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Begriffsschrift

From Biocrawler, the free encyclopedia.

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. Frege's motivation for developing his formal logical system was similar to Leibniz's desire for a calculus ratiocinator.

Frege defines a logical calculus to support his research in the foundations of mathematics. Begriffsschrift is both the name of the book and the calculus defined therein.

The calculus introduced quantifiers, and is essentially classical predicate logic, albeit in an idiosyncratic two-dimensional notation: connectives and quantifiers are written using lines connecting formulas, rather than the symbols ¬, and ∧, and ∀ in use today. For example, the implication between the B and A judgements, i.e. $B \rightarrow A$ is designated with .

In the first chapter of his work Frege determines basic ideas and signs, 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 (they are formalized statements proved semantically).

He gives a definition of condititional (chapter 1. §5.):

"Let A and B refer to judgeable contents, then the four possibilities are potential:

(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

sign that the third of those possibilities does not obtain but one of the three others. So if we negate 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

Frege declares nine tautologic judgements as axioms. He proves them in a semantical way, and all other tautologic judgements are proved by syntactical deduction.

$\vdash \ \ A \rightarrow \left( B \rightarrow A \right)$
$\vdash \ \ \left[ \ A \rightarrow \left( B \rightarrow C \right) \ \right] \ \rightarrow \ \left[ \ \left( A \rightarrow B \right) \rightarrow \left( A \rightarrow C \right) \ \right]$
$\vdash \ \ \left[ \ D \rightarrow \left( B \rightarrow A \right) \ \right] \ \rightarrow \ \left[ \ B \rightarrow \left( D \rightarrow A \right) \ \right]$
$\vdash \ \ \left( B \rightarrow A \right) \ \rightarrow \ \left( \lnot A \rightarrow \lnot B \right)$
$\vdash \ \ \lnot \lnot A \rightarrow A$
$\vdash \ \ A \rightarrow \lnot \lnot A$
$\vdash \ \ \left( c=d \right) \rightarrow \left( f(c) = f(d) \right)$
$\vdash \ \ c = c$
$\vdash \ \ \left( \ \forall a : f(a) \ \right) \ \rightarrow \ f(c)$

Frege numbered all propositions he formalized in the second chapter; to this his axioms are the 1^st, 2^nd, 8^th, 28^th, 31^st, 41^st, 52^nd, 54^th, 58^th propositions.

He also declares two inference rules in this chapter: these are the modus ponens, the law of substitution. In the first chapter he speaks about a convention, "the law of generalization". That means if a "free" variable can be found in a judgement, then it should be considered as a universally quantified, fixed variable, because Frege's laws behind the $\vdash$ sign ("judging sign") are judgements, not "opened" formulae, i.e. predicates.

Frege proves more than one hundred formal statements syntactically in tha second and third chapter. The third chapter ("Parts from a general series theory") is the introduction of his latter works on founding arithmetics.

Some vestige of his notation survives: the symbol $\vdash$ that logicians informally call "turnstile" is derived from Frege's "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 the Tractatus Logico Philosophicus, Ludwig Wittgenstein pays homage to Frege in that he uses the term Begriffsschrift as a synonym for logical formalism.

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A quote

"If the task of philosophy is to break the domination of words above human mind [...], then my concept notation, being developed for these purposes, can be a useful instrument for philosophers [...] I think, matter of logics has been forwarded just merely by the invention of this concept notation."

Begriffsschrift [Preface]

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