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Begriffsschrift

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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

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" Frege's motivation for developing his formal approach to logic resembled 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

Contents

  • 1 Notation and the system
  • 2 The calculus in Frege's work
  • 3 Influence on other works
  • 4 Quotations
  • 5 See also
  • 6 References
  • 7 Further reading
  • 8 External links

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 it allows quantification over both The modifier "with identity" specifies that the language includes the identity function, =

Frege presents his calculus using 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, ie B → 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 and the "sign for identity of content" ≡ which he used to indicate both material equivalence and identity proper; in the second chapter he declares nine formalized propositions as axioms

Basic concept Frege's notation Modern notations
Judging ⊢ A , ⊩ A p A = 1

p A = i

Negation ¬ A , ∼ A
Conditional implication B → A

B ⊃ A

Universal quantification ∀ x : F x
Existential quantification ∼ ∀ x : ∼ F x

∃ x : F x

Content identity equivalence/identity A ≡ B A ↔ B

A ≡ B
A = B

In chapter 1, §5, Frege defines the conditional as follows:

"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

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, ie 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:

  1. ⊢     A → B → A
  2. ⊢     [   A → B → C   ]   →   [   A → B → A → C   ]
  3. ⊢     [   D → B → A   ]   →   [   B → D → A   ]
  4. ⊢     B → A   →   ¬ A → ¬ B
  5. ⊢     ¬ ¬ A → A
  6. ⊢     A → ¬ ¬ A
  7. ⊢     c = d → f c = f d
  8. ⊢     c = c
  9. ⊢     ∀ a f a →   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, 7 and 8 identity, and 9 the universal quantifier 7 expresses Leibniz's indiscernibility of identicals, and 8 asserts that identity is a reflexive relation

All other propositions are deduced from 1–9 by invoking any of the following inference rules:

  • Modus ponens allows us to infer ⊢ B from ⊢ A → B and ⊢ A ;
  • The rule of generalization allows us to infer ⊢ P → ∀ x A x from ⊢ P → 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 of a relation R "a is an R-ancestor of b" is written "aRb"

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 0Ry is the predicate "y is a natural number" 133 says that if x, y, and z are natural numbers, then one of the following must hold: x < y, x = y, or y < x This is the so-called "law of trichotomy"

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 ⊢ derived from his "Urteilsstrich" judging/inferring stroke │ and "Inhaltsstrich" ie 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" ie definition double stroke │├─ indicates that a proposition is a definition Furthermore, the negation sign ¬ can be read as a combination of the horizontal Inhaltsstrich with a vertical negation stroke This negation symbol was reintroduced by Arend Heyting 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, "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

Quotations

"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

See also

  • Ancestral relation
  • Frege's propositional calculus

References

  1. ^ Arend Heyting: "Die formalen Regeln der intuitionistischen Logik," in: Sitzungsberichte der preußischen Akademie der Wissenschaften, phys-math Klasse, 1930, pp 42–65

Further reading

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

Translations:

  • 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:

  • George Boolos, 1985 "Reading the Begriffsschrift", Mind 94: 331-44
  • Ivor Grattan-Guinness, 2000 In Search of Mathematical Roots Princeton University Press
  • Risto Vilkko, 1998, "The reception of Frege's Begriffsschrift," Historia Mathematica 254: 412-22

External links

  • Zalta, Edward N "Frege's Logic, Theorem, and Foundations for Arithmetic" Stanford Encyclopedia of Philosophy 
  • Begriffsschrift as facsimile for download 25 MB

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    Begriffsschrift beatiful post thanks!

    29.10.2014


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