Henry M. Sheffer
Encyclopedia
Henry Maurice Sheffer was an American
logician
.
Sheffer was a Polish
Jew born in the western Ukraine
, who immigrated to the USA in 1892 with his parents and six siblings. He studied at the Boston Latin School
before entering Harvard University
, learning logic from Josiah Royce
, and completing his undergraduate degree in 1905, his master's in 1907, and his Ph.D. in philosophy in 1908. After holding a postdoctoral position at Harvard, Henry traveled to Europe on a fellowship. Upon returning to the United States, he became an academic nomad, spending one year each at the University of Washington
, Cornell, the University of Minnesota
, the University of Missouri
, and City College of New York
. In 1916 he returned to Harvard as a faculty member in the philosophy department. He remained at Harvard until his retirement in 1952. Scanlan (2000) is a study of Sheffer's life and work.
Sheffer proved in 1913 that Boolean algebra could be defined using a single primitive binary operation, "not both . . . and . . .", now abbreviated NAND, or its dual NOR
, (in the sense of "neither . . . nor"). Likewise, the propositional calculus
could be formulated using a single connective, having the truth table
either of the logical NAND, usually symbolized with a vertical line called the Sheffer stroke
, or its dual logical NOR
(usually symbolized with a vertical arrow or with a dagger
symbol). Charles Peirce had also discovered these facts in 1880, but the relevant paper was not published until 1933. Sheffer also proposed axioms formulated solely in terms of his stroke.
Sheffer introduced what is now known as the Sheffer stroke in 1913; it became well known only after its use in the 1925 edition of Whitehead
and Russell's
Principia Mathematica
. Sheffer's discovery won great praise from Bertrand Russell, who used it extensively to simplify his own logic, in the second edition of his Principia Mathematica. Because of this comment, Sheffer was something of a mystery man to logicians, especially because Sheffer, who published little in his career, never published the details of this method, only describing it in mimeographed notes and in a brief published abstract. W. V. Quine's Mathematical Logic also made much of the Sheffer stroke.
A Sheffer connective, subsequently, is any connective in a logical system that functions analogously: one in terms of which all other possible connectives in the language can be expressed. For example, they have been developed for quantificational and modal logics as well.
Sheffer was a dedicated teacher of mathematical logic. He liked his classes to be small and did not like auditors. When strangers appeared in his classroom, Sheffer would order them to leave, even his colleagues or distinguished guests visiting Harvard. Sheffer was barely five feet tall; he was noted for his wit and vigor, as well as for his nervousness and irritability. Although widely liked, he was quite lonely. He is noted for a quip he spoke at his retirement: "Old professors never die, they just become emeriti." Sheffer is also credited with coining the term "Boolean algebra". Sheffer was briefly married and lived most of his later life in small rooms at a hotel packed with his logic books and vast files of slips of paper he used to jot down his ideas. Unfortunately, Sheffer suffered from severe depression during the last two decades of his life.
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
logician
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
.
Sheffer was a Polish
Poland
Poland , officially the Republic of Poland , is a country in Central Europe bordered by Germany to the west; the Czech Republic and Slovakia to the south; Ukraine, Belarus and Lithuania to the east; and the Baltic Sea and Kaliningrad Oblast, a Russian exclave, to the north...
Jew born in the western Ukraine
Ukraine
Ukraine is a country in Eastern Europe. It has an area of 603,628 km², making it the second largest contiguous country on the European continent, after Russia...
, who immigrated to the USA in 1892 with his parents and six siblings. He studied at the Boston Latin School
Boston Latin School
The Boston Latin School is a public exam school founded on April 23, 1635, in Boston, Massachusetts. It is both the first public school and oldest existing school in the United States....
before entering Harvard University
Harvard University
Harvard University is a private Ivy League university located in Cambridge, Massachusetts, United States, established in 1636 by the Massachusetts legislature. Harvard is the oldest institution of higher learning in the United States and the first corporation chartered in the country...
, learning logic from Josiah Royce
Josiah Royce
Josiah Royce was an American objective idealist philosopher.-Life:Royce, born in Grass Valley, California, grew up in pioneer California very soon after the California Gold Rush. He received the B.A...
, and completing his undergraduate degree in 1905, his master's in 1907, and his Ph.D. in philosophy in 1908. After holding a postdoctoral position at Harvard, Henry traveled to Europe on a fellowship. Upon returning to the United States, he became an academic nomad, spending one year each at the University of Washington
University of Washington
University of Washington is a public research university, founded in 1861 in Seattle, Washington, United States. The UW is the largest university in the Northwest and the oldest public university on the West Coast. The university has three campuses, with its largest campus in the University...
