Max Black: The Philosopher of Metaphor, Vagueness and Two Identical Spheres

Max Black (24 February 1909 – 27 August 1988) was a Russian-born British-American philosopher and one of the central figures of analytic philosophy in the decades after the Second World War. He spent most of his career at Cornell University in Ithaca, New York. Black wrote on an unusually wide range of subjects: the foundations of mathematics, logic, the philosophy of language, the philosophy of science, induction, decision theory and the philosophy of art. He was also a translator and interpreter of other philosophers, above all Gottlob Frege and Ludwig Wittgenstein.

Many philosophy students meet Black through three short pieces of work that are still taught everywhere. The first is his 1937 paper on vagueness, which proposed a way of measuring how far a word applies to borderline cases and is often described as a forerunner of fuzzy set theory. The second is his 1952 paper "The Identity of Indiscernibles", which imagines a universe containing nothing but two identical iron spheres and asks whether they could really be two. The third is his 1954 essay "Metaphor", which launched the "interaction" theory of metaphor and helped convince philosophers that metaphors are a way of thinking, not just a decoration of speech.

Black described himself, in a phrase quoted by the MacTutor history of mathematics archive, as a "lapsed mathematician, addicted reasoner, and devotee of metaphor and chess." This article explains his life and his main ideas in plain terms, and sets out why they still matter.

Quick facts

Early life: from Baku to London

Max Black was born Max Tcherny in Baku, on the Caspian Sea, then part of the Russian Empire and now the capital of Azerbaijan. His family was Jewish and, according to the Oxford Dictionary of National Biography as summarised on Wikipedia, well off. MacTutor names his parents as Lionel Black, a businessman, and Sophia Davinska, and says the family suffered from the anti-Semitism widespread in the region at the time.

Soon after Max's birth the family left, first for Paris and then for London, where they settled in 1912. The surname was changed around 1911 or 1912 from Tcherny to Black, a translation: the Russian word chorny means "black". Max was about three when the family arrived in England, so he grew up entirely within English culture. His two younger brothers also became well known in Britain. Misha Black became a leading industrial designer and architect, and Sam Black a public relations manager.

Black attended a free school in north London and then, from the age of nine, a scholarship school, Dame Alice Owen's School, until he was eighteen. He was gifted in both mathematics and music. MacTutor reports that he played the violin to a high standard and at school thought seriously about a career in music. He was also a strong chess player, and chess remained a lifelong pleasure.

Cambridge, Göttingen and the philosophy of mathematics

In the end Black chose mathematics, which he studied at Queens' College, Cambridge. The Cambridge of the late 1920s was an extraordinary place for anyone interested in logic and the foundations of mathematics. Bertrand Russell, Ludwig Wittgenstein, G. E. Moore and Frank Ramsey were all there or nearby during Black's undergraduate years, and their influence pulled him toward philosophy. He graduated in 1930 and then spent a year on a research fellowship at the University of Göttingen in Germany, at that time one of the great centres of mathematics.

The Nature of Mathematics (1933)

At Göttingen Black worked on his first book, The Nature of Mathematics: A Critical Survey, published in 1933 (with a second edition in 1950). Its purpose was to explain the three great rival schools in the foundations of mathematics.

• Logicism, the view of Frege and of Russell and Whitehead's Principia Mathematica, holds that mathematics can be reduced to pure logic.
• Formalism, associated with David Hilbert, treats mathematics as the manipulation of symbols according to rules, and aims to prove that these systems are free of contradiction.
• Intuitionism, developed by L. E. J. Brouwer, holds that mathematical objects are mental constructions and rejects some standard methods of proof, such as proving that something exists by showing its non-existence leads to contradiction.

According to MacTutor, Black's aims were to give a careful critical account of Principia Mathematica and then to present formalism and intuitionism in enough detail for readers to go on to the original papers. MacTutor singles out his presentation of Brouwer's intuitionism as particularly fine. The book was reprinted more than a quarter of a century after it first appeared, a sign of how long it stayed useful.

