Charles Sanders Peirce

How to use this course

Peirce described himself first as a logician, but for thirty years he earned his living as a physical scientist, swinging pendulums for the United States Coast Survey. He invented quantifiers independently of Gottlob Frege, built a calculus of relations, designed a diagrammatic logic, founded the modern study of signs and launched the philosophy called pragmatism. He published little of it in finished form, and much of his work was read only after his death. This course treats the logic as mathematics, with theorems and exercises, and the physics and philosophy as the methods of a working scientist.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Modern notation is used throughout, with Peirce’s own notation shown where it helps.

Timeline

A Cambridge childhood, 1839–1855

Charles Sanders Peirce was born on 10 September 1839 at 3 Phillips Place, Cambridge, Massachusetts. His father, Benjamin Peirce, was professor of mathematics and astronomy at Harvard, the most prominent American mathematician of his day and a leading figure in the Coast Survey and the new American scientific establishment. His mother was Sarah Hunt Mills, daughter of a senator. His elder brother James Mills Peirce became a Harvard mathematician; his younger brother Herbert became a diplomat.

Benjamin Peirce took charge of his second son’s education. He set problems and let the boy find the methods himself, and he trained him in long sessions of concentration. Charles set up a chemistry laboratory as a child. As a boy he also read Richard Whately’s Elements of Logic, and he later said that logic had held him ever since.

He grew up among the scientists and writers who visited his father’s house, including the naturalist Louis Agassiz and the astronomers and mathematicians of the Harvard circle. He was also proud, impatient and hard to manage. These traits, together with recurrent painful facial neuralgia, would shape his career as much as his gifts.

Harvard, chemistry and the Survey, 1855–1872

Peirce entered Harvard College in 1855 and graduated A.B. in 1859. In July 1859 he began to work for the U.S. Coast Survey, the federal agency responsible for mapping and geodesy, where his father was influential. He studied classification with Agassiz, received the Harvard A.M. in 1862, and in 1863 took the Sc.B. in chemistry at the Lawrence Scientific School, the first summa cum laude in chemistry at Harvard.

In October 1863 he married Harriet Melusina Fay, called Zina, a writer and later a campaigner for cooperative housekeeping. In 1865 he lectured at Harvard on the logic of science, and in 1866 he gave the Lowell Institute lectures. In 1867 he was elected to the American Academy of Arts and Sciences and read to it a series of papers: improvements to George Boole’s logical algebra, and “On a New List of Categories”, the seed of his theory of signs (Lesson 08).

From 1869 to 1872 he was an assistant at the Harvard College Observatory, measuring the brightness of stars. In 1870 he published “Description of a Notation for the Logic of Relatives”, his first major work in logic (Lesson 05), and travelled to Europe for the Survey to observe a solar eclipse in Sicily. In the early 1870s he met regularly with William James, Oliver Wendell Holmes Jr., Chauncey Wright and others in a discussion group he later called the Metaphysical Club. There, he said, the ideas of pragmatism first took shape.

The scientist in Europe, 1872–1879

In November 1872 Benjamin Peirce, now Superintendent of the Coast Survey, put his son in charge of the Survey’s pendulum experiments. The aim was to measure the acceleration of gravity at many sites and so determine the shape of the Earth. Peirce spent April 1875 to August 1876 in Europe learning the reversible pendulum, comparing instruments in Berlin, Geneva and Paris, and meeting scientists including George Gabriel Stokes and George Biddell Airy in England.

At the meeting of the European geodetic association in Paris in 1875 he argued that the stands used to hold the reversible pendulums were not rigid. They swayed slightly with the swing, and this flexure biased every European measurement of gravity. His report on the effect was distributed at the geodetic conference in Stuttgart in 1877 and read at Geneva that October. The Europeans came to accept the correction, and the episode established Peirce’s reputation as a research geodesist (Lesson 09).

Honours followed. On 20 April 1877 he was elected to the National Academy of Sciences. In 1877 he proposed defining the metre by a wavelength of light, and in 1878 he published Photometric Researches, his observatory work on stellar brightness and the shape of the Milky Way. In 1877–78 a series of six papers in Popular Science Monthly, “Illustrations of the Logic of Science”, set out his theory of inquiry and the principle later called pragmatism (Lesson 10).

