Begriffsschrift

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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. 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.

1 Notation and the system
2 The calculus in Frege's work
3 Influence on other works
4 A quote
5 References
6 See also
7 External links

[edit] Notation and the system

The calculus contains the first appearance of quantified variables, and is essentially classical bivalent second-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 .

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" $\equiv$ ; in the second chapter he declares nine formalized propositions as axioms.

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

├────┬── A
     │
     └── B

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, i.e. we negate A and assert B."

[edit] The calculus in Frege's work

Frege declares nine of his propositions axioms. He verifies them by arguing informally that, as they are intended to be understood, they express truths. The axioms, expressed in a more modern notation, are:

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

These are propositions 1, 2, 8, 28, 31, 41, 52, 54, and 58. (1)-(3) govern material implication. (4)-(6) govern negation. (7) and (8) govern identity; (7) is Leibniz's indiscernibles; (8) asserts that identity is reflexive. (9) governs the universal quantifier. All other propositions are proved by deduction from these.

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. Modus ponens allows us to infer $\vdash B$ from $\vdash A \to B$ and $\vdash A$ . The rule of generalization, which allows us to infer $\vdash P \rightarrow \forall x : A(x)$ from $\vdash P \to A(x)$ if the variable 'x' does not occur in 'P'. The rule of substitution is far more complex, and Frege applies it in ways that are not obviously legitimate.

The third chapter is titled "Parts from a general series theory". The main results concern what we now call the "ancestral" of a relation. Given a relation R, Frege says that a property F is "R-hereditary" if, whenever x is F and xRy, y too is F. Frege then defines b to be an R-ancestor of a if, and only if, b has every R-hereditary property that all objects x such that aRx have. That is, writing "b is an R-ancestor of a" as "aR*b", we have:

76: $\Vdash aR*b \equiv \forall F [\forall x (aRx \to Fx) \wedge \forall x \forall y (Fx \wedge xRy \to Fy) \to Fb]$ .

Frege's first result is then that this relation is transitive:

98: $\vdash aR*b \wedge bR*c \to aR*c$

His second result applies only to cases where R is what Frege calls "many-one", that is, function-like:

115: $\Vdash I(R) \equiv \forall x \forall y \forall z (xRy \wedge xRz \to y=z)$

The result is that the ancestral of R is, as we would say today, "connected" if R is many-one:

133: $\vdash I(R) \wedge aR*b \wedge aR*c \to (bR*c \vee b=c \vee cR*b)$

Although Frege does not make these applications here, it is clear that these results are intended to be applied in his later work on the foundations of arithmetic. Thus, if we take xRy to be the relation y=x+1, then 0R*y is the predicate: y is a natural number, and, in this case, (133) tells us that if x, y, and z are natural numbers, then either x<y or x=y or y<x, this being the so-called "law of trichotomy".

[edit] Influence on other works

The formulation of second-order logic in Begriffsschrift was the first formalization of any logic capable of dealing with a reasonable fragment of mathematics, or of human language. All later work in formal logic owes significantly to Begriffsschrift.

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 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 Begriffschrifft about identity (denoted in mathematics by the = sign).

[edit] 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)

[edit] References

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 25(4): 412-22.