoretical domain, or by adopting the typicality perspective. If accepted, the next step is to explain why \(X\) is necessary for \(Y\). This involves developing a causal explanation. During the formulation of the preliminary necessity theory including its hypothesis and during the process of conducting thought experiments, the analyst implicitly used a causal reasoning about necessity. In the last step before empirical testing of the hypothesis, the causal reasoning is explicitly articulated and further deepened. Since causality is a human interpretation rather than a directly observable phenomenon (Chapter 2), the analyst must construct a plausible and coherent narrative to justify the proposed necessity-based causal link.
Thus, describing the causal explanation is a creative process that can be based on insights obtained during the thought experiments, and can be enriched with causal reasoning from the academic literature, from practice, and from the analyst’s own experience (Sections 7.4.1 and 7.4.2).
The causal narrative can be inspired by answering three questions that are positioned in the \(XY\)-plot shown in Figure 7.5:
Explaining why \(X\) is an enabler for \(Y\) (why the presence of \(X\) allows the presence of \(Y\)).
Explaining why \(X\) is a constraint for \(Y\) (why the absence of \(X\) leads to the absence of \(Y\)).
Explaining why there is no substitute for the absence of \(X\) (why no other paths without \(X\) can lead to the outcome \(Y\)).
The first question builds on conventional causal reasoning, exploring how one factor allows an outcome and in this way helps to produce it. The second and third, however, go beyond standard approaches: they introduce the need to reason counterfactually and to explore the uniqueness of \(X\) in the causal structure. These perspectives encourage creative and rigorous thinking about necessity and irreplaceability.
One core idea of necessity causality is that the absence of \(X\) causes the absence of \(Y\). More precisely, if the necessity cause is reduced or removed, the present or high outcome should disappear. This thought experiment resembles a true ‘loss of function’, see the necessity experiment discussed in Section 8.2.4.
Figure 7.5: Questions for explanation why \(X\) is necessary for \(Y\). \(\neg X\) and \(\neg Y\) mean absence of \(X\) and \(Y\).
The three guiding questions are part of Step 3 of the hypothesis development tool (Figure 7.6).
Figure 7.6: Hypothesis development tool for Step 3 of the development process of a formal necessity hypothesis: Adding a causal explanation.
Q10. Is it explained why \(X\) enables \(Y\)? The accepted hypothesis from the thought experiments is first considered from the perspective that \(X\) is an enabler for \(Y\). To explain why a condition enables an outcome, the starting point is the presence of \(X\). The explanation then focuses on how this presence can plausibly contribute to producing the outcome, possibly through a chain of necessity (Sections 3.3 and 4.6.4). Common narratives about the causal mechanisms through which \(X\) contributes to \(Y\) can support this explanation. While such narratives are used to describe probabilistic sufficiency where the presence of \(X\) increases the likelihood of \(Y\), the same reasoning can be used to illustrate \(X\)’s role as a contributor: when \(X\) is present, \(Y\) may occur, but it is not guaranteed; \(Y\) may still be absent. The contributor becomes only an enabler when also the next two questions apply: if \(X\) is absent, \(Y\) will not occur.
Q11 Is it explained why \(X\) constrains \(Y\)? Now \(X\) is considered from the perspective of being a constraint for \(Y\). To explain why a condition constrains an outcome, the starting point is the absence of \(X\). The explanation then focuses on how this absence disrupts the causal mechanism through which \(X\) contributes to the outcome as described in the answer to Q10, thereby preventing the outcome from occurring.
Q12. Is it explained why there is no substitute for the absence of \(X\)? But even if \(X\) is not present, it may be that its absence is compensable by another factor or path. Therefore, the hypothesis is considered from the perspective of the possibility of substitutes. To explain why there is no substitute for the absence of a condition, the starting point is again the absence of \(X\). The explanation then focuses on whether any alternative path (mechanism) could plausibly exist to still produce the outcome. If no viable substitutes exist, the absence of \(X\) guarantees the absence of \(Y\). If an alternative path is discovered that was not identified in Step 2, the analyst may return to Q7 of Step 2.
Q13. Are temporal aspects considered? If Q10-Q12 are answered affirmatively, the three components (enabler, constraint, substitute) provide the basis for a narrative for explaining why \(X\) is a necessary cause for \(Y\). In addition to the explanation that \(X\) is essential to produce \(Y\), and that its absence cannot be substituted, also three temporal aspects need to be considered. First, one fundamental requirement for establishing causality is temporal order: \(X\) must precede \(Y\). If this sequence is not self-evident, the expected temporal ordering should be explicitly stated and justified. Doing so helps to address the risk of reverse causality (Section 5.7). Second, it is important to clarify whether \(X\) is necessary for the onset of \(Y\) or for its continuation. For instance, a match is necessary to ignite a fire (onset), but not to keep it burning (continuation). In contrast, oxygen is necessary for both the onset and continuation of the fire. This distinction can be incorporated into the causal explanation or into the definition of the outcome concept itself. Third, the necessity of \(X\) for \(Y\) may be time-dependent. That is, \(X\) may only be necessary during certain time periods. For example, in the past, attention to sustainability may not have been necessary for a company’s success, whereas it may be essential today. This time-sensitive aspect of necessity can be included either in the causal narrative or in the specification of the theoretical domain.
