1On the background of Frege’s Begriffsschrift, see Kreiser (), in particular Couturat’s contribution appeared in an English translation. Reproduktion in Begriffsschrift (). [Vortrag, gehalten in der Sitzung vom Juli der Jenaischen Gesellschaft für Medizin und Naturwissenschaft.]. In , Frege published his first book Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens (Concept.
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Friedrich Ludwig Gottlob Frege b. Frege then demonstrated that one could bdgriffsschrift 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 Habilitationsschrifthe was born on November 8, in Wismar, a town then in Mecklenburg-Schwerin but now begriffsschrify Mecklenburg-Vorpommern.
His father, Alexander, a headmaster of a secondary school for girls, and his mother, Auguste nee Bialloblotzkybrought him up in the Lutheran faith.
Frege attended the local Gymnasium for 15 years, and after graduation inentered the University of Jena see Fregetranslation in McGuinness ed. At Jena, Frege attended lectures by Ernst Karl Abbe, who subsequently became Frege’s mentor and who had a significant intellectual and personal influence on Frege’s life. Frege explains the project in his thesis as follows: 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 This course of Frege’s reading and lectures during the period of — dovetailed quite naturally with the interests he displayed in his Habilitationsschrift.
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 begriffssxhrift to intuition in the development of pure number theory. Indeed, some recent scholars have a shown how Frege’s work in logic was informed in part by his understanding of the analogies and disanalogies between geometry and number theory Wilsonand b shown that Frege was begriffsscbrift familiar with the division among late 19th century mathematicians doing complex analysis who split over whether it is better to use the analytic methods of Weierstrass or the intuitive geometric methods of Riemann Tappenden Weierstrass’s paper, describing a real-valued function that is continuous everywhere but differentiable nowhere, [ 4 ] was well known and provided an example of an ungraphable functions that places limits on intuition.
Yet, at the same time, Frege clearly accepted Riemann’s practice and methods derived from taking functions as fundamental, as opposed to Weierstrass’s focus on functions that can begriffscshrift represented or analyzed in terms of other mathematical objects e. InFrege published his first book Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens Concept Notation: Begriffsschrrift the Begriffsschrift constituted a major advance in logic, it was neither widely understood nor well-received.
Some scholars have suggested that this was due to the facts that the notation was 2-dimensional instead of linear and that he didn’t build upon the work of others but rather presented something radically new e. Frege’s next really significant work was his second book, Die Grundlagen der Arithmetik: 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: Immediately after that, inhe published the first volume of the technical work previously mentioned, Grundgesetze der Arithmetik.
Inhe 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 Grundgesetzehe received a letter from Betriffsschrift 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. In the latter, Frege criticized Hilbert’s understanding and use of the axiomatic method see the entry on the Frege-Hilbert controversy. 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 Inhe retired from the University of Jena.
After that, however, we have only fragments of philosophical works. Enblish, his last years saw begruffsschrift 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: We discuss these developments in the following subsections. In an attempt to realize Leibniz’s ideas for a language of thought and a rational calculus, Frege developed a formal notation for regimenting thought and reasoning.
Frege’s two systems are best characterized as term logics, since all of the complete expressions are denoting terms. Frege analyzed dnglish 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. In Frege’s term logic, all of begtiffsschrift terms and well-formed formulas begriffaschrift denoting expressions.
Even the sentences of Frege’s mature logical system are complex denoting terms; they are terms that denote truth-values. 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. Thus, the number 2 falls under the concept that which when squared is identical to 4.
Concept Script: Frege
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 begridfsschrift 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. This function takes a pair of arguments x and y and maps them to The True if x loves y and maps all other pairs of arguments to The False. Although it is a descendent of Frege’s system, the modern predicate calculus analyzes loves as a two-place relation Lxy rather than a function; some objects stand in fregw relation and others do not.
The difference between Frege’s understanding of predication and the one manifested by the modern predicate calculus is simply this: By contrast, Frege took functions to be more basic than relations.
His logic is based on functional application rather than predication; so, a binary relation is analyzed as a binary function that maps a pair of arguments to a truth-value. Thus, a 3-place relation like gives would be analyzed in Frege’s logic as a function that maps arguments xyand z to an appropriate truth-value depending on whether x gives y to z ; the 4-place relation buys would be analyzed as a function that maps the arguments xyzand u to an appropriate truth-value depending on whether x buys y from z for amount u ; etc.
Though we no longer use his notation for representing complex and general statements, it is important to see how the notation in Frege’s term logic already contained all the expressive power of the modern predicate calculus.
There are four special functional expressions which are used in Frege’s system to express complex and general statements:. The best way to understand this notation is by way of some tables, which show some specific examples of statements and how those are rendered in Frege’s notation and in the modern predicate calculus.
Begriffsschrift – Wikipedia
The first table shows how Frege’s logic can express the truth-functional connectives such as not, if-then, and, or, and if-and-only-if. 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. The table below compares statements of generality in Frege’s notation and in the modern predicate calculus. 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. As one can see from the table above, Frege didn’t use an existential quantifier. This means it allows quantification over functions as well as quantification over objects; i. In what follows, however, we shall continue to use the notation of the modern predicate calculus instead of Frege’s notation. 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. That’s because the subject John and the direct object Mary are both considered on a logical par, as arguments of the function loves.
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 englis valid inference rule. To see this more clearly, here are the formal representations of the above informal arguments:. This logical axiom tells us that from a simple predication involving an n -place relation, one can existentially generalize on any argument, and validly derive a existential statement.
Indeed, this axiom can be made even more general. There is one other consequence of Frege’s logic of quantification that should be mentioned. He suggested that existence is not a concept under which objects fall but rather a second-level concept under which first-level concepts fall.
A concept F falls under this second-level concept just in case F maps at least one object to The True. Many philosophers have thought that this analysis validates Kant’s view that existence is not frefe real predicate. 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 wnglish expression of language which can begrifcsschrift 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.