Begriffsschrift

 ( 1879 Non-fiction Books ) 

Begriffsschrift (German for, roughly, "concept-writing") 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, for pure thought." Frege's motivation for developing his formal approach to logic resembled Leibniz's motivation for his calculus ratiocinator (despite that, in the foreword Frege clearly denies that he achieved this aim, and also that his main aim would be constructing an ideal language like Leibniz's, which Frege declares to be a quite hard and idealistic—though 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. This is the first work in Analytical Philosophy, a field that later British and Anglo philosophers such as Bertrand Russell further developed.

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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.

In the first chapter, Frege defines basic ideas and notation: judgement, conditionality, negation, identity of content, functions and generality.

Frege presents his calculus in an idiosyncratic two-dimensional notation, based on negation, material conditional and universal quantification. Other connectives and existential quantification are provided as definitions. Parentheses are not needed.

The conditional ($B\; \backslash to\; A$) is expressed by . Regarding its meaning Frege wrote:

"If A and B stand for contents that can become judgments, there are the following four possibilities:

1. A is affirmed and B is affirmed;
2. A is affirmed and B is denied;
3. A is denied and B is affirmed;
4. A is denied and B is denied.

Now

stands for the judgment that the third of those possibilities does not take place, but one of the three others does."

The building blocks are:

Judging	$\backslash vdash\; A,\backslash Vdash\; A$		$p(A)=1,$ $p(A)=i$ $\backslash vdash\; A,\; \backslash Vdash\; A$
Negation		basic	$\backslash neg\; A$
Material conditional		basic	$B\backslash to\; A$
Logical conjunction		$\backslash lnot\; (B\; \backslash to\; \backslash lnot\; A)$	$A\; \backslash land\; B$
Logical disjunction		$\backslash lnot\; B\; \backslash to\; A$	$A\; \backslash lor\; B$
Universal quantification		basic	$\backslash forall\; x\backslash ,\; F(x)$
Existential quantification		$\backslash lnot\; \backslash forall\; x\backslash ,\; \backslash lnot\; F(x)$	$\backslash exists\; x\backslash ,\; F(x)$
Material equivalence	$A\backslash equiv\; B$		$A\; \backslash leftrightarrow\; B$
Identity	$A\backslash equiv\; B$		$A\; =\; B$

In hindsight one can say that in Begriffsschrift, formulas are represented by their .

Example

Proposition 59 is written in modern notation as

$\backslash vdash\; g\; \backslash left(b\; \backslash right)\; \backslash to\; \backslash left(\backslash lnot\; f\; \backslash left(b\backslash right)\; \backslash to\; \backslash lnot\; \backslash left(\backslash forall\; a\backslash right)\backslash left(g\; \backslash left(a\backslash right)\; \backslash to\; f\; \backslash left(a\backslash right)\backslash right)\backslash right)$.

The parse tree is

     
    →
   / \
     
g(b)   →
     
     / \
    ¬   ¬
    |   |
 f(b)   ∀a
        |
        →
       / \
   g(a)   f(a)
     
     

... in left-to-right horizontal layout

→ ─── → ─── ¬ ─── ∀a ─── → ─── f(a)
     
\     \                  \
 g(b)  ¬                  g(a)
        \
         f(b)
     
     

In Begriffsschrift, proposition 59 is represented as

├─┬─┬─┬─a̲─┬─── f(a)
     
 │ │     └─── g(a)
 │ └──────┬── f(b)
 └─────────── g(b)
     
     

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The calculus in Frege's work In the second chapter Frege declared nine of his propositions to be , 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. $\backslash vdash\; A\; \backslash to\; (B\; \backslash to\; A)$
2. $\backslash vdash\; \backslash to$
3. $\backslash vdash\; \backslash to$
4. $\backslash vdash\; (B\; \backslash to\; A)\; \backslash to\; (\backslash lnot\; A\; \backslash to\; \backslash lnot\; B)$
5. $\backslash vdash\; \backslash lnot\; \backslash lnot\; A\; \backslash to\; A$
6. $\backslash vdash\; A\; \backslash to\; \backslash lnot\; \backslash lnot\; A$
7. $\backslash vdash\; (c\; =\; d)\; \backslash to\; (f(c)\; =\; f(d))$
8. $\backslash vdash\; c\; =\; c$
9. $\backslash vdash\; \backslash forall\; a\; \backslash ,\; f(a)\; \backslash to\; f(c)$

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 :

• Modus ponens allows to infer $\backslash vdash\; B$ from $\backslash vdash\; A\; \backslash to\; B$ and $\backslash vdash\; A$;
• The rule of generalization allows to infer $\backslash vdash\; P\; \backslash to\; \backslash forall\; x\; A(x)$ from $\backslash vdash\; P\; \backslash to\; A(x)$ 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.

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 " aR* b".

Frege applied the results from the Begriffsschrift, 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 0 R* y is the predicate " y is a natural number." (133) says that if x, y, and z are , then one of the following must hold: x < y, x = y, or y < x. This is the so-called "law of trichotomy".

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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 $\backslash vdash$ derived from his "Urteilsstrich" ( judging/inferring stroke) │ and "Inhaltsstrich" (i.e. 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" (i.e. definition double stroke) │├─ indicates that a proposition is a definition. Furthermore, the negation sign $\backslash neg$ 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 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.

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Editions Translations:

• Gottlob Frege (2025). 9780198243595, Oxford University Press. . ISBN 9780198243595

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See also

• Ancestral relation
• Calculus of equivalent statements
• First-order logic
• Prior Analytics
• The Laws of Thought
• Principia Mathematica

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Notes

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Bibliography

• Richard L. Mendelsohn (2025). 9780511109775, Cambridge University Press. ISBN 9780511109775

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External links

• Esoteric programming language: