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Encyclopedia > Begriffsschrift

Begriffsschrift is the title of a short book on logic by Gottlob Frege, published in 1879, and is also the name of the formal system set out in that book. Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of arguments, although the exact definition of logic is a matter of controversy among philosophers. ... Friedrich Ludwig Gottlob Frege Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar – 26 July 1925, Bad Kleinen) was a German mathematician who evolved into a logician and philosopher. ... 1879 was a common year starting on Wednesday (see link for calendar). ...


Begriffsschrift is usually translated as concept writing or concept notation; the full title of the book identifies it as "a formula language, modelled on that of arithmetic, of pure thought." The Begriffsschrift was arguably the most simportant publication in logic since Aristotle founded the subject. Frege's motivation for developing his formal approach to logic resembled Leibniz's motivation for his calculus ratiocinator. Frege went on to employ his logical calculus in his research on the foundations of mathematics, carried out over the next quarter century. In mathematics and in the sciences, a formula is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. ... Arithmetic or arithmetics (from the Greek word αριθμός = number) in common usage is a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numerals, though professional mathematicians often treat arithmetic as a synonym for number theory. ... Thought or thinking is a mental process which allows beings to model the world, and so to deal with it effectively according to their goals, plans, ends and desires. ... Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of arguments, although the exact definition of logic is a matter of controversy among philosophers. ... Aristotle, marble copy of bronze by Lysippos. ... Gottfried Leibniz Gottfried Wilhelm von Leibniz (July 1, 1646 in Leipzig - November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ... There are two different understandings of Leibnizs Calculus Ratiocinator in the history of ideas. ... The term foundations of mathematics is sometimes used for certain fields of mathematics itself, namely for mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory. ...

Contents


Notation and the system

The calculus contains the first appearance of quantified variables, and is essentially classical bivalent first order logic with identity, albeit presented using a highly idiosyncratic two-dimensional 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. B rightarrow A is written as Image:Kondicionaliskis_wb.png. First-order predicate calculus or first-order logic (FOL) is a theory in symbolic logic that permits the formulation of quantified statements such as there is at least one X such that. ... The term notation can be used in several contexts. ... Image File history File links Kondicionaliskis_wb. ...


In the first chapter, Frege defines basic ideas and notation, like proposition ("judgement"), the universal quantifier ("the generality"), the implication ("the conditionality"), the negation and the equal sign equiv; in the second chapter he declares nine formalized propositions as axioms (statements verified semantically). In language and logic, quantification is a construct that specifies the extent of validity of a predicate, that is the extent to which a predicate holds over a range of things. ... In propositional calculus, or logical calculus in mathematics, the logical conditional is a binary logical operator connecting two statements, if p then q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). ...

In chapter 1, §5, Frege defines the conditional as follows: Image File history File links On this picture Freges notation system can be seen for basic ideas and signs of mathematical logic, in his work Begriffsschrift. ...

"Let A and B refer to judgeable contents, then the four possibilities are:
(1) A is asserted, B is asserted;
(2) A is asserted, B is negated;
(3) A is negated, B is asserted;
(4) A is negated, B is negated.

Let

 ├────┬── A │ └── B 

signify that the third of those possibilities does not obtain, but one of the three others does. So if we negate Image:Kondicionaliskis_wb.png, that means the third possibility is valid, i.e. we negate A and assert B." Image File history File links Kondicionaliskis_wb. ...


The calculus in Frege's work

Frege declares nine tautologic judgements as axioms. He verifies them in a semantical way; all other tautological judgements are proved by syntactical deduction. In logic, a tautology is a statement that is true by its own definition. ...

  1. vdash   A rightarrow left( B rightarrow A right)
  2. vdash   left[  A rightarrow left( B rightarrow C right)  right]  rightarrow  left[  left( A rightarrow B right) rightarrow left( A rightarrow C right)  right]
  3. vdash   left[  D rightarrow left( B rightarrow A right)  right]  rightarrow  left[  B rightarrow left( D rightarrow A right)  right]
  4. vdash   left( B rightarrow A right)  rightarrow  left( lnot A rightarrow lnot B right)
  5. vdash   lnot lnot A rightarrow A
  6. vdash   A rightarrow lnot lnot A
  7. vdash   left( c=d right) rightarrow left( f(c) = f(d) right)
  8. vdash   c = c
  9. vdash   left(  forall a : f(a)  right)  rightarrow  f(c)

All propositions formalized in the second chapter are numbered. His axioms are the propositions 1st, 2nd, 8th, 28th, 31st, 41st, 52nd, 54th, and 58th.


(1)-(3) govern material implication. (4)-(6) govern negation. (7) and (8) govern identity; (7) is Leibniz's identity of indiscernibles; (8) asserts that identity is reflexive. (9) governs the universal quantifier. The identity of indiscernibles, also known as Leibnizs law, is an ontological principle first forumlated by German philosopher Göttfried Wilhelm Leibniz. ...