Teaching in England

Back in England, Black taught mathematics at the Royal Grammar School in Newcastle from 1931 to 1936, and then lectured in mathematics at the Institute of Education in London from 1936 to 1940. In 1933 he married Michal Landesberg (sources also give the spelling Landsberg); they had a daughter and a son. Alongside his teaching he took a PhD at the University of London, awarded in 1939 for a dissertation on theories of logical positivism. In the same period he published papers on "The Evolution of Positivism" (1938) and the "Relations between Logical Positivism and the Cambridge School of Analysis" (1939).

Vagueness (1937): a forerunner of fuzzy sets

The problem

Most ordinary words are vague. There is no exact number of hairs that makes someone bald, no exact height that makes someone tall, and no exact point at which a pile of sand becomes a heap. Logic, by contrast, traditionally assumes that every statement is either true or false. If "Sam is tall" must be one or the other, where exactly is the line? This is the ancient sorites paradox, and it remains one of the hardest problems in the philosophy of language.

Black's paper "Vagueness: An Exercise in Logical Analysis" appeared in the journal Philosophy of Science in 1937. It looked at two questions: what vagueness is and how it can be observed, and what difference vagueness makes to logic.

The museum of chairs

Black's most memorable illustration asks us to imagine an exhibition in some "unlikely museum of applied logic": a long row of objects, starting with a perfectly ordinary chair, such as a Chippendale, and ending with a shapeless lump of wood, with each object differing from its neighbours by an amount too small to notice. Somewhere along the row, "chair" stops applying, but there is no single exhibit where it obviously does. In the paper Black wrote that "the vagueness of the word chair is typical of all terms whose application involves the use of the senses. In all such cases 'borderline cases' and 'doubtful objects' are easily found to which we are unable to say either that the class name does or does not apply."

Consistency profiles

Black's key move was to treat vagueness as something about how people actually use words, and therefore as something that could in principle be measured. Imagine asking many speakers whether each object in the row is a chair, or asking the same speaker many times. For objects near the chair end, nearly everyone says yes. Near the lump end, nearly everyone says no. In between, the answers are mixed. Plotting the proportion of "yes" answers against position along the row gives a curve, which Black called a "consistency profile". A precise word would have a profile that jumps suddenly from all yes to all no. A vague word has a profile that slopes.

This was, as MacTutor puts it, the first attempt to give a precise mathematical treatment of sets with a membership curve. Black's "vague sets" are close in spirit to what the engineer Lotfi Zadeh called "fuzzy sets" in 1965, in which membership comes in degrees between 0 and 1. MacTutor suggests that if Black's paper had made more impact at the time, today's terminology might be different. A paper by Filippone, La Rosa and Tabacchi argues that Black's search for a formal expression of vagueness probably influenced Zadeh, particularly through the idea of the consistency profile, and notes that the two men later corresponded about vagueness. That claim of influence is a scholarly interpretation, not something Zadeh is recorded as stating.

Black did not abandon the topic. He reprinted and extended the vagueness work in Language and Philosophy (1949), where he argued that the rules of ordinary discourse can never be followed with perfect precision, and he returned to it in a 1963 paper, "Reasoning with Loose Concepts". According to Filippone and colleagues, Black there argued against treating vagueness with many-valued logic.

America: Illinois and Cornell

In 1940 Black moved to the United States to join the philosophy department at the University of Illinois at Urbana–Champaign. After six years there he became a professor of philosophy at Cornell University in 1946. He became a US citizen in 1948, and in 1954 he was appointed Susan Linn Sage Professor of Philosophy and Humane Letters, a post he held until he retired. Sources differ slightly on the year of retirement, giving either 1977 or 1978. He continued to lecture widely after retiring.

