﻿ Begriffsschrift

# Begriffsschrift

The term Scripture is a narrow, only about eighty -page book of the Jena mathematician and philosopher Gottlob Frege on logic. It was published in 1879 with the subtitle " A replica of the arithmetic formula language of pure thought " and is generally considered the most important publication in the field of logic since Aristotle's Organon.

Frege succeeded in this book for the first time a formalization of classical predicate logic and thus the first formalization of a logic in which a sufficiently large part of mathematics, but also the natural language could express. Together with George Boole's Mathematical Analysis of Logic of 1847 marked the Begriffsschrift therefore the beginning of modern formal logic. The term conceptual notation is also used for the defined logical calculus by Frege and Frege's logical notation for. Frege designed the concept-script to support his research on the foundations of mathematics.

Frege's calculus introduced for the first time a universal quantifier and the more -place predicates (relations). It is a classical predicate logic calculus second with identity, albeit in a compared to the common spellings idiosyncratic, two-dimensional notation.

## Position of the Begriffsschrift in the complete works of Frege

Despite their epochal significance the term font is not Frege's main work. It was followed in 1884 The Foundations of Arithmetic, and in 1893 and 1903, the two volumes of the basic laws of arithmetic, which can be regarded as Frege's main work well due to their size.

Frege's primary objective was to expel mathematics as part of the logic, ie to show that all mathematical sentences of a few purely logical axioms can be derived (see logicism ). This company was only promising if an agent was available, with which the imperviousness of a chain of conclusions could check any doubt. Since the traditional Aristotelian logic ( syllogistic ) turned out to be useless for this purpose, Frege initially participated in the task of creating a new, more appropriate logic. This took the form of logical symbolism. For Frege Begriffsschrift was therefore only the first step on the path to full formalization of mathematics as a whole, which he carried out in the basic laws of arithmetic for the number theory part. Frege logizistisches program failed initially (before the appearance of the second volume of the fundamental laws ) to Russell's antinomy, but it was continued by Bertrand Russell, Rudolf Carnap and others.

The Begriffsschrift was by no means intended exclusively for use in mathematics. On the contrary, Frege presented his writing in the preface explicitly in the context of Leibniz's idea of ​​a lingua characterica universalis, a universal language that should be an orderly system of all terms according to a mathematical model. Frege 1879 submitted Form should form the logical core of such a universal language. It can be assumed that the term Begriffsschrift of a treatise Friedrich Adolf Trendelenburg on Leibniz ' design of this universal language is borrowed, quoted in the preface to Frege. In addition, there was the word " conceptual notation " around the turn of the 20th century as a Germanization of " ideography " in common usage.

## Notation

Frege used in the Begriffsschrift a specially created by him notation (notation ) for expressions of propositional and predicate logic. Although it was the first formalized notation for unrestricted predicate logic, it has not been enforced.

The notation of the Begriffsschrift is a graphical, two-dimensional representation, are connected in the formulas by horizontal and vertical lines with each other. It uses the universal quantifier as propositional basic elements of character for negation and the conditional, as in predicate logic element. How much later that - however, linear, one-dimensional and therefore much more space-efficient - Polish Notation is the Begriffsschrift notation without parantheses.

### Syntax

The term Scripture knows only two syntactic primitives: function expressions and proper names, both of which may be represented by variables. All syntactic operations follow the pattern function - argument - value, by applying a function with n vacancies on n arguments we obtain a certain value of the function.

For details on the concept of function: When replacing, for example, in the complex expression '1 × 1 ' both incidents the numeral '1' by the variable ' n ' or ' m', we obtain the function expression 'n × m'. The variables make it clear that the term " unsaturated", as Frege says: He referred in this form no object, but requires the completion of two arguments. By re- substitution of digits for the variables we obtain a number of arithmetic terms, such as x 1 '1 ', '1 x 2 ', '2 x 1 ', etc. The various possible substitutions for argument variables are expressions. The Designated by the complex expression is the value of the function. The value of the function to the arguments n × m 2 and 3, for example the number 6