, Cornell, the University of Minnesota
University of Minnesota
The University of Minnesota, Twin Cities is a public research university located in Minneapolis and St. Paul, Minnesota, United States. It is the oldest and largest part of the University of Minnesota system and has the fourth-largest main campus student body in the United States, with 52,557...
, the University of Missouri
University of Missouri
The University of Missouri System is a state university system providing centralized administration for four universities, a health care system, an extension program, five research and technology parks, and a publishing press. More than 64,000 students are currently enrolled at its four campuses...
, and City College of New York
City College of New York
The City College of the City University of New York is a senior college of the City University of New York , in New York City. It is also the oldest of the City University's twenty-three institutions of higher learning...
. In 1916 he returned to Harvard as a faculty member in the philosophy department. He remained at Harvard until his retirement in 1952. Scanlan (2000) is a study of Sheffer's life and work.
Sheffer proved in 1913 that Boolean algebra could be defined using a single primitive binary operation, "not both . . . and . . .", now abbreviated NAND, or its dual NOR
Logical NOR
In boolean logic, logical nor or joint denial is a truth-functional operator which produces a result that is the negation of logical or. That is, a sentence of the form is true precisely when neither p nor q is true—i.e. when both of p and q are false...
, (in the sense of "neither . . . nor"). Likewise, the propositional calculus
Propositional calculus
In mathematical logic, a propositional calculus or logic is a formal system in which formulas of a formal language may be interpreted as representing propositions. A system of inference rules and axioms allows certain formulas to be derived, called theorems; which may be interpreted as true...
could be formulated using a single connective, having the truth table
Truth table
A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—to compute the functional values of logical expressions on each of their functional arguments, that is, on each combination of values taken by their...
either of the logical NAND, usually symbolized with a vertical line called the Sheffer stroke
Sheffer stroke
In Boolean functions and propositional calculus, the Sheffer stroke, named after Henry M. Sheffer, written "|" , "Dpq", or "↑", denotes a logical operation that is equivalent to the negation of the conjunction operation, expressed in ordinary language as "not both"...
, or its dual logical NOR
Logical NOR
In boolean logic, logical nor or joint denial is a truth-functional operator which produces a result that is the negation of logical or. That is, a sentence of the form is true precisely when neither p nor q is true—i.e. when both of p and q are false...
(usually symbolized with a vertical arrow or with a dagger
Dagger (typography)
A dagger, or obelisk. is a typographical symbol or glyph. The term "obelisk" derives from Greek , which means "little obelus"; from meaning "roasting spit"...
symbol). Charles Peirce had also discovered these facts in 1880, but the relevant paper was not published until 1933. Sheffer also proposed axioms formulated solely in terms of his stroke.
Sheffer introduced what is now known as the Sheffer stroke in 1913; it became well known only after its use in the 1925 edition of Whitehead
Whitehead
-With common given names on further disambiguation pages:*Alan Whitehead *George Whitehead *Henry Whitehead *James Whitehead *John Whitehead...
and Russell's
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
Principia Mathematica
Principia Mathematica
The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913...
. Sheffer's discovery won great praise from Bertrand Russell, who used it extensively to simplify his own logic, in the second edition of his Principia Mathematica. Because of this comment, Sheffer was something of a mystery man to logicians, especially because Sheffer, who published little in his career, never published the details of this method, only describing it in mimeographed notes and in a brief published abstract. W. V. Quine's Mathematical Logic also made much of the Sheffer stroke.
A Sheffer connective, subsequently, is any connective in a logical system that functions analogously: one in terms of which all other possible connectives in the language can be expressed. For example, they have been developed for quantificational and modal logics as well.
Sheffer was a dedicated teacher of mathematical logic. He liked his classes to be small and did not like auditors. When strangers appeared in his classroom, Sheffer would order them to leave, even his colleagues or distinguished guests visiting Harvard. Sheffer was barely five feet tall; he was noted for his wit and vigor, as well as for his nervousness and irritability. Although widely liked, he was quite lonely. He is noted for a quip he spoke at his retirement: "Old professors never die, they just become emeriti." Sheffer is also credited with coining the term "Boolean algebra". Sheffer was briefly married and lived most of his later life in small rooms at a hotel packed with his logic books and vast files of slips of paper he used to jot down his ideas. Unfortunately, Sheffer suffered from severe depression during the last two decades of his life.