His private life was less settled. Zina left him in 1876, and he was soon living with a woman known as Juliette Froissy Pourtalès, whose origins remain uncertain. In 1879 he took a lectureship in logic at the new Johns Hopkins University in Baltimore, while keeping his Survey post.

The algebra of logic

For the general reader. George Boole showed in 1847 that reasoning about classes obeys an algebra. Peirce reworked that algebra, found that all of it can be built from a single operation, and saw that it could be carried out by electrical switches. These three observations are the ancestors of every logic gate in a modern computer.

Improving Boole

Boole’s union was exclusive and his algebra used ordinary subtraction and division. In 1867 Peirce, following William Stanley Jevons, made union inclusive and replaced Boole’s arithmetic by a lattice-like algebra with an order relation. Peirce took inclusion (he called it illation, written \(\prec\)) as primitive. In 1880 he wrote down the laws of what we now call a Boolean lattice and claimed that the distributive law follows from the others. Ernst Schröder showed that it does not, and Peirce accepted the correction.

One connective is enough

In a manuscript of about 1880, published only in 1933, Peirce showed that a single binary connective suffices. Write \(p\downarrow q\) for “neither \(p\) nor \(q\)”, the NOR connective, now called the Peirce arrow.

Proof sketch. \(\neg p=p\downarrow p\), \(p\vee q=(p\downarrow q)\downarrow(p\downarrow q)\), and \(p\wedge q=(p\downarrow p)\downarrow(q\downarrow q)\). Since \(\{\neg,\vee,\wedge\}\) is complete (by disjunctive normal form), so is \(\{\downarrow\}\). Henry Sheffer published the corresponding result for NAND in 1913. For the uniqueness claim, a sole sufficient connective must map \((1,1)\) to 0 and \((0,0)\) to 1 (otherwise it preserves truth or falsity), and it must not be self-dual or a function of one variable; only NOR and NAND remain.

Peirce’s law

In his 1885 paper Peirce gave a set of principles for propositional logic using implication as the main connective. One of them is

It is a classical tautology, but it is not provable in intuitionistic logic. Added to the intuitionistic implicational axioms \(p\to(q\to p)\) and \((p\to(q\to r))\to((p\to q)\to(p\to r))\) with modus ponens, it yields every classical tautology built from \(\to\) alone. Under the Curry–Howard correspondence, Peirce’s law is the type of the control operator call/cc.

The same 1885 paper evaluates formulas by assigning the values true and false to their parts and checking every case, an early form of the truth-table method.

Switching circuits

In 1886 Peirce wrote to his former student Allan Marquand, who had built a mechanical logic machine, that logical operations could be carried out by electrical circuits. Switches in series realize conjunction and switches in parallel realize disjunction. Claude Shannon developed the idea into a theory of circuit design in 1937.

The logic of relatives

For the general reader. Boole’s logic handles properties: “is a man”, “is mortal”. It cannot express “every man loves some woman”, which involves a relation between two things. In 1870 Peirce extended the algebra to relations, with operations for chaining them together. The result, revived by Alfred Tarski in 1941, is the calculus of relations used today in databases and program semantics.

Peirce called a relation a “relative” term, such as “lover of”, and treated it as a class of ordered pairs. For relations \(R,S\) on a set \(U\) define

Peirce wrote the relative product by juxtaposition: if \(l\) is “lover of” and \(s\) is “servant of”, then \(ls\) is “lover of a servant of”. He also applied a relative to a class: \(lw\) is “lover of a woman”, the class \(\{x:\exists y\,(y\in w\wedge xly)\}\). For “lover of every woman” he used an exponent, \(l^w=\{x:\forall y\,(y\in w\to xly)\}\). These two constructions are an existential and a universal quantifier in disguise.

Proof. Each condition says that there are no \(x,y,z\) with \(xRy\), \(ySz\) and \(xTz\). For the second, a witness \(y\,(R^{\smile};T)\,z\) with \(ySz\) means some \(x\) has \(xRy\) and \(xTz\). For the third, a witness \(x\,(T;S^{\smile})\,y\) with \(xRy\) means some \(z\) has \(xTz\) and \(ySz\). All three describe the same forbidden triangle.

Ernst Schröder built the third volume of his Vorlesungen über die Algebra der Logik (1895) on Peirce’s calculus. Tarski axiomatized it as relation algebra in 1941, and with Steven Givant showed in 1987 that it is as expressive as first-order logic restricted to three variables.

