Indeed, for each condition defined above, the concepts that satisfy the condition are all pairwise equinumerous to one another. This extension contains all the concepts that satisfy Condition 0 above, and so the number of all such concepts is 0. Frege, however, had a deep idea about how to do this. Note that the last conjunct is true because there is exactly 1 object namely, Bertrand Russell that falls under the concept author of Principia Mathematica other than Whitehead.

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Important Secondary Works 1. His full christened name was Friedrich Ludwig Gottlob Frege. Little is known about his youth. Both were also principals of the school at various points: Karl held the position until his death , when Auguste took over until her death in Frege probably lived in Wismar until ; in the years from he is known to have studied at the Gymnasium in Wismar.

In Spring , Frege began studies at the University of Jena. In , with the recommendation of Ernst Abbe, Frege received a lectureship at the University of Jena, where he stayed the rest of his intellectual life. His position was unsalaried during his first five years, and he was supported by his mother. Frege had a heavy teaching load during his first few years at Jena.

Therein, Frege presented for the first time his invention of a new method for the construction of a logical language. Upon the publication of the Begriffsschrift, he was promoted to ausserordentlicher Professor, his first salaried position. Sometime after the publication of the Begriffsschrift, Frege was married to Margaret Lieseburg They had at least two children, who unfortunately died young.

Years later they adopted a son, Alfred. Frege had aimed to use the logical language of the Begriffsschrift to carry out his logicist program of attempting to show that all of the basic truths of arithmetic could be derived from purely logical axioms. However, on the advice of Carl Stumpf, and given the poor reception of the Begriffsschrift, Frege decided to write a work in which he would describe his logicist views informally in ordinary language, and argue against rival views. Soon thereafter, Frege began working on his attempt to derive the basic laws of arithmetic within his logical language.

However, his work was interrupted by changes to his views. In the late s and early s Frege developed new and interesting theories regarding the nature of language, functions and concepts, and philosophical logic, including a novel theory of meaning based on the distinction between sense and reference. However, in , Frege finally finished a revised volume, employing a slightly revised logical system.

In the first volume, Frege presented his new logical language, and proceeded to use it to define the natural numbers and their properties. His aim was to make this the first of a three volume work; in the second and third, he would move on to the definition of real numbers, and the demonstration of their properties. Nevertheless, he was promoted once again in , now to the position of Honorary Ordinary Professor. It is likely that Frege was offered a position as full Professor, but turned it down to avoid taking on additional administrative duties.

His new position was unsalaried, but he was able to support himself and his family with a stipend from the Carl Zeiss Stiftung, a foundation that gave money to the University of Jena, and with which Ernst Abbe was intimately involved. Because of the unfavorable reception of his earlier works, Frege was forced to arrange to have volume II of the Grundgesetze published at his own expense.

It was not until that Frege was able to make such arrangements. However, while the volume was already in the publication process, Frege received a letter from Bertrand Russell, informing him that it was possible to prove a contradiction in the logical system of the first volume of the Grundgesetze, which included a naive calculus for classes.

He was forced to quickly prepare an appendix in response. For the next couple years, he continued to do important work.

He produced very little work between and his retirement in However, he continued to influence others during this period. Russell had included an appendix on Frege in his Principles of Mathematics. It is from this that Frege came be to be a bit wider known, including to an Austrian student studying engineering in Manchester, England, named Ludwig Wittgenstein. Frege invited him to Jena to discuss his views.

Wittgenstein did so in late However, these were not wholly new works, but later drafts of works he had initiated in the s. In , a year before his death, Frege finally returned to the attempt to understand the foundations of arithmetic. However, by this time, he had completely given up on his logicism, concluding that the paradoxes of class or set theory made it impossible. He instead attempted to develop a new theory of the nature of arithmetic based on Kantian pure intuitions of space. However, he was not able to write much or publish anything about his new theory.

Frege died on July 26, at the age of He did not live to see the profound impact he would have on the emergence of analytic philosophy, nor to see his brand of logic—due to the championship of Russell—virtually wholly supersede earlier forms of logic. Take care that nothing gets lost.

Unfortunately, however, they were destroyed in an Allied bombing raid on March 25, Although he was a fierce, sometimes even satirical, polemicist, Frege himself was a quiet, reserved man.

He was right-wing in his political views, and like many conservatives of his generation in Germany, he is known to have been distrustful of foreigners and rather anti-semitic. Himself Lutheran, Frege seems to have wanted to see all Jews expelled from Germany, or at least deprived of certain political rights.

Mill , who thought that arithmetic was grounded in observation. In other words, Frege subscribed to logicism. His logicism was modest in one sense, but very ambitious in others. Indeed, Frege himself set out to demonstrate all of the basic laws of arithmetic within his own system of logic. Frege concurred with Leibniz that natural language was unsuited to such a task.

Although there had been attempts to fashion at least the core of such a language made by Boole and others working in the Leibnizian tradition, Frege found their work unsuitable for a number of reasons.

Frege found this unacceptable for a language which was to be used to demonstrate mathematical truths, because the signs would be ambiguous. It analyzed propositions in terms of subject and predicate concepts, which Frege found to be imprecise and antiquated. Frege saw the formulae of mathematics as the paradigm of clear, unambiguous writing.

