Begriffsschrift

Template:Short description Template:Use dmy dates Template:Infobox book 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.

Notation and the systemEdit

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.Template:Sfn

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

Frege presents his calculus in an idiosyncraticTemplate:Sfn 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 (<math>B \to A</math>) is expressed by File:Begriffsschrift connective2.svg. Regarding its meaning Frege wrote:Template:Sfn

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

A is affirmed and B is affirmed;
A is affirmed and B is denied;
A is denied and B is affirmed;
A is denied and B is denied.

Now

File:BS-05-Kondicionaliskis-svg.svg

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:Template:Refn

Basic concept	Frege's notation	The diagram shows (in modern notation)	Modern notation
Judging	<math>\vdash A,\Vdash A</math>		<math>p(A)=1,</math> <math>p(A)=i</math> <math>\vdash A, \Vdash A</math>
Negation	File:Begriffsschrift connective1.svg	basic	<math>\neg A</math>
Material conditional	File:Begriffsschrift connective2.svg	basic	<math>B\to A</math>
Logical conjunction	File:Begriffsschrift connective3.svg	<math>\lnot (B \to \lnot A)</math>	<math>A \land B</math>
Logical disjunction	File:Begriffsschrift connective4.svg	<math>\lnot B \to A</math>	<math>A \lor B</math>
Universal quantification	File:BS-12-Begriffsschrift Quantifier1-svg.svg	basic	<math>\forall x\, F(x)</math>
Existential quantification	File:BS-14-Begriffsschrift Quantifier3-svg.svg	<math>\lnot \forall x\, \lnot F(x)</math>	<math>\exists x\, F(x)</math>
Material equivalence	<math>A\equiv B</math>		<math>A \leftrightarrow B</math>
Identity	<math>A\equiv B</math>		<math>A = B</math>

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

Example

Proposition 59Template:Sfn is written in modern notation asTemplate:Sfn

<math>\vdash g \left(b \right) \to \left(\lnot f \left(b\right) \to \lnot \left(\forall a\right)\left(g \left(a\right) \to f \left(a\right)\right)\right)</math>.

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 asTemplate:Sfn

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

The calculus in Frege's workEdit

In the second chapter 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:

<math>\vdash A \to (B \to A) </math>
<math>\vdash [ A \to (B \to C) ] \to [ (A \to B) \to (A \to C) ] </math>
<math>\vdash [ D \to (B \to A) ] \to [ B \to (D \to A) ] </math>
<math>\vdash (B \to A) \to (\lnot A \to \lnot B) </math>
<math>\vdash \lnot \lnot A \to A </math>
<math>\vdash A \to \lnot \lnot A </math>
<math>\vdash (c = d) \to (f(c) = f(d)) </math>
<math>\vdash c = c </math>
<math>\vdash \forall a \, f(a) \to f(c) </math>

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 to infer <math>\vdash B</math> from <math>\vdash A \to B</math> and <math>\vdash A</math>;
The rule of generalization allows to infer <math>\vdash P \to \forall x A(x)</math> from <math>\vdash P \to A(x)</math> 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 0R*y 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 worksEdit

For a careful recent study of how the Begriffsschrift was reviewed in the German mathematical literature, see Vilko (1998).Template:Sfn 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 <math>\vdash</math> 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 <math>\neg</math> can be read as a combination of the horizontal Inhaltsstrich with a vertical negation stroke. This negation symbol was reintroduced by Arend Heyting Template:Sfn 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, "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.