Quantification, 1883–1885

For the general reader. “Everyone loves someone” and “someone is loved by everyone” use the same words in a different order and say different things. Logic needs a way to track this, and the way is quantifiers with variables. Frege introduced them in 1879; Peirce and his student Oscar Howard Mitchell arrived at them independently a few years later, and it was Peirce’s notation that spread.

The notation

In his paper in Studies in Logic (1883), drawing on Mitchell’s contribution to the same volume, Peirce wrote \(\Sigma_i\) for “some \(i\)” and \(\Pi_i\) for “every \(i\)”, by analogy with sums and products. In “On the Algebra of Logic: A Contribution to the Philosophy of Notation” (1885) he set out the theory. Writing \(l_{ij}\) for “\(i\) loves \(j\)”,

Over a finite domain, \(\Sigma\) is a disjunction and \(\Pi\) a conjunction, which is the reason for the notation. Peirce distinguished the logic of quantifiers over individuals from a logic that also quantifies over qualities and relations, the distinction now drawn between first-order and second-order logic. In the second, identity can be defined: \(i\) and \(j\) are identical when every quality of one is a quality of the other. Ernst Schröder adopted Peirce’s notation, and through Schröder it reached Giuseppe Peano, Leopold Löwenheim and Thoralf Skolem. Peirce seems not to have known Frege’s Begriffsschrift.

Proof of 1. If some \(j_0\) satisfies \(\phi(i,j_0)\) for every \(i\), then for each \(i\) the witness \(j=j_0\) works. For the converse failure, take the domain \(\{1,2\}\) with \(l=\{(1,1),(2,2)\}\): everyone loves someone (themselves), but nobody is loved by both.

Peirce’s 1885 paper puts formulas in prenex form, with the quantifier string first and a quantifier-free “Boolian” matrix after it. Löwenheim (1915) and Skolem (1920) worked in exactly this format.

Existential graphs

For the general reader. Late in life Peirce designed a logic made of drawings. Writing a statement on a sheet asserts it; drawing an oval around it denies it. With a handful of rules for erasing and copying, these pictures prove exactly the same things as ordinary logic. Peirce called them his masterpiece.

Peirce developed existential graphs from 1896, after an earlier system of entitative graphs, and published accounts in a 1903 syllabus and in The Monist in 1906. There are three parts. Alpha is propositional logic, beta is first-order logic with identity, and gamma, never completed, treats modality and higher-order notions.

Alpha

Graphs are written on a sheet of assertion. Writing two graphs side by side asserts both. A closed curve, called a cut, denies the graph inside it. An area is even if it lies inside an even number of cuts (the sheet itself is even) and odd otherwise. Writing \((\ldots)\) for a cut, \(P\ Q\) means \(P\wedge Q\), \((P)\) means \(\neg P\), and the scroll \((P\ (Q))\) means \(\neg(P\wedge\neg Q)\), that is \(P\to Q\).

Why the rules are sound. Erasing in an even area weakens what is asserted, and inserting in an odd area strengthens what is denied, which again weakens the whole. Iteration and deiteration replace \(A\wedge B\) by \(A\wedge B'\) where \(B'\) differs from \(B\) only by copies of \(A\), which are true given \(A\). Double cut is \(\neg\neg X\equiv X\). Completeness for alpha and the equivalence of beta with first-order logic with identity were established by Don Roberts and Jay Zeman in the 1960s and 1970s.

Beta

In beta a heavy line, the line of identity, asserts that something exists. Predicates attached to the same line are said of the same thing. A line whose outermost part lies in an even area is read existentially; one whose outermost part lies in an odd area is read universally. In the hero figure, the line runs from “man” in one cut to “mortal” in a second cut inside it: there is nothing that is a man and is not mortal, that is, every man is mortal.

Signs and categories

For the general reader. Smoke means fire, a map means a country, and the word “tree” means trees, but each in a different way. Peirce classified the ways a thing can stand for another, and he insisted that meaning always involves three terms, never two. His theory of signs, or semiotics, underlies much of modern linguistics, philosophy of language and the study of communication.

