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''Begriffsschrift'' (German for, roughly, "concept-script") is a book on
logic 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 ...
by
Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic phil ...
, published in 1879, and the
formal 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 form ...
set out in that book. ''Begriffsschrift'' is usually translated as ''concept writing'' or ''concept notation''; the full title of the book identifies it as "a
formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwee ...
language Language is a structured system of communication. The structure of a language is its grammar and the free components are its vocabulary. Languages are the primary means by which humans communicate, and may be conveyed through a variety of met ...
, modeled on that of
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
, for pure
thought In their most common sense, the terms thought and thinking refer to conscious cognitive processes that can happen independently of sensory stimulation. Their most paradigmatic forms are judging, reasoning, concept formation, problem solving, a ...
." Frege's motivation for developing his formal approach to logic resembled
Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ma ...
's motivation for his ''
calculus ratiocinator The ''calculus ratiocinator'' is a theoretical universal logical calculation framework, a concept described in the writings of Gottfried Leibniz, usually paired with his more frequently mentioned ''characteristica universalis'', a universal conce ...
'' (despite that, in the foreword Frege clearly denies that he achieved this aim, and also that his main aim would be constructing an ideal language like Leibniz's, which Frege declares to be a quite hard and idealistic—though not impossible—task). Frege went on to employ his logical calculus in his research on the
foundations of mathematics Foundations of mathematics is the study of the philosophy, philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the natu ...
, carried out over the next quarter century. This is the first work in
Analytical Philosophy Analytic philosophy is a branch and tradition of philosophy using analysis, popular in the Western world and particularly the Anglosphere, which began around the turn of the 20th century in the contemporary era in the United Kingdom, United Sta ...
, a field that future British and Anglo philosophers such as
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ...
further developed.


Notation and the system

The calculus contains the first appearance of quantified variables, and is essentially classical bivalent
second-order logic In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory. First-order logic quantifies onl ...
with identity. It is bivalent in that sentences or formulas denote either True or False; second order because it includes relation variables in addition to object variables and allows quantification over both. The modifier "with identity" specifies that the language includes the identity relation, =. Frege stated that his book was his version of a
characteristica universalis The Latin term ''characteristica universalis'', commonly interpreted as ''universal characteristic'', or ''universal character'' in English, is a universal and formal language imagined by Gottfried Leibniz able to express mathematical, scientif ...
, a Leibnizian concept that would be applied in mathematics. Frege presents his calculus using idiosyncratic two-dimensional
notation In linguistics and semiotics, a notation is a system of graphics or symbols, characters and abbreviated expressions, used (for example) in artistic and scientific disciplines to represent technical facts and quantities by convention. Therefore, ...
: connectives and quantifiers are written using lines connecting formulas, rather than the symbols ¬, ∧, and ∀ in use today. For example, that judgement ''B'' materially implies judgement ''A'', i.e. B \rightarrow A is written as . In the first chapter, Frege defines basic ideas and notation, like proposition ("judgement"), the universal quantifier ("the generality"), the conditional,
negation In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...
and the "sign for identity of content" \equiv (which he used to indicate both
material equivalence Material is a substance or mixture of substances that constitutes an object. Materials can be pure or impure, living or non-living matter. Materials can be classified on the basis of their physical and chemical properties, or on their geolog ...
and identity proper); in the second chapter he declares nine formalized propositions as axioms. In chapter 1, §5, Frege defines the conditional as follows: :"Let A and B refer to judgeable contents, then the four possibilities are: # A is asserted, B is asserted; # A is asserted, B is negated; # A is negated, B is asserted; #A is negated, B is negated. Let : signify that the third of those possibilities does not obtain, but one of the three others does. So if we negate , that means the third possibility is valid, i.e. we negate A and assert B."