At Cornell Black was part of one of the strongest philosophy departments in the United States. MacTutor reports that lists of his publications contain more than 200 items. His books from this period include:

He served as President of the Eastern Division of the American Philosophical Association in 1958–1959, was elected a Fellow of the American Academy of Arts and Sciences in 1963, and from 1981 to 1984 was President of the International Institute of Philosophy. MacTutor notes he was only the second American to hold that office.

Frege and Wittgenstein: Black as translator and interpreter

Bringing Frege into English

The German mathematician and philosopher Gottlob Frege (1848–1925) is now regarded as a founder of modern logic and of analytic philosophy, but for decades after his death his work was little read in the English-speaking world. In 1952 Black and the British philosopher Peter Geach published Translations from the Philosophical Writings of Gottlob Frege. The volume made available in English the essays that are now standard reading, including Frege's distinction between the "sense" and the "reference" of an expression. Wikipedia calls the translation "a classic text", and it shaped how generations of English-speaking philosophers understood Frege.

A guide to the Tractatus

Wittgenstein's Tractatus Logico-Philosophicus (1921) is a short, numbered series of compressed remarks on logic, language and the world, and it is notoriously difficult. Black's A Companion to Wittgenstein's Tractatus (1964) works through the text in detail, explaining the background and offering interpretations of individual remarks. It became one of the main reference works for students of the book.

The identity of indiscernibles (1952): two spheres in an empty universe

The principle

The "identity of indiscernibles" is a principle associated with the German philosopher Gottfried Wilhelm Leibniz. It says that if two things share all their properties, they are not really two things at all: they are one and the same. Put the other way round, any two distinct things must differ in at least one respect. The principle seems plausible. It is hard to see what could make two things two if absolutely nothing distinguished them.

The Stanford Encyclopedia of Philosophy (SEP) distinguishes several versions of the principle, depending on which properties count. A strong version counts only "pure" qualities such as size, shape, colour and temperature, plus relations to things described purely qualitatively. Weaker versions also count properties that mention particular individuals, such as "being identical to this sphere", which makes the principle trivially true.

Black's thought experiment

Black's paper "The Identity of Indiscernibles" appeared in the journal Mind in 1952. It is written as a dialogue between two speakers, one defending the principle and one attacking it. The attacker invites us to imagine a universe that contains only two iron spheres. They have the same diameter, the same colour, the same temperature and the same chemical make-up, and they sit a fixed distance apart. Nothing else exists: no observers, no landmarks, no other objects.

In this universe, every pure quality one sphere has, the other has too. Each is a sphere of a certain size. Each is a certain distance from a sphere of that size. Neither is to the left or right of anything except relative to the other, and there is no outside point of view to break the symmetry. So the two spheres share all their pure properties, both intrinsic and relational. Yet, the attacker argues, it seems perfectly conceivable that there are two of them. If this situation is possible, the strong version of the principle is false.

Why it mattered

The SEP calls Black's spheres "the most influential argument against the Identity of Indiscernibles in the history of philosophy", and says the paper "set much of the agenda for discussion of it since the 1950s onwards." The argument relies on one key assumption: that if we can coherently conceive a situation, it is possible. Much of the debate since has been about whether we really can conceive Black's world as containing two spheres.

Replies

Philosophers have answered Black in several ways, as summarised in the SEP.

• One sphere, not two. Ian Hacking argued in 1975 that the imagined scenario leaves open whether what we picture is two spheres at a distance from each other or merely one sphere at a distance from itself. If so, the example does not clearly describe two things.
• One scattered object. Katherine Hawley suggested in 2009 that what we imagine might be a single scattered object occupying two spherical regions.
• Weak discernibility. Simon Saunders and others have argued that the spheres can be told apart in a weak sense: each is at a distance from the other, but neither is at a distance from itself. The relation "is some distance from" holds between them but not of either with itself. The SEP notes that this rescues a version of the principle but does not answer the claim that Black's spheres break the strongest version.
• Bundle theory. Some defenders of the "bundle theory", the view that objects are nothing more than bundles of properties, have tried other ways of resisting the conclusion, since the bundle theory seems to require the identity of indiscernibles.

