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Gottlob Frege (18481925)

Gottlob Frege was a German logician, mathematician and philosopher

who played a crucial role in the emergence of modern logic and analytic

philosophy. Freges logical works were revolutionary, and are oftentaken to represent the fundamental break between contemporary

approaches and the older, Aristotelian tradition. He invented modern

quantificational logic, and created the first fully axiomatic system for

logic, which was complete in its treatment of propositional and first-

order logic, and also represented the first treatment ofhigher-order

logic. In thephilosophy ofmathematics,he was one ofthe most ardent

proponents of logicism, the thesis that mathematical truths are logical

truths, and presented influential cr iticisms of riv al views such as

psychologism and formalism. His theory of meaning, especially his

distinction between the sense and reference of linguistic expressions,

was groundbreaking in semantics and the philosophy o f language . He

had a profound and direct influence on such thinkers as Russell, Carnap and Wittgenstein. Frege is often

called the founder of modern logic, and he is sometimes even heralded as the founder of analytic

philosophy.

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Table of Contents

1. Life and Works

2. Contributions to Logic

3. Contributions to the Philosophy of Mathematics

4. The Theory of Sense and Reference

5. References and Further Reading

a. Freges Own Works

b. Important Secondary Works

1. Life and Works

Frege was born on November 8, 1848 in the coastal city of Wismar in Northern Germany. His full

christened name was Friedrich Ludwig Gottlob Frege. Little is known about his youth. His father, Karl

Alex ander Frege, and his mother, Auguste (Bialloblotzsky ) Frege, both worked at a girls private school

founded in part by Karl. Both were also principals of the school at various points: Karl held the positionuntil his death 1866, when Auguste took over until her death in 1878. The German writer Arnold Frege,

born in Wismar in 1852, may have been Freges younger brother, but this has not been confirmed. Frege

probably lived in Wismar until 1869; in the years from 1864-1869 he is known to have studied at the

Gymnasium in Wismar.

In Spring 1869, Frege began studies at the University of Jena. There, he studied chemistry, philosophy

and mathematics, and must have solidly impressed Ernst Abbe in mathematics, who later become of

Freges benefactors. After four semesters, Frege transferred to the University of Gttingen, where he

studied mathematics and physics, as well as philosophy of religion under Hermann Lotze. (Lotze is

sometimes thought to have had a profound impact on Freges philosophical views.) In late 1873, Frege

finished his doctoral dissertation, under the guidance of Ernst Schering, entitled ber eine geometrische

Darstellung der imaginren Gebilde in der Ebene (On a Geometrical Representation of Imaginary

Figures in a Plane), and received his Ph.D.

In 1874, with the rec ommendation 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 f ive years, and

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he was supported by his mother. FregesHabilitationsschrift, entitledRechnungsmethoden, die auf eine

Erweiterung des Grssenbegriffes grnden (Methods of Calculation Based upon An Amplification of the

Concept of Magnitude,), was included with the material submitted to obtain the position. It involves the

theory of complex mathematical functions, and contains seeds of Freges advances in logic and the

philosophy of mathematics.

Frege had a heavy teaching load during his first few years at Jena. However, he still had time to work on

his first major work in logic, which was published in 1879 under the title Begriffsschrift, eine der

arithmetischen nachgebildete Formelsprache des reinen Denkens (Concept-Script: A Formula Language

for Pure T hought Modeled on T hat of Arithmetic). 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. However,

the book was not well-reviewed by Freges contemporaries, who apparently found its two-dimensional

logical notation difficult to comprehend, and failed to see its advantages over previous approaches, such

as that of Boole.

Sometime after the publication of theBegriffsschrift, Frege was married to Margaret Lieseburg (1856-1905). They had at least two children, who unfortunately died young. Years later they adopted a son,

Alfred. However, little else is known about Freges family life.

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 theBegriffsschrift, Frege decided

to write a work in which he would describe his logicist views informally in ordinary language, and argue

against riv al views. The result was hisDie Grundlagen der Arithmetik (The Foundations of Arithmetic),

published in 1884. However, this work seems to have been virtually ignored by most of Fregescontemporaries.

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 1880s and early

1890s 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. These views were published in influential articles such as Funktion und Begriff

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(Function and Concept, 1891), ber Sinn und Bedeutung (On Sense and Reference, 1892) and ber

Begriff und Gegenstand (On Concept and Object, 1892). This maturation of Freges semantic and

philosophical views lead to changes in his logical language, forcing him to abandon an almost completed

draft of his work in logic and the foundations of mathematics. However, in 1893, Frege finally finished a

revised volume, employing a slightly revised logical system. This was his magnum opus, Grundgesetze

der Arithmetik (Basic Laws of Arithmetic), volume I. 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 v olume work; in the second and third, he would move on to the definitionof real numbers, and the demonstration of their properties.

Again, however, Freges work was unfavorably reviewed by his contemporaries. Nevertheless, he was

promoted once again in 1894, 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 invo lved.

Because of the unfavorable reception of his earlier works, Frege was forced to arrange to have vo lume II

of the Grundgesetze published at his own expense. It was not until 1902 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 c alculus for c lasses. For more information,

see the article on Russells Paradox. Frege was, in his own words, thunderstruck. He was forced to

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

A series of articles entitled ber die Grundlagen der Geometrie, (On the Foundations o f Geometry)

was published between 1903 and 1906, representing Freges side of a debate with Dav id Hilbert ov er the

nature of geometry and the proper construction and understanding of axiomatic systems within

mathematics.

However, around 1906, probably due to some combination of poor health, the early loss of his wife in

1905, frustration with his failure to find an adequate solution to Russells paradox, and disappointment

over the c ontinued poor reception of his work, Frege seems to have lost his intellectual steam. He

produced very little work between 1906 and his retirement in 1918. However, he continued to influence
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others during this per iod. Russell had included an appendix on Frege in his 1903 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. Wittgenstein studied the

work of Frege and Russell c losely, and in 191 1, he wrote to both of them concerning his own solution to

Russells paradox. Frege invited him to Jena to discuss his views. Wittgenstein did so in late 1911. The two

engaged in a philosophical debate, and while Wittgenstein reported that Frege wiped the floor with him,

Frege was sufficiently impressed with Wittgenstein that he suggested that he go to Cambridge to study

with Russella suggestion that had profound importance for the history of philosophy. Moreover ,RudolfCarnap was one of Freges students from 1910 to 1913, and doubtlessly Frege had significant influence on

Carnaps interest in logic and semantics and his subsequent intellectual development and successes.

After his retirement in 191 8, Frege moved to Bad Kleinen, near Wismar, and managed to publish a

number of important articles, Der Gedanke (The Thought, 1918), Der Verneinung (Negation,

1918), and Gedankengefge (Compound Thoughts, 1923). However, these were not wholly new

works, but later drafts of works he had initiated in the 1890s. In 1924 , 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 itimpossible. 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, 1925 at the age of 76.

At the time of his death, Freges own works were still not very widely known. He did not live to see the

profound impact he would have on the emergence o f analytic philosophy, nor to see his brand of logic

due to the championship of Russellvirtually wholly supersede earlier forms of logic. However, in

bequeathing his unpublished work to his adopted son, Alfred, he wrote prophetically , I believ e there are

things here which will one day be prized much more highly than they are now. Take care that nothing gets

lost. Alfred later gave Freges papers to Heinrich Scholz of the University of Mnster for safekeeping.

Unfortunately, however, they were destroyed in an Allied bombing raid on March 25, 1945. Although

Scholz had made copies of some of the more important pieces, a good portion of Freges unpublished

works were lost.

Although he was a fierc e, sometimes ev en satiric al, 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
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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 cer tain political r ights. This

distasteful feature of Freges personality has grav ely disappointed some of Freges intellectual progeny.

2. Contributions to Logic

Trained as a mathematician, Freges interests in logic grew out of his interests in the foundations ofarithmetic. Early in his career, Frege became convinced that the truths of arithmetic are logical, analytic

truths, agreeing with Leibniz, and disagreeing with Kant, who thought that arithmetical knowledge was

grounded in pure intuition, as well as more empiricist thinkers such as J. S. 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. Freges logicism was limited to arithmetic; unlike other

important historical logicists, such as Russell, Frege did not think that geometry was a branch of logic.

However, Freges logicism was very ambitious in another regard, as he believed that one couldprove all

of the truths of arithmetic deductively from a limited number of logical axioms. 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. Thus, Frege sought to

create a language that would combine the tasks of what Leibniz called a calculus ratiocinator and

lingua characterica, that is, a logically perspicuous language in which logical relations and possible

inferences would be clear and unambiguous. Freges own term for such a language, Begriffsschrift was

likely borrowed from a paper on Leibnizs ideas written by Adolf T rendelenburg. Although there had been

attempts to fashion at least the c ore o f such a language made by Boole and others working in the

Leibnizian tradition, Frege found their work unsuitable for a number of reasons. Booles logic used some

of the same signs used in mathematics, except with different logical meanings. Frege found this

unacceptable for a language which was to be used to demonstrate mathematical truths, because the signs

would be ambiguous. Booles logic, though innovativ e in some respects, was weak in others. It was

divided into a primary logic and secondary logic, bifurcating its propositional and categorical

elements, and could not deal adequately with multiple generalities. 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. Freges brand of
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logical language was modeled upon the international language of arithmetic, and it replaced the

subject/predicate sty le of logical analysis with the notions of function and argument. In mathematics, an

equation such as f(x) =x2 + 1 states thatfis a function that takesxas argument and yields as value the

result of multiplying x by itself and adding one. I n 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. See below for more on

Freges understanding of concepts, functions and objects. The concept being human is understood as afunction that has the True as value for any argument that is human, and the False as value for anything

else. Suppose that H( ) stands for this concept, and a is a c onstant for Aristotle, and b is a constant

for the city of Boston. Then H(a) stands for the True, while H(b) stands for the False. In Freges

terminology, an object for which a concept has the True as v alue is said to fall under the concept.

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. However, let us instead replace Freges own notation with more

contemporary notation. For Frege, the conditional function, is understood as a function the value ofwhich is the False if its first argument is the True and the second argument is anything other than the

True, and is the True otherwise. Therefore, H(b) H(a) stands for the True, while H(a) H(b)

stands for the False. The negation sign ~ stands for a function whose value is the True for every

argument except the True, for which its value is the False. Conjunction and disjunction signs could then

be def ined from the negation and conditional signs. Frege also introduced an identity sign, standing for a

function whose value is the True if the two arguments are the same object, and the False otherwise, and a

sign, which he called the horizontal, namely , that stands for a function that has the True as value

for the True as argument, and has the False as value for any other argument.

Variables and quantifiers are used to express generalities. Frege understands quantifiers as second-level

concepts. The distinction between levels of functions involves what kind of arguments the functions

take. In Freges view, unlike objects, all functions are unsaturated insofar as they require arguments to

yield values. But different sorts of functions require different sorts of arguments. Functions that take

objects as argument, such as those referred to by ( ) + ( ) or H( ), are called first-level functions.

Functions that take first-level functions as argument are called second-level functions. The quantifier,

x(x), is understood as standing for a function that takes a first-level function as argument, and
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yields the True as value if the argument-function has the True as value for all values ofx, and has the

False as value otherwise. Thus, xH(x) stands for the False, since the conceptH( ) does not have the

True as value for all arguments. However, x[H(x)H(x)] stands for T rue, since the complex concept

H( ) H( ) does have the T rue as v alue for all arguments. The existential quantifier, now written

x(x) is defined as ~x~(x).

Those familiar with modern predicate logic will recognize the parallels between it and Freges logic. Frege

is often credited with having founded predicate logic. However, Freges logic is in some ways differentfrom modern predicate logic. As we have seen, a sign such as H( ) is a sign for afunction in the strictest

sense, as are the conditional and negation connectives. Freges conditional is not, like the modern

connective, something that flanks statements to form a statement. Rather, it flanks terms for truth-values

to form a term for a truth-value. Freges H(b)H(a) is simply a name for the True, by itself it does not

assert anything. Therefore, Frege introduces a sign he called the judgment stroke, , used to assert thatwhat follows it stands for the True. Thus, while H(b)H(a) is simply a term for a truth-value, H(b)H(a) asserts that this truth-value is the True, or in this case, that if Boston is human, then Aristotle is

human. Moreover, Freges logical system was second-order. In addition to quantifiers ranging over

objects, it also contained quantifiers ranging over first-level functions. Thus, xF[F(x)] asserts thatevery object falls under at least one concept.

