CHAPTER 1

Preliminaries

Math is a formal language useful in clarifying and exploring connections betweenconcepts. Like any language, it has a syntax that must be understood beforeits meaning can be parsed. We discuss the building blocks of this syntax in thischapter. The first is the variables that translate concepts into mathematics, andwe begin here. Next we cover groupings of these variables into sets, and thenoperators on both variables and sets. Most data in political science are ordered,and relations, the topic of our fourth section, provide this ordering. In the fifthsection we discuss the level of measurement of variables, which will aid us inconceptual precision. In the sixth section we offer an array of notation thatwill prove useful throughout the book; the reader may want to bookmark thissection for easy return. Finally, the seventh section discusses methods of proof,through which we learn new things about our language of mathematics. Thissection is the most difficult, is useful primarily to those doing formal theory ordevising new methods in statistics, and can be put aside for later reading orskipped entirely.

1.1 VARIABLES AND CONSTANTS

Political scientists are interested in concepts such as participation, voting, democracy,party discipline, alliance commitment, war, etc. If scholars are to communicatemeaningfully, they must be able to understand what one another isarguing. In other words, they must be specific about their theories and theirempirical evaluation of the hypotheses implied by their theories.

A theory is a set of statements that involve concepts. The statements compriseassumptions, propositions, corollaries, and hypotheses. Typically, assumptionsare asserted, propositions and corollaries are deduced from these assumptions,and hypotheses are derived from these deductions and then empiricallychallenged. Concepts are inventions that human beings create to help themunderstand the world. They can generally take different values: high or low,present or absent, none or few or many, etc.

Throughout the book we use the term "concept," not "variable," when discussingtheory. Theories (and the hypotheses they imply) concern relationshipsamong abstract concepts. Variables are the indicators we develop to measureour concepts. Current practice in political science does not always honor thisdistinction, but it can be helpful, particularly when first developing theory, tospeak of concepts when referring to theories and hypotheses, and reserve theterm variables for discussion of indicators or measures.

We assign variables and constants to concepts so that we may use them informal mathematical expressions. Both variables and constants are frequentlyrepresented by an upper- or lowercase letter. Y or y is often used to representa concept that one wishes to explain, and X or x is often used to represent aconcept that causes Y to take different values (i.e., vary). The letter one choosesto represent a concept is arbitrary—one could choose m or z or h, etc. There aresome conventions, such as the one about x and y, but there are no hard-and-fastrules here.

Variables and constants can be anything one believes to be important toone's theory. For example, y could represent voter turnout and x the level ofeducation. They differ only in the degree to which they vary across some setof cases. For example, students of electoral politics are interested in the gendergap in participation and/or party identification. Gender is a variable in the USelectorate because its value varies across individuals who are typically identifiedas male or female. In a study of voting patterns among US Supreme Courtjustices between 1850 and 1950, however, gender is a constant (all the justiceswere male).

More formally, a constant is a concept or a measure that has a singlevalue for a given set. We define sets shortly, but the sets that interest politicalscientists tend to be the characteristics of individuals (e.g., eligible voters), collectives(e.g., legislatures), and countries. So if the values for a given concept(or its measure) do not vary across the individuals, collectives, or countries, etc.,to which it applies, then the value is a constant. A variable is a concept or ameasure that takes different values in a given set. Coefficients on variables (i.e.,the parameters that multiply the variables) are usually constants.

1.1.1 Why Should I Care?

Concepts and their relationships are the stuff of...