Gottlob Frege (1848-1925) was a
German logician, mathematician and philosopher who played a crucial
role in the emergence of modern logic and analytic philosophy.
Frege's logical works were revolutionary, and are often taken
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 of higher-order logic. In the philosophy of
mathematics, he was one of the most ardent proponents of
logicism, the thesis that mathematical truths are logical
truths, and presented influential criticisms of rival 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 of 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.
Table of Contents (Clicking on the links below will take you to those parts of this article)
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 Alexander Frege, and his mother, Auguste (Bialloblotzsky) Frege,
both worked at a girl's private school founded in part by Karl. Both
were also principals of the school at various points: Karl held the
position until 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 Frege's 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
Frege's benefactors. After four semesters, Frege transferred to the
University of Göttingen, 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 Frege's philosophical views.) In late 1873, Frege finished
his doctoral dissertation, under the guidance of Ernst Schering,
entitled Über eine geometrische Darstellung der
imaginären Gebilde in der Ebene ("On a Geometrical
Representation of Imaginary Figures in a Plane"), and received his
Ph.D.
In 1874, with the recommendation 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
five years, and he was supported by his mother. Frege's
Habilitationsschrift, entitled Rechnungsmethoden, die auf
eine Erweiterung des Grössenbegriffes gründen ("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 Frege's 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 Thought Modeled on That 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 Frege's 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 the Begriffsschrift,
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 Frege's
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 the Begriffsschrift,
Frege decided to write a work in which he would describe his logicist
views informally in ordinary language, and argue against rival views.
The result was his Die Grundlagen der Arithmetik ("The
Foundations of Arithmetic"), published in 1884. However, this work
seems to have been virtually ignored by most of Frege's
contemporaries.
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" ("Function and
Concept", 1891), "Über Sinn und Bedeutung" ("On Sense and
Reference", 1892) and "Über Begriff und Gegenstand" ("On Concept
and Object", 1892). This maturation of Frege's 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 volume work; in the second and third, he would move on to the
definition of real numbers, and the demonstration of their
properties.
Again, however, Frege's 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 involved.
Because of the unfavorable reception of his earlier works,
Frege was forced to arrange to have volume 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 calculus for classes.
For more information, see the article on "Russell's
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 "Über die Grundlagen der Geometrie," ("On the
Foundations of Geometry") was published between 1903 and 1906,
representing Frege's side of a debate with David Hilbert over 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 Russell's paradox, and
disappointment over the continued 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 others during this period. 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 closely, and in 1911, he wrote to both of them concerning his
own solution to Russell's 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
Russell--a suggestion that had profound importance for the history of
philosophy. Moreover, Rudolf
Carnap was one of Frege's students from 1910 to 1913, and
doubtlessly Frege had significant influence on Carnap's interest in
logic and semantics and his subsequent intellectual development and
successes.
After his retirement in 1918, 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 "Gedankengefüge" ("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 it
impossible. 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, Frege's own works were still not very
widely known. He did not live to see the profound impact he would
have on the emergence of analytic philosophy, nor to see his brand of
logic--due to the championship of Russell--virtually wholly supersede
earlier forms of logic. However, in bequeathing his unpublished work
to his adopted son, Alfred, he wrote prophetically, "I believe 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
Frege's papers to Heinrich Scholz of the University of Münster
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
Frege's unpublished works were lost.
Although he was a fierce, sometimes even satirical, 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 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 certain
political rights. This distasteful feature of Frege's personality has
gravely disappointed some of Frege's intellectual progeny.
2. Contributions to Logic
Trained as a mathematician, Frege's interests in logic grew out
of his interests in the foundations of arithmetic. 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.
Frege's 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, Frege's logicism was very
ambitious in another regard, as he believed that one could
prove 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. Frege's own term
for such a language, "Begriffsschrift" was likely borrowed from a
paper on Leibniz's ideas written by Adolf Trendelenburg. Although
there had been attempts to fashion at least the core of such a
language made by Boole and others working in the Leibnizian
tradition, Frege found their work unsuitable for a number of reasons.
Boole's 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. Boole's logic, though
innovative 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. Frege's brand of logical language was modeled
upon the international language of arithmetic, and it replaced the
subject/predicate style of logical analysis with the notions of
function and argument. In mathematics, an equation such as
"f(x) = x2 + 1" states that f
is a function that takes x as argument and yields as value the
result of multiplying x by itself and adding one. In 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
Frege's understanding of concepts, functions and objects. The concept
being human is understood as a function 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 constant 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 Frege's
terminology, an object for which a concept has the True as value 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 Frege's
own notation with more contemporary notation. For Frege, the
conditional function, " ">" is understood as a
function
the value of which 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 defined 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", "—",
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
Frege's 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 yields the True as value if the
argument-function has the True as value for all values of x,
and has the False as value otherwise. Thus, "xH(x)"
stands for the False, since the concept H( ) does not have the
True as value for all arguments. However, "x[H(x)
H(x)]"
stands for True, since the complex concept H( )
H( ) does have the True as value 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 Frege's logic. Frege is often credited with
having founded predicate logic. However, Frege's logic is in some
ways different from modern predicate logic. As we have seen, a sign
such as "H( )" is a sign for a function in the
strictest sense, as are the conditional and negation connectives.
Frege's 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. Frege's
"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 that what 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, Frege's 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 that every object falls under at least one concept.
Frege's 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. Frege's first logical system, that
of the 1879 Begriffsschrift, 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 Frege's
later system, the system of the Begriffsschrift was fully
consistent. (It has since been proven 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 the Begriffsschrift by adding
axioms governing what Frege called the "value-ranges"
(Werthverlaüfe) of functions, understood as objects
corresponding to the complete argument-value mappings generated
by functions. In the case of concepts, their value-ranges were
identified with their extensions. While Frege did sometimes
also refer to the extensions of concepts as "classes", he did
not conceive of such classes as aggregates or collections. They were
simply understood as objects corresponding to the complete
argument-value mappings generated by concepts considered as
functions. Frege then introduced 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 function F being
identical to the value-range of function G is the same as the
truth-value of F and G having the same value for 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 object x, that object is in the
extension of concept F if and only if the value of F
for x as argument is the True. 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 Russell's Paradox. See
the entry on Russell's
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 the
Begriffsschrift, is complete in its treatment of propositional
logic and 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 saying roughly the same as, "for all values of
x, if x is a student, then x is hardworking".
This made it possible to capture the logical connection between
statements such as "either all students are hardworking or all
students are intelligent" 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, Frege's 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, Frege's "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-order "predicate" logic (the latter of which
he invented outright), attempted the first formulation of
higher-order logic, presented the first 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 Frege's "philosophy of 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. Thus, Frege denied the popular view that logic
is without content 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 anyone's psychology or anyone's beliefs.
To think otherwise is to confuse something's being true with
something's 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 anyone's mental images, nor truths of
mathematics with psychological truths. Mathematical truths are
objective, not subjective. Frege was also a critic of Mill's 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 Mill's view, numbers must be taken to be conglomerations
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 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 contains fifty 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 Frege's 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 Frege's 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 that numbers 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 of debate and discussion to what extent and
how this principle coheres with Frege's 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 times a certain concept is instantiated. Consider,
for example, "I have six cards in my hand" or "There are 11 members
of congress from Wisconsin." These propositions seem to tell us how
many times the concepts of being a card in my hand and
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 Frege's 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 concept F 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 x that instantiates F, and that for all
objects y, either y does not instantiate F or
y is x. To say that F is instantiated twice is
to say that there are two objects, x and y, each of
which instantiates F, but which are not the same as each
other, and for all z, either z does not instantiate
F, or z is x or z is 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 two Fs. For this
purpose, Frege appeals to his theory of the value-ranges of concepts.
On the notion of a value-range, see above.
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 of Fs as there are
Gs. It is to say that there is a one-one mapping between the
objects that instantiate F and the objects instantiating
G, i.e. that there is some function f from entities
that instantiate F onto entities that instantiate G
such that there is a different F for every G, and a
different G for every F, with none left over. (In this,
Frege's views on the nature of cardinality were in part anticipated
by Cantor.) However, we must bear in mind that the
propositions:
(1) There are equally many Fs as there are
Gs.
(2) The number of Fs = the number of Gs
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 of Fs" 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. Frege's suggestion
is that "the number of Fs" 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 of Fs
is a certain value-range, containing value-ranges, and in particular,
all those value-ranges that have as many members as there are
Fs. 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
concept being a value-range of a concept instantiated equally many
times as G", which will be true if and only if there are equally
many Fs as Gs, i.e. if every value-range of a concept
instantiated equally many times as F is also a value-range of
a concept instantiated equally many times as G.
To give some examples, if there are zero Fs, then the
number of Fs, 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 class. If there is one
F, then the number of Fs, 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 case 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 Frege's 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 then defines a function that takes a
value-range as argument and yields 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 defined as the value-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
all natural 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 of
being 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 ancestral of a relation. To understand the
ancestral of a relation, consider the example of the relation
of being the child of. A person x bears this relation
to y just in case x is y's child. However,
x falls in the ancestral of this relation with respect
to y just in case x is the child of y, or is the
child of y's child, or is the child of y's child's
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 on ad
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.
Frege's 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, Frege's 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 Russell's paradox. This presents a
serious problem for Frege's logicist approach. Another heavy blow
came after Frege's death. In 1931, Kurt Gödel discovered his
famous incompleteness proof to the effect that there can be no
consistent formal 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 Frege's, which aimed to provide precisely the sort
of system Gödel showed impossible. Nevertheless, it cannot be
denied that Frege's work in the philosophy of mathematics was
important and insightful.
4. The Theory of Sense and Reference
Frege's influential theory of meaning, the theory of sense
(Sinn) and reference (Bedeutung) was first outlined,
albeit briefly, in his article, "Funktion und Begriff" of 1891, and
was expanded and explained in greater detail in perhaps his most
famous work, "Über 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
Frege's 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 Frege's
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 favor 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 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 "Über 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 the expressions "the
morning star" and "the evening star". Both of these expressions refer
to the planet Venus, yet they obviously denote Venus in virtue of
different properties that it has. Thus, Frege claims that these two
expressions have the same reference but different senses. The
reference of an expression is the actual thing corresponding to it,
in the case of "the morning star", the reference is the planet Venus
itself. The 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 two claims:
(1) the morning star = the morning star
(2) the morning star = the evening star
The first appears to be a trivial case of the law of
self-identity, knowable a priori, while the second seems to be
something that was discovered a posteriori by astronomers.
However, if "the morning star" means the same thing as "the evening
star", then the two statements themselves would also seem to have the
same meaning, both involving a thing's relation of identity to
itself. However, it then becomes to difficult to explain why (2)
seems informative while (1) does not. Frege's 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 object's relation of identity to itself. However, because the
senses of these expressions are different--in (1) the object is
presented the same way twice, and in (2) it is presented in two
different ways--it 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 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 the component 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
expressions, 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 True 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 "Über 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 clear that the
distinction can also be applied to "incomplete expressions", which
include functional expressions and grammatical
predicates. These expressions 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 root of ( )"
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 (Gegenstände), in Frege's
terminology, are self-standing, complete entities, while functions
are essentially incomplete, or as Frege says, "unsaturated"
(ungesättigt) in that they must take something else as
argument in order to yield 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 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 expression, "( ) is a planet"
has as its reference a function that yields as value the True 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 True 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 between functions 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 Leibniz's law with regard to identity. This law was
stated by Leibniz as, "those things are the same of which one can be
substituted for another 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.
to the conclusion:
(4) The evening star is a planet.
in virtue of (2) above and Leibniz's law is unproblematically
valid. However, there seem to be some serious counterexamples to this
principle. We know for example that "the morning star" and "the
evening star" have the same customary reference. However, it is not
always true that they can replace one another without changing the
truth of a sentence. For example, if we consider the
propositions:
(5) Gottlob believes that the morning star is a planet.
(6) Gottlob believes that the evening star is a planet.
If we assume that Gottlob does not know that the morning star
is the same heavenly body as the evening star, (5) may be true while
(6) false or vice versa.
Frege meets this challenge to Leibniz's law by making a
distinction between what he calls the primary and
secondary references of expressions. Frege suggests that when
expressions appear in certain unusual contexts, they have as their
references what is customarily their senses. In such cases, the
expressions are said to have their secondary references.
Typically, such cases involve what Frege calls "indirect speech" or
"oratio obliqua", as in the case of statements of beliefs,
thoughts, desires and other so-called "propositional attitudes", such
as the examples of (5) and (6). However, expressions also have their
secondary references (for reasons which should already be apparent)
in contexts such as "it is informative that..." or "... is
analytically true".
Let us consider the examples of (5) and (6) more closely. To
Frege's mind, these statements do not deal directly with the morning
star and the evening star itself. Rather, they involve a relation
between a believer and a thought believed. Thoughts, as we have seen,
are the senses of complete propositions. Beliefs depend for their
make-up on how certain objects and concepts are presented, not only
on the objects and concepts themselves. The truth of belief claims,
therefore, will depend not on the customary references of the
component expressions of the stated belief, but their senses. Since
the truth-value of the whole belief claim is the reference of that
belief claim, and the reference of any proposition, for Frege,
depends on the references of its component expressions, we are lead
to the conclusion that the typical senses of expressions that appear
in oratio obliqua are in fact the references of those
expressions when 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
Leibniz's 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 each 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 every 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 person's psychology. A
thought, for example, has a truth-value regardless of whether or not
anyone believes it and even whether or not anyone has grasped it at
all. Moreover, 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 Russell's theory of
descriptions, a Fregean sense is often interpreted as a set of
descriptive information or criteria that picks out its reference in
virtue of the reference alone satisfying or fitting 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 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 descriptive phrase, 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 converging
series" or "the present King of France" express senses, insofar as
they lay out criteria 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 claims that many of us seem to be able to use a name to
refer to an individual even if we are unaware of any
properties uniquely held by that individual. For example, many of us
don't know enough about the physicist Richard Feynman to be able to
identify a property differentiating him from other prominent
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 Frege's notion of a sense to circumvent Kripke's worries.
This has lead to a very important debate in the philosophy of
language, which, unfortunately, we cannot fully discuss here.
5. Suggestions for Further Reading
Frege's own works:
"Antwort auf die Ferienplauderei des Herrn Thomae."
Jahresbericht der Deutschen Mathematiker-Vereinigung 15
(1906): 586-90. Translated as "Reply to Thomae's Holiday
Causerie." 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.
"Über Begriff und Gegenstand." Vierteljahrsschrift
für wissenschaftliche Philosophie 16 (1892): 192-205.
Translated as "On Concept and Object." In >CP
182-94. 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, 1879.
Translated as Begriffsschrift, a Formula Language, Modeled
upon that of Arithmetic, for Pure Thought. In From Frege to
Gödel, edited by Jean van Heijenoort. Cambridge, MA:
Harvard University Press, 1967. Also as Conceptual Notation and
Related Articles. Edited and translated by Terrell W. Bynum.
London: Oxford University Press, 1972.
"Über die Begriffsschrift des Herrn Peano und meine
eigene." Verhandlungen der Königlich Sächsischen
Gesellschaft der Wissenschaften zu Leipzig 48 (1897): 362-8.
Translated as "On Mr. Peano's Conceptual Notation and My Own." In
CP 234-48.
"Über formale Theorien der Arithmetik."
Sitzungsberichte der Jenaischen Gesellschaft für Medizin
und Naturwissenschaft 19 (1885): 94-104. Translated as "On
Formal Theories of Arithmetic." In CP
112-21.
Funktion und Begriff. Jena: Hermann Pohle, 1891.
Translated as "Function and Concept." In CP
137-56, TPW 21-41 and
FR 130-48.
"Der Gedanke." Beträge zur Philosophie des
deutschen Idealismus 1 (1918-9): 58-77. Translated as
"Thoughts." In CP 351-72.
Also as part I of Logical Investigations
[LI], edited by P. T. Geach. Oxford: Blackwell, 1977.
And as "Thought." In FR
325-45.
"Gedankengefüge." Beträge zur Philosophie des
deutschen Idealismus 3 (1923): 36-51. Translated as "Compound
Thoughts." In CP
390-406, and as part III of LI.
Über eine geometrische Darstellung der
imaginären Gebilde in der Ebene. Ph. D. Dissertation:
University of Göttingen, 1873. Translated as "On a
Geometrical Representation of Imaginary Forms in the Plane." In
CP 1-55.
Grundgesetze der Arithmetik. 2 vols. Jena: Hermann
Pohle, 1893-1903. Translated in part as The Basic Laws of
Arithmetic: Exposition of the System. Edited and translated by
Montgomery Furth. Berkeley: University of California Press,
1964.
"Über die Grundlagen der Geometrie." Jahresbericht
der Deutschen Mathematiker-Vereinigung 12 (1903): 319-24,
368-75, 15 (1906): 293-309, 377-403, 423-30. Translated as "On the
Foundations of Geometry." In CP
273-340. 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 über 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. Translated by J. L. Austin. Oxford: Blackwell,
1953.
"Kritische Beleuchtung einiger Punkte in E. Schröders
Vorlesungen über die Algebra der Logik." Archiv
für systematsche Philosophie 1 (1895): 433-56. Translated
as "A Critical Elucidation of Some Points in E. Schröder,
Vorlesungen über die Algebra der Logik." In
CP 210-28, and
TPW 86-106.
Nachgelassene Schriften. Hamburg: Felix Meiner,
1969. Translated as Posthumous Writings. Translated by
Peter Long and Roger White. Chicago: University of Chicago Press,
1979.
