* ( edited, with Max Black ) Translations from the Philosophical Writings of Gottlob Frege, 1952 / 1960 / 1966

** Frege, G., 1960, " A critical elucidation of some points in E. Schröder's Vorlesungen über die Algebra der Logik ", translated by Geach, in Geach & Black, Translations from the philosophical writings of Gottlob Frege.

In a letter dated May 24, 1891, Frege thanked Husserl for sending him a copy of the Philosophy of Arithmetic and Husserl's review of Ernst Schröder's Vorlesungen über die Algebra der Logik.

In the same letter, Frege used the review of Schröder's book to analyze Husserl's notion of the sense of reference of concept words.

Schröder's influence on the early development of the predicate calculus, mainly by popularising C. S. Peirce's work on quantification, is at least as great as that of Frege or Peano.

Frege developed a similar view ( though later ) in his great work The Foundations of Arithmetic, as did Charles Sanders Peirce ( but Peirce held that the possible and the real are not limited to the actually, individually existent ).

The syllogism was superseded by first-order predicate logic following the work of Gottlob Frege, in particular his Begriffsschrift ( Concept Script ) ( 1879 ), but syllogisms remain useful.

This method of inquiry is largely indebted to the work of philosophers such as Gottlob Frege, Bertrand Russell, G. E.

The German mathematician Gottlob Frege ( 1848 – 1925 ) presented an independent development of logic with quantifiers in his Begriffsschrift ( formula language ) published in 1879, a work generally considered as marking a turning point in the history of logic.

The group considered themselves logical positivists because they believed all knowledge is either derived through experience or arrived at through analytic statements, and they adopted the predicate logic of Frege, as well as the early work of Ludwig Wittgenstein ( 1889 – 1951 ) as foundations to their work.

Although the formalisation of logic was much advanced by the work of such figures as Gottlob Frege, Giuseppe Peano, Bertrand Russell, and Richard Dedekind, the story of modern proof theory is often seen as being established by David Hilbert, who initiated what is called Hilbert's program in the foundations of mathematics.

Serious metamathematical reflection began with the work of Gottlob Frege, especially his Begriffsschrift.

Husserl had reacted strongly to Gottlob Frege's criticism of his first work on the philosophy of arithmetic and was investigating the sense of mathematical and other structures, which Frege had distinguished from empirical reference.

In the years 1925-1926, the Thursday night group discussed recent work in the foundations of mathematics by Gottlob Frege, Bertrand Russell, and Ludwig Wittgenstein.

He made contributions to the philosophy of language, the philosophy of mathematics and science, and the philosophy of art, also publishing studies of the work of philosophers such as Frege.

This book reprints much of Boolos's work on the rehabilitation of Frege, as well as a number of his papers on set theory, second-order logic and nonfirstorderizability, plural quantification, proof theory, and three short insightful papers on Gödel's Incompleteness Theorem.

However, today most students of logic are more familiar with the works of Frege, who actually published his work several years prior to Peirce but whose works remained in obscurity until Bertrand Russell and Alfred North Whitehead made them famous.

The oldest definition of the cardinality of a set X ( implicit in Cantor and explicit in Frege and Principia Mathematica ) is as the set of all sets which are equinumerous with X: this does not work in ZFC or other related systems of axiomatic set theory because this collection is too large to be a set, but it does work in type theory and in New Foundations and related systems.

Symbolic computational approaches to creating intelligent machines had long been the focus of AI since the days of Alan Turing, directly tracing back to the work of Gottlob Frege.

He also defends a reading of Frege, derived in part from Michael Dummett's work, according to which Frege's notion of sense is not equivalent to a description, and indeed remains essential to a theory of reference that abandoned descriptivism ( 1982, § 1. 3 ).

In most years, Kaplan teaches an upper division course on philosophy of language, focusing on the work of either Gottlob Frege, Bertrand Russell, or P. F.

Hintikka's work can be seen as a continuation of analytic tendency in philosophy founded by Brentano and Peirce, advanced by Frege and Bertrand Russell, and continued by Carnap, Quine, and by Hintikka's teacher Georg Henrik von Wright.

In the tradition of analytical philosophy, according to Michael Dummett the linguistic movement first took shape in Gottlob Frege's 1884 work The Foundations on Arithmetic, specifically paragraph 62 where Frege explores the identity of a numerical proposition.

In philosophical logic, Martin-Löf has wrestled with the philosophy of logical consequence and judgment, partly inspired by the work of Brentano, Frege, and Husserl.

Gottlob Frege, founder of the analytic tradition in philosophy, famously argued for the analysis of language in terms of sense and reference.

According to Frege the reference of a sentence is a truth value ; for Husserl it is a " state of affairs.

Logical empiricism ( aka logical positivism or neopositivism ) was an early 20th century attempt to synthesize the essential ideas of British empiricism ( e. g. a strong emphasis on sensory experience as the basis for knowledge ) with certain insights from mathematical logic that had been developed by Gottlob Frege and Ludwig Wittgenstein.

Besides Frege and Russell, others credited for having preceding ideas of truth-tables include Philo, Boole, Charles Sanders Peirce, and Ernst Schröder.

This would not have been enough for Frege because ( to paraphrase him ) it does not exclude the possibility that the number 3 is in fact Julius Caesar.

The " Prolegomena " is considered a more concise, fair, and thorough refutation of psychologism than the criticisms made by Frege, and also it is considered today by many as being a memorable refutation for its decisive blow to psychologism.

However, the methods developed by Frege and Tarski for the study of mathematical language have been extended greatly by Tarski's student Richard Montague and other linguists working in formal semantics to show that the distinction between mathematical language and natural language may not be as great as it seems.

Already in the 1879 Begriffsschrift important preliminary theorems, for example a generalized form of law of trichotomy, were derived within what Frege understood to be pure logic.

He is best known for the lambda calculus, Church – Turing thesis, proving the undecidability of the Entscheidungsproblem, Frege – Church ontology, and the Church – Rosser theorem.

This fact is sometimes thought to have severe consequences for the program of logicism proposed by Gottlob Frege and Bertrand Russell, which aimed to define the natural numbers in terms of logic ( Hellman 1981, p. 451 – 468 ).

Frege is typically translated as saying that an expression " expresses its sense " and " stands for or designates its reference ".

One application Frege saw for the distinction concerns what are called nonreferring, nondenoting, or empty, expressions.

Bertrand Russell famously rejected Frege's sense-reference distinction, though there is some possibility that the two were misinterpreting and arguing past one another: Frege talks about ( for example ) sentences, which have both a sense ( a proposition ) and a reference ( a truth value ); Russell on the other hand deals directly with propositions, but construes these not as abstract para-linguistic items but as tuples, or sets, of objects and concepts.

For example, Joyce ( 1949 ), written for use in Catholic seminaries, made no mention of Frege or Bertrand Russell.

* Gottlob Frege publishes Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens (" Concept-Script: A Formal Language for Pure Thought Modeled on that of Arithmetic ") in Halle, a significant text in the development of mathematical logic.

See also Gottlob Frege 1895 for a critique of an earlier view defended by Ernst Schroeder.

At the time of Kripke's lectures, the dominant theories of reference in Analytic philosophy ( associated with the theories of Gottlob Frege and Bertrand Russell ) held that the meaning of sentences involving proper names could be given by substituting a contextually appropriate description for the name.

Begriffsschrift ( German for, roughly, " concept-script ") is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book.

Frege's motivation for developing his formal approach to logic resembled Leibniz's motivation for his calculus ratiocinator ( despite that, in his Foreword Frege clearly denies that he reached this aim, and also that his main aim would be constructing an ideal language like Leibniz's, what Frege declares to be quite hard and idealistic, however, not impossible task ).

In the first chapter, Frege defines basic ideas and notation, like proposition (" judgement "), the universal quantifier (" the generality "), the conditional, negation and the " sign for identity of content " ( which he used to indicate both material equivalence and identity proper ); in the second chapter he declares nine formalized propositions as axioms.