Begriffsschrift German for, roughly, "concept-script" is a book on logic by Gottlob Frege , published in , 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 , of pure thought. Frege went on to employ his logical calculus in his research on the foundations of mathematics , carried out over the next quarter century. The calculus contains the first appearance of quantified variables, and is essentially classical bivalent second-order logic with identity.

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Friedrich Ludwig Gottlob Frege b. Frege then demonstrated that one could use his system to resolve theoretical mathematical statements in terms of simpler logical and mathematical notions. One of the axioms that Frege later added to his system, in the attempt to derive significant parts of mathematics from logic, proved to be inconsistent.

Nevertheless, his definitions e. To ground his views about the relationship of logic and mathematics, Frege conceived a comprehensive philosophy of language that many philosophers still find insightful. However, his lifelong project, of showing that mathematics was reducible to logic, was not successful.

According to the curriculum vitae that the year old Frege filed in with his Habilitationsschrift , he was born on November 8, in Wismar, a town then in Mecklenburg-Schwerin but now in Mecklenburg-Vorpommern. His father, Alexander, a headmaster of a secondary school for girls, and his mother, Auguste nee Bialloblotzky , brought him up in the Lutheran faith. Frege attended the local Gymnasium for 15 years, and after graduation in , entered the University of Jena see Frege , translation in McGuinness ed.

Interestingly, one section of the thesis concerns the representation of complex numbers by magnitudes of angles in the plane. Immediately after submitting this thesis, the good offices of Abbe led Frege to become a Privatdozent Lecturer at the University of Jena. Library records from the University of Jena establish that, over the next 5 years, Frege checked out texts in mechanics, analysis, geometry, Abelian functions, and elliptical functions Kreiser , Kratzsch Here we can see the beginning of two lifelong interests of Frege, namely, 1 in how concepts and definitions developed for one domain fare when applied in a wider domain, and 2 in the contrast between legitimate appeals to intuition in geometry and illegitimate appeals to intuition in the development of pure number theory.

Although the Begriffsschrift constituted a major advance in logic, it was neither widely understood nor well-received. Frege begins this work with criticisms of previous attempts to define the concept of number, and then offers his own analysis. The Grundlagen contains a variety of insights still discussed today, such as: a the claim that a statement of number e. Immediately after that, in , he published the first volume of the technical work previously mentioned, Grundgesetze der Arithmetik.

In , he was promoted to ordentlicher Honorarprofessor regular honorary professor. Six years later on June 16, , as he was preparing the proofs of the second volume of the Grundgesetze , he received a letter from Bertrand Russell, informing him that one could derive a contradiction in the system he had developed in the first volume. Frege, in the Appendix to the second volume, rephrased the paradox in terms of his own system. Frege never fully recovered from the fatal flaw discovered in the foundations of his Grundgesetze.

His attempts at salvaging the work by restricting Basic Law V were not successful. From this time period, we have the lecture notes that Rudolf Carnap took as a student in two of his courses see Reck and Awodey In , he retired from the University of Jena. After that, however, we have only fragments of philosophical works. Unfortunately, his last years saw him become more than just politically conservative and right-wing — his diary for a brief period in show sympathies for fascism and anti-Semitism see Frege [], translated by R.

Frege provided a foundations for the modern discipline of logic by developing a more perspicuous method of formally representing the logic of thoughts and inferences.

He did this by developing: a a system allowing one to study inferences formally, b an analysis of complex sentences and quantifier phrases that showed an underlying unity to certain classes of inferences, c an analysis of proof and definition , d a theory of extensions which, though seriously flawed, offered an intriguing picture of the foundations of mathematics, e an analysis of statements about number i.

We discuss these developments in the following subsections. Frege analyzed ordinary predication in these systems, and so they can also be conceived as predicate calculi. A predicate calculus is a formal system a formal language and a method of proof in which one can represent valid inferences among predications, i.

These are the statements involving function applications and the simple predications which fall out as a special case. Frege distinguished two truth-values, The True and The False, which he took to be objects. And so on, for functions of more than two variables. If we replace a complete name appearing in a sentence by a placeholder, the result is an incomplete expression that signifies a special kind of function which Frege called a concept. Concepts are functions which map every argument to one of the truth-values.

Frege would say that any object that a concept maps to The True falls under the concept. The preceding analysis of simple mathematical predications led Frege to extend the applicability of this system to the representation of non-mathematical thoughts and predications.

This move formed the basis of the modern predicate calculus. Thus, a simple predication is analyzed in terms of falling under a concept, which in turn, is analyzed in terms of functions which map their arguments to truth values. By contrast, in the modern predicate calculus, this last step of analyzing predication in terms of functions is not assumed; predication is seen as more fundamental than functional application.

In the modern predicate calculus, functional application is analyzable in terms of predication, as we shall soon see. By contrast, Frege took functions to be more basic than relations. His logic is based on functional application rather than predication; a binary relation is analyzed as a binary function that maps a pair of arguments to a truth-value.

