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| [[File:Begriffsschrift Titel.png|thumb|200px|The title page of the original 1879 edition]]
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| '''''Begriffsschrift''''' (German for, roughly, "concept-script") is a book on [[logic]] by [[Gottlob Frege]], published in 1879, and the [[formal system]] set out in that book. It is generally considered the work that marks the birth of modern logic.{{citation needed|date=November 2010}}
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| ''Begriffsschrift'' is usually translated as ''concept writing'' or ''concept notation''; the full title of the book identifies it as "a [[formula]] [[language]], modeled on that of [[arithmetic]], of pure [[thought]]." The ''Begriffsschrift'' was arguably the most important publication in [[logic]] since [[Aristotle]] founded the subject.{{fact|date=January 2014}} Frege's motivation for developing his formal approach to logic resembled [[Gottfried Wilhelm Leibniz|Leibniz]]'s motivation for his [[calculus ratiocinator]] (despite that, in his ''Foreword'' Frege clearly denies that he reached this aim, and also that his main aim would be constructing an ideal language like Leibniz's, what Frege declares to be quite hard and idealistic, however, not impossible task). Frege went on to employ his logical calculus in his research on the [[foundations of mathematics]], carried out over the next quarter century.
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| == Notation and the system ==
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| The calculus contains the first appearance of quantified variables, and is essentially classical bivalent [[second-order logic]] with identity,{{clarification needed|date=January 2014}} albeit presented using a highly idiosyncratic two-dimensional [[mathematical notation|notation]]: 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. <math> B \rightarrow A </math> is written as [[File:Kondicionaliskis wb.png]].
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| In the first chapter, Frege defines basic ideas and notation, like proposition ("judgement"), the [[universal quantifier]] ("the generality"), the [[material conditional|conditional]], [[negation]] and the "sign for identity of content" <math> \equiv </math> (which he used to indicate both [[material equivalence]] and identity proper); in the second chapter he declares nine formalized propositions as axioms.
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| {| class="wikitable" style="margin:0.5em auto;"
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| |-
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| ! scope="col" |Basic concept
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| ! scope="col" |Frege's notation
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| ! scope="col" |Modern notations
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| |-
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| ||Judging
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| |style="text-align:center;"|<math>\vdash A,\Vdash A</math>
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| |style="text-align:center;"|<math>p(A)=1</math>
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| <math>p(A)=i</math>
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| |-
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| ||Negation
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| |style="text-align:center;"|[[File:Begriffsschrift connective1.svg|60px]]
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| |style="text-align:center;"|<math>\neg A, \sim A</math>
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| |-
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| ||Conditional (implication)
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| |style="text-align:center;"|[[File:Begriffsschrift connective2.svg|80px]]
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| |style="text-align:center;"|<math>B\rightarrow A</math>
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| <math>B\supset A</math>
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| |-
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| ||Universal quantification
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| |style="text-align:center;"|[[File:Begriffsschrift Quantifier1.png]]
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| |style="text-align:center;"|<math>\forall x\colon F(x)</math>
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| |-
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| ||[[Existential quantification]]
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| |style="text-align:center;"|[[File:Begriffsschrift Quantifier3.png]]
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| |style="text-align:center;"|<math>\sim \forall x \sim F(x)</math><br>
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| <math>\exists x\colon F(x)</math>
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| |-
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| ||Content identity (equivalence/identity)
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| |style="text-align:center;"|<math>A\equiv B</math>
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| |style="text-align:center;"|A ↔ B<br>
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| <math>A \equiv B</math><br>
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| <math>A = B</math>
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| |}
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| In chapter 1, §5, Frege defines the conditional as follows:
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| :"Let A and B refer to judgeable contents, then the four possibilities are:
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| # A is asserted, B is asserted;
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| # A is asserted, B is negated;
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| # A is negated, B is asserted;
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| #A is negated, B is negated.
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| Let
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| :[[File:Kondicionaliskis wb.png]]
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| signify that the third of those possibilities does not obtain, but one of the three others does. So if we negate [[File:Begriffsschrift connective2.svg|69x55px]],
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| that means the third possibility is valid, i.e. we negate A and assert B."
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| == The calculus in Frege's work ==
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| Frege declared nine of his propositions to be [[axiom]]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:
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| # <math> \vdash \ \ A \rightarrow \left( B \rightarrow A \right) </math>
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| # <math> \vdash \ \ \left[ \ A \rightarrow \left( B \rightarrow C \right) \ \right] \ \rightarrow \ \left[ \ \left( A \rightarrow B \right) \rightarrow \left( A \rightarrow C \right) \ \right] </math>
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| # <math> \vdash \ \ \left[ \ D \rightarrow \left( B \rightarrow A \right) \ \right] \ \rightarrow \ \left[ \ B \rightarrow \left( D \rightarrow A \right) \ \right] </math>
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| # <math> \vdash \ \ \left( B \rightarrow A \right) \ \rightarrow \ \left( \lnot A \rightarrow \lnot B \right) </math>
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| # <math> \vdash \ \ \lnot \lnot A \rightarrow A </math>
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| # <math> \vdash \ \ A \rightarrow \lnot\lnot A </math>
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| # <math> \vdash \ \ \left( c=d \right) \rightarrow \left( f(c) \rightarrow f(d) \right) </math>
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| # <math> \vdash \ \ c = c </math>
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| # <math> \vdash \ \ \forall a f(a) \rightarrow \ f(c) </math>
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| These are propositions 1, 2, 8, 28, 31, 41, 52, 54, and 58 in the ''Begriffschrifft''. (1)–(3) govern [[material conditional|material implication]], (4)–(6) [[negation]], (7) and (8) identity, and (9) the [[universal quantifier]]. (7) expresses [[Gottfried Wilhelm Leibniz|Leibniz]]'s [[identity of indiscernibles|indiscernibility of identicals]], and (8) asserts that identity is a [[reflexive relation]].
