3.1 Preliminaries

In this section, I demonstrate that, contrary to claims by QCA proponents, quantitative methods are firmly grounded in sets and Boolean algebras. As conventional statistics are based on a more general Boolean algebra, the tools available to quantitative researchers allow much richer ways to understand and manipulate sets. The fact that QCA proponents are unaware of the connection between sets, Boolean algebras, and statistics is not altogether surprising; scholars who have not made a concerted study of statistics may fail to realize the deep connection. Goertz and Mahoney (Reference Goertz and Mahoney2012, 16–17) go so far as to claim erroneously that there are “virtually” no books on logic and set theory in the social sciences!

Econometrics textbooks commonly used by graduate students in political science (e.g., Gujarati and Porter (Reference Gujarati and Porter2008), Greene (Reference Greene2011), and Wooldridge (Reference Wooldridge2012)) usually relegate the discussion of set theory to a paragraph in an appendix. Maddala and Lahiri (Reference Maddala and Lahiri2010) include a brief paragraph in an early chapter.Footnote ⁶ Students looking for a rigorous discussion of set theory need to turn either to a serious probability text (e.g., DeGroot and Schervish (Reference DeGroot and Schervish2010) or Casella and Berger (Reference Casella and Berger2002)) or an intermediate level econometrics text (e.g., Amemiya (Reference Amemiya1994), Poirier (Reference Poirier1995), Spanos (Reference Spanos1999), or Bierens (Reference Bierens2005)).Footnote ⁷ These latter books, however, are often inaccessible to political science students without stronger-than-average backgrounds in mathematics.

To keep the exposition concrete, consider a classic random experiment: spinning a roulette wheel.Footnote ⁸ An American roulette wheel has thirty-eight pockets labeled 1–36 and 0 and 00. Eighteen of the pockets are red, eighteen are black, and the 0 and 00 pockets are green (see Figure 1).

Figure 1.

An American roulette wheel with two green pockets (0, 00). French roulette wheels have only a single green pocket.

The outcomes set comprises the possible outcomes of a spin of the wheel

The 1 pocket is an elementary element in $S$ , $1\in S$ , and the outcome “landed on red” ( $R$ ) is an event, which is a subset of the outcomes set, $R\subset S$ .Footnote ⁹ The complement of $R$ , $\bar{R}$ , is the set of outcomes that are in $S$ , but not in $R$ (that is, the outcomes that are black or green).

The above definitions demonstrate that there is no difference between the notion of a set used by quantitative scholars and the definition used by QCA scholars. A set for quantitative researchers does not represent “a property or a group of properties” (Schneider and Wagemann Reference Schneider and Wagemann2012, 24); it is “a collection of objects” (Binmore Reference Binmore1982, 1). In the next sections, I show the relationship between sets and random variables and sets and probability as well as the connection between random variables, probability, and Boolean algebras.

3.2 Random variables and Boolean algebras

The world does not produce data; it produces outcomes. When we spin a roulette wheel, we might be interested in the events “landed on red” ( $R$ ), “landed on black” ( $B$ ), and “landed on green” ( $G$ ). We need to assign numbers to these events in order to talk about notions such as expectation and variation. That is the job of a random variable.

Graduate students (and professors) in political science often have difficulty answering the question “What is a random variable?” Answers range from a variable that can take multiple values to a variable that is random. A random variable in actuality is just a function, and like all functions, it is a mapping between two sets. Consider a function that maps from a set $A$ to a set $B$ , $f:A\rightarrow B$ . The function assigns to every element in the set $A$ a unique element in the set $B$ .

In the case of a random variable, the set of outcomes produced by the world is mapped onto the set of real numbers, $\mathbb{R}$ .Footnote ¹⁰ Denoted by a capital letter, a random variable is the function $X(.):S\rightarrow \mathbb{R}$ . We can think of a random variable as a numerical representation of a set.

Let us assume that we are interested in the event “landed on red” when we spin the roulette wheel. The outcomes set is then $S=\{R,B,G\}$ . We can represent the event “landed on red” as 1 and “did not land on red” as 0:

Note that $X(.)$ is a function; it assigns a unique value in $\mathbb{R}$ to every outcome in $S$ .

Not every function, however, is a random variable. The outcomes of our random experiment, a spin of a roulette wheel, have probabilities associated with them. We know that the probability of the event “landed on red” is the number of possible red pockets divided by the total number of pockets or 18/38. The random variable we defined above assigns the number 1 to the event “landed on red,” and we need to ensure that we can assign the probability 18/38 to the number 1. The way we do that is with a field, which is also known as an algebra.

A collection $\Im$ of subsets of $S$ is a field or algebra if it meets three conditions:

• ∙ the outcomes set $S$ is in the collection, $S\in \Im$ ;

• ∙ if an event $A$ is in the collection, $A\in \Im$ , then its complement, $\bar{A}$ , must also be in it, $\bar{A}\in \Im$ ;

• ∙ if two events $A$ and $B$ are in the collection, $A,B\in \Im$ , then their union is also in it, $A\cup B\in \Im$ .

The definition of a field means that $\Im$ is nonempty and is closed under complementation, unions, and intersections.Footnote ¹¹

If we demand that our function preserve the field, we can have no problems assigning the correct probabilities, which are derived straight from the experiment (see Section 3.3). The function we defined above is a random variable relative to the field $\Im =\{S,\emptyset ,R,\bar{R}\}$ . We can easily check that this collection of subsets meets the three conditions of being an algebra. First, the outcomes set $S$ is in the collection. Second, for every event in the collection, its complement is also included. The complement of $S$ is the empty set, $\emptyset$ , and vice versa. The complement of $R$ is $\bar{R}$ and vice versa. Third, the union of any two events must be in the collection. The union of $R$ and $\bar{R}$ is $S$ . The union of $S$ and any other event is $S$ , and the same is true of $\emptyset$ . We therefore meet the conditions for a random variable.Footnote ¹²

The notion of a field is important because fields are Boolean algebras. The difference is that fields do not have to be 0–1 Boolean algebras. Consider, for example, the field described above: $\Im =\{S,\emptyset ,R,\bar{R}\}$ . $\Im$ is not a 0–1 Boolean algebra (it comprises four elements), but it is a Boolean algebra with the following translations:

In fact, a straightforward way to form a Boolean algebra is through the power set ${\mathcal{P}}(\cdot )$ of a set $S$ . Consider the set $S=\{R,B,G\}$ from above. The power set of $S$ , ${\mathcal{P}}(S)$ , is the set of all subsets of $S$ , or

${\mathcal{P}}(S)$ is closed under the set-theoretic operations of union, intersection, and complementation just as fields are (see Section 3.2). Although a full proof that the power set is a Boolean algebra is beyond the scope of this paper, we can see the connection by substituting any one of the subsets of ${\mathcal{P}}(S)$ for $p$ , $q$ , and $r$ in the Boolean laws above. Consider the subset $(R)$ :

It should be clear that ${\mathcal{P}}(S)$ properly translated is a Boolean algebra. There is a deep connection between random variables and Boolean algebras.Footnote ¹³

3.3 Probability and Boolean algebras

So far, we have a random variable, $X$ , to translate the set of outcomes, $S$ , and the associated algebra, $\Im$ , into the set of real numbers, $\mathbb{R}_{X}$ . We now want to assign probabilities to those numbers that are consistent with the probabilities from the roulette wheel. The first step is to define probability, just like a random variable, as a function that maps sets into numbers. In this case, the function maps from the field, $\Im$ , defined above, to the 0–1 interval. The function $\Pr (.):\Im \rightarrow [0,1]$ is a probability set function if it meets three conditions:

• ∙ the probability of the outcomes set is 1, $\Pr (S)=1$ ;

• ∙ the probability of any event $A$ in the field is greater than or equal to 0, $\Pr (A)\geqslant 0,\text{ for }A\in \Im$ ;

• ∙ if $A_{1}$ and $A_{2}$ are mutually exclusive events in $\Im$ , the probability of their union is the sum of their probabilities, $\Pr (A_{1}\cup A_{2})=\Pr (A_{1})+\Pr (A_{2})$ .

The random variable assigns numbers to subsets of the outcomes set, and the probability function assigns probabilities (values in the 0–1 interval) to those same subsets. We connect these two sets of numbers using a density function, $f_{x}(.)$ , which assigns the probabilities from the probability function to the value of the random variable. The density function tells us the probability that the random variable, $X$ , takes on a particular value,

In our example, the probability of “lands on red” is equivalent to $\Pr (X=1)$ , which is 18/38. The probability of “does not land on red” is equivalent to $\Pr (X=0)$ , which is 20/38. The result is a distribution,

The probability distribution is a representation of the set of outcomes of the random experiment plus the probabilities associated with those outcomes. Instead of working with the outcomes set, $S$ , the algebra, $\Im$ , and the known probabilities, we can now work with numbers that represent the outcomes set, $\mathbb{R}_{X}$ , and the density function, $f_{x}(x)$ .Footnote ¹⁵ The important thing to keep in mind is that we are still dealing with sets.

