Boole and the Algebraic Tradition

In 1847, George Boole published The Mathematical Analysis of Logic. Its central proposal was that inference could be investigated through algebraic operations. Instead of examining each argument solely through its verbal form, a logician could translate its conditions into equations and calculate their consequences.

This was a new mathematical treatment of an old subject. Aristotle's syllogistic, Stoic propositional inference, and medieval theories of consequence had already provided systematic accounts of reasoning. Boole's achievement was to connect logical operations with an algebra whose laws could themselves become objects of mathematical investigation. The earlier-history companions explain those previous traditions on their own terms.

The algebraic approach subsequently grew beyond classes. De Morgan investigated relations, Peirce developed an algebra of relatives and a notation for quantification, and Schröder organized much of the resulting theory. Understanding that development prevents modern logic from appearing to have emerged fully formed in a single work.

Classes as algebraic objects

Let a fixed universe contain the objects under discussion. In a modern reconstruction of Boole's notation, let $x$ denote a class and $xy$ the intersection of the classes $x$ and $y$. Let $1$ denote the whole universe and $0$ the empty class. Then $1-x$ denotes the complement of $x$.

The assertion “Every member of $x$ is a member of $y$” becomes

or, equivalently,

The second equation says that nothing belongs to $x$ while failing to belong to $y$. Its logical content is the absence of a counterexample.

These equations already represent a significant change in perspective. The same form can be interpreted using classes of animals, geometric figures, or numbers. The operation matters independently of the subject matter.

Boole's original system should nevertheless be distinguished from the Boolean algebra now taught in mathematics and computer science. In particular, his use of addition was subject to restrictions when interpreted as the union of classes: the classes were ordinarily required to be disjoint. Later algebraic logicians developed operations and conventions that made unrestricted union more straightforward. Reading every occurrence of Boole's addition as the modern join operation obscures part of the historical development.

Boole also allowed intermediate algebraic expressions whose immediate logical interpretation was problematic. His project involved investigating how interpretable consequences could be recovered from a calculation. The polished axiomatic theory of Boolean algebras was a later achievement, not simply the notation of his first book.

A syllogism as a calculation

Suppose every planet is a body and every body has mass. Write $p$, $b$, and $m$ for the corresponding classes. The premises become

Using associativity, substitution, and commutativity, we obtain

The conclusion is that every planet belongs to the class of things with mass.

This calculation is useful precisely because its details are modest. It shows what the algebra contributes: the validity of the inference is carried by general laws governing operations. We no longer need a separate verbal argument about planets.

One characteristic law is

Intersecting a class with itself changes nothing. This distinguishes the algebra from ordinary numerical calculation, in which $x^2=x$ is not true for every number. When the logical symbols are represented by the numbers $0$ and $1$, however, both satisfy that equation.

The numerical representation connects class operations with propositional reasoning. If $p$ and $q$ take truth values in $\{0,1\}$, multiplication represents conjunction, and $1-p$ represents negation. A modern expression for inclusive disjunction is

The subtraction corrects the double counting when both propositions are true. This is a modern explanatory reconstruction; it should not be mistaken for a complete description of Boole's own treatment of propositional symbols.

What class equations leave implicit

An algebra of classes handles inclusion, exclusion, and intersections effectively. Existential claims require additional care.

For example, the equation $xy=x$ does not assert that any $x$ exists. It remains true when $x=0$. Consequently, a translation of a traditional syllogism must state whether its terms are assumed to have members. Otherwise, a calculation may preserve a different interpretation from the one intended by the historical argument.

The equations above can now be tested on a small universe. In particular, an empty class distinguishes an inclusion claim from a claim of existence.

The issue becomes more substantial with relations. Consider:

Every mathematician has a teacher.

We need to relate each mathematician to another object. A class of people who have teachers can record the resulting property, but it does not expose who teaches whom or how that relation combines with another relation. Merely placing a long relational phrase into a single class symbol hides the structure needed for further inference.

For example, from “Ada teaches Beatrice” and “Beatrice teaches Clara,” we may infer that Ada teaches someone who teaches Clara. We cannot infer that Ada teaches Clara. A useful relational calculus must represent the intermediate object and distinguish composition from transitivity.

This is a limitation of a particular formal representation. It does not mean that earlier logicians lacked the concept of a relation.

De Morgan and the composition of relations

Augustus De Morgan made relational reasoning a subject of explicit investigation. His examples showed that ordinary arguments could depend on structures not adequately represented by standard categorical syllogisms.

One operation is the converse of a relation. If $R(x,y)$ means that $x$ is a parent of $y$, its converse relates a child to a parent. Another operation is composition. In modern notation, define

If $R$ means “is a parent of” and $S$ means “is a sibling of,” then $R;S$ relates a person to someone who is a sibling of that person's child. The interpretation depends on the order of the relations.

Composition is therefore generally noncommutative. Reversing the operations changes the claim. This is an important departure from an algebra consisting only of intersecting classes, where the order of the factors does not matter.

De Morgan's work opened a systematic problem: which algebraic laws govern such relational operations, and how can they be combined with the existing calculus of classes?

Peirce: from relatives to quantification

Charles Sanders Peirce's 1870 work on the logic of relatives extended the algebraic approach to relations and their combinations. His later writings also introduced a flexible notation for quantification.

The choice of notation expressed an algebraic idea. Universal quantification could be understood by analogy with a product, and existential quantification by analogy with a sum. For a finite universe $\{a,b,c\}$,

amounts to

whereas existential quantification amounts to the corresponding disjunction. Peirce's product-and-sum notation extended that perspective beyond a finite enumeration.

The conceptual gain is most visible when quantifiers alternate. Consider a universe containing Ada and Beatrice, where each admires herself and neither admires the other. Then

is true: each person admires someone. But

is false: there is no person admired by everyone.

The difference cannot be expressed by treating “admiration” as one undifferentiated class property. Variables and quantifier order expose how one choice depends on another.

Peirce's work developed in an intellectual community rather than in isolation. Oscar Howard Mitchell's contribution to the 1883 Studies in Logic, edited by Peirce, belongs to the contemporary development of quantificational notation. Care is needed with priority claims: the collection contains related but distinct contributions, and later notation should not be assigned wholesale to one participant.

Peirce also investigated normal forms and truth-functional methods. These developments provided ways to reorganize logical expressions systematically, an idea that would become central to later decision procedures and automated reasoning.

Schröder, representation, and transmission

Ernst Schröder's Vorlesungen über die Algebra der Logik, published in volumes during the 1890s, assembled and extended the algebraic tradition. This consolidation mattered. A mathematical method becomes influential partly through texts that make its results available for further work.

The relational tradition subsequently provided a setting for Löwenheim's investigations. Modern first-order logic therefore has an important genealogy through algebraic logic as well as through Frege and the foundational projects associated with him.

Other nineteenth-century developments addressed how calculations could be represented and organized. William Stanley Jevons simplified aspects of logical algebra and explored mechanical inference. John Venn's diagrams systematically represented combinations of class membership. These were contributions to the intelligibility and operation of a calculus, even though they did not by themselves provide the full expressive resources of quantified relational logic.

A diagram of three intersecting classes can display which regions are empty. It cannot, without additional machinery, display every dependence between arbitrarily many quantified variables. Different representations solve different problems.

From logical algebra to later mathematics

The tradition did not end when predicate notation became standard. Boolean algebra became a mathematical subject in its own right. Marshall Stone's representation theorem in the 1930s connected abstract Boolean algebras with algebras of sets and topological spaces. The theorem showed that the familiar set interpretation had a deep structural relationship to the abstract laws.

Claude Shannon's 1938 analysis of relay and switching circuits supplied a different application. Logical operations could describe combinations of switches. This helped connect the algebra with engineering, but it should not be read backward as Boole's original purpose. A method developed to investigate reasoning acquired applications its originator had not designed it to serve.

The central historical lesson is that calculation and expression developed together. Algebra made logical operations manipulable; relational and quantificational extensions made the resulting language capable of displaying more complicated dependencies.

Frege approached those dependencies from a different direction. His problem was how to express complete mathematical arguments without leaving inferential steps concealed in ordinary language. The next note examines that project and why its aims were not identical to those of logical algebra.

Sources and further reading

• George Boole, The Mathematical Analysis of Logic (1847), and An Investigation of the Laws of Thought (1854). The class calculus should be read with attention to Boole's conventions for addition and interpretation.
• Augustus De Morgan, “On the Syllogism: IV, and on the Logic of Relations” (1864).
• Charles S. Peirce, “Description of a Notation for the Logic of Relatives” (1870); the collection Studies in Logic (1883), including Mitchell's contribution; and “On the Algebra of Logic: A Contribution to the Philosophy of Notation” (1885).
• Ernst Schröder, Vorlesungen über die Algebra der Logik (1890–1905, including posthumous material).
• Marshall H. Stone, “The Theory of Representations for Boolean Algebras” (1936); Claude E. Shannon, “A Symbolic Analysis of Relay and Switching Circuits” (1938).
• The Emergence of First-Order Logic, for the relationship among the algebraic tradition, Frege, and later metamathematics.