Freges logic took the form of an axiomatic system. 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. Freges first logical system, that of the 1879Begriffsschrift, had nine axioms (one of which

was not independent), one explicit inference rule, and also employed a second and third inference rule

implicitly. It represented the first axiomatization of logic, and was complete in its treatment of both

propositional logic and first-order quantified logic. Unlike Freges later sy stem, the system of the

Begriffsschrift was fully consistent. (It has since been prov en 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 1893 logical system of the Grundgesetze, Frege used fewer

axioms and more inference rules: seven and twelve, respectively, this time leaving nothing implicit. The

Grundgesetze also expanded upon the system of theBegriffsschrift by adding axioms governing what


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Frege called the value-ranges (Werthverlafe) o f functions, understood as objects corresponding to the

complete argument-value mappings generated by functions. In the c ase of concepts, their value-ranges

were identified with their extensions.While Frege did sometimes also refer to the ex tensions of c oncepts

as classes, he did not conceive of such classes as aggregates or collections. They were simply

understood as objects c orresponding to the complete argument-value mappings generated by c oncepts

considered as functions. Frege then introduc ed 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 functionFbeing

identical to the v alue-range of function G is the same as the truth-value ofFand G having the same valuefor 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 objectx, that object is in the extension of conceptFif and only if the value ofF

forxas argument is the T rue. 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. Therefore, the logical system of the Grundgesetze

was inconsistent due to Russells Paradox. See the entry on Russells Paradox for more details. 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 theBegriffsschrift, is complete in its treatment of propositional logicand 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. Frege

was the first to understand a statement such as all students are hardworking as say ing roughly the same

as, for all values ofx, ifxis a student, thenxis hardworking. This made it possible to c apture the logical

connection between statements such as either all students are hardworking or all students areintelligent and all students are either hardworking or intelligent (for example, that the first implies the

second). In earlier logical systems such as that of Boole, in which the propositional and quantificational

elements were bifurcated, the connection was wholly lost. Moreover, Freges logical system was the first

to be able to capture statements of multiple generality, such as every person loves some city by using

multiple quantifiers in the same logical formula. This too was impossible in all earlier logical systems.

Indeed, Freges firsts in logic are almost too numerous to list. We have seen here that he invented

modern quantification theory, presented the first complete axiomatization of propositional and first-
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order predicate logic (the latter of which he invented outright), attempted the first formulation of

higher-order logic, presented the f irst coherent and full analysis of variables and functions, first showed it

possible to reduce all truth-functions to negation and the conditional, and made the first clear distinction

between axioms and inference rules in a formal system. 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.

On Freges philosophy o f logic, logic is made true by a realm of logical entities. 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. T hus, Frege denied the popular v iew that logic is without c ontent 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. If humans were genetically designed to use

regularly the so-called inference rule of affirming the consequent, etc., this would not make it logically

valid. What is true or false, valid of invalid, does not depend on anyones psychology or anyones beliefs.To think otherwise is to confuse somethings being true with somethings being-taken-to-be-true.

3. 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. He also presented significant criticisms against rival views. We have seen that Frege was a harsh

critic of psychologism in logic. He thought similarly about psychologism in mathematics. Numbers

cannot be equated with anyones mental images, nor truths of mathematics with psychological truths.

Mathematical truths are objectiv e, not subjective. Frege was also a c ritic of Mills view that arithmetical

truths are empirical truths, based on observation. Frege pointed out that it is not just observable things

that can be counted, and that mathematical truths seem to apply also to these things. On Mills view,

numbers must be taken to be c onglomerations of objects. Frege rejects this view for a number of reasons.

Firstly, is one conglomeration of two things the same as a different conglomeration of two things, and if


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not, in what sense are they equal? Secondly, a conglomeration can be seen as made up of a different

number of things, depending on how the parts are counted. One deck of cards containsfifty two cards,

but each card consists of a multitude of atoms. There is no one uniquely determined number of the

whole conglomeration. He also reiterated the arguments of others: that mathematical truths seem

apodictic and knowable a priori. He also argued against the Kantian view that arithmetic truths are based

on the pure intuition of the succession of time. His main argument against this view, however, was simply

his own work in which he showed that truths about the nature of succession and sequence can be proven

purely from the axioms of logic.

Frege was also an opponent of formalism, the view that arithmetic can be understood as the study of

uninterpreted formal systems. While Freges logical language represented a kind of formal system, he

insisted that his formal system was important only because of what its signs represent and its propositions

mean. The signs themselves, independently of what they mean, are unimportant. To suggest that

mathematics is the study simply of the formal system, is, in Freges eyes, to confuse the sign and thing

signified. To suggest that arithmetic is the study of formal systems also suggests, absurdly, that the

formula 5 + 7 = 12, written in Arabic numerals, is not the same truth as the formula, V + VII = XII,

written in Roman numerals. Frege suggests also that this confusion would have the absurd result thatnumbers simply are the numerals, the signs on the page, and that we should be able to study their

properties with a microscope.

Frege suggests that rival views are often the result of attempting to understand the meaning of number

terms in the wrong way, for example, in attempting to understand their meaning independently of the

contexts in which they appear in sentences. If we are simply asked to consider what two means

independently of the context of a sentence, we are likely to simply imagine the numeral 2, or perhaps

some conglomeration of two things. Thus, in the Grundlagen, Frege espouses his famous context

principle, to never ask for the meaning of a word in isolation, but only in the context of a proposition.The Grundlagen is an earlier work, written before Frege had made the distinction between sense and

reference (see below). It is an active matter o f debate and discussion to what extent and how this

principle coheres with Freges later theory of meaning, but what is clear is that it plays an important role

in his own philosophy of mathematics as described in the Grundlagen.

According to Frege, if we look at the contexts in which number words usually occur in a proposition, they

appear as part of a sentence about a concept, specifically, as part of an expression that tells us how many


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times a certain concept is instantiated. Consider, for example, I have six cards in my hand or T here are

11 members of congress from Wisconsin. T hese propositions seem to tell us how many times the

concepts ofbeing a card in my handand being a member of congress from Wisconsin are instantiated.

Thus, Frege concludes that statements about numbers are statements about concepts. This insight was

very important for Freges case for logicism, as Frege was able to show that it is possible to define what it

means for a concept to be instantiated a certain number of times purely logically by making use of

quantifiers and identity. To say that the conceptF is instantiated zero times is to say that there are no

objects that instantiate F, or, equivalently, that everything does not instantiate F. To say that F is

instantiated one time is to say there is an object xthat instantiatesF, and that for allobjects y, either y

does not instantiateFor y isx. To say thatFis instantiated twice is to say that there are two objects,xand

y, each of which instantiatesF, but which are not the same as each other, and for allz, eitherzdoes not

instantiateF, orzisxorzis y. One could then consider numbers as second-level concepts, or concepts

of concepts, which can be defined in purely logical terms. (For more on the distinction of levels of

concepts, see above.)

Frege, however, does not leave his analysis of numbers there. Understanding number-claims as involving

second-level concepts does give us some insight into the nature of numbers, but it cannot be left at this.Mathematics requires that numbers be treated as objects, and that we be able to provide a definition of

the number two simpliciter, without having to speak of twoFs. For this purpose, Frege appeals to his

theory of the value-ranges of concepts. On the notion of a value-range, seeabove. We saw above that we

can gain some understanding of number claims as involving second-level concepts, or concepts of

concepts. In order to find a definition of numbers as objects, Frege treats them instead as value-ranges of

value-ranges. Exactly , however, are they to be understood?

Frege notes that we have an understanding of what it means to say that there are the same number ofFs as

there are Gs. It is to say that there is a one-one mapping between the objects that instantiate Fand theobjects instantiating G, i.e. that there is some functionffrom entities that instantiateFonto entities that

instantiate G such that there is a differentFfor everyG, and a different G for everyF, with none left over.

(In this, Freges v iews on the nature of cardinality were in part anticipated by Georg Cantor.) However,

we must bear in mind that the propositions:

(1) There are equally manyFs as there are Gs.

(2) The number ofFs = the number ofGs


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must obviously have the same truth-value, as they seem to express the same fact. We must, therefore,

look for a way of understanding the phrase the number ofFs that occurs in (2) that makes clear how and

why the whole proposition will be true or false for the same reason as (1) is true or false. Freges

suggestion is that the number ofFs means the same as the value-range of the concept being a value-

range of a concept instantiated equally many times as F. This means that the number ofFs is a certain

value-range, containing value-ranges, and in particular , all those value-ranges that have as many

members as there areFs. Then (2) is understood as saying the same as the value-range of the concept

being a value-range of a concept instantiated equally many times as F = the value-range of the conceptbeing a value-range of a concept instantiated equally many times as G, which will be true if and only if

there are equally manyFs as Gs, i.e. if every value-range of a concept instantiated equally many times as

Fis also a value-range of a concept instantiated equally many times as G.

To give some examples, if there are zeroFs, then the number ofFs, i.e. zero, is the value-range consisting

of all value-ranges with no members. Recall that for Frege, classes are identified with value-ranges of

concepts. (See above.) To rephrase the same point in terms of classes, zero is the class of all classes with

no members. Since there is only one such class, zero is the class containing only the empty c lass. If thereis one F, then the number ofFs, i.e. one, is the class consisting of all classes with one member (the

extensions of concepts instantiated once). Here we can see the connection with the understanding of

number expressions as being statements about concepts. Rather than understanding zero as the concept a

concept has just in case it is not instantiated, zero is understood as the value-range consisting of value-

ranges of concepts that are not instantiated. Rather than understanding one as the concept a concept has

just in c ase it is instantiated by a unique object, it is understood as the value-range consisting of value-

ranges of concepts instantiated by unique objects. This allows us to understand numbers as abstract

objects, and provide a clear definition of the meaning of number signs in arithmetic such as 1, 2, 3,

etc.

Some of Freges most brilliant work came in providing definitions of the natural numbers in his logical

language, and in proving some of their properties therein. After laying out the basic laws of logic, and

defining axioms governing the truth-functions and value-ranges, etc., Frege begins by defining a relation

that holds between two value-ranges just in case they are the value-ranges of concepts instantiated

equally many times. This relation holds between value-ranges just in case they are the same size, i.e. just

in case there is one-one correspondence between the entities that fall under their concepts. Using this, he

th d fi f ti th t t k l t d i ld l th l


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then defines a function that takes a value-range as argument and y ields as value the value-range

consisting of all value-ranges the same size as it. The number zero is then defined as the value-range

consisting of all value-ranges the same size as the value-range of the concept being non-self-identical.

Since this concept is not instantiated, zero is def ined as the v alue-range of all value-ranges with no

members, as described above. There is only one such number zero. Since this is true, then the concept of

being identical to zero is instantiated once. Frege then uses this to define one. One is defined as the value-

range of all value-ranges equal in size to the value-range of the concept being identical to zero. Having

defined one is this way, Frege is able to define two. He has already defined one and zero; they are each

unique, but different from each other. Therefore, two can be defined as the value-range of all value-

ranges equal in size to the value-range of the concept being identical to zero or identical to one. Frege is

able to define allnatural numbers in this way, and indeed, prove that there are infinitely many of them.

Each natural number can be defined in terms of the previous one: for each natural number n, its

successor (n + 1) can be defined as the value-range of all value-ranges equal in size to the value-range of

the concept ofbeing identical to one of the numbers between zero and n.

In the Begriffsschrift, Frege had already been able to prove certain results regarding series and

sequences, and was able to define the ancestralof a relation. To understand the ancestral of a relation,consider the example of the relation ofbeing the child of. A personxbears this relation to y just in casex

is ys child. However,xfalls in the ancestralof this relation with respect to y just in casexis the child ofy,

or is the child ofys child, or is the child ofys childs child, etc. Frege was able to define the ancestral of

relations logically even in his early work. He put this to use in the Grundgesetze to define the natural

numbers. We have seen how the notion of successorship can be defined for Frege, i.e. the relation n + 1

bears to n. The natural numbers can be defined as the value-range of all value-ranges that fall under the

ancestral of the successor relation with respect to zero. The natural numbers then consist of zero, the

successor of zero (one), the successor of the successor of zero (two), and so onad infinitum. Frege was

then able to use this definition of the natural numbers to provide a logical analysis of mathematical

induction, and prove that mathematical induction can be used validly to demonstrate the properties of

the natural numbers, an extremely important result for making good on his logicist ambitions. Frege

could then use mathematical induction to prove some of the basic laws of the natural numbers. Frege next

turned his logicist method to an analysis of integers (including negative numbers) and then to the real

numbers, defining them using the natural numbers and certain relations holding between them. We need

not dwell on the details of this work here.