A related argument by Robert Adams in 1979 imagined two spheres that differ only in a tiny way, say by one degree of temperature, and argued that if that world is possible, it is hard to see why the world where even that difference vanishes is impossible.

The two spheres also matter outside metaphysics. In quantum physics, particles of the same type, such as two electrons, are often described as indistinguishable in a deep sense, and debates about whether they are "individuals" use the same tools that Black's example made standard.

Metaphor (1954): the interaction view

The background

For most of the history of Western thought, metaphor was treated as a matter of style. On this view, when a poet writes "Juliet is the sun", she could just as well have said something literal, such as "Juliet is radiant and Romeo cannot live without her", and the metaphor merely makes the point more vividly. Metaphor was decoration, useful for rhetoric and poetry but not for serious thought.

Black's essay "Metaphor", published in the Proceedings of the Aristotelian Society in 1954, attacked that picture. Building on earlier work by the literary critic I. A. Richards in The Philosophy of Rhetoric (1936), Black set out three views and argued for the third.

Three views of metaphor

The substitution view. A metaphor is simply a replacement for some literal expression that would say the same thing. Understanding a metaphor means working out which literal phrase it stands in for. On this view, metaphors could in principle always be replaced without loss.

The comparison view. This is a special case of the substitution view. A metaphor is a condensed comparison, a simile with the word "like" left out. "Man is a wolf" means "man is like a wolf", and understanding it means spotting the similarities that already exist.

Black argued that both views were inadequate, except perhaps for "dead" metaphors and cases where a language simply lacks a literal word (what rhetoricians call catachresis, as in the "leg" of a table). Good metaphors cannot be paraphrased without loss, and they often do not rest on similarities we had noticed before.

The interaction view. Black's alternative was that a metaphor brings two ideas into active interaction, and that this interaction produces a new meaning.

How interaction works

Black introduced a vocabulary that is still used.

• Focus and frame. In a metaphorical sentence, one word is used metaphorically and the rest is used literally. Black called the metaphorical word the focus and the surrounding sentence the frame. In "The chairman plowed through the discussion", one of Black's examples, "plowed" is the focus and the rest is the frame.
• Two subjects. A metaphor has a principal subject (what the sentence is really about, here the chairman's conduct) and a subsidiary subject (the idea brought in from elsewhere, here plowing). In his later essay he called these the primary and secondary subjects.
• The system of associated commonplaces. What a metaphor brings in is not the dictionary meaning of the subsidiary subject but the set of beliefs, images and associations that speakers commonly share about it, whether or not they are true. In "Man is a wolf", the relevant commonplaces about wolves are that they are fierce, predatory, hungry and engaged in constant struggle.
• The filter. The commonplaces about the subsidiary subject act as a filter or screen through which we see the principal subject. Thinking of man as a wolf brings forward the features of human life that fit the wolf picture, such as competition and aggression, and pushes others, such as kindness, into the background. The metaphor organises our view of its subject.
• Two-way change. The interaction runs both ways. A metaphor does not leave either idea untouched. If man is a wolf, wolves begin to seem more human as well.

Notice what "The chairman plowed through the discussion" leaves out. Most of what defines plowing, such as farming, soil and sowing, plays no part. What carries across is a pattern of forceful, steady, obstacle-ignoring progress. On the interaction view, understanding a metaphor is a matter of working out which part of the subsidiary system applies, and that is an active, open-ended process for the listener.

Metaphor as a cognitive tool

The most radical part of Black's view was the claim that metaphors can do real intellectual work. Rather than reporting similarities that already existed and could have been stated literally, a good metaphor can create a new way of seeing its subject, bringing similarities into view that no one had noticed before. If that is right, metaphors have cognitive content: they can be insightful or misleading, and they cannot be replaced by a literal paraphrase without losing something.

