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Begriffsschrift
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{{Short description|1879 book on logic by Gottlob Frege}}- the content below is remote from Wikipedia
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Notation and the system
The calculus contains the first appearance of quantified variables, and is essentially classical bivalent second-order logic 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, a Leibnizian concept that would be applied in mathematics.JOURNAL, Korte, Tapio, Frege's Begriffsschrift as a lingua characteristica, Synthese,weblink 2008-10-22, 174, 2, 283â294, 10.1007/s11229-008-9422-7, 20587814, Frege presents his calculus using idiosyncratic two-dimensional notation: connectives and quantifiers are written using lines connecting formulas, rather than the symbols ¬, â§, and â in use today. For example, that judgement B materially implies judgement A, i.e. B rightarrow A is written as (File:BS-05-Kondicionaliskis-svg.svg|60px).In the first chapter, Frege defines basic ideas and notation, like proposition ("judgement"), the universal quantifier ("the generality"), the conditional, negation and the "sign for identity of content" equiv (which he used to indicate both material equivalence and identity proper); in the second chapter he declares nine formalized propositions as axioms.{| class="wikitable" style="margin:0.5em auto;"Judging | vdash A,Vdash A | p(A)=1,p(A)=ivdash A, Vdash A |
Negation | (File:Begriffsschrift connective1.svg|60px) | neg A{sim} A |
Conditional (implication) | (File:Begriffsschrift connective2.svg|80px) | Brightarrow ABsupset A |
Universal quantification | (File:BS-12-Begriffsschrift Quantifier1-svg.svg |85px) | forall x, F(x) |
Existential quantification | (file:BS-14-Begriffsschrift Quantifier3-svg.svg|95px) | exists x, F(x) |
Content identity (equivalence/identity) | Aequiv B | A leftrightarrow BA equiv BA = B |
"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.
(File:Kondicionaliskis wb.png)
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), 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 axioms, 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 lnotlnot A
- vdash left( c=d right) rightarrow left( fleft(cright) = fleft(dright) right)
- vdash c = c
- vdash forall a f(a) rightarrow f(c)
- Modus ponens 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.
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" symbol vdash derived from his "Urteilsstrich" (judging/inferring stroke) â and "Inhaltsstrich" (i.e. content stroke) ââ. Frege used these symbols in the Begriffsschrift in the unified form ââ for declaring that a proposition is true. In his later "Grundgesetze" he revises slightly his interpretation of the ââ symbol.In "Begriffsschrift" the "Definitionsdoppelstrich" (i.e. definition double stroke) âââ 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 HeytingArend Heyting: "Die formalen Regeln der intuitionistischen Logik," in: Sitzungsberichte der preuÃischen Akademie der Wissenschaften, physikalisch-mathematische Klasse, 1930, pp. 42â65. in 1930 to distinguish intuitionistic from classical negation. It also appears in Gerhard Gentzen's doctoral dissertation.In the Tractatus Logico Philosophicus, Ludwig Wittgenstein pays homage to Frege by employing the term Begriffsschrift as a synonym for logical formalism.Frege's 1892 essay, "On 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 a relationship between the objects that are denoted by those names.Editions
- Gottlob Frege. Begriffsschrift: eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle an der Saale: Verlag von Louis Nebert, 1879.
- weblink" title="web.archive.org/web/20061229202121weblink">Bynum, Terrell Ward, translated and edited, 1972. Conceptual notation and related articles, with a biography and introduction. Oxford University Press.
- Bauer-Mengelberg, Stefan, 1967, "Concept Script" in Jean van Heijenoort, ed., From Frege to Gödel: A Source Book in Mathematical Logic, 1879â1931. Harvard University Press.
- Beaney, Michael, 1997, "Begriffsschrift: Selections (Preface and Part I)" in The Frege Reader. Oxford: Blackwell.
See also
- Ancestral relation
- Calculus of equivalent statements
- First-order logic
- Frege's propositional calculus
- Prior Analytics
- The Laws of Thought
- Principia Mathematica
References
{{reflist}}Bibliography
- George Boolos, 1985. "Reading the Begriffsschrift", Mind 94: 331â344.
- Ivor Grattan-Guinness, 2000. In Search of Mathematical Roots. Princeton University Press.
- Risto Vilkko, 1998, "weblink" title="web.archive.org/web/20090720231814weblink">The reception of Frege's Begriffsschrift," Historia Mathematica 25(4): 412â422.
External links
{{Commons category|Begriffsschrift}}- SEP, frege-logic, Frege's Logic, Theorem, and Foundations for Arithmetic, Zalta, Edward N.,
- Begriffsschrift as facsimile for download (2.5 MB)
- Esoteric programming language: WEB, 2020-03-27, Gottlob: Write Code in Frege's Concept Notation,weblink 2022-06-19, esoteric.codes, en,
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