Math Origins: The Logical Symbols

In our last article, we explored the struggles of 17th- and 18th-century mathematicians who sought to devise a coherent system for describing logical ideas, and a few novel attempts at establishing a system of symbols to express those ideas. Contributions ranged from Gottfried Wilhelm Leibniz's vision of a "universal calculus" that could encode any logical statement with mathematical precision to Ludovico Richeri's binary-tree system for classifying logical statements in terms of their philosophical qualities.

By the early 19th century, then, mathematicians possessed a number of different methods for representing logical statements, including Leibniz's numerical system and Johann Lambert's appropriation of arithmetic symbols in Sechs Versuche einer Zeichenkunst in der Vernunftlehre (Six attempts at a drawing-art in the theory of reason). In this article, we continue the narrative through the 19th and 20th centuries, during which time the logical ideas of past centuries were tamed: not only through a symbol system, but also through a system for manipulating them. Eventually, the familiar symbols of today's mathematical logic emerged, alongside the terminologies typically used to describe them.

The British Duo: Boole and De Morgan

With the start of the 19th century, our narrative shifts from the European continent to Britain, where Augustus De Morgan and George Boole were to make lasting contributions to logic and (eventually) computer science. In fact, the two men maintained a long-term correspondence with each other, beginning in 1842. As we will see, however, their approaches to the study of logic were quite different.

Born in India to British parents, De Morgan returned to Britain as an infant, later joining Trinity College at Cambridge University at age 16. Remarkably, at the age of 22 he found himself appointed chair of mathematics at London University (now University College London). While not directly connected to our goals for this article, this happens to be the time at which De Morgan provided a modern definition (see [DeM1]) of the method of proof that we today call "mathematical induction," as part of a series of articles for the Penny Cyclopedia. Until then, the word "induction" was often used to refer to what today's mathematicians would call "proof by example"—an erroneous method of proving mathematical truth that relies on an observed pattern.

In the decades following his 1828 appointment to London University, De Morgan's interest in logic continued, culminating with his publication of Formal Logic: or, The Calculus of Inference, Necessary and Probable, in 1847. De Morgan's book advertised a "simple notation" to express any proposition by using only three letters, and a "subordinate notation" to abbreviate it. Chapter 1, titled "First Notions," begins with a discourse on propositions, which are described as a subject and predicate linked by a copula. Properly quantified, there are four species of proposition in De Morgan's telling.

Figure 1. De Morgan's presentation of the quantified logic statements denoted by the vowels \(A\), \(E\), \(I\), \(O\) in Formal Logic (1847). Image courtesy of Archive.org [DeM2].

Using the language of set theory, we may translate De Morgan's vowels \(A\), \(E\), \(I\), \(O\) in the following way:

• \(A\):  For all \(x\in X\), it follows that \(x\in Y\).
• \(E\):  For all \(x\in X\), it follows that \(x\not\in Y\).
• \(I\):  There exists \(x\in X\) for which \(x\in Y\).
• \(O\):  There exists \(x\in X\) for which \(x\not\in Y\).

Aside from the fact that De Morgan used the word "particular" in place of "existential," the classification system is essentially the same as in any logic textbook today. In fact, the entire chapter is a pleasant tour of the fundamental concepts in predicate calculus, with many familiar concepts wearing nearly the same clothing as today—the interested reader is highly encouraged to read it for its own sake! For our purposes, we move to Chapter 4, "On Propositions," in which De Morgan introduced his system for expressing propositions symbolically. On page 60, we find the following notations to write these quantified statements in a fully symbolic form:

• \(X)Y\) – "Every \(X\) is \(Y\),"
• \(X.Y\) – "No \(X\) is \(Y\),"
• \(X:Y\) – "Some \(X\)s are not \(Y\)s,"
• \(XY\) – "Some \(X\)s are \(Y\)s."

This new notational system allowed De Morgan to characterize the six distinct "modes" of proposition that can be built from \(X\) and \(Y\):

Figure 2. De Morgan's six "modes" of proposition formed from simple propositions \(X\) and \(Y\), from Formal Logic (1847). Image courtesy of Archive.org.