"More about Metaphor" (1977)

Black returned to the subject in "More about Metaphor", published in the journal Dialectica in 1977 and widely reprinted. He refined the theory, replacing talk of "associated commonplaces" with a more flexible notion of an "implication complex" of ideas that the secondary subject suggests, and he defended the view that metaphors can be cognitive instruments. He also noted that critics of the interaction view tended to focus on trivial examples such as "Man is a wolf", which are the kind of case the old substitution view handles best, and he urged attention to richer, more creative metaphors.

Criticism and influence

The interaction view has been criticised as well as admired. A frequent objection, made for example by the media theorist Hartmut Winkler, is that the idea of "associated commonplaces" is unclear: are they part of the language, or general knowledge about the world? If the latter, it is hard to say where the meaning of the word ends and the knowledge begins. The most famous opposing view came from the American philosopher Donald Davidson in his 1978 essay "What Metaphors Mean". Davidson argued that metaphors have no special metaphorical meaning at all. A metaphorical sentence means just what it literally says, usually something plainly false, and its effects come from what it leads us to notice, not from a hidden second meaning. The SEP classes Black, Richards and Monroe Beardsley together as early "semantic twist" theorists and presents Davidson as their sharpest critic.

Even so, Black's essay is widely seen as the moment when philosophers began treating metaphor as a central topic. Winkler describes it as marking "the breakthrough to a completely new understanding", in which metaphor is seen not as an ornament of speech but as one of the basic mechanisms of language. Later work on conceptual metaphor in linguistics and cognitive science, and on the role of metaphor in science, builds on the questions Black made respectable.

Models and Archetypes: metaphor in science

Black's 1962 book Models and Metaphors collected his metaphor essay alongside a chapter called "Models and Archetypes", which applied similar ideas to science. Scientists use the word "model" in many senses, and Black tried to sort them out.

Black's point was that theoretical models work much as metaphors do. Describing a gas as a collection of bouncing billiard balls brings a familiar system of ideas to bear on an unfamiliar one, and the model suggests questions, predictions and extensions that would not otherwise occur to anyone. A good model, like a good metaphor, is not just a teaching aid. It is a tool for discovery.

Black also introduced the idea of a "conceptual archetype": a broad, often unspoken set of ideas and images drawn from one area and used to organise thinking in another, sometimes across a whole field. This connected his work to later discussions of how scientific paradigms shape research. His remark that a science may begin with metaphor and end with algebra, moving from suggestive pictures to precise mathematics, is often quoted as a summary of the view.

Induction: self-supporting arguments

Induction is the kind of reasoning that moves from observed cases to general conclusions or predictions: the sun has risen every morning so far, so it will rise tomorrow. The Scottish philosopher David Hume argued in the eighteenth century that induction cannot be justified without circularity, since any argument that induction is reliable will itself depend on induction.

Black discussed induction in Problems of Analysis (1954). In his 1958 paper "Self-Supporting Inductive Arguments", published in the Journal of Philosophy, he defended a controversial response to Hume. An inductive rule might be supported by an inductive argument: the rule has worked well in the past, so, by that same rule, it will probably work well in the future. Critics say this is plainly circular. Black, in an approach also associated with R. B. Braithwaite, argued that it is not viciously circular in the usual sense, because the conclusion is not assumed among the premises. The argument uses the rule but does not assume, as a premise, that the rule is reliable. Philosophers now call this "rule-circularity", as distinct from "premise-circularity".

The view was immediately challenged. A note in the journal Analysis in 1962 argued that Black's argument was circular after all, and the debate has continued, with later philosophers such as David Papineau defending rule-circular justifications and others such as Andrew Cling arguing they cannot create justification. Black's general attitude, as MacTutor describes it, was pragmatic: inductive reasoning can be trusted when it is seen to work, and common sense must have its place.

