Function and Concept

Function and concept is beside On Sense and Reference, and over concept and object of one of the three in quick succession published essays by Gottlob Frege, in which he explains the basic terms of its logic and philosophy of language. "Function and concept " was released in 1891, so by the above texts first. Frege here presented, inter alia, a construction with which one predicates ( in Frege's terminology terms ) may be regarded as functions. This idea laid one of the foundations for the modern analysis of natural language by means of formal logic, as it was carried out, inter alia, by Max Cresswell and Richard Montague ( Montague grammar ).

Summary

Functions

Frege first clarifies the concept of function. To get a " function expression ", you have to remove the "sign of the argument " in a " statement expression". If, for example, in the calculation expression

The "X" removed, one containing the functional expression

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

And

.

These functions give the same value, no matter which argument is used for x.

You can buy a values ​​history as an assignment of the respective subjects, here to introduce numbers. In the case of the latter functions, therefore, the one assigned to the 6, 2 of the 8 etc. Graphically illustrate the progression of values ​​could be here with a coordinate system.

Terms

Concepts in mathematics

Frege is now expanding the range of characters " used to form the functional expression " by it "to take " characters like " =", " >" and " <". He can now " speak of the function". If, here's one for "x" numbers, it is observed that the expression for 1 and -1 for true and all other numbers is wrong. According to Frege, the value of this function " a truth value ," ie, one of the two values ​​" truth " and " falsity ". Functions, " whose value is always a truth value is ", Frege also called " terms". The above function is thus equated with the term " square root of 1".

By allowing the truth of performance expressions ( statements ) as a function expressions and the simultaneous introduction of truth values ​​Frege has paved the way for a treatment of mathematics with means of logic. This is the basic idea of ​​the logicist program, which Frege formulated in " The Foundations of Arithmetic ." The Frege's " terms" are referred to as " predicates " in today's common usage of logic.

Frege calls the value sequence of such a function a " term scope ". The term scope of the term " square root of 1" so it can be thought of as the assignment of the truth value of " truth " to 1 and -1 and the value imagine " the wrong thing " to all the other numbers. The Fregean extensions of concepts are called in modern mathematics " characteristic functions ". Each characteristic function and thus each term corresponds exactly to scope a lot, namely the set of objects to which the function of the "true" value assigns. The term scope of the term " square root of 1" thus corresponds to the set { 1, -1 }. Due to this equivalence of term perimeters and volumes Frege's theory as a continuation and at the same time clarifying the cantor between set theory can be considered.

Natural language terms

Frege now goes one step further and also natural language expressions as a function expressions. The sentence " Caesar conquered Gaul ", for example, in the expression " Caesar " and the function, more specifically, the term " x conquered Gaul " be dismantled. The "x conquered Gaul " function so gives the truth value of the true, if it is applied to the argument Caesar and what is false when it is for instance applied to Hannibal. It will therefore also spatio-temporal objects ( people) passed as arguments. Such may also be the function values ​​, such as the " the capital of x ". Receives this function Germany as an argument, it provides the city of Berlin.

In general, a subject for Frege " everything that is not a function whose expression is no empty space brings with it ". Examples are numbers that spatio-temporal objects such as places (cities) and people, as well as the logical objects and logical values ​​and value trends.

Logical constants as functions

Frege introduces some basic notations of his Begriffsschrift, that the developed him of predicate logic level 2, as a function. The " horizontal " is a function that, when applied to truth, the truth and otherwise provides the wrong thing. Similarly, the negation can be regarded as functional: it returns applied to the false truth and falsity to truth. The quantifiers are functions, but those that are applied to functions, Frege calls them " second-level functions". The universal quantifier, for example, provides the truth when it is applied to a function that applied to any object in turn provides the true, otherwise the wrong thing.

Frege also deals with functions with several arguments, such as "x > y". He calls these "relationships ". These include the logical function of the subjunction that delivers the wrong thing, if its first argument the truth, her second is the falsehood, otherwise the truth. The arguments of a function with several arguments can " of the same or of different stages to be ", so for example all objects or both objects and features.

• Language Philosophical works
• Philosophical logic
• Gottlob Frege