In his notation, \(x\) and \(y\) are the negations of \(X\) and \(Y\). Since each pair of letters may be combined into six distinct simple propositions, there are at most \({4\choose 2}\cdot 6 = 6\cdot 6 = 36\) possible modes. De Morgan did not consider contradictory statements such as \(X)x\) or tautologies such as \(y:Y\), so he only counted 24 possible modes. A little thought reduces the list to eight distinct propositions:

Figure 3. In Formal Logic (1847), De Morgan noted eight distinct modes of proposition based on the simple propositions \(X\) and \(Y\) and their negations \(x\) and \(y\). Image courtesy of Archive.org.

In De Morgan's system, then, any of eight distinct modes of expression may be referred to with one of \(A\), \(E\), \(I\), \(O\), and an accompanying superscript or subscript mark. While the notation obscures the underlying logical concepts (for example, the letters \(A\) and \(E\) give no indication that their referents are both universally-quantified statements), it is certainly comprehensive, providing a self-contained system for writing statements logically.

In the same year that De Morgan's Formal Logic appeared in print, George Boole made the first of several contributions to logic with The Mathematical Analysis of Logic, Being an Essay Towards a Calculus of Deductive Reasoning [Boo1]. In fact, as the logician Clarence Lewis related in A Survey of Symbolic Logic (1918), Boole and De Morgan's works were published on the very same day in 1847. Along with Lewis, we observe that Boole's approach to logic "allows operations which have no direct logical interpretation, and is obviously more at home in mathematics than in logic. It is probably the great advantage of Boole's work that he either neglected or was ignorant of these refinements of logical theory which hampered his predecessors" [Lew, p. 51].

Unlike De Morgan, Boole had difficulty obtaining a university appointment. This was due in part to the fact that he was largely self-educated in mathematics. He was eventually appointed, in 1849, to Queen's College, Cork. In 1854 he produced a second, more comprehensive work on logic titled An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities [Boo2]. As a youth, Boole had demonstrated an early propensity for language, having taught himself Greek, French, and German. Unsurprisingly for one with Boole's background in languages, he began Chapter 2 of his Laws of Thought with a definition of a "sign":

Figure 4. Boole began his Laws of Thought (1854) by reflecting on the denotative meaning of written language and giving a definition of a sign. Image courtesy of Archive.org.

In Boole's view, any mark with a fixed interpretation counts as a "sign," and—crucially—the laws for manipulating signs are independent of any particular symbol system. This frees the logician to choose among any existing symbols (or create new symbols) to describe logical ideas. Naturally enough, Boole's first proposition establishes one way to reappropriate the symbols and rules of algebra for use in mathematical logic:

Figure 5. Boole's catalog of symbols from Laws of Thought (1854). Image courtesy of Archive.org.

In this way, Boole may give a more granular accounting of the ideas in logic, more like an "atomic" form of reasoning, when compared to De Morgan's approach in Formal Logic. Moreover, while his symbolic choices were similar to Lambert's approach from the previous article, Boole's overall framework is more systematic than that of Lambert. In particular, Boole chose to reserve different types of symbols to refer to "subjects" and others to refer to "operations of the mind." For an example, let us skip ahead to page 33:

Figure 6. An example of Boole's symbol system in use, from Laws of Thought (1854). Image courtesy of Archive.org.

Here, \(x\), \(y\), and \(z\) are subjects of our conception, while addition and multiplication are operations of the mind. This allows Boole to conceive of the phrase "European men and women" as being identical to "European men and European women." (As an aside: mathematics was not Boole's only interest in this work; take a look at this selection from Boole in the Mathematical Treasures series in which he gives a proof of the existence of a Aristotelian-style "prime mover.")