Paradoxes of the infinite

Black also wrote on the paradoxes of the Greek philosopher Zeno, who argued that motion is impossible because, for example, a runner must first cover half the distance, then half the remainder, and so on without end. In "Achilles and the Tortoise", published in Analysis in 1951, Black argued that the standard mathematical answer, that an infinite series of distances can add up to a finite sum, does not by itself explain how anyone could actually complete infinitely many separate acts. To make the point he imagined "infinity machines" that perform an infinite series of tasks in a finite time. This discussion helped launch the modern philosophical literature on "supertasks", including James Thomson's well-known thought experiment of a lamp switched on and off infinitely many times in a minute.

Language, relativity and humbug

Whorf and linguistic relativity

In 1959 Black published "Linguistic Relativity: The Views of Benjamin Lee Whorf", a critical examination of the claim that the language a person speaks shapes or even determines how they think and perceive the world. Whorf's ideas, drawn from his studies of Hopi and other languages, were highly influential at the time. Black's paper subjected them to careful analytic scrutiny and is often cited in the history of the debate. In Models and Metaphors he also argued, as MacTutor quotes, that "the conception of language as a mirror of reality is radically mistaken", while holding that language should try to "conform to the discovered regularities of experience."

The prevalence of humbug

One of Black's best-known late essays gave its title to his 1985 collection The Prevalence of Humbug and Other Essays. In it he offered a careful definition of humbug as "deceptive misrepresentation, short of lying, especially by pretentious word or deed, of somebody's own thoughts, feelings, or attitudes." The essay found an unexpected second life. The philosopher Harry Frankfurt began his famous essay "On Bullshit" by working through Black's definition, using humbug as a close relative and stepping-stone to his own account. Frankfurt noted that Black's "especially" shows pretentiousness is not essential to humbug, and puzzled over what "short of lying" could mean. Some commentators have since argued that Frankfurt read Black too narrowly, since Black limited his definition to deliberate "first-degree" humbug and allowed for self-deluded forms.

Decision theory

Mathematics was never far from Black's mind. In a 1985 paper, "Making Intelligent Choices: How Useful Is Decision Theory?", he surveyed Bayesian decision theory, the mathematical theory of rational choice under uncertainty, and pointed out that real choosers often do not behave like ideal Bayesians. According to a review by P. L. Quinn quoted by MacTutor, Black argued that rational people do not simply take their options as given but actively structure them, that inconsistent preferences need not be a sign of irrationality, and that intelligent choice might be better thought of as "an informal, practical art" than as the application of a mathematical calculus.

Black's style and approach

Several features run through Black's work.

Clarity and caution. The historian J. Wilson-Quayle, writing in the American National Biography and quoted by MacTutor, said Black was known for "a commonsense, pragmatic approach to those theoretical issues that he knew required clarity." He was "highly sceptical of those who offered facile classifications" and "ever mindful of the tentative nature that characterised most philosophical investigations."

Breadth. Black moved easily between mathematics, logic, language, science and art. He wrote major papers in each area rather than staying inside one specialism.

Short, sharp papers. Many of his most influential contributions are essays rather than books: the vagueness paper, the spheres dialogue, the metaphor essay. Each sets up a problem in a vivid way that others could not ignore.

Teaching. Black was a prominent teacher at Cornell. His students included the philosopher of language Keith Donnellan, known for work on how names and descriptions refer, and John Perry, known for work on indexicals and personal identity. He also supervised the doctoral dissertation of William H. Gass, who became a major American novelist and essayist.

Death

Black's wife Michal died in 1985. Max Black died in Ithaca, New York, on 27 August 1988, aged 79, of a heart attack following surgery for cancer. A collection of his essays, Perplexities, appeared in 1990. His papers are held at Cornell University Library.