In order to make his logical language suitable for purposes other than arithmetic, Frege expanded the notion of function to allow arguments and values other than numbers. He defined a concept Begriff as a function that has a truth-value, either of the abstract objects the True or the False, as its value for any object as argument.

The concept being human is understood as a function that has the True as value for any argument that is human, and the False as value for anything else. The values of such concepts could then be used as arguments to other functions. In his own logical systems, Frege introduced signs standing for the negation and conditional functions.

His own logical notation was two-dimensional. Conjunction and disjunction signs could then be defined from the negation and conditional signs. Variables and quantifiers are used to express generalities. The distinction between levels of functions involves what kind of arguments the functions take. But different sorts of functions require different sorts of arguments. Functions that take first-level functions as argument are called second-level functions. Frege is often credited with having founded predicate logic.

Rather, it flanks terms for truth-values to form a term for a truth-value. In addition to quantifiers ranging over objects, it also contained quantifiers ranging over first-level functions.

In fact, Frege was the first to take a fully axiomatic approach to logic, and the first even to suggest that inference rules ought to be explicitly formulated and distinguished from axioms. He began with a limited number of fixed axioms, introduced explicit inference rules, and aimed to derive all other logical truths including, for him, the truths of arithmetic from them.

It represented the first axiomatization of logic, and was complete in its treatment of both propositional logic and first-order quantified logic. It has since been proven impossible to devise a system for higher-order logic with a finite number of axioms that is both complete and consistent. In order to make deduction easier, in the logical system of the Grundgesetze, Frege used fewer axioms and more inference rules: seven and twelve, respectively, this time leaving nothing implicit.

In the case of concepts, their value-ranges were identified with their extensions. They were simply understood as objects corresponding to the complete argument-value mappings generated by concepts considered as functions.

Frege then introduced two axioms dealing with these value-ranges. Most infamous was his Basic Law V, which asserts that the truth-value of the value-range of function F being identical to the value-range of function G is the same as the truth-value of F and G having the same value for every argument. If one conceives of value-ranges as argument-value mappings, then this certainly seems to be a plausible hypothesis. However, from it, it is possible to prove a strong theorem of class membership: that for any object x, that object is in the extension of concept F if and only if the value of F for x as argument is the True.

Given that value-ranges themselves are taken to be objects, if the concept in question is that of being a extension of a concept not included in itself, one can conclude that the extension of this concept is in itself just in case it is not. However, the core of the system of the Grundgesetze, that is, the system minus the axioms governing value-ranges, is consistent and, like the system of the Begriffsschrift, is complete in its treatment of propositional logic and first-order predicate logic.

Given the extent to which it is taken granted today, it can be difficult to fully appreciate the truly innovative and radical approach Frege took to logic. Frege was the first to attempt to transcribe the old statements of categorical logic in a language employing variables, quantifiers and truth-functions.

In earlier logical systems such as that of Boole, in which the propositional and quantificational elements were bifurcated, the connection was wholly lost. This too was impossible in all earlier logical systems. As we shall see, he also made advances in the logic of mathematics. It is small wonder that he is often heralded as the founder of modern logic. Logical functions, value-ranges, and the truth-values the True and the False, are thought to be objectively real entities, existing apart from the material and mental worlds.

As we shall see below , Frege was also committed to other logical entities such as senses and thoughts. Logical axioms are true because they express true thoughts about these entities. Thus, Frege denied the popular view that logic is without content and without metaphysical commitment.

Frege was also a harsh critic of psychologism in logic: the view that logical truths are truths about psychology. While Frege believed that logic might prescribe laws about how people should think, logic is not the science of how people do think. Logical truths would remain true even if no one believed them nor used them in their reasoning. Contributions to the Philosophy of Mathematics Frege was an ardent proponent of logicism, the view that the truths of arithmetic are logical truths.

Perhaps his most important contributions to the philosophy of mathematics were his arguments for this view.

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## Gottlob Frege

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. Frege applied the results from the Begriffsschrifft, including those on the ancestral of a relation, in his later work The Foundations of Arithmetic. This is the so-called "law of trichotomy ". Influence on other works[ edit ] For a careful recent study of how the Begriffsschrift was reviewed in the German mathematical literature, see Vilko 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.

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## Gottlob Frege (1848—1925)

In breve, Frege voleva dimostrare che, con la teoria degli insiemi e una buona definizione di logica, si potessero derivare tutti gli insiemi numerici. Riguardo a molti problemi particolari trovo nella sua opera discussioni, distinzioni e definizioni che si cercano invano nelle opere di altri logici. Specialmente per quel che riguarda le funzioni cap. Lei afferma p. Sto finendo un libro sui principi della matematica e in esso vorrei discutere la sua opera in tutti i dettagli. Molto rispettosamente suo Bertrand Russell Ho scritto a Peano di questo fatto, ma non ho ancora ricevuto risposta. Due funzioni hanno lo stesso decorso di valori, se con gli stessi argomenti assumono gli stessi valori.