Note the last row of the table — when Frege wants to assert that two conditions are materially equivalent, he uses the identity sign, since this says that they denote the same truth-value. Frege used a special typeface Gothic for variables in general statements. Note the last line. Here again, Frege uses the identity sign to help state the material equivalence of two concepts. This means it allows quantification over functions as well as quantification over objects; i.

In particular, we adopt the following conventions:. In traditional Aristotelian logic, the subject of a sentence and the direct object of a verb are not on a logical par. The rules governing the inferences between statements with different but related subject terms are different from the rules governing the inferences between statements with different but related verb complements.

The rule governing the first inference is a rule which applies only to subject terms whereas the rule governing the second inference governs reasoning within the predicate, and thus applies only to the transitive verb complements i.

In Aristotelian logic, these inferences have nothing in common. In effect, Frege treated these quantified expressions as variable-binding operators.

Thus, Frege analyzed the above inferences in the following general way:. Both inferences are instances of a single valid inference rule. To see this more clearly, here are the formal representations of the above informal arguments:.

Indeed, this axiom can be made even more general. He suggested that existence is not a concept under which objects fall but rather a second-level concept under which first-level concepts fall.

The latter consisted of a set of logical axioms statements considered to be truths of logic and a set of rules of inference that lay out the conditions under which certain statements of the language may be correctly inferred from others.

Frege made a point of showing how every step in a proof of a proposition was justified either in terms of one of the axioms or in terms of one of the rules of inference or justified by a theorem or derived rule that had already been proved. In essence, he defined a proof to be any finite sequence of statements such that each statement in the sequence either is an axiom or follows from previous members by a valid rule of inference.

These are essentially the definitions that logicians still use today. Frege was extremely careful about the proper description and definition of logical and mathematical concepts.

He developed powerful and insightful criticisms of mathematical work which did not meet his standards for clarity. For example, he criticized mathematicians who defined a variable to be a number that varies rather than an expression of language which can vary as to which determinate number it may take as a value.

More importantly, however, Frege was the first to claim that a properly formed definition had to have two important metatheoretical properties. Let us call the new, defined symbol introduced in a definition the definiendum , and the term that is used to define the new term the definiens. Then Frege was the first to suggest that proper definitions have to be both eliminable a definendum must always be replaceable by its definiens in any formula in which the former occurs and conservative a definition should not make it possible to prove new relationships among formulas that were formerly unprovable.

In the Grundgesetze der Arithmetik, II , Sections 56—67 Frege criticized the practice of defining a concept on a given range of objects and later redefining the concept on a wider, more inclusive range of objects. In that same work , Sections — , Frege criticized the mathematical practice of introducing notation to name unique entities without first proving that there exist unique such entities. Creative definitions fail to be conservative, as this was explained above.

This distinguishes them from objects. The course-of-values of a function is a record of the value of the function for each argument. Using this notation, Frege formally represented Basic Law V in his system as:. Actually, Frege used an identity sign instead of the biconditional as the main connective of this principle, for reasons described above.

Thus Basic Law V applies equally well to the extensions of concepts. Then Frege would use the expression:. Unfortunately, Basic Law V implies a contradiction, and this was pointed out to Frege by Bertrand Russell just as the second volume of the Grundgesetze was going to press.

Russell recognized that some extensions are elements of themselves and some are not; the extension of the concept extension is an element of itself, since that concept would map its own extension to The True. But now what about the concept extension which is not an element of itself? Few philosophers today believe that mathematics can be reduced to logic in the way Frege had in mind.

Mathematical theories such as set theory seem to require some non-logical concepts such as set membership which cannot be defined in terms of logical concepts, at least when axiomatized by certain powerful non-logical axioms such as the proper axioms of Zermelo-Fraenkel set theory.

Despite the fact that a contradiction invalidated a part of his system, the intricate theoretical web of definitions and proofs developed in the Grundgesetze nevertheless offered philosophical logicians an intriguing conceptual framework. See T. In what has come to be regarded as a seminal treatise, Die Grundlagen der Arithmetik , Frege began work on the idea of deriving some of the basic principles of arithmetic from what he thought were more fundamental logical principles and logical concepts.

Philosophers today still find that work insightful. In the second case, the second level claim asserts that the first-level concept being an author of Principia Mathematica falls under the second-level concept being a concept under which two objects fall. This sounds circular, since it looks like we have analyzed. But despite appearances, there is no circularity, since Frege analyzes the second-order concept being a concept under which two objects fall without appealing to the concept two.

Frege then took his analysis one step further. Indeed, for each condition defined above, the concepts that satisfy the condition are all pairwise equinumerous to one another. Thus, the number 0 becomes defined as the extension of all the concepts equinumerous to the concept not being self-identical. 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. Using this definition as a basis, Frege later derived many important theorems of number theory.

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## Gottlob Frege

Friedrich Ludwig Gottlob Frege b. Frege then demonstrated that one could use his system to resolve theoretical mathematical statements in terms of simpler logical and mathematical notions. One of the axioms that Frege later added to his system, in the attempt to derive significant parts of mathematics from logic, proved to be inconsistent. Nevertheless, his definitions e. To ground his views about the relationship of logic and mathematics, Frege conceived a comprehensive philosophy of language that many philosophers still find insightful.

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