The calculus in Frege's work

Frege declared nine of his propositions to be
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
s, and justified them by arguing informally that, given their intended meanings, they express self-evident truths. Re-expressed in contemporary notation, these axioms are: # \vdash \ \ A \rightarrow \left( B \rightarrow A \right) # \vdash \ \ \left \ A \rightarrow \left( B \rightarrow C \right) \ \right\ \rightarrow \ \left \ \left( A \rightarrow B \right) \rightarrow \left( A \rightarrow C \right) \ \right # \vdash \ \ \left \ D \rightarrow \left( B \rightarrow A \right) \ \right\ \rightarrow \ \left \ B \rightarrow \left( D \rightarrow A \right) \ \right # \vdash \ \ \left( B \rightarrow A \right) \ \rightarrow \ \left( \lnot A \rightarrow \lnot B \right) # \vdash \ \ \lnot \lnot A \rightarrow A # \vdash \ \ A \rightarrow \lnot\lnot A # \vdash \ \ \left( c=d \right) \rightarrow \left( f\left(c\right) = f\left(d\right) \right) # \vdash \ \ c = c # \vdash \ \ \forall a \ f(a) \rightarrow \ f(c) These are propositions 1, 2, 8, 28, 31, 41, 52, 54, and 58 in the ''Begriffschrifft''. (1)–(3) govern material implication, (4)–(6)
negation In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...
, (7) and (8) identity, and (9) the universal quantifier. (7) expresses
Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ma ...
's indiscernibility of identicals, and (8) asserts that identity is a
reflexive relation In mathematics, a binary relation ''R'' on a set ''X'' is reflexive if it relates every element of ''X'' to itself. An example of a reflexive relation is the relation " is equal to" on the set of real numbers, since every real number is equal to ...
. All other propositions are deduced from (1)–(9) by invoking any of the following
inference rule In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of ...
s: *
Modus ponens In propositional logic, ''modus ponens'' (; MP), also known as ''modus ponendo ponens'' (Latin for "method of putting by placing") or implication elimination or affirming the antecedent, is a deductive argument form and rule of inference. ...
allows us to infer \vdash B from \vdash A \to B and \vdash A; *The rule of generalization allows us to infer \vdash P \to \forall x A(x) from \vdash P \to A(x) if ''x'' does not occur in ''P''; *The rule of substitution, which Frege does not state explicitly. This rule is much harder to articulate precisely than the two preceding rules, and Frege invokes it in ways that are not obviously legitimate. The main results of the third chapter, titled "Parts from a general series theory," concern what is now called the
ancestral An ancestor, also known as a forefather, fore-elder or a forebear, is a parent or (recursively) the parent of an antecedent (i.e., a grandparent, great-grandparent, great-great-grandparent and so forth). ''Ancestor'' is "any person from whom ...
of a relation ''R''. "''a'' is an ''R''-ancestor of ''b''" is written "''aR''*''b''". Frege applied the results from the ''Begriffsschrifft'', including those on the ancestral of a relation, in his later work ''
The Foundations of Arithmetic ''The Foundations of Arithmetic'' (german: Die Grundlagen der Arithmetik) is a book by Gottlob Frege, published in 1884, which investigates the philosophical foundations of arithmetic. Frege refutes other theories of number and develops his own t ...
''. Thus, if we take ''xRy'' to be the relation ''y'' = ''x'' + 1, then 0''R''*''y'' is the predicate "''y'' is a natural number." (133) says that if ''x'', ''y'', and ''z'' are
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
s, then one of the following must hold: ''x'' < ''y'', ''x'' = ''y'', or ''y'' < ''x''. This is the so-called "law of trichotomy".


Philosophy


Influence on other works

For a careful recent study of how the ''Begriffsschrift'' was reviewed in the German mathematical literature, see Vilko (1998). Some reviewers, especially Ernst Schröder, were on the whole favorable. All work in formal logic subsequent to the ''Begriffsschrift'' is indebted to it, because its second-order logic was the first formal logic capable of representing a fair bit of mathematics and natural language. Some vestige of Frege's notation survives in the "
turnstile A turnstile (also called a turnpike, gateline, baffle gate, automated gate, turn gate in some regions) is a form of gate which allows one person to pass at a time. A turnstile can be configured to enforce one-way human traffic. In addition, a t ...
" symbol \vdash derived from his "Urteilsstrich" (''judging/inferring stroke'') │ and "Inhaltsstrich" (i.e. ''content stroke'') ──. Frege used these symbols in the ''Begriffsschrift'' in the unified form ├─ for declaring that a proposition is true. In his later "Grundgesetze" he revises slightly his interpretation of the ├─ symbol. In "Begriffsschrift" the "Definitionsdoppelstrich" (i.e. ''definition double stroke'') │├─ indicates that a proposition is a definition. Furthermore, the negation sign \neg can be read as a combination of the horizontal ''Inhaltsstrich'' with a vertical negation stroke. This negation symbol was reintroduced by
Arend Heyting __NOTOC__ Arend Heyting (; 9 May 1898 – 9 July 1980) was a Dutch mathematician and logician. Biography Heyting was a student of Luitzen Egbertus Jan Brouwer at the University of Amsterdam, and did much to put intuitionistic logic on a foot ...
Arend Heyting: "Die formalen Regeln der intuitionistischen Logik," in: ''Sitzungsberichte der preußischen Akademie der Wissenschaften, physikalisch-mathematische Klasse'', 1930, pp. 42–65. in 1930 to distinguish
intuitionistic In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of f ...
from classical negation. It also appears in Gerhard Gentzen's doctoral dissertation. In the ''
Tractatus Logico Philosophicus The ''Tractatus Logico-Philosophicus'' (widely abbreviated and Citation, cited as TLP) is a book-length philosophical work by the Austrian philosopher Ludwig Wittgenstein which deals with the relationship between language and reality and aims to ...
'',
Ludwig Wittgenstein Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrian-British philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. He is considere ...
pays homage to Frege by employing the term ''Begriffsschrift'' as a synonym for logical formalism. Frege's 1892 essay, "
On Sense and Reference In the philosophy of language, the distinction between sense and reference was an idea of the German philosopher and mathematician Gottlob Frege in 1892 (in his paper "On Sense and Reference"; German: "Über Sinn und Bedeutung"), reflecting the ...
," recants some of the conclusions of the ''Begriffsschrifft'' about identity (denoted in mathematics by the "=" sign). In particular, he rejects the "Begriffsschrift" view that the identity predicate expresses a relationship between names, in favor of the conclusion that it expresses a relationship between the objects that are denoted by those names.