The Begriffschrifft has three inference rules. Two of them, modus ponens and "the law of generalization" are explicit, while the law of substitution is invoked but not stated explicitly. Generalization means that if a "free" (unquantified) variable appears in a judgement, then it should be deemed universally quantified, fixed variable, because Frege's laws governing the vdash sign ("judging sign") govern judgements, not "open" formulae, i.e. predicates. In logic, especially in mathematical logic, a rule of inference is a scheme for constructing valid inferences. ... In Logic, Modus ponens (Latin: mode that affirms) is a valid, simple argument form (often abbreviated to MP): If P, then Q. P. Therefore, Q. or in logical operator notation: P → Q P ⊢ Q where ⊢ represents the logical assertion. ... In computer science and mathematics, a variable is a symbol denoting a quantity or symbolic representation. ...


Frege proves more than one hundred formal statements syntactically in the second and third chapter. The third chapter ("Parts from a general series theory") is the gateway to his later work on the foundations of arithmetic.


Influence on other works

Some vestige of Frege's notation survives in the "turnstile" symbol vdash derived from his "Inhaltsstrich" ── and "Urteilsstrich" │. Frege used these symbols in the Begriffsschrift in the unified form ├─ for declaring that a proposition is (tautologically) true, not simply speaking about that. He used the "Definitionsdoppelstrich" │├─ as a sign that a proposition is a definition. In logic, a tautology is a statement that is true by its own definition. ...


In the Tractatus Logico Philosophicus, Ludwig Wittgenstein pays homage to Frege by employing the term Begriffsschrift as a synonym for logical formalism. Tractatus Logico-Philosophicus is the only book-length work published by the philosopher Ludwig Wittgenstein in his lifetime. ... Meaning is determined by use, in the context of a language-game {later} Meaning is determined by use, in the context of a language-game {later} Ludwig Josef Johann Wittgenstein (IPA: ) (April 26, 1889 – April 29, 1951) was an Austrian philosopher who contributed several ground-breaking works to modern philosophy...


Frege's 1892 essay, "Sense and reference" recants some of the conclusions of the Begriffschrifft about identity (denoted in mathematics by the = sign). The distinction between Sinn and Bedeutung (usually but not always translated sense and reference, respectively) was an innovation of the German philosopher and mathematician Gottlob Frege in his 1892 paper Ãœber Sinn und Bedeutung (On Sense and Reference), which is still widely read today. ... // Computer programming In object-oriented programming, object identity is a mechanism for distinguishing different objects from each other. ...


A quote

"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)

References

  • Gottlob Frege. Begriffsschrift: eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle, 1879.

Translations: Friedrich Ludwig Gottlob Frege Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar – 26 July 1925, Bad Kleinen) was a German mathematician who evolved into a logician and philosopher. ...

  • Bynum, Terrell Ward, trans. and ed., 1972. Conceptual notation and related articles, with a biography and introduction. Oxford Uni. Press.
  • 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.

Secondary literature: Jean van Heijenoort (prounounced highenort) (July 23, 1912, Creil France - March 29, 1986, Mexico City) was a pioneer historian of mathematical logic. ...

  • Ivor Grattan-Guinness, 2000. In Search of Mathematical Roots. Princeton Uni. Press.
  • Risto Vilkko, 1998, "The reception of Frege's Begriffsschrift", Historia Mathematica 25(4): 412-22.

Ivor Grattan-Guiness is a prolific contemporary historian of mathematics and logic. ...

See also

Friedrich Ludwig Gottlob Frege Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar – 26 July 1925, Bad Kleinen) was a German mathematician who evolved into a logician and philosopher. ... In mathematical logic Freges propositional calculus was the first axiomatization of propositional calculus. ...

External links


  Results from FactBites:
 
Begriffsschrift: Information from Answers.com (906 words)
Begriffsschrift is the title of a short book on logic by Gottlob Frege, published in 1879, and is also the name of the formal system 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 language, modelled on that of arithmetic, of pure thought." The Begriffsschrift was arguably the most important publication in logic since Aristotle founded the subject.
Risto Vilkko, 1998, "The reception of Frege's Begriffsschrift", Historia Mathematica 25(4): 412-22.
Begriffsschrift (555 words)
Begriffsschrift is the name of a book on Logic by Gottlob Frege published in 1879.
The name Begriffsschrift is usually translated as concept writing or concept notation; the full title of the book identifies it as "a Formula Language, modelled on that of Arithmetic, of pure Thought." This little book is arguably the most significant publication in Logic since Aristotle.
Begriffsschrift is both the name of the book and the calculus defined therein.
  More results at FactBites »

 
 

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