F h t idi l i l l i f di lit th t l b i fi it d


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Freges approach to providing a logical analysis of cardinality, the natural numbers, infinity and

mathematical induction were groundbreaking, and have had a lasting importance within mathematical

logic. Indeed, prior to 1902, it must have seemed to him that he had been completely successful in

showing that the basic laws of arithmetic could be understood purely as logical truths. However, as we

have seen, Freges definition of numbers heavily involves the notion of classes or value-ranges, but his

logical treatment of them is shown to be impossible due to Russells paradox. This presents a serious

problem for Freges logicist approach. Another heavy blow came after Freges death. In 1931 , Kurt Gdel

discovered his famous incompleteness proof to the effect that there can be no consistentformal system

with a finite number of axioms in which it is possible to derive all of the truths of arithmetic. This

presents a serious blow to more ambitious forms of logicism, such as Freges, which aimed to provide

precisely the sort of system Gdel showed impossible. Nevertheless, it cannot be denied that Freges work

in the philosophy of mathematics was important and insightful.

4. The Theory of Sense and Reference

Freges influential theory of meaning, the theory of sense (Sinn) and reference (Bedeutung) was firstoutlined, albeit briefly, in his article, Funktion und Begriff of 1891, and was expanded and explained in

greater detail in perhaps his most famous work, ber Sinn und Bedeutung of 1892. In Funktion und

Begriff, the distinction between the sense and reference of signs in language is first made in regard to

mathematical equations. During Freges time, there was a widespread dispute among mathematicians as to

how the sign, =, should be understood. If we consider an equation such as, 4 x 2 = 11 3, a number of

Freges contemporaries, for a variety of reasons, were wary of viewing this as an expression of an

identity, or, in this case, as the claim that 4 x 2 and 11 3 are one and the same thing. Instead, they

posited some weaker form of equality such that the numbers 4 x 2 and 11 3 would be said to be equal

in number or equal in magnitude without thereby constituting one and the same thing. In opposition to

the view that = signifies identity, such thinkers would point out that 4 x 2 and 11 3 cannot in all ways

be thought to be the same. The former is a product, the latter a difference, etc.

In his mature period, however , Frege was an ardent opponent of this view, and argued in fav or of

understanding = as identity proper, accusing rival views of confusing form and content. He argues

instead that expressions such as 4 x 2 and 11 3 can be understood as standing for one and the same

thing the number eight but that this single entity is determined or presented differently by the two


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thing, the number eight, but that this single entity is determined or presented differently by the two

expressions. Thus, he makes a distinction between the actual number a mathematical expression such as

4 x 2 stands for, and the way in which that number is determined or picked out. The former he called

the reference (Bedeutung) of the expression, and the latter was called the sense (Sinn) of the expression.

In Fregean terminology, an expression is said to express its sense, and denote or refer to its reference.

The distinction between reference and sense was expanded, primarily in ber Sinn und Bedeutung as

holding not only for mathematical expressions, but for all linguistic expressions (whether the language in

question is natural language or a formal language). One of his primary examples therein involves theexpressions the morning star and the evening star. Both of these expressions refer to the planet

Venus, yet they obviously denote Venus in virtue of dif ferent properties that it has. T hus, Frege claims

that these two expressions have the same reference but different senses. The reference of an ex pression

is the actual thing corresponding to it, in the case of the morning star, the reference is the planet Venus

itself. T he sense of an expression, however, is the mode of presentation or cognitive content associated

with the expression in virtue of which the reference is picked out.

Frege puts the distinction to work in solving a puzzle concerning identity claims. If we consider the twoclaims:

(1) the morning star = the morning star

(2) the morning star = the evening star

The first appears to be a triv ial case of the law of self-identity, knowable a priori, while the second seems

to be something that was discov ered a posterioriby astronomers. However, if the morning star means

the same thing as the evening star, then the two statements themselves would also seem to have thesame meaning, both involving a things relation of identity to itself. However, it then becomes to difficult

to ex plain why (2) seems informativ e while (1) does not. Freges response to this puzzle, given the

distinction between sense and reference, should be apparent. Because the reference of the evening star

and the morning star is the same, both statements are true in virtue of the same objects relation of

identity to itself. However, because the senses of these expressions are differentin (1) the object is

presented the same way twice, and in (2) it is presented in two different waysit is informative to learn of

(2). While the truth of an identity statement involves only the references of the component expressions,

the informativity of such statements involves additionally the way in which those references are


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the informativity of such statements involves additionally the way in which those references are

determined, i.e. the senses of the component expressions.

So far we have only considered the distinction as it applies to expressions that name some object

(including abstract objects, such as numbers). For Frege, the distinction applies also to other sorts of

expressions and even whole sentences or propositions. If the sense/reference distinction can be applied

to whole propositions, it stands to reason that the reference of the whole proposition depends on the

references of the parts and the sense of the proposition depends of the senses of the parts. (At some

points, Frege even suggests that the sense of a whole proposition is composed of the senses of thecomponent expressions.) In the example considered in the previous paragraph, it was seen that the truth-

value of the identity claim depends on the references of the component ex pressions, while the

informativity of what was understood by the identity claim depends on the senses. For this and other

reasons, Frege concluded that the reference of an entire proposition is its truth-value, either the T rue or

the False. The sense of a complete proposition is what it is we understand when we understand a

proposition, which Frege calls a thought (Gedanke). Just as the sense of a name of an object determines

how that object is presented, the sense of a proposition determines a method of determination for a truth-

value. The propositions, 2 + 4 = 6 and the Earth rotates, both have the True as their references,though this is in virtue of very different conditions holding in the two cases, just as the morning star and

the evening star refer to Venus in virtue of different properties.