Boole was to exert a heavy influence on the next several decades of research in logic, including with a young Ernst Schröder, who wrote in 1877 that, in logic, "it is now possible to obtain the utmost attainable simplicity and perfection—a goal whose realization, after Boole's preliminary work, requires only fairly straightforward observations" [Sch1, p. IV]. In this work, Der Operationskreis des Logikkalkuls, Schröder developed a Boolean-style system for the calculus of logic, observing that "Nothing prevents naming those 4 basic [arithmetic] operations with the same names and expressing them with the same symbols as are used in arithmetic. After all, the object of the operations is a completely different one—there they are numbers, but here they are arbitrary concepts!" [Sch1, p. 2]. This may seem an unnecessary observation to a modern reader, but this is likely because of Boole's influence rather than in spite of it. We now turn to the intellectual descendants of both Boole and De Morgan as the 19th century reached its conclusion.

The Lightning Round: Symbol Systems from the late 1800s

The number of works in logic increased greatly during the late 19th century, so here we will take a brief tour of a few notable works and symbol choices from this period. First, we take a peek at Schröder's Vorlesungen über die Algebra der Logik (Lectures on the Algebra of Logic), which appeared in 1890. Notably, it introduces the \(\subset\) symbol to indicate "subordination," which Schröder compared to the \(<\) symbol used with numbers.

Figure 7. Ernst Schröder's introduction of the \(\subset\) symbol in Vorlesungen über die Algebra der Logik (Lectures on the Algebra of Logic),  published in 1890 [Sch2, p. 129]. Image courtesy of Göttingen State and University Library.

While the \(\subset\) symbol is used today to denote the containment of sets, Schröder's intent was more general—the containment can refer to the content of either sets or logical ideas. Interestingly, Schröder overlaid an \(=\) over the \(\subset\) to indicate the "subordinate or equal to" relationship, in contrast to today's \(\subseteq\) symbol.

Figure 8. Ernst Schröder's use of \(\subset\) and  \(\subset\!\!\!\!\!=\) symbols in Vorlesungen über die Algebra der Logik (Lectures on the Algebra of Logic), published in 1890 [Sch2, p. 132]. Image courtesy of Göttingen State and University Library.

At the end of this passage, the "subordinate or equal to" symbol is used as a way to show how the subject of a proposition is either "contained in" or "equivalent to" the predicate. On page 130, we find an appraisal of some alternative symbols in use at the time, including Paul Du Bois-Reymond's \(\prec\) (seen in a previous article in this series) and C. S. Peirce's \(-\!\!\!<\). Peirce, an American mathematician, was a founder of the pragmatist school of philosophy and made significant contributions to logic throughout the late 19th century. Familiar with the work of De Morgan and Boole, a young Peirce published "On an Improvement in Boole's Calculus of Logic" in the 1868 Proceedings of the American Academy of Arts and Sciences. In this short paper—appearing some 20 years before Schröder's Vorlesungen über die Algebra der Logik—Peirce proposed that Boole's symbol usage from Laws of Thought be adapted by placing a comma atop the various arithmetic symbols used for logical analysis.

Figure 9. C. S. Peirce used a comma to indicate when the symbols of arithmetic were used to denote logical concepts. From his 1868 work, "On an Improvement in Boole's Calculus of Logic" [Pei, p. 402]. Image courtesy of Biodiversity Heritage Library.

Having established his alternative symbol system, Peirce then laid out a set of familiar theorems in his new system, including reflexivity (Theorem I) and transitivity (Theorem II).

Figure 10. Peirce's first set of theorems using his comma notation. From his 1868 work, "On an Improvement in Boole's Calculus of Logic" [Pei, p. 403]. Image courtesy of Biodiversity Heritage Library.

In later years, Peirce's interests grew to include astronomy, the philosophy of science, and statistics. As a philosopher, he became attracted to symbols in a more abstract sense, developing a three-category system to describe them. In The Math Gene [Dev], Keith Devlin describes the three categories as icon, index, and symbol. The last of these sits at the apex of this mental edifice; in Devlin's words,

An object, action, event, or entity X is called a symbol for some other object, action, event, or entity Y if there is an agreed convention whereby X represents (or symbolizes) Y, irrespective of the physical nature of X and Y. For example, a wedding ring is symbolic of the fact that the wearer is married. Nothing about a band of gold worn around the third finger of the left hand intrinsically represents a married state. The meaning is purely by convention [Dev, p. 208].