This basic scheme is not limited in its applicability to the area of ​​mathematics: Substituting for example, in ' the conqueror of x' the variable ' x ' with ' Gaul ', the function takes the value of Julius Caesar. Also predicates are after Frege functions: the relation expressed by ' x conquered Gaul ' function takes the argument for Julius Caesar to true to, for the argument Hannibal returns false. The replacement of the subject-predicate form by the function-argument form of judgment was already a significant improvement over the traditional logic because it makes it possible to formulate a logic of relations: Modern logic knows (other than the syllogism ) also two - and more -place predicates ( relational expressions ) as ' x loves y ', ' x is between y and z', etc. ( See also logic - classical logic. )

### Truth functionality

Frege now summarized all compound expressions as results of applying a function to arguments; in particular he treated even those expressions as a function expressions that are commonly known as connectives today. Their arguments are statements that arise as values ​​the truth values ​​true and false, which are called in Frege " the truth " and " falsity ". To specify the meaning of a connective, it is sufficient to define the conditions under which a statement with this connective is true or false. Today, this relationship is called the truth functionality, and you are the truth conditions usually in the form of so-called truth tables. The truth functionality is essential for the establishment of an extensional semantics, as developed by Alfred Tarski in the 1930s.

### Content dash and judgment stroke

The horizontal " content line" states in the Begriffsschrift, that what follows him to a ( to truth or falsity back ) " beurtheilbarer content " is, in modern terminology, a statement that can be true or false. The contents of stroke is not found over the veracity of a statement; it is not alleged, but only as potentially true or " placed in the room " as it were wrong:

Mind you would be an absurd connection appears as "- 2" in the concept-script syntax is not unconstitutional; its value would be the wrong thing. This is due to the fact that Frege's Begriffsschrift is a pure Term logic; also statements are singular terms, to a certain extent different names for the two truth values ​​.

The vertical " judgment stroke" before the contents line stating that the contents are true:

Frege said this, the content will expressed with " say graduated force ".

### Connectives

Frege used by today's conventional five connectives ' not ', ' and', ' or', ' if - then ', ' if and only if ' only two: 'not' ( negation), and ' if - then ' ( implication or the conditional). The negation is represented by attaching a small vertical line on the contents line. The negation of " not A" ( ) is expressed as follows:

The value of this function is exactly then the truth when the truth value of ' - A' not the truth, otherwise the wrong thing.

The implication (read: ' if B, then A') is in the conceptual notation by

Expressed. For the meaning of these signs connection Frege wrote:

"If A and B are beurtheilbare content, so it gives the following four options:

These are today unfamiliar appearing formulation, the truth conditions of material implication: The implication is false only when the antecedent is true and the consequent is false.

Disjunction ( ' or') and conjunction ( 'and') can be expressed by compounds of these two connectives: disjunction is

- Expressed by the conjunction

-. Since Frege's logic is a term logic, in which statements are also singular terms, the "mark of identity of content '(same characters) also serves as an expression of substantive equivalence.

### Quantifiers

As a universal quantifier Frege uses an indentation ( " cavity " ) in the content line, in which the variable is written to binding ( see chart ). Due to the force in the classical predicate logic equivalence

Is its own existential not required; its contents can be expressed by universal quantifier and inverter.

The following example shows the statement " for every x with the property F, there is a y to the x in the relation R " (eg, " every person has a mother "). It illustrates the two main achievements of the concept-script, which sets them apart against both the traditional syllogistic and against the contemporary algebra: nested quantifiers ( " for all x there is a y" ) and more -place predicates ( "R (x, y) " ).

From this statement follows by Axiom 9 (see below):

This, in conjunction with the statement " F (c) " using the rule Modus Ponens (see below) the statement " there is a y to the c in R " can be derived:

The quantifier allows (under the condition that the term subject is not empty ), all the circuits of the traditional logic. The following figure shows on the left the "logical square " from the original edition of the Begriffsschrift, a right in modern notation for comparison:

The fact that Frege on the bottom line " contraries " instead is " subconträr " apparently is a mistake.

See also the reference table for the notation at the end of the article.

## The axiom system of the Begriffsschrift

According to the explanation on the spelling in the first chapter, Frege goes in the second chapter, entitled "Representation and derivation of some judgments of pure thought " to prove some logically true propositions on the basis of a few axioms.