Editions

*
Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic phil ...
. ''Begriffsschrift: eine der arithmetischen nachgebildete Formelsprache des reinen Denkens''. Halle an der Saale: Verlag von Louis Nebert, 1879. Translations:
Bynum, Terrell Ward
translated and edited, 1972. ''Conceptual notation and related articles'', with a biography and introduction.
Oxford University Press Oxford University Press (OUP) is the university press of the University of Oxford. It is the largest university press in the world, and its printing history dates back to the 1480s. Having been officially granted the legal right to print books ...
. * Bauer-Mengelberg, Stefan, 1967, "Concept Script" in
Jean van Heijenoort Jean Louis Maxime van Heijenoort (; July 23, 1912 – March 29, 1986) was a historian of mathematical logic. He was also a personal secretary to Leon Trotsky from 1932 to 1939, and an American Trotskyist until 1947. Life Van Heijenoort was born ...
, ed., ''From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931''.
Harvard University Press Harvard University Press (HUP) is a publishing house established on January 13, 1913, as a division of Harvard University, and focused on academic publishing. It is a member of the Association of American University Presses. After the retirem ...
. * Beaney, Michael, 1997, "Begriffsschrift: Selections (Preface and Part I)" in ''The Frege Reader''. Oxford: Blackwell.


See also

* Ancestral relation *
Calculus of equivalent statements Hugh MacColl (before April 1885 spelled as Hugh McColl; 1831–1909) was a Scottish mathematician, logician and novelist. Life MacColl was the youngest son of a poor Highland family that was at least partly Gaelic-speaking. Hugh's father died w ...
*
First-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
*
Frege's propositional calculus In mathematical logic, Frege's propositional calculus was the first axiomatization of propositional calculus. It was invented by Gottlob Frege, who also invented predicate calculus, in 1879 as part of his second-order predicate calculus (although ...
*''
Prior Analytics The ''Prior Analytics'' ( grc-gre, Ἀναλυτικὰ Πρότερα; la, Analytica Priora) is a work by Aristotle on reasoning, known as his syllogistic, composed around 350 BCE. Being one of the six extant Aristotelian writings on logic ...
'' *''
The Laws of Thought ''An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities'' by George Boole, published in 1854, is the second of Boole's two monographs on algebraic logic. Boole was a professor of mathe ...
'' *''
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. ...
''


References


Bibliography

*
George Boolos George Stephen Boolos (; 4 September 1940 – 27 May 1996) was an American philosopher and a mathematical logician who taught at the Massachusetts Institute of Technology. Life Boolos is of Greek-Jewish descent. He graduated with an A.B. i ...
, 1985. "Reading the ''Begriffsschrift''", ''Mind'' 94: 331–344. * Ivor Grattan-Guinness, 2000. ''In Search of Mathematical Roots''. Princeton University Press. * Risto Vilkko, 1998,
The reception of Frege's ''Begriffsschrift''
" ''Historia Mathematica 25(4)'': 412–422.


External links

*
''Begriffsschrift'' as facsimile for download (2.5 MB)
*
Esoteric programming language An esoteric programming language (sometimes shortened to esolang) is a programming language designed to test the boundaries of computer programming language design, as a proof of concept, as software art, as a hacking interface to another language ...
: {{Authority control 1879 non-fiction books Books by Gottlob Frege Logic books Diagram algebras Analytic philosophy literature Philosophy of logic Classical logic Predicate logic