In Peirce's view, all human language is symbolic—words themselves have meaning only via the consensus of those who speak them. This takes us beyond the scope of this article, though it should be clear how Boole's use of the "sign" in logic lurks in the background of Peirce's philosophy of "symbol".

We choose to end the century in 1901, with Giuseppe Peano's Formulaire de Mathématiques (The Form of Mathematics). Peano adopted Boole's convention of using letters to represent "objects of the mind" and Schröder's \(\subset\) for logical containment, while also distinguishing between propositions (statements which are either true or false) and classes (collections of instances) in his notational system. Especially note the use of \(x\varepsilon a\) in Rule 4 to denote that \(x\) is an instance of the class \(a\).

Figure 11. Giuseppe Peano's 1901 definition of "classes" in Formulaire de Mathématiques [Pea, p. 1]. Image courtesy of Archive.org.

Rule 7 illustrates the form taken in Peano's logic: "Given classes \(a\) and \(b\), \(a\supset b\) means 'every \(a\) is \(b\).' Given propositions \(p\) and \(q\) which contain a variable \(x\); [we write] \(p.\supset x.q\) to mean 'from \(p\) we deduce, regardless of \(x\), that \(q\) [is true]'." While the universal quantification is not spelled out in the notation, we have here a nice melding of Boole's conventions with Schröder's notion of logical containment.

While this ends the lightning round, we note that there are many valuable teaching resources on the history of logic that expand on these authors' contributions, including Jerry Lodder's Deduction through the Ages: A History of Truth (2013), published in Convergence. In particular, interested readers can learn more about the work of Boole, Venn, and Peirce in Origins of Boolean Algebra in the Logic of Classes (2013) by Janet Barnett, also published in Convergence.

Epilogue: The Principia Mathematica

Around the time Peano was working on Formulaire de Mathématiques, he attended the International Congress of Philosophy in 1900, where he met a young Bertrand Russell. Writing in his autobiography many years later, Russell recalled this meeting:

I already knew [Peano] by name and had seen some of his work, but had not taken the trouble to master his notation. In discussions at the Congress I observed that he was always more precise than anyone else, and that he invariably got the better of any argument on which he embarked. As the days went by, I decided that this must be owing to his mathematical logic. ... It became clear to me that his notation afforded an instrument of logical analysis such as I had been seeking for years... [Rus, p. 144].

A decade later, Russell and Alfred North Whitehead produced the Principia Mathematica, an ambitious work published in three volumes from 1910 to 1913. In it, Russell and Whitehead sought to give a systematic and rigorous accounting of mathematical truths using symbolic logic. As Russell wrote in the introduction to the second edition in 1963, the Principia was "framed with a view to the perfectly precise expression, in its symbols, of mathematical propositions" [WR, p. 1]. As one would expect, a goal as lofty as this one required a notational system equal to the task. This is the task of Chapter 1, which the authors begin by observing their debt to Peano: "The notation adopted in the present work is based upon that of Peano, and the following explanations are to some extent modelled on those which he prefixes to his Formulario Mathematico." The notational choices are spread throughout this chapter, but we list some of them here (all page numbers correspond to the second edition of Volume I):

• Page 6: "The Contradictory Function with argument \(p\) ... is denoted by \(\sim p\)."

• Page 6: "The Logical Sum ... is denoted by \(p\lor q\)."

• Page 6: "The Logical Product ... is denoted by \(p.q\), or—to make the dots act as brackets in a way to be explained immediately—by \(p:q\), or by \(p:.q\), or by \(p::q\)."

• Page 7: The equivalence of two statements \(p\) and \(q\) is "denoted by \(p\equiv q\)."

• Page 8: "The sign '\(\vdash\),' called the 'assertion sign,' means that what follows is asserted." (This was borrowed from Gottlob Frege.)