The categories

In “On a New List of Categories” (1867) Peirce argued that everything that can be thought falls under three universal categories, which he later named Firstness (quality, possibility), Secondness (reaction, brute fact) and Thirdness (mediation, law, habit). He derived them from the logic of relations: monadic, dyadic and triadic predicates.

Icon, index, symbol

Classified by the relation of sign to object:

• an icon signifies by resemblance or shared quality: a portrait, a diagram, an algebraic formula viewed as a pattern of relations;
• an index signifies by a real, causal or existential connection: smoke for fire, a weathercock for the wind, a pointing finger, a variable bound to an individual;
• a symbol signifies by convention or habit of interpretation: words, most mathematical notation.

Peirce combined this with two other trichotomies, of the sign itself (qualisign, sinsign, legisign) and of the interpretant (rheme, dicent, argument), to obtain ten classes of signs in 1903. He later projected finer classifications into 28 and 66 classes. He held that logic, in its broad sense, is the formal theory of signs.

The reduction thesis

Peirce claimed that every relation, of any number of places, can be built from monadic, dyadic and triadic relations by joining them on shared places, while some triadic relations, such as “\(A\) gives \(B\) to \(C\)”, cannot be built from dyadic relations alone. The claim depends on what counts as a construction. With his graphical joins, in which three lines of identity may meet at a point (“teridentity”), versions of the thesis were proved by Hans Herzberger (1981) and Robert Burch (1991).

Pendulums, gravity and the metre

For the general reader. A pendulum swings more slowly where gravity is weaker. By timing pendulums carefully at many places, geodesists measured how gravity varies over the Earth, and from that its flattening at the poles. Peirce found that the best European instruments had a hidden flaw: their stands swayed. He also proposed measuring length itself with light.

The reversible pendulum

A rigid body of mass \(M\) swinging about a horizontal axis at distance \(h\) from its centre of mass, with radius of gyration \(k\) about the centre of mass, has small-oscillation period

Proof. Equal periods mean \(h_1+k^2/h_1=h_2+k^2/h_2\), so \((h_1-h_2)(1-k^2/(h_1h_2))=0\), and \(h_1\ne h_2\) gives \(h_1h_2=k^2\). Then \(L=h_1+h_2\). The point is that \(g=4\pi^2\ell/T^2\) needs only a distance between two edges and a time, with no knowledge of the mass distribution. Friedrich Wilhelm Bessel refined the method; the Hamburg firm of Repsold built the instruments the Coast Survey bought.

Flexure of the stand

Peirce saw that the stand holding the knife edges yields slightly to the horizontal force of the swinging pendulum. A simple model shows the effect. Let the support move horizontally by \(x_s\) under a horizontal force \(F\), with \(F=\kappa x_s\). For small swings the pendulum pulls the support with force about \(Mg\theta\), so \(x_s=(Mg/\kappa)\theta\). The bob of a simple pendulum then sits at horizontal position \(x_s+\ell\theta=(\ell+\delta)\theta\), with \(\delta=Mg/\kappa\), and

The pendulum behaves as if longer by \(\delta\), the static displacement of the stand under a horizontal load equal to the pendulum’s weight. The same correction applies to a reversible pendulum about each knife edge, so the equal-period condition still holds, but the observer who uses \(g=4\pi^2\ell/T^2\) obtains \(g\,\ell/(\ell+\delta)\), too small. Peirce measured \(\delta\) by loading the stand statically and also by swinging pendulums of different weights. Émile Plantamour at Geneva confirmed the effect experimentally, and the international geodetic association accepted the correction.

Gravity and the figure of the Earth

Measuring \(g\) at many latitudes determines \(\beta\) and hence \(f\). Peirce’s pendulum work also showed how local gravity departs from this formula. The Coast Survey’s measurements, including his, fed into Friedrich Robert Helmert’s determinations of the Earth ellipsoid around 1900.

Light as a standard of length

In 1877 Peirce proposed defining the metre as a fixed number of wavelengths of a spectral line, so that the standard could be recovered if the platinum bar were lost. Around 1879, working with Lewis Morris Rutherfurd’s diffraction gratings, he compared a wavelength of light with the metre. The metre was defined by a wavelength of krypton light from 1960 to 1983.

Pragmatism as method

For the general reader. What does a word really mean? Peirce’s answer was practical: work out what difference it would make to experience if the claim were true, and that difference is the whole of its meaning. He offered this as a method for scientists, a way to clear away empty disputes. He also gave a theory of how inquiry moves from guesses to tested beliefs, and he ran one of the first randomized experiments.