After providing the temporal aspects, the hypothesis is complete. The formal hypothesis is now fully specified and is ready for empirical testing:
Hypothesis: \(X\) is necessary for \(Y\). (add direction if needed)
Focal unit = …
Concept \(X\) = …
Concept \(Y\) = …
Theoretical domain = …
Causal explanation = …
7.5.3.1 Example Step 3
The hypothesis development tool for Step 3 (Figure 7.6) can be applied to the example about the necessity of an Internet connection for attending an online meeting. The accepted hypothesis after the thought experiments is ‘Internet connection is necessary for Public online meeting attendance’.
Q10. Is it explained why Internet connection enables Online meeting attendance? First, it is evaluated if an Internet connection can be considered as an enabler for Public online meeting attendance. This enabling role can be explained as follows: the presence of a communication pathway makes it possible for a participant to connect to a platform and to communicate remotely. The platform provides the necessary technical infrastructure for video, audio, and chat functionalities that allow a public online meeting to occur71 \(\rightarrow\) yes.
Q11. Is it explained why Internet connection constrains Public online meeting attendance? Next, it is considered if an Internet connection is a constraint for a public online meeting. The constraining role of an Internet connection can be explained as follows. Without a communication pathway it is also not possible to connect to a digital platform, such that participants are unable to connect remotely. The essential infrastructure for real-time communication is missing. As a result, the conditions required for a public online meeting are not met, and the person cannot attend the meeting. The absence of the communication pathway thus blocks the occurrence of the outcome \(\rightarrow\) yes.
Q12. Is it explained why there is no substitute for the absence of \(X\)? Next, it is considered if, when a digital platform is absent, other technologies for example, email or satellite communication could serve as substitutes. Email is an asynchronous communication tool that may support collaboration in a broader sense, but does not enable the real-time interaction required for a public online meeting. Satellite communication may offer connectivity for specific groups or in remote areas, but does not facilitate general public online meetings. If no substitute exists in the theoretical domain, with the absence of an Internet connection the outcome cannot be achieved, confirming that \(X\) is irreplaceable in this context \(\rightarrow\) yes.
Q13. Are temporal aspects considered? An online meeting cannot begin without a working Internet connection already in place. Thus, the Internet connection must exist before or at least at the onset of the meeting. The Internet connection is necessary both for the onset of the online meeting (to initiate the call or session), and the continuation of the meeting (to maintain the live connection throughout). The necessity of an Internet connection for an online meeting is time-dependent in a technological sense: Before Internet-based communication tools existed (e.g., before the 1990s), an Internet connection was not necessary; online meetings did not exist as a concept, so only after 1990 the hypothesis is non-trivial. In the future, alternative technologies might make traditional Internet connections obsolete and become substitutes \(\rightarrow\) yes.
If all questions are answered affirmatively, the hypothesis is now a formal necessity hypothesis, and is ready to be tested with empirical data.
References
Factors that are not important on average may still be necessary (Table 11.5).↩︎
Elements of such a reasoning can be found in Tynan et al. (2020). In this study the necessity-in-kind analysis showed that class attendance is necessary for high grades (\(d\) = 0.28, \(p\) < 0.001), and the subsequent necessity-in-degree analysis found that students needed to attend at least 50% of classes to achieve a grade of 90% or higher.↩︎
In the context of formulating a formal hypothesis for testing, the term ‘hypothesis’ instead of ‘proposition’ (Chapter 3) is used to describe the relationship between the theory’s concepts.↩︎
Narrowing the condition \(X\) weakens the theory because it applies to fewer cases from reality. Broadening the outcome \(Y\) and broadening the theoretical domain adds new counterexamples.↩︎
It is, of course, possible that during empirical testing, counterexamples emerge that were not anticipated during the thought experiment. In the NCA method, counterexamples are empirically defined as uncommon cases in the expected empty space, far from the ceiling, see Section 4.5.4. Such findings result in an empirical rejection of deterministic necessity. After reporting this result, the analyst may then reflect on whether the original deterministic interpretation of necessity was overly strict. However, adopting the typicality perspective after the results are known, and suggesting that this causal perspective was adopted before the results were known, should be avoided (HARKING, see Hollenbeck & Wright, 2017).↩︎
However, the mere presence of the Internet connection does not guarantee that the meeting will be attended. Other conditions must also be met, such as the participant’s willingness to join, availability, and the scheduling of the meeting itself. Thus, the Internet connection enables the possibility of the outcome (a public online meeting), but it does not ensure its occurrence. Ensuring the outcome would mean that corner 4 is also empty. However, the hypothesis is only about necessity, and does not make a statement whether or not the condition is also sufficient. For the hypothesized necessity, corner 4 is not relevant.↩︎