Frege justified his nine axioms non- formal, by he explained why they are true in their intended interpretation. In modern notation translated, are the axioms:

These are in Frege's own numbering the sets 1, 2, 8, 28, 31, 41, 52, 54 and 58 (1 ) - (3 ) relate to the material implication, ( 4) - ( 6) negation. (7) and (8) relating to the identity: (7) is the identity of Leibniz principle. (8 ) requires the reflexivity of identity. (9 ) allows the transition from a allquantifizierten statement at any instance. All remaining sets are derived from these axioms.

The term Scripture has three inference rules. Two of these, the modus ponens and the Generalisierungsregel be named explicitly. The modus ponens allows the transition to and from. The Generalisierungsregel allows the transition from when the variable "x" does not occur in P. The third, not explicitly mentioned rule is a principle of substitution.

The corresponding one of the first -order predicate logic fragment as specified in the conceptual notation calculus is complete and consistent. It was the expansion of the system to a theory of extensions of concepts that Frege undertook in the basic laws of arithmetic later, led to inconsistency.

The third chapter is entitled " Some of a general theory of series ". The main results concern the heritability of a trait in a row and the succession in a row. If a relation R is given, a property F by Frege is hereditary in the R- series if and only if:

Then Frege defined: b follows in the R- series on a iff b has any hereditary property in the R- series, all of which have x with ARX. If we write R * for this relation of following in the R- series, as can be Frege's definition of play as follows:

Less, where " ERBL (F, R) " is intended to mean "F is hereditary in the R- series":

About this R * - relation Frege proves in the following few sentences, showing that it is an order relation. These considerations are obviously intended as preparatory work on the two successor plants to the fundamentals of number theory. When the relation of y = a xRy considered x 1, then is 0R * y ( or 1R * y), the characteristic of y being a natural number.

## Reception and impact

The term font first found a remarkably cool reception. Not least because of their unusual and hard to read symbolism seems to have first taken little notice of her broad professional public. The tenor of contemporary reviews were mostly critical to behave. Unanimously expressed concerns about the large-scale, difficult to handle spelling. Above all, accused the critics Frege to ignore the algebraic approach to symbolic logic (Ernst Schröder, Giuseppe Peano, George Boole ). The criticism is justified: It is striking that Frege the dominant flow at the time of formal logic completely merges and his own work is not set to the other contemporary researchers in relationship. This failure, he made ​​up for in some direct effect on the Begriffsschrift following essays.

Among the few who recognized the significance of the concept-script early on, the British philosopher and mathematician Bertrand Russell, the phenomenologist Edmund Husserl, Frege's student Rudolf Carnap and the Austro- British philosopher Ludwig Wittgenstein, who in the preface to his famous Tractatus Logico-Philosophicus included ( 1921) wrote: " I ​​will only mention that I owe to the great works of Frege and the writings of my friend Mr Bertrand Russell a large part of the stimulation of my thoughts. "

The logicist program to which the concept of writing was only the prelude was continued especially by Russell and Alfred North Whitehead in their monumental Principia Mathematica ( 1910ff. ), which were for some time as the canonical standard work on logic. Russell and Whitehead used essentially already one of the usual logical notations now that is leaning on the spelling of the algebra so-called Peano - Russell notation.

From Frege's symbolism survived (probably through the mediation of the Principia Mathematica), the characters, the combination of his own judgment and content line, but usually in a generalized meaning as a derivation relation. Furthermore, in today's conventional negation sign Arend Heyting introduced in 1930 (originally to distinguish the intuitionistic inverter from the classic ), are considered as content bar with attached negation stroke.

Even if Frege's idiosyncratic spelling not met with great success, is based almost any job in modern logic, albeit indirectly, to the idea of concept-script. Since the logic further auxiliaries and basic discipline, inter alia, mathematics, linguistics and computer science, the indirect effects of Frege's work are hard to look at. In philosophy recognized personalities refer to the very recent past coming back to ideas from the Begriffsschrift, including, for example, Michael Dummett and Robert Brandom.

## Expenditure

• Begriffsschrift. One of the simulated arithmetic formula language of pure thought, Halle, 1879. [ Original edition ]
• Begriffsschrift and other essays, edited by Ignacio Angelelli, Hildesheim 1964 passim ISBN 978-3-487-00623-9 [ This Reprintausgabe has some small but meaningful part disturbing misprint; particularly lacking equal to p 1 the judgment bar. See the note from Angelelli / Bynum in the bibliography. ]
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