"Le nombre entier." Revue de Métaphysique et de
Morale 3 (1895): 73-8. Translated as "Whole Numbers." In
CP 229-33.
Rechnungsmethoden, die auf eine Erweiterung des
Grössenbegriffes gründen. Habilitationsschrift:
University of Jena, 1874. Translated as "Methods of Calculation
based on an Extension of the Concept of Quantity." In
CP 56-92.
Review of Zur Lehre vom Transfiniten, by Georg
Cantor. Zeitschrift für Philosophie und philosophische
Kritik 100 (1892): 269-72. Translated in CP
178-181.
Review of Philosophie der Arithmetik, by Edmund
Husserl. Zeitschrift für Philosophie und philosophische
Kritik 103 (1894): 313-32. Translated in CP
195-209.
"Über Sinn und Bedeutung." Zeitschrift für
Philosophie und philosophische Kritik 100 (1892): 25-50.
Translated as "On Sense and Meaning." In CP
157-77. As "On Sinn and Bedeutung." In <FR
151-71. And as "On Sense and Reference." In TPW
56-78.
"Über das Trägheitsgesetz." Zeitschrift
für Philosophie und philosophische Kritik 98 (1891):
145-61. Translated as "On the Law of Inertia." In CP
123-36.
"Die Unmöglichkeit 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. Thomae's Formal
Arithmetic." In CP
346-50.
"Der Verneinung." Beträge zur Philosophie des
deutschen Idealismus 1 (1918-9): 143-57. Translated as
"Negation." In CP 373-89,
part II of LI, and
FR 346-61.
"Was ist ein Funktion?" In Festschrift Ludwig Boltzmann
gewidmet zum sechzigsten Geburtstage, 656-66. Leipzig:
Amrosius Barth, 1904. Translated as "What is a Function?" In
CP 285-92, and
TPW 285-92.
Wissenschaftlicher Briefwechsel. Hamburg: Felix
Meiner, 1976. Translated as Philosophical and Mathematical
Correspondence. Translated by Hans Kaal. Chicago: University
of Chicago Press, 1980.
Über die Zahlen des Herrn H. Schubert. Jena:
Hermann Pohle, 1899. Translated as "On Mr. H. Schubert's Numbers."
In CP 249-72.
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. Frege's Theory of Judgment. New York:
Oxford University Press, 1979.
Bynum, Terrell W. "On the Life and Work of Gottlob Frege. "
Introduction to Conceptual Notation and Related Articles,
by Gottlob Frege. London: Oxford University Press, 1972.
Carl, Wolfgang. Frege's 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." In Structure, Method and Meaning: Essays in
Honor of Henry M. Sheffer, edited by P. Henle, H. Kallen and
S. Langer, 3- 24. New York: Liberal Arts Press, 1951.
Currie, Gregory. Frege: An Introduction to His
Philosophy. Totowa, 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 Frege's
Philosophy. Cambridge, MA: Harvard University Press,
1981.
Geach, Peter T. "Frege." In Three Philosophers,
edited by G. E. M. Anscombe and P. T. Geach, 127-62. Oxford:
Oxford University Press, 1961.
Gödel, Kurt. "On Formally Undecidable Propositions of
Principia Mathematica and Related Systems I." In From
Frege to Gödel, edited by Jan van Heijenoort, 596-616.
Cambridge, MA: Harvard University Press, 1967. Originally
published as "Über formal unentscheidbare Sätze der
Principia Mathematica und verwandter Systeme I."
Monatshefte für Mathematik und Physik 38 (1931):
173-98.
Grossmann, Reinhardt. Reflections on Frege's
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 in Semantics 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, Thomas G., ed. The Cambridge Companion to
Frege. Cambridge: Cambridge University Press,
forthcoming.
Russell, Bertrand. "The Logical and Arithmetical Doctrines
of Frege." In The Principles of Mathematics, Appendix A.
1903. 2d. ed. Reprint, New York: W. W. Norton & Company,
1996.
Russell, Bertrand. "On Denoting." Mind 14 (1905):
479-93.
Salmon, Nathan. Frege's 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 University Press,
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. Frege's Logical Theory.
Carbondale: Southern Illinois University Press, 1966.
Thiel, Christian. Sense and Reference in Frege's
Logic. Translated by T. J. Blakeley. Dordrecht: D. Reidel,
1968.
Tichý, Pavel. The Foundations of Frege's
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. Frege's Conception of Numbers as
Objects. Aberdeen: Aberdeen University Press, 1983.
Wright, Crispin. Frege: Tradition and
Influence. Oxford: Blackwell, 1984.
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