Though his education and early work were mathematical, especially geometrical, Frege's thought soon turned to logic. His

*Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens* (Halle a/S: Verlag von Louis Nebert, 1879) (

*Concept-Script: A Formal Language for Pure Thought Modeled on that of Arithmetic*) marked a turning point in the history of logic. The

*Begriffsschrift* broke new ground, including a rigorous treatment of the ideas of function and variables. Frege wanted to show that mathematics grows out of logic, but in so doing he devised techniques that took him far beyond the Aristotelian syllogistic and Stoic propositional logic that had come down to him in the logical tradition. In effect, Frege invented axiomatic predicate logic, in large part thanks to his invention of quantified variables, which eventually became ubiquitous in mathematics and logic, and which solved the problem of multiple generality. Previous logic had dealt with the logical constants

*and*,

*or*,

*if ... then ...*,

*not*, and

*some* and

*all*, but iterations of these operations, especially "some" and "all", were little understood: even the distinction between a pair of sentences like "every boy loves some girl" and "some girl is loved by every boy" was able to be represented only very artificially, whereas Frege's formalism had no difficulty expressing the different readings of "every boy loves some girl who loves some boy who loves some girl" and similar sentences, in complete parallel with his treatment of, say, "every boy is foolish".

It is frequently noted that Aristotle's logic is unable to represent even the most elementary inferences in Euclid's geometry, but Frege's "conceptual notation" can represent inferences involving indefinitely complex mathematical statements. The analysis of logical concepts and the machinery of formalization that is essential to

*Principia Mathematica* (3 vols., 1910—1913) (by Bertrand Russell, 1872-1970, and Alfred North Whitehead, 1861—1947), to Russell's theory of descriptions, to Kurt Gödel's (1906—1978) incompleteness theorems, and to Alfred Tarski's (1901—1983) theory of truth, is ultimately due to Frege.

One of Frege's stated purposes was to isolate genuinely logical principles of inference, so that in the proper representation of mathematical proof, one would at no point appeal to "intuition". If there was an intuitive element, it was to be isolated and represented separately as an axiom: from there on, the proof was to be purely logical and without gaps. Having exhibited this possibility, Frege's larger purpose was to defend the view that arithmetic is a branch of logic, a view known as logicism: unlike geometry, arithmetic was to be shown to have no basis in "intuition", and no need for non-logical axioms. Already in the 1879

*Begriffsschrift* important preliminary theorems, for example a generalized form of mathematical induction, were derived within what Frege understood to be pure logic.

This idea was formulated in non-symbolic terms in his

*Die Grundlagen der Arithmetik* (1884) (

*The Foundations of Arithmetic*). Later, in his

*Grundgesetze der Arithmetik* (

*Basic Laws of Arithmetic*) (vol. 1, 1893; vol. 2, 1903) (vol. 2 of which was published at his own expense), Frege attempted to derive, by use of his symbolism, all of the laws of arithmetic from axioms he asserted as logical. Most of these axioms were carried over from his

*Begriffsschrift*, though not without some significant changes. The one truly new principle was one he called the Basic Law V: the "value-range" of the function

*f*(

*x*) is the same as the "value-range" of the function

*g*(

*x*) if and only if ?

*x*[

*f*(

*x*) =

*g*(

*x*)].

The crucial case of the law may be formulated in modern notation as follows. Let {

*x*|

*Fx*} denote the extension of the predicate

*Fx*, i.e., the set of all Fs, and similarly for

*Gx*. Then Basic Law V says that the predicates

*Fx* and

*Gx* have the same extension iff ?x[

*Fx* ?

*Gx*]. The set of Fs is the same as the set of Gs just in case every F is a G and every G is an F. (The case is special because what is here being called the extension of a predicate, or a set, is only one type of "value-range" of a function.)

In a famous episode, Bertrand Russell wrote to Frege, just as Vol. 2 of the

*Grundgesetze* was about to go to press in 1903, showing that Russell's paradox could be derived from Frege's Basic Law V. It is easy to define the relation of

*membership* of a set or extension in Frege's system; Russell then drew attention to "the set of things

*x* that are such that

*x* is not a member of

*x*". The system of the

*Grundgesetze* entails that the set thus characterised

*both* is

*and* is not a member of itself, and is thus inconsistent. Frege wrote a hasty, last-minute Appendix to Vol. 2, deriving the contradiction and proposing to eliminate it by modifying Basic Law V. Frege opened the Appendix with the exceptionally honest comment: "Hardly anything more unfortunate can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished. This was the position I was placed in by a letter of Mr. Bertrand Russell, just when the printing of this volume was nearing its completion." (This letter and Frege's reply are translated in Jean van Heijenoort 1967.)

Frege's proposed remedy was subsequently shown to imply that there is but one object in the universe of discourse, and hence is worthless (indeed, this would make for a contradiction in Frege's system if he had axiomatized the idea, fundamental to his discussion, that the True and the False are distinct objects; see, for example, Dummett 1973), but recent work has shown that much of the program of the

*Grundgesetze* might be salvaged in other ways:

- Basic Law V can be weakened in other ways. The best-known way is due to philosopher and mathematical logician George Boolos (1940—1996), who was an expert on the work of Frege. A "concept"
*F* is "small" if the objects falling under *F* cannot be put into one-to-one correspondence with the universe of discourse, that is, if: ?*R*[*R* is 1-to-1 & ?*x*?*y*(*xRy* & *Fy*)]. Now weaken V to V*: a "concept" *F* and a "concept" *G* have the same "extension" if and only if neither *F* nor *G* is small or ?*x*(*Fx* ? *Gx*). V* is consistent if second-order arithmetic is, and suffices to prove the axioms of second-order arithmetic.
- Basic Law V can simply be replaced with Hume's Principle, which says that the number of
*F*s is the same as the number of *G*s if and only if the *F*s can be put into a one-to-one correspondence with the *G*s. This principle, too, is consistent if second-order arithmetic is, and suffices to prove the axioms of second-order arithmetic. This result is termed Frege's Theorem because it was noticed that in developing arithmetic, Frege's use of Basic Law V is restricted to a proof of Hume's Principle; it is from this, in turn, that arithmetical principles are derived. On Hume's Principle and Frege's Theorem, see "Frege's Logic, Theorem, and Foundations for Arithmetic". Frege's Logic, Theorem, and Foundations for Arithmetic, *Stanford Encyclopedia of Philosophy* at plato.stanford.edu
- Frege's logic, now known as second-order logic, can be weakened to so-called predicative second-order logic. However, this logic, although provably consistent by finitistic or constructive methods, can interpret only very weak fragments of arithmetic.

Frege's work in logic had little international attention until 1903 when Russell wrote an appendix to

*The Principles of Mathematics* stating his differences with Frege. The diagrammatic notationthat Frege used had no antecedents (and has had no imitators since). Moreover, until Russell and Whitehead's

*Principia Mathematica* (3 vols.) appeared in 1910—13, the dominant approach to mathematical logic was still that of George Boole (1815—1864) and his intellectual descendants, especially Ernst Schröder (1841—1902). Frege's logical ideas nevertheless spread through the writings of his student Rudolf Carnap (1891—1970) and other admirers, particularly Bertrand Russell and Ludwig Wittgenstein (1889—1951).