The fixation of belief

In “The Fixation of Belief” (1877) Peirce described inquiry as the struggle to pass from the irritation of doubt to a settled belief. He compared four ways of settling belief: tenacity (holding on to what one already believes), authority, the a priori method (what seems agreeable to reason) and the method of science, which lets beliefs be fixed by something independent of us. Only the last, he argued, corrects itself.

The pragmatic maxim

In “How to Make Our Ideas Clear” (1878) he stated a rule for attaining the highest grade of clarity of a concept. Paraphrased: consider what effects, with conceivable practical bearings, we conceive the object of the concept to have; our conception of those effects is our whole conception of the object. His example was hardness: to call a diamond hard is to say that it would not be scratched by many other substances. In 1905, after William James and others had broadened “pragmatism” to a theory of truth, Peirce renamed his own version “pragmaticism”, a word he judged ugly enough to be safe from kidnappers.

Three kinds of inference

In “Deduction, Induction, and Hypothesis” (1878) Peirce used a bag of beans to separate three forms. Take the rule “all beans from this bag are white”, the case “these beans are from this bag”, and the result “these beans are white”.

Randomization

In 1884 Peirce and his student Joseph Jastrow tested whether people can perceive differences in weight too small to be consciously noticed. Subjects judged which of two weights was heavier and stated their confidence. The order of the stimuli was chosen at random by drawing playing cards, and the experimenter did not reveal it. The study, published in 1885, is one of the first randomized, blinded experiments in psychology, decades before Ronald Fisher made randomization a principle of design. Even at the lowest confidence the answers were correct more often than chance. Peirce argued that random sampling is what makes induction self-correcting.

Johns Hopkins and the fall, 1879–1891

At Johns Hopkins, from 1879, Peirce taught logic in the mathematics department, where James Joseph Sylvester was professor. He gathered a remarkable group of graduate students. Christine Ladd wrote a dissertation on the algebra of logic under him, completed in 1882; the university awarded her the degree only in 1926. Oscar Howard Mitchell contributed the quantifier ideas Peirce built on. Allan Marquand built logic machines, Benjamin Ives Gilman worked on probability, and Joseph Jastrow ran the experiments of Lesson 10. John Dewey attended his logic course. Their work appeared in Studies in Logic by Members of the Johns Hopkins University (1883), which Peirce edited.

His divorce from Zina became final on 24 April 1883, and six days later he married Juliette. The astronomer Simon Newcomb, a long-standing adversary, informed a trustee that Peirce had lived and travelled with Juliette before they were married. In January 1884 the trustees ended his appointment. He never held another academic post, though friends tried for decades to find him one.

The death of his father in 1880 had removed his protector at the Coast Survey. Peirce’s reports were slow, long and costly, and his manner offended superiors. In 1887 he and Juliette moved to Milford, Pennsylvania, where he used an inheritance to buy land and remodel a farmhouse, which he named Arisbe. His major report on gravity was returned for revision in 1890, and at the end of 1891 the Survey’s superintendent, Thomas Mendenhall, required his resignation. He was left without a regular income.

Arisbe, last years, legacy

From 1891 Peirce lived by writing: book reviews for The Nation, thousands of definitions in logic, mathematics, metaphysics and other fields for the Century Dictionary, and entries for James Mark Baldwin’s Dictionary of Philosophy and Psychology. He wrote a large treatise on logic and other books that found no publisher. Expensive schemes and lawsuits left the couple in debt. At times they had no fuel or food, and friends led by William James organized a fund to support them.

William James kept Peirce’s name alive. In 1898 he arranged a series of lectures in Cambridge, published much later as Reasoning and the Logic of Things, and in a lecture at Berkeley that year he credited Peirce with the principle of pragmatism, which made the word famous. In 1903 Peirce gave the Harvard lectures on pragmatism and a series at the Lowell Institute. In 1905 he restated his position as pragmaticism in The Monist. He worked to the end on logic and existential graphs, and he died of cancer at Arisbe on 19 April 1914, aged 74.

Juliette sold his manuscripts to Harvard. The Collected Papers appeared in 1931–1958, and a chronological critical edition, the Writings of Charles S. Peirce, has been in progress since 1982.