In ber Sinn und Bedeutung, Frege limits his discussion of the sense/reference distinction to complete

expressions such as names purporting to pick out some object and whole propositions. However, in

other works, Frege makes it quite c lear that the distinction can also be applied to incomplete

expressions, which include functional expressions and grammatical predicates. T hese ex pressions are

incomplete in the sense that they contain an empty space, which, when filled, yields either a complex

name referring to an object, or a complete proposition. Thus, the incomplete expression the square rootof ( ) contains a blank spot, which, when completed by an expression referring to a number, yields a

complex expression also referring to a number, e.g., the square root of sixteen. The incomplete

expression, ( ) is a planet contains an empty place, which, when filled with a name, yields a complete

proposition. According to Frege, the references of these incomplete expressions are not objects but

functions. Objects (Gegenstnde), in Freges terminology, are self-standing, complete entities, while

functions are essentially incomplete, or as Frege says, unsaturated (ungesttigt) in that they must take

something else as argument in order to y ield a value. The reference of the expression square root of ( )

is thus a function which takes numbers as arguments and yields numbers as values The situation may


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is thus a function, which takes numbers as arguments and yields numbers as values. The situation may

appear somewhat different in the case of grammatical predicates. However, because Frege holds that

complete propositions, like names, have objects as their references, and in particular, the truth-values

the True or the False, he is able to treat predicates also as having functions as their references. In

particular, they are functions mapping objects onto truth-values. The ex pression, ( ) is a planet has as

its reference a function that yields as value the T rue when saturated by an object such as Saturn or Venus,

but the False when saturated by a person or the number three. Frege calls such a function of one

argument place that yields the T rue or False for every possible argument a concept (Begriff), and calls

similar functions of more than one argument place (such as that denoted by ( ) > ( ), which is doubly in

need of saturation), relations.

It is clear that functions are to be understood as the references of incomplete expressions, but what of the

senses of such expressions? Here, Frege tells us relatively little save that they exist. There is some

amount of controversy among interpreters of Frege as to how they should be understood. It suffices here

to note that just as the same object (e.g. the planet Venus), can be presented in different ways, so also can

a function be presented in different ways. While identity, as Frege uses the term, is a relation holding

only between objects, Frege believes that there is a relation similar to identity that holds betweenfunctions just in case they always share the same value for every argument. Since all and only those

things that have hearts have kidneys, strictly speaking, the concepts denoted by the expressions ( ) has a

heart, and ( ) has a kidney are one and the same. Clearly, however, these expressions do not present

that concept in the same way. For Frege, these expressions would have different senses but the same

reference. Frege also tells us that it is the incomplete nature of these senses that provides the glue

holding together the thoughts of which they form a part.

Frege also uses the distinction to solve what appears to be a difficulty with Leibnizs law with regard to

identity. This law was stated by Leibniz as, those things are the same of which one can be substituted foranother without loss of truth, a sentiment with which Frege was in full agreement. As Frege understands

this, it means that if two expressions have the same reference, they should be able to replace each other

within any proposition without changing the truth-value of that proposition. Normally , this poses no

problem. The inference from:

(3) The morning star is a planet.


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senses of ex pressions that appear in oratio obliqua are in fact the references of those expressions when


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p pp q p

they appear in that context. Such contexts can be referred to as oblique contexts, contexts in which the

reference of an expression is shifted from its customary reference to its customary sense.

In this way, Frege is able to actually retain his commitment in Leibnizs law. The expressions the morning

star and the evening star have the same primary reference, and in any non-oblique context, they can

replace each other without changing the truth-value of the proposition. However, since the senses of

these expressions are not the same, they cannot replace eac h other in oblique contexts, because in such

contexts, their references are non-identical.

Frege ascribes to senses and thoughts objective existence. In his mind, they are objects ev ery bit as real

as tables and chairs. Their existence is not dependent on language or the mind. Instead, they are said to

exist in a timeless third realm of sense, existing apart from both the mental and the physical. Frege

concludes this because, although senses are obviously not physical entities, their existence likewise does

not depend on any one persons psychology. A thought, for example, has a truth-value regardless of

whether or not anyone believ es it and even whether or not any one has grasped it at all. Moreov er, senses

are interpersonal. Different people are able to grasp the same senses and same thoughts and communicate

them, and it is even possible for expressions in different languages to express the same sense or thought.

Frege concludes that they are abstract objects, incapable of full causal interaction with the physical

world. They are actual only in the very limited sense that they can have an effect on those who grasp

them, but are themselves incapable of being changed or acted upon. They are neither created by our uses

of language or acts of thinking, nor destroyed by their cessation.

Unfortunately, Frege does not tell us very much about exactly how these abstract objects pick out or

present their references. Exactly what is it that makes a sense a way of determining or mode of

presenting a reference? In the wake of Russells theory of descriptions, a Fregean sense is ofteninterpreted as a set of descriptive information or criteria that picks out its reference in virtue of the

reference alone satisfying or f itting that descriptive information. In giving examples, Frege implies that a

person might attach to the name Aristotle the sense the pupil of Plato and teacher of Alexander the

Great. This sense picks out Aristotle the person because he alone matches this description. Here, care

must be taken to avoid misunderstanding. The sense of the name Aristotle is not the words the pupil of

Plato and teacher of Alexander the Great; to repeat, senses are not linguistic items. It is rather that the

sense consists in some set of descriptive information, and this information is best described by a

descriptive phrase of this form. The property of being the pupil of Plato and teacher of Alexander is


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unique to Aristotle, and thus, it may be in virtue of associating this information with the name Aristotle

that this name may be used to refer to Aristotle. As certain commentators have noted, it is not even

necessary that the sense of the name be expressible by some descriptivephrase, because the descriptive

information or properties in virtue of which the reference is determined may not be directly nameable in

any natural language.

From this standpoint, it is easy to understand how there might be senses that do not pick out any

reference. Names such as Romulus or Odysseus, and phrases such as the least rapidly convergingseries or the present King of France express senses, insofar as they lay out c riteria that things would

have to satisfy if they were to be the references of these expressions. However, there are no things which

do in fact satisfy these criteria. Therefore, these expressions are meaningful, but do not have references.

Because the sense of a whole proposition is determined by the senses of the parts, and the reference of a

whole proposition is determined by the parts, Frege claims that propositions in which such expressions

appear are able to express thoughts, but are neither true nor false, because no references are determined

for them.

This interpretation of the nature of senses makes Frege a forerunner to what has since been come to be

known as the descriptivist theory of meaning and reference in the philosophy of language. The view

that the sense of a proper name such as Aristotle could be descriptive information as simple as the pupil

of Plato and teacher of Alexander the Great, however, has been harshly criticized by many philosophers,

and perhaps most notably by Saul Kripke. Kripke points out that this would make a claim such as

Aristotle taught Alexander seem to be a necessary and analytic truth, which it does not appear to be.

Moreover, he c laims that many of us seem to be able to use a name to refer to an individual even if we are

unaware ofany properties uniquely held by that individual. For example, many of us dont know enough

about the physicist Richard Feynman to be able to identify a property differentiating him from otherprominent physicists such as Murray Gell-Mann, but we still seem to be able to refer to Feynman with the

name Feynman. John Searle, Michael Dummett and others, however, have proposed ways of expanding

or altering Freges notion of a sense to circumvent Kripkes worries. This has lead to a very important

debate in the philosophy of language, which, unfortunately, we c annot fully discuss here.

