Function and concept

In addition to About Meaning and Meaning and About Concept and Object, function and concept is one of three essays by Gottlob Frege that appeared in quick succession , in which he explains basic concepts of his logic and philosophy of language . Function and Term appeared in 1891, the first of the texts mentioned. Frege presents here u. a. a construction with which one can understand predicates (in Frege's terminology terms ) as functions . This idea laid one of the foundations for the modern analysis of natural language by means of formal logic, as it is u. a. by Max Cresswell and Richard Montague ( Montague grammar ).

Table of contents

Functions

Frege first clarifies the term function. To get a " function expression ", you have to remove the " character of the argument " in a " calculation expression ". Used, for example, in the invoice printout

${\displaystyle 2\cdot \mathrm {x} ^{3}+\mathrm {x} }$

Removing the "x" contains the function expression

${\displaystyle 2\cdot ()^{3}+()}$

In contrast to the argument, a function is "to be called incomplete, in need of completion or unsaturated ". If a function is supplemented by an argument, Frege calls the result the " value " of the function. If you add the number 2 to the above function, you get the value 18. If two functions give the same value for each argument, then according to Frege they have the same " value curve ". An example of functions with the same value curve would therefore be the functions

${\displaystyle 2(\mathrm {x} +2)}$

and

${\displaystyle 2\mathrm {x} +4}$.

These functions always give the same value, regardless of which argument you use for x.

A course of values ​​can be imagined as an assignment of the corresponding objects, here numbers. In the case of the last-mentioned functions, 1 is assigned to 6, to 2 to 8, etc. The course of values ​​can be graphically illustrated here with a coordinate system .

Terms

Terms in mathematics

Frege is now expanding the range of characters that " serve to form the function expression " by adding " characters such as" = ","> "and" <" ". He can now " speak of the function ${\displaystyle x^{2}=1}$ ". If you substitute numbers for "x" here, you can see that the expression becomes true for 1 and −1 and false for all other numbers. According to Frege, the value of this function is " a truth value "; H. one of the two values ​​" the true " and " the false ". Frege also calls functions " whose value is always a truth value " " concepts ". The above function can therefore be equated with the term "square root of 1".

By allowing truthful expressions ( statements ) as function expressions and the simultaneous introduction of truth values, Frege paved the way for a treatment of mathematics with the means of logic. This is the basic idea of ​​the logistic program that Frege formulated in " The Basics of Arithmetic ". The Fregeschen "terms" are called "predicates" in today's common usage of logic.

Frege calls the value curve of such a function a " conceptual scope ". The scope of the term "square root of 1" can thus be imagined as the assignment of the truth value "the true" to 1 and −1 and the value "the false" to all other numbers. The Fregeschen term ranges are called " characteristic functions " in modern mathematics . Each characteristic function and thus each scope corresponds exactly to one set , namely the set of those objects to which the function assigns the value "true". The set {1, -1} corresponds to the scope of the term "square root of 1". Because of this equivalence of the scopes of terms and sets, Frege's theory can be viewed as a continuation and, at the same time, a more precise specification of Cantor's set theory .

Natural language terms

Frege now goes one step further and also allows natural language expressions as functional expressions. For example, the phrase "Caesar conquered Gaul" can be broken down into the expression "Caesar" and the function, more precisely, the term "x conquered Gaul". The function "x conquered Gaul" thus gives the truth value the true if it is applied to the argument Caesar and the false if it is applied, for example, to Hannibal. So here spatiotemporal objects (people) are also allowed as arguments. Such values ​​can also be function values, e.g. B. the function "the capital of x". If Germany receives this function as an argument, the city of Berlin supplies it.

In general, an object for Frege is " anything that is not a function, the expression of which therefore does not have a blank space ". Examples are numbers , space-time objects such as places (cities) and people, as well as the logical objects truth values ​​and value processes.

Logical constants as functions

Frege introduces some of the basic notations of his conceptual writing , i.e. the second level predicate logic he developed , as functions. The " horizontal " is a function which, when applied to the true, yields the true and otherwise the false. Similarly, negation can be understood as a function: Applied to the false, it delivers the true and the false to the true. The quantifiers are also functions, but those that are applied to functions, Frege calls them " second level functions ". The universal quantifier, for example, delivers the true if it is applied to a function which, applied to any object, in turn delivers the true, otherwise the false.

Frege also handles functions with multiple arguments such as "x> y". He calls these " relationships ". One of these is the logical function of the subjunction , which delivers false if its first argument is true, its second is false, otherwise true. The arguments of a function with multiple arguments can be " of the same or different levels ", for example all objects or both objects and functions.

literature

• Thank God Frege: Function and Concept. Lecture given at the meeting of January 9, 1891 of the Jena Society for Medicine and Science. Verlag Hermann Pohle, Jena 1891 ( digitized version , accessed on June 29, 2017).