Legacy

• Logic: quantifiers and prenex form passed through Schröder to Löwenheim and Skolem; the calculus of relations through Schröder to Tarski; Peirce’s law and the single connective NOR remain textbook results. He also experimented with three-valued truth tables in his notebooks of 1909.
• Computing: the switching-circuit idea and the Peirce arrow anticipate digital logic; existential graphs inspired conceptual graphs in artificial intelligence.
• Semiotics: the icon–index–symbol distinction is standard in linguistics, philosophy of mind and design.
• Philosophy of science: pragmatism, through James and Dewey, became the distinctive American philosophy; abduction is now a central term in the study of scientific discovery and explanation.
• Science: his pendulum corrections improved international gravity measurement, and his proposal of a light standard for length anticipated the definition of the metre adopted in 1960.

Connections

People linked to Charles Sanders Peirce across this collection. A name without a link has no page here yet. See Charles Sanders Peirce in the network of all 337 lives →

Teachers and advisors

• Louis Agassiz

Students

• Allan Marquand 1879-1881
• Benjamin Ives Gilman
• Christine Ladd-Franklin 1879-1882
• John Dewey
• Joseph Jastrow 1882-1885
• Oscar Howard Mitchell 1879-1883

Influenced by

• Augustus De Morgan
• George Boole

Influenced

• Alfred Tarski
• Bertrand Russell
• Claude Shannon
• Ernst Schröder

Colleagues and collaborators

• James Joseph Sylvester 1879-1883
• Émile Plantamour

Rivals

• Gottlob Frege
• Simon Newcomb

Friends

• William James 1861-1910

Family

• Benjamin Peirce
• James Mills Peirce

Also mentioned in this life

• George Biddell Airy
• Friedrich Wilhelm Bessel
• Alexis Clairaut
• Ronald Fisher
• Isaac Newton
• George Gabriel Stokes

Fields

• logic 8 lives
• philosophy of science 7 lives
• geodesy 12 lives
• metrology 5 lives
• statistics 7 lives

On CoursesHub

• Philosophy of Science

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Charles Sanders Peirce, MacTutor History of Mathematics.
2. Charles Sanders Peirce, Wikipedia.
3. R. Burch, Charles Sanders Peirce, Stanford Encyclopedia of Philosophy.
4. M. H. Fisch, C. Eisele and others, Peirce, Charles Sanders, Dictionary of Scientific Biography, via encyclopedia.com.
5. J. Brent, Charles Sanders Peirce: A Life, Indiana University Press, revised edition 1998.
6. Grupo de Estudios Peirceanos, Peirce’s publications on gravity determination, Universidad de Navarra.
7. R. Robin, Annotated catalogue of the papers of C. S. Peirce: geodesy and metrology, Peirce Edition Project.

Logic, signs and method

1. C. S. Peirce, Description of a notation for the logic of relatives, Memoirs of the American Academy of Arts and Sciences 9 (1870).
2. C. S. Peirce, On the algebra of logic, American Journal of Mathematics 3 (1880).
3. C. S. Peirce (ed.), Studies in Logic by Members of the Johns Hopkins University, Little, Brown, 1883.
4. C. S. Peirce, On the algebra of logic: a contribution to the philosophy of notation, American Journal of Mathematics 7 (1885).
5. C. S. Peirce, The fixation of belief (1877) and How to make our ideas clear (1878), Popular Science Monthly.
6. C. S. Peirce and J. Jastrow, On small differences of sensation, Memoirs of the National Academy of Sciences 3 (1885).
7. C. S. Peirce, Collected Papers, 8 vols., Harvard University Press, 1931–1958; Writings of Charles S. Peirce: A Chronological Edition, Indiana University Press, 1982–.
8. I. H. Anellis and others, Peirce’s deductive logic, Stanford Encyclopedia of Philosophy.
9. A. Atkin, Peirce’s theory of signs, Stanford Encyclopedia of Philosophy.
10. Existential graph and Peirce’s law, Wikipedia.
11. D. D. Roberts, The Existential Graphs of Charles S. Peirce, Mouton, 1973.
12. A. Tarski, On the calculus of relations, Journal of Symbolic Logic 6 (1941).
13. R. W. Burch, A Peircean Reduction Thesis, Texas Tech University Press, 1991.