5. References and Further Reading


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a. Freges Own Works

Antwort auf die Ferienplauderei des Herrn Thomae.Jahresbericht der Deutschen Mathematiker-

Vereinigung 15 (1906): 586-90. Translated as Reply to T homaes HolidayCauserie. In Collected

Papers on Mathematics, Logic and Philosophy [CP], 341-5. Translated by M. Black, V. Dudman, P.

Geach, H. Kaal, E.-H. W. Kluge, B. McGuinness and R. H. Stoothoff. New York: Basil Blackwell,

1984.

ber Begriff und Gegenstand. Vierteljahrsschrift fr wissenschaftliche Philosophie 16 (1892): 192-

205. Translated as On Concept and Object. In >CP182-94. Also in The Frege Reader [FR], 181-

93. Edited by Michael Beaney. Oxford: Blackwell, 1997. And In Translations from the

Philosophical Writings of Gottlob Frege [TPW], 42-55. 3d ed. Edited by Peter Geach and Max

Black. Oxford: Blackwell, 1980.

Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle: L.

Nebert, 1 879. Translated asBegriffsschrift, a Formula Language, Modeled upon that of

Arithmetic, for Pure Thought. InFrom Frege to Gdel, edited by Jean van Heijenoort. Cambridge,MA: Harvard University Press, 1967. A lso asConceptual Notation and Related Articles. Edited and

translated by T errell W. Bynum. London: Oxford University Press, 1972.

ber die Begriffsschrift des Herrn Peano und meine eigene. Verhandlungen der Kniglich

Schsische n Gesellschaft der Wissenschaften zu Leipzig 48 (1897): 362-8. Translated as On Mr.

Peanos Conceptual Notation and My Own. In CP234-48.

ber formale Theorien der Arithmetik.Sitzungsberichte der Jenaischen Gesellschaft fr Medizin

und Naturwissenschaft19 (1885): 94-104 . Translated as On Formal Theories of Arithmetic. In

CP112-21.Funktion und Begriff. Jena: Hermann Pohle, 1891. Translated as Function and Concept. In CP137-

56, TPW21-41 andFR 130-48.

Der Gedanke.Betrge zur Philosophie des deutschen Idealismus 1 (191 8-9): 58-77. T ranslated as

Thoughts. In CP351-7 2. Also as part I ofLogical Investigations [LI], edited by P. T . Geach.

Oxford: Blackwell, 1977. And as Thought. InFR 325-45.

Gedankengefge.Betrge zur Philosophie des deutschen Idealismus 3 (1923): 36-51. T ranslated as

Compound Thoughts. In CP390-406, and as part III ofLI.

ber eine geometrische Darstellung der imaginren Gebilde in der Ebene. Ph. D. Dissertation:


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g g g

University of Gttingen, 1873. Translated as On a Geometrical Representation of Imaginary Forms

in the Plane. In CP1-55.

Grundgesetze der Arithmetik. 2 vols. Jena: Hermann Pohle, 1893-1903. T ranslated in part as The Basic

Laws of Arithmetic: Exposition of the System. Edited and translated by Montgomery Furth.

Berkeley: University of California Press, 1964.

ber die Grundlagen der Geometrie.Jahresbericht der Deutschen Mathematiker-Vereinigung 12

(1903): 319-24, 368-75, 1 5 (1906): 293-309, 377 -403, 4 23-30. Translated as On the Foundationsof Geometry. In CP273-340. Also as On the Foundations of Geometry and Formal Theories of

Arithmetic. Translated by Eike-Henner W. Kluge. New York: Yale University Press, 1971 .

Die Grundlagen der Arithmetik, eine logisch mathematische Untersuchung ber den Begriff der Zahl.

Breslau: W. Koebner, 1884. Translated as The Foundations of Arithmetic: A Logico-Mathematical

Enquiry into the Concept of Number. 2d ed. T ranslated by J. L. Austin. Oxford: Blackwell, 1953.

Kritische Beleuchtung einiger Punkte in E. Schrders Vorlesungen ber die Algebra der Logik.

Archiv fr systematsche Philosophie 1 (1 895): 433-56. T ranslated as A Critical Elucidation of

Some Points in E. Schrder, Vorlesungen ber die Algebra der Logik. In CP210-28, and TPW86-

106.

Nachgelassene Schriften. Hamburg: Felix Meiner, 1969. Translated asPosthumous Writings.

Translated by Peter Long and Roger White. Chicago: University of Chicago Press, 1979.

Le nombre entier.Revue de Mtaphysique et de Morale 3 (1895): 73-8. Translated as Whole

Numbers. In CP229-33.

Rechnungsmethoden, die auf eine Erweiterung des Grssenbegriffes grnden. Habilitationsschrift:

University of Jena, 1874. Translated as Methods of Calculation based on an Extension of the

Concept of Quantity. In CP56-92.

Review ofZur Lehre vom Transfiniten, by Georg Cantor.Zeitschrift fr Philosophie und

philosophische Kritik100 (1892): 269-72. T ranslated in CP178-181.

Review ofPhilosophie der Arithmetik, by Edmund Husserl.Zeitschrift fr Philosophie und

philosophische Kritik103 (1 894): 313-32. Translated inCP195-209.

ber Sinn und Bedeutung.Zeitschrift fr Philosophie und philosophische Kritik 100 (1892): 25-50.

Translated as On Sense and Meaning. In CP157-77 . As OnSinn andBedeutung. In FR 151-71 .

And as On Sense and Reference. In TPW56-78.

ber das Trgheitsgesetz.Zeitschrift fr Philosophie und philosophische Kritik98 (1891): 145-61.


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Translated as On the Law of Inertia. In CP123-36.

Die Unmglichkeit der Thomaeschen formalen Arithmetik aus Neue nachgewiesen.Jahresbericht

der Deutschen Mathematiker-Vereinigung 17 (1908): 52-5. Translated as Renewed Proof of the

Impossibility of Mr. T homaes Formal Arithmetic. In CP346-50.

Der Verneinung.Betrge zur Philosophie des deutschen Idealismus 1 (191 8-9): 143-57 . Translated as

Negation. In CP373-89, part II ofLI, andFR 346-61.

