This is a file in the archives of the Stanford Encyclopedia of Philosophy. |

- Frege's Life
- Frege's Advances in Logic
- Frege's Ontology and Philosophy of Language
- Frege's Writings
- Bibliography
- Other Internet Resources
- Related Entries

- 1848, born November 8 in Wismar (Mecklenburg-Schwerin)
- 1869, entered the University of Jena
- 1871, entered the University of Göttingen
- 1873, awarded Ph. D. in Mathematics (Geometry), University of Göttingen
- 1874, earned a Habilitation in Mathematics, University of Jena
- 1874, became Privatdozent, University of Jena
- 1879, became Professor Extraordinarius, University of Jena
- 1896, became ordentlicher Honorarprofessor, University of Jena
- 1917, retired from the University of Jena
- 1925, died July 26 in Bad Kleinen (now in Mecklenburg-Vorpommern)

**The Predicate Calculus**. In an attempt to realize Leibniz's
ideas for a language of thought and a rational calculus, Frege
developed a formal notation for regimenting thought and reasoning (see
his *Begriffsschrift* ). Though we no longer use his notation,
Frege's in effect developed the first predicate calculus. A predicate
calculus is a formal system with two components: a formal language and
a logic. The formal language Frege designed was capable of: (a)
expressing predicational statements of the form `*x* falls under
the concept *F*' and `*x* bears relation *R* to
*y*', etc., (b) expressing complex statements such as `it is not
the case that ...' and `if ... then ...', and (c) expressing
`quantified' statements of the form `Some *x* is such that
...*x*...' and `Every *x* is such that ...*x*...'. The
logic of Frege's calculus was a set of rules that govern when some
statements of the language may be correctly inferred from others.

Frege's system was powerful enough to resolve the essential logic of
mathematical reasoning. That was partly due to the fact that his
predicate calculus was a `second-order' predicate calculus,
allowing statements of the form `Some concept *F* is such that
...*F*...' and `Every concept *F* is such that
...*F*...'. However, the most important insight underlying
Frege's calculus was his `function-argument' analysis of sentences.
This freed him from the limitations of the `subject-predicate'
analysis of sentences that formed the basis of Aristotelian logic and
it made it possible for him him to develop a general treatment of
quantification.

**The Analysis of Atomic Sentences and Quantifier Phrases**. In
traditional Aristotelian logic, the subject of a sentence and the
direct object of a verb are not on a logical par. The rules governing
the inferences between statements with different but related subject
terms are different from the rules governing the inferences between
statements with different but related verb complements. For example,
in Aristotelian logic, the rule which permits the valid inference from
`John loves Mary' to `Something loves Mary' is different from the rule
which permits the valid inference from `John loves Mary' to `John
loves something'. The rule governing the first inference is a rule
which applies only to the subject terms `John' and `Something'. The
rule governing the second inference applies only to the transitive verb
complements `Mary' and `something'. In Aristotelian logic, these
inferences have nothing in common.

In Frege's logic, a single rule governs both the inference from
`John loves Mary' to `Something loves Mary' and the inference from
`John loves Mary' to `John loves something'. This was made possible
by Frege's analysis of atomic and quantified sentences. Frege took
intransitive verb phrases such as `is happy' to be functions of one
variable (`*x* is happy'), and resolved the sentence "John is
happy" in terms of the application of the function denoted by `is
happy' to the argument denoted by `John'. In addition, Frege took the
verb phrase `loves' to be a function of two variables (`*x* loves
*y*') and resolved the sentence `John loves Mary' as the
application of the function denoted by `*x* loves *y*' to
the objects denoted by `John' and `Mary' respectively. In effect,
Frege saw no distinction between the subject `John' and the direct
object `Mary'. What is logically important is that `loves' denotes a
function of 2 arguments, that `gives' denotes a function of 3
arguments (*x* gives *y* to *z*), that `bought' denotes
a function of 4 arguments (*x* bought *y* from *z* for
amount *u*), etc.

This analysis allowed Frege to develop a more systematic treatment of
quantification than that offered by Aristotelian logic. No matter
whether the quantified expression `something' appears within a subject
("Something loves Mary") or within a predicate ("John loves
something"), it is to be resolved in the same way. In effect, Frege
treated quantified expressions as variable-binding operators. The
variable-binding operator `some *x* is such that' can bind the
variable `*x*' in the expression `*x* loves Mary' as well as
the variable `*x*' in the expression `John loves *x*'.
Thus, Frege analyzed the above inferences in the following general
way:

- John loves Mary. Therefore, some x is such that x loves Mary.
- John loves Mary. Therefore, some x is such that John loves x.

**Proof**. As part of his predicate calculus, Frege developed a
strict definition of a `proof'. In essence, he defined a proof to be
any finite sequence of well-formed statements such that each statement in the
sequence either is an axiom or follows from previous members by a
valid rule of inference. A proof of the statement B from the premises
A_{1},...,A_{n} is any finite sequence of statements (with B
the final statement in the sequence) such that each member of the
sequence: (a) is one of the premises A_{1},...,A_{n},
or (b) is an axiom, or (c) follows from previous members of the
sequence by a rule of inference. This is essentially the definition
of a proof that logicians still use today.

**A Foundation for Mathematics**. Frege attempted to construct a
foundation for mathematics. His most comprehensive logical system was
developed in his landmark work *Grundgesetze der Arithmetik*,
in which he attempted to validate the philosophical doctrine known as
*logicism*, i.e., the idea that mathematical concepts can be
defined in terms of purely logical concepts and that mathematical
axioms can be derived from the laws of logic alone. Unfortunately,
Frege employed a principle in the *Grundgesetze* (Basic Law V)
which turned out to be subject to Russell's Paradox. This paradox caused him to question the truth of logicism,
and few philosophers today believe that mathematics can be reduced to
logic. Mathematics seems to require some non-logical notions (such
set membership) and some non-logical axioms (such as the non-logical
axioms of set theory). Despite the fact that a contradiction
invalidated a part of his system, the intricate theoretical web of
definitions and proofs developed in the *Grundgesetze* produced
a conceptual framework for mathematical logic that was nothing short
of revolutionary. There is no doubt that the logical system and maze
of definitions developed by Bertrand Russell
and Alfred North Whitehead in *Principia Mathematica* owe a huge debt to the work found in Frege's
*Grundgesetze*.

**Definition**. Frege was extremely careful about the proper
description and definition of logical and mathematical concepts. He
developed powerful and insightful criticisms of mathematical work
which did not meet his standards for clarity. For example, he
criticized mathematicians who defined a variable to be a number that