Legacy

Black's influence is spread across many fields, which is one reason his name is less familiar than those of some contemporaries who concentrated on a single big system. Yet in several areas his work remains a starting point.

• Metaphor. The interaction view is the standard reference point for philosophical theories of metaphor. Almost every survey starts with Richards and Black, and Davidson's rival view was formulated against theirs.
• Vagueness. The 1937 paper is cited in histories of fuzzy logic and as an early empirical, quantitative approach to vagueness.
• Metaphysics. Black's spheres remain the classic test case for the identity of indiscernibles and for theories of what makes one thing distinct from another.
• Philosophy of science. His account of models anticipated the large later literature on models, analogies and scientific representation.
• Frege scholarship. The Geach and Black translations were the English-speaking world's main route into Frege for decades.
• Students and colleagues. Through Cornell, his teaching and his offices in professional bodies, he shaped the profession in the United States and internationally.

Frequently asked questions

Where was Max Black born, and why is his name "Black"?

He was born Max Tcherny in Baku, then in the Russian Empire and now the capital of Azerbaijan. When the family moved west, around 1911 or 1912, they translated the surname into English. The Russian word for "black" is chorny.

What is the interaction theory of metaphor, in one paragraph?

It is the view that a metaphor works by bringing two ideas together so that each affects how we understand the other. In "Man is a wolf", the ideas commonly associated with wolves act as a filter that highlights some features of humans and hides others. The result is a new way of seeing the subject that cannot be fully captured by a literal paraphrase. Black developed it in 1954 from earlier work by I. A. Richards.

What is the "two spheres" argument?

It is a thought experiment from Black's 1952 paper "The Identity of Indiscernibles". Imagine a universe containing only two exactly similar iron spheres a fixed distance apart. They share every pure property, yet there seem to be two of them. If that is possible, Leibniz's principle that things sharing all properties must be identical is false, at least in its strong form.

Did Max Black invent fuzzy logic?

No. Fuzzy sets were introduced by Lotfi Zadeh in 1965. But Black's 1937 paper on vagueness proposed "consistency profiles", curves describing how far a vague word applies along a range of cases, and this is often cited as an important forerunner. Some scholars think it probably influenced Zadeh.

What did Black do with Frege's work?

With Peter Geach, he translated Frege's key philosophical essays into English in 1952. The collection became the standard source for English-speaking readers and helped make Frege central to analytic philosophy.

How did Black's work influence Harry Frankfurt's "On Bullshit"?

Frankfurt began his essay by analysing Black's definition of humbug, "deceptive misrepresentation, short of lying, especially by pretentious word or deed, of somebody's own thoughts, feelings, or attitudes", and used it as a starting point for his own, different account of bullshit.

Where did Max Black teach?

Mainly at Cornell University, from 1946 until his retirement in the late 1970s, where he held the Susan Linn Sage chair from 1954. Before that he taught at the University of Illinois (1940–1946) and, in England, at the Royal Grammar School in Newcastle and the Institute of Education in London.

Who were Max Black's brothers?

Misha Black, an influential designer and architect, and Sam Black, a public relations manager.

Sources

• Max Black – Wikipedia
• Max Black – MacTutor History of Mathematics, University of St Andrews
• Peter Forrest, "The Identity of Indiscernibles" – Stanford Encyclopedia of Philosophy
• Metaphor – Stanford Encyclopedia of Philosophy
• Hartmut Winkler, "Metaphor, context, discourse, system" (University of Paderborn)
• Max Black, "Vagueness: An Exercise in Logical Analysis", Philosophy of Science 4 (1937) – Cambridge Core
• Filippone, La Rosa & Tabacchi, paper on Max Black and a logic of vagueness (University of Palermo)
• Max Black, "The Identity of Indiscernibles", Mind (1952) – JSTOR
• Harry Frankfurt, "On Bullshit" (1986/1988)
• Max Black, The Prevalence of Humbug (ditext)