Henry Maurice Sheffer (1 September 1882 – 17 March 1964) was an American

logician Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...

Life and career

Sheffer was a Polish

Jew Jews ( he, יְהוּדִים, , ) or Jewish people are an ethnoreligious group and nation originating from the Israelites Israelite origins and kingdom: "The first act in the long drama of Jewish history is the age of the Israelites""T ...

born in the western

Ukraine Ukraine ( uk, Україна, Ukraïna, ) is a country in Eastern Europe. It is the second-largest European country after Russia, which it borders to the east and northeast. Ukraine covers approximately . Prior to the ongoing Russian inva ...

, who immigrated to the USA in 1892 with his parents and six siblings. He studied at the

Boston Latin School The Boston Latin School is a public exam school in Boston, Massachusetts. It was established on April 23, 1635, making it both the oldest public school in the British America and the oldest existing school in the United States. Its curriculum f ...

before entering

Harvard University Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of highe ...

, learning logic from

Josiah Royce Josiah Royce (; November 20, 1855 – September 14, 1916) was an American objective idealist philosopher and the founder of American idealism. His philosophical ideas included his version of personalism, defense of absolutism, idealism and his ...

, and completing his undergraduate degree in 1905, his master's in 1907, and his Ph.D. in philosophy in 1908. Sheffer was a postdoctoral fellow at Harvard, and then taught

University of Washington The University of Washington (UW, simply Washington, or informally U-Dub) is a public research university in Seattle, Washington. Founded in 1861, Washington is one of the oldest universities on the West Coast; it was established in Seatt ...

Cornell Cornell University is a private statutory land-grant research university based in Ithaca, New York. It is a member of the Ivy League. Founded in 1865 by Ezra Cornell and Andrew Dickson White, Cornell was founded with the intention to tea ...

, the

University of Minnesota The University of Minnesota, formally the University of Minnesota, Twin Cities, (UMN Twin Cities, the U of M, or Minnesota) is a public land-grant research university in the Twin Cities of Minneapolis and Saint Paul, Minnesota, United States. ...

, the

University of Missouri The University of Missouri (Mizzou, MU, or Missouri) is a public land-grant research university in Columbia, Missouri. It is Missouri's largest university and the flagship of the four-campus University of Missouri System. MU was founded in ...

, and

City College of New York The City College of the City University of New York (also known as the City College of New York, or simply City College or CCNY) is a public university within the City University of New York (CUNY) system in New York City. Founded in 1847, Cit ...

for one year each. In 1916, he was hired by Harvard as a philosophy professor, where he stayed until he retired in 1952. Scanlan (2000) is a study of Sheffer's life and work. Sheffer proved in 1913 that

Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...

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 Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...

could be formulated using a single connective, having the

truth table A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional arg ...

either of the

logical NAND In Boolean functions and propositional calculus, the Sheffer stroke denotes a logical operation that is equivalent to the negation of the conjunction operation, expressed in ordinary language as "not both". It is also called nand ("not and") or ...

, usually symbolized with a vertical line called the

Sheffer stroke In Boolean functions and propositional calculus, the Sheffer stroke denotes a logical operation that is equivalent to the negation of the conjunction operation, expressed in ordinary language as "not both". It is also called nand ("not and") ...

, or its dual logical NOR (usually symbolized with a vertical arrow or with a

dagger A dagger is a fighting knife with a very sharp point and usually two sharp edges, typically designed or capable of being used as a thrusting or stabbing weapon.State v. Martin, 633 S.W.2d 80 (Mo. 1982): This is the dictionary or popular-use def ...

symbol).

Charles Peirce Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American philosopher, logician, mathematician and scientist who is sometimes known as "the father of pragmatism". Educated as a chemist and employed as a scientist for ...

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 The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by mathematician–philosophers 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 A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A for ...

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.

Notes

* Scanlan, Michael, 2000, "The Known and Unknown H. M. Sheffer," ''The Transactions of the C.S. Peirce Society 36'': 193–224. * Rosen, Kenneth, 2005, "Discrete Mathematics and its Applications" ''The Foundations: Logic and Proofs 1'': 28.

References

{{DEFAULTSORT:Sheffer, Henry M. 1880s births 1964 deaths American people of Polish-Jewish descent American people of Ukrainian-Jewish descent Boston Latin School alumni Harvard University alumni Harvard University faculty American logicians Ukrainian Jews Austro-Hungarian emigrants to the United States Jewish philosophers