Was ist ein Funktion? InFestschrift Ludwig Boltzmann gewidmet zum sechzigsten Geburtstage,

656-66. Leipzig: Amrosius Barth, 1904. Translated as What is a Function? In CP285-92, and

TPW285-92.

Wissenschaftlicher Briefwechsel. Hamburg: Felix Meiner, 1976. T ranslated asPhilosophical and

Mathematical Correspondence. Translated by Hans Kaal. Chicago: University of Chicago Press,

1980.

ber die Zahlen des Herrn H. Schubert. Jena: Hermann Pohle, 1899. Translated as On Mr. H.

Schuberts Numbers. In CP249-72.

b. Important Secondary Works

Angelelli, Ignacio.Studies on Gottlob Frege and Traditional Philosophy. Dordrecht: D. Reidel, 1967.

Baker, G. P. and P. M. S. Hacker.Frege: Logical Excavations. New York: Oxford University Press,

1984.

Beaney, Michael.Frege: Making Sense. London: Duckworth, 1996.

Beaney, Michael. Introduction to The Frege Reader, by Gottlob Frege. Oxford: Blackwell, 1997.

Bell, David.Freges Theory of Judgment. New York: Oxford University Press, 1979.Bynum, Terrell W. On the Life and Work of Gottlob Frege. I ntroduction to Conceptual Notation and

Related Articles, by Gottlob Frege. London: Oxford University Press, 1972.

Carl, Wolfgang.Freges Theory of Sense and Reference. Cambridge: Cambridge University Press,

1994.

Carnap, Rudolph.Meaning and Necessity. 2d ed. Chicago: University of Chicago Press, 1956.

Church, Alonzo. A Formulation of the Logic of Sense and Denotation. InStructure,Method and

Meaning: Essays in Honor of Henry M. Sheffer, edited by P. Henle, H. Kallen and S. Langer, 3- 24.

N Y k Lib l A P


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New York: Liberal Arts Press, 1951 .

Currie, Gregory.Frege: An Introduction to His Philosophy. T otowa, NJ: Barnes and Noble, 1982.

Dummett, Michael.Frege: Philosophy of Language. 2d ed. Cambridge, MA: Harvard University Press,

1981.

Dummett, Michael.Frege: Philosophy of Mathematics. Cambridge, MA: Harvard University Press,

1991.

Dummett, Michael.Frege and Other Philosophers. Oxford: Oxford University Press, 1991.

Dummett, Michael. The Interpretation of Freges Philosophy. Cambridge, MA: Harvard University

Press, 1981.

Geach, Peter T. Frege. In Three Philosophers, edited by G. E. M. Anscombe and P. T. Geach, 1 27-62.

Oxford: Oxford University Press, 1961.

Gdel, Kurt. On Formally Undecidable Propositions ofPrincipia Mathematica and Related Systems

I. InFrom Frege to Gdel, edited by Jan van Heijenoort, 5 96-616. Cambridge, MA: Harvard

University Press, 1967 . Originally published as ber formal unentscheidbare Stze derPrincipia

Mathematica und verwandter Systeme I.Monatshefte fr Mathematik und Physik 38 (1931): 173-

98.

Grossmann, Reinhardt.Reflections on Freges Philosophy. Evanston: Northwestern University Press,

1969.

Haaparanta, Leila and Jaakko Hintikka, eds.Frege Synthesized. Boston: D. Reidel, 1986.

Kaplan, David. Quantifying In.Synthese 19 (1968): 178-214.

Klemke, E. D., ed.Essays on Frege. Urbana: University of Illinois Press, 1968.

Kluge, Eike-Henner W. The Metaphysics of Gottlob Frege. Boston: Martinus Nijhoff, Boston, 1980.

Kneale, William and Martha Kneale. The Development of Logic. London: Oxford University Press,

1962.

Kripke, Saul.Naming and Necessity. Cambridge, MA: Harvard University Press, 1980. First published

inSemantics of Natural Languages. Edited by Donald Davidson and Gilbert Harman. Dordrecht: D.

Reidel, 1972.

Linsky, Leonard. Oblique Contexts. Chicago: University of Chicago Press, 1983.

Resnik, Michael D.Frege and the Philosophy of Mathematics. Ithaca: Cornell University Press, 1980.

Ricketts, T homas G., ed. The Cambridge Companion to Frege. Cambridge: Cambridge University

Press, forthcoming.

R ll B d Th L i l d A i h i l D i f F I Th P i i l f


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Russell, Bertrand. The Logical and Arithmetical Doctrines of Frege. InThe Principles of

Mathematics, Appendix A . 1903. 2d. ed. Reprint, New York: W. W. Norton & Company, 1 996.

Russell, Bertrand. On Denoting.Mind14 (1905): 479-93.

Salmon, Nathan.Freges Puzzle. Cambridge: MIT Press, 1986.

Schirn. Matthias, ed.Logik und Mathematik: Frege Kolloquium 1993. Hawthorne: de Gruyter, 1995.

Schirn. Matthias, ed.Studien zu Frege. 3 vols. Stuttgart-Bad Cannstatt: Verlag-Holzboog, 1976.

Searle, John R.Intentionality: An Essay in the Philosophy of Mind. Cambridge: Cambridge UniversityPress, 1983.

Sluga, Hans. Frege and the Rise of Analytic Philosophy.Inquiry 18 (1975): 471-87.

Sluga, Hans. Gottlob Frege. Boston: Routledge & Kegan Paul, 1980.

Sluga, Hans. The Philosophy of Frege. 4 vols. New York: Garland Publishing, 1993.

Sternfeld, Robert.Freges Logical Theory. Carbondale: Southern Illinois University Press, 1966.

Thiel, Christian.Sense and Reference in Freges Logic. Translated by T . J. Blakeley. Dordrecht: D.

Reidel, 1968.

Tich, Pavel. The Foundations of Freges Logic. New York: Walter de Gruyter, 1988.Walker , Jeremy D. B.A Study of Frege. London: Oxford University Press, 1965.

Weiner, Joan.Frege in Perspective. Ithaca: Cornell University Press, 1990.

Wright, Crispin.Freges Conception of Numbers as Objects. Aberdeen: Aberdeen University Press,

1983.

Wright, Crispin.Frege: Tradition and Influence. Oxford: Blackwell, 1984.

Author InformationKevin C. Klement

Email: [email protected]

University o f Massachusetts, Amherst

Last updated: July 8, 2005 | Originally published: April/16/2001

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