QCA scholars have little choice but concede that random variables are based on Boolean algebras, but once the sets are mapped to the real line and become numbers, the same scholars may well claim that the “setness” disappears. That is, the numbers do not have to obey the laws of Boolean algebra although the sets upon which they are based do. It is now time to link probability to Boolean algebra. Recall that the reason for defining a random variable on a field is to ensure that probabilities are assigned that do not violate the “setness” of the outcomes.

Consider a random variable $X$ with density:

It is certainly true that if we sum 0 and “not 0,” we get 6, not 1. However, if we add the probability of 0 and the probability of “not 0,” we do get 1,

By the same token, it must be the case that $p\wedge p^{\prime }=0$ . Well, the probability of 0 and “not 0” occurring together is 0. These results suggest that there is a connection between probability and Boolean algebra and, in fact, probability is defined as a function on a Boolean algebra.Footnote ¹⁶ This particular fact is fairly esoteric, but the point is that all aspects of classical statistics are based on sets.

Figure 2.

The relationship between fields, random variables, and probability. The functions $X(.)$ and $\Pr (.)$ map the real line $\mathbb{R}$ and the 0–1 interval, respectively, while preserving the field, $\Im$ . The density function $f(.)$ connects them.

Consider Figure 2. A random variable $X(.)$ maps subsets of a set (a Boolean algebra) to a subset of the real line while preserving the field. Probability $\Pr (.)$ maps subsets of a set (a Boolean algebra) to the [0,1] interval while preserving the field. Finally, the density function $f(.)$ connects the random variable to the [0,1] interval. At no point in the structure depicted in Figure 2 does the “setness” of the field get lost.

4.1 Preliminaries

There are many different types of logics, including propositional logic, syllogistic logic, predicate logic, modal logic, as well as nonclassical logics such as intuitionistic or constructive logic. QCA is solely concerned with propositional (also known as classical or sentential) logic, and as we shall see, only some Boolean algebras model propositional logic.

A proposition (or sentence) is a statement that is either true or false, but not both. Commands and questions are not propositions, and neither are statements such as $X\geqslant 5$ , because whether the statement is true depends on the value of $X$ . Examples of propositions include:

• ∙ David Collier studies American politics;

• ∙ the United States of America is a democracy;

• ∙ the Red Sox won the 2019 World Series.

All three statements are either true or false. Propositional logic concerns these kinds of simple sentences and their connectives (and, or, not). The sentences or propositions are denoted by symbols such as $P$ , $Q$ , and $S$ , and the user defines what they mean (the semantics). So I could assign the proposition “David Collier studies American politics” to the letter $P$ . These propositional symbols are known as atoms or atomic formulas because they contain no connectives. Thus, the proposition represented by $P$ is the smallest unit that the logic can analyze. Propositions in propositional logic do not have internal structure (van Benthem et al. Reference van Benthem, van Ditmarsch, van Eijck and Jaspars2016); there are no properties. Molecules or molecular formulas are formed out of atoms and the connectives (and, or, not). If I assign the letter $Q$ to the proposition “The United States of America is a democracy,” then “ $P$ and $Q$ ” ( $P\wedge Q$ ) is a molecular formula.

A well-formed formula in propositional logic obeys the following rules:

• ∙ $P$ is a well-formed formula;

• ∙ if $P$ is a well-formed formula, then so is $\neg P$ ;

• ∙ if $P$ and $Q$ are well-formed formulas, then so are $P\wedge Q$ , $P\vee Q$ , $P\rightarrow Q$ , and $P\leftrightarrow Q$ .

Propositional logic is truth-functional, which means that the truth of a molecular formula is a function of the truth values of the atomic propositions in it. That is, every proposition is either true or false, and every connective is truth-functional. We can therefore assess the truth value of a formula such as $(P\vee Q)\wedge S\rightarrow P$ by considering the truth values of the atoms $P$ , $Q$ , and $S$ . Truth functions are also known as Boolean functions.