This is neatly summarized on page 12:

Figure 12. Whitehead and Russell's summary of the basic concepts and symbols in their propositional calculus. From Volume I of Principia Mathematica (1910). Image courtesy of HathiTrust Digital Library.

In this way, Whitehead and Russell gave three descriptive concepts (negation, disjunction, and assertion) and three rigorously-defined concepts (conjunction, implication, and equivalence), each with their own symbolic shorthand. Continuing further into the work, we find the symbol \(\exists\) used for existentially-quantified statements, and De Morgan's notation \(p)q\) borrowed occasionally to denote implications. There is much more in this work, so perhaps the best suggestion is for readers to explore it for themselves, either by browsing this Mathematical Treasure or paging through the second edition of Volume I on Archive.org.

Much more could be said of Whitehead and Russell's work, not to mention the many contributors whose works were left out of this survey. For now, we conclude with an observation from Florian Cajori, whose A History of Mathematical Notations formed the backbone of this article:

No topic which we have discussed approaches closer to the problem of a uniform and universal language in mathematics than does the topic of symbolic logic. The problem of efficient and uniform notations is perhaps the most serious one facing the mathematical public. No group of workers has been more active in the endeavor to find a solution of that problem than those who have busied themselves with symbolic logic—Leibniz, Lambert, De Morgan, Boole, C. S. Peirce, Schröder, Peano, E. H. Moore, Whitehead, Russell. ... Each proposed a list of symbols, with the hope, no doubt, that mathematicians in general would adopt them. That expectation has not been realized. What other mode of procedure is open for the attainment of the end which all desire? [Caj, p. 314]

Cajori made his observation in the 1920s, before Whitehead and Russell's symbol system took hold in the English-speaking world. Today, a century later, the symbols used in logic and computer science are largely in keeping with the system in Principia Mathematica. As we have seen, the Principia owes much of its notation to Boole and Peano, though many others made their own contributions in the preceding century. 

Last, we end with a "what if?" moment in the history of mathematics: Gottlob Frege's Grundgesetze der Arithmetik. This book, published in 1893, contained a fascinating diagrammatic method for representing logical ideas. It did not catch on in the way that Whitehead and Russell's method did, but it is fun to imagine what the world of logic would look like if it had. Anyone who is curious to learn more should take a look at Chapter 1 of the Grundgesetze.

References

[Boo1] Boole, George. The Mathematical Analysis of Logic. Cambridge, UK: Macmillan, Barclay, & Macmillan, 1847.

[Boo2] Boole, George. An Investigation of the Laws of Thought. New York: Dover Publications, 1854.

[Caj] Cajori, Florian. A History of Mathematical Notations (Volume II). Chicago: The Open Court Publishing Co., 1929.

[Dev] Devlin, Keith. The Math Gene. Basic Books, 2000.

[DeM1] De Morgan, Augustus. "Induction (Mathematics)," in Volume XII of the Penny Cyclopedia. London: Charles Knight and Co., 1838.

[DeM2] De Morgan, Augustus. Formal Logic: or, The Calculus of Inference, Necessary and Probable. London: Taylor & Walton, 1847.

[Lew] Lewis, Clarence. A Survey of Symbolic Logic. Berkeley, CA: University of California Press, 1918.

[Pea] Peano, Giuseppe. Formulaire de Mathématiques. Paris: G. Carré and C. Naud, 1901.

[Pei] Peirce, C. S. "On an Improvement in Boole's Calculus of Logic," Proceedings of the American Academy of Arts and Sciences, Vol. VII (1868), 250–261.

[Sch1] Schröder, Ernst. Der Operationskreis des Logikkalkuls. Leipzig: B. G. Teubner, 1877.

[Sch2] Schröder, Ernst. Vorlesungen über die Algebra der Logik (Volume I). Leipzig: B. G. Teubner, 1890.

[WR] Whitehead, Alfred N. and Russell, Bertrand. Principia Mathematica. Cambridge, UK: Cambridge University Press, 1910.