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| All other propositions are deduced from (1)–(9) by invoking any of the following [[inference rule]]s:
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| *[[Modus ponens]] allows us to infer <math>\subset B</math> from <math>\subset A \to B</math> and <math>\subset A</math>;
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| *The [[generalization (logic)|rule of generalization]] allows us to infer <math>\subset P \subset \mathcal x : A(x)</math> from <math>\vdash P \to A(x)</math> if ''x'' does not occur in ''P'';
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| *The [[first-order logic|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.
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| The main results of the third chapter, titled "Parts from a general series theory," concern what is now called the [[ancestral relation|ancestral]] of a relation ''R''. "''a'' is an ''R''-ancestor of ''b''" is written "''aR''*''b''".
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| Frege applied the results from the ''Begriffsschrifft'', including those on the ancestral of a relation, in his later work ''[[The Foundations of Arithmetic]]''. 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]]s, then one of the following must hold: ''x'' < ''y'', ''x'' = ''y'', or ''y'' < ''x''. This is the so-called "law of [[trichotomy (mathematics)|trichotomy]]".
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| == Influence on other works ==
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| 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.
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| Some vestige of Frege's notation survives in the "[[Turnstile (symbol)|turnstile]]" symbol <math>\vdash</math> derived from his "Inhaltsstrich" (i.e. ''content stroke'') ── and "Urteilsstrich" (''judging/infering 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.
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| In "Begriffsschrift" the "Definitionsdoppelstrich" (i.e. ''definition double stroke'') │├─ indicates that a proposition is a definition. Furthermore, the negation sign <math>\neg</math> can be read as a combination of the horizontal ''Inhaltsstrich'' with a vertical negation stroke. This negation symbol was reintroduced by [[Arend Heyting]]<ref>Arend Heyting: "Die formalen Regeln der intuitionistischen Logik," in: ''Sitzungsberichte der preußischen Akademie der Wissenschaften, phys.-math. Klasse'', 1930, S. 42-65.</ref> in 1930 to distinguish [[intuitionistic]] from classical negation. It also appears in Gerhard Gentzen's doctoral dissertation.
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| In the ''[[Tractatus Logico Philosophicus]]'', [[Ludwig Wittgenstein]] pays homage to Frege by employing the term ''Begriffsschrift'' as a synonym for logical formalism.
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| Frege's 1892 essay, ''[[Sense and reference]]'', 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 [[equality (mathematics)|a relationship between the objects]] that are [[denotation (semiotics)|denoted]] by those names.
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| == A quotation ==
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| {{Copy to Wikiquote}}
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| <blockquote>"If the task of philosophy is to break the domination of words over the human mind [...], then my concept notation, being developed for these purposes, can be a useful instrument for philosophers [...] I believe the cause of logic has been advanced already by the invention of this concept notation." (Preface to the ''Begriffsschrift'')</blockquote>
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| ==See also==
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| *[[Ancestral relation]]
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| *[[Frege's propositional calculus]]
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| ==References==
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| {{Reflist}}
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| == Further reading ==
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| *[[Gottlob Frege]]. ''Begriffsschrift: eine der arithmetischen nachgebildete Formelsprache des reinen Denkens''. Halle, 1879.
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| Translations:
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| * [http://www.southernct.edu/organizations/rccs/staff.html Bynum, Terrell Ward], trans. and ed., 1972. ''Conceptual notation and related articles'', with a biography and introduction. Oxford Uni. Press.
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| *Bauer-Mengelberg, Stefan, 1967, "Concept Script" in [[Jean Van Heijenoort]], ed., ''From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931''. Harvard Uni. Press.
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| Secondary literature:
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| * [[George Boolos]], 1985. "Reading the ''Begriffsschrift''", ''Mind'' 94: 331-44.
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| *[[Ivor Grattan-Guinness]], 2000. ''In Search of Mathematical Roots''. Princeton University Press.
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| * Risto Vilkko, 1998, "[http://www.sciencedirect.com/science?_ob=PublicationURL&_cdi=6817&_pubType=J&_acct=C000007858&_version=1&_urlVersion=0&_userid=103118&md5=cdca08d0984650f66659ab072801d527&jchunk=25#25 The reception of Frege's ''Begriffsschrift'',]" ''Historia Mathematica 25(4)'': 412-22.
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| == External links ==
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| {{Commons category|Begriffsschrift}}
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| *{{sep entry|frege-logic|Frege's Logic, Theorem, and Foundations for Arithmetic|Edward N. Zalta}}
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| *[http://gallica.bnf.fr/ark:/12148/bpt6k65658c] ''Begriffsschrift'' as facsimile for download (2.5 MB)
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| {{computable knowledge}}
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| [[Category:1879 books]]
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| [[Category:Books by Gottlob Frege]]
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| [[Category:Logic books]]
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| [[Category:Diagram algebras]]
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| [[Category:Analytic philosophy literature]]
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| [[Category:Philosophy of logic]]
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| {{Link GA|de}}
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