The two-element Boolean algebra is a model of propositional logic because we can interpret “true” as 1, “false” as 0, and the connectives (and, or, not) as meet, join, and complement. To see the connection more completely, remember that two-element Boolean algebras have special properties. These are

We can write these special properties in what is known as an arithmetic table for the join operator $\vee$ and the meet operator $\wedge$ :

We can compare these arithmetic tables to the truth tables for the logical “or” and “and” in propositional logic:

In the arithmetic table for $\vee$ , whenever a 1 appears on top or the left side, the outcome of the join operator is 1. In the truth table for “or,” whenever a T appears under $P$ or $Q$ , a T appears under $P\vee Q$ . The operations take on the value 0 or F only when both inputs are 0 or F. In the arithmetic table for $\wedge$ , only when a 1 appears on top and on the left side is the outcome 1. In the truth table for “and,” only when T appears under $P$ and $Q$ does a T appear under $P\wedge Q$ . The operations take on the value 1 or T only when both inputs are 1 or T. The correspondence is one to one.

A less simple Boolean algebra, such as the power set ${\mathcal{P}}(S)$ discussed above, is not a model of propositional logic. We can interpret $S$ as “true” and $\emptyset$ as “false,” but what of the other values in the set? QCA is inextricably tied to propositional logic because it uses two-element Boolean algebra.

4.2 The limitations of propositional logic

Propositional logic, although of immense importance to computer design (0s and 1s) and electrical engineering (on and off), has serious limitations, and contrary to the claims of QCA proponents, social theory is often too complex to be rendered in propositional logic. Consider, for example, the following perfectly sound argument (known as a syllogism):

This argument cannot be expressed in propositional logic. Recall that the smallest unit of analysis, the atom, is a true or false sentence, and we assign a unique letter to each sentence. The logic ignores the structure of the atom. That is, propositional logic cannot “see” inside the sentences (Jeffrey Reference Jeffrey1991). In the second premise, there is an object (the United States) and a property of that object (a legislature). Both, however, are hidden when we assign a letter. So we might assign $L$ to the first premise, $D$ to the second premise, and $U$ to the conclusion. The result, $L$ , $D$ , and therefore $U$ , is not valid (if the premises are true, the conclusion must be true). $U$ does not necessarily follow from $L\,,D$ . We can see this from the left-hand truth table in Table 1, which lists all possible combinations of true and false for the three propositions.

Table 1.

Two propositional logics, only one of which is valid. The logic on the left is invalid because the conclusion is not necessarily true when both premises are true. The logic on the right is valid because the conclusion is true only when both premises are true.

In the second line of the left-hand table, we see that both premises are true, but the conclusion is false. The takeaway is not that the natural language argument is false, but that the natural language argument cannot be proven in propositional logic.Footnote ¹⁸ The only valid rendering of the argument requires making a different argument. If the premises were, “If the United States is a democracy, then the United States has a legislature” ( $D\rightarrow L$ ) and “the United States is a democracy” ( $D$ ), we could validly conclude that “the United States has a legislature” ( $L$ ), or $D\rightarrow L,D\vdash L$ . The validity of this logic is demonstrated in the right-hand truth table in Table 1. The conclusions is true only when both premises are true.

We cannot use propositional logic to identity traits or properties that units may have in common such as “democracies have legislatures.” That, in turn, makes it difficult, if not impossible, to generalize or talk of patterns (Lemmon Reference Lemmon1992). The implications for QCA are dramatic. Goertz and Mahoney (Reference Goertz and Mahoney2012, 193), for example, begin their discussion of “qualitative generalizations” by pointing to statements that “All $A$ are $B$ ,” and arguing that such generalizations have the form of “covering laws.” The example they give matches the democracies have legislatures example nicely,

The same problems exist, however. These statements cannot be connected using propositional logic.

The weaknesses of propositional logic led to the development of predicate logic, which introduced quantifiers (the universal $\forall$ and the existential $\exists$ ), variables, and relations. Whereas propositional logic ignores the internal structure of propositions, predicate logic considers internal structure including properties and relations. A predicate is a generalization of a proposition, and predicate logic provides a “more powerful language than expressions involving only propositions” (Aho and Ullman Reference Aho and Ullman1994, 733). Using quantifiers and properties, a generalization such as “All democracies have a legislature” is easily translated as $\forall x(democracy(x)\rightarrow legislature(x))$ . More generally, covering law explanations of the kind Goertz and Mahoney (Reference Goertz and Mahoney2012) write about have the form (Tan Reference Tan1997, 361)

which is written in predicate logic, not propositional logic. Two-element Boolean algebra, however, is not a model for predicate logic, which requires Halmos algebra or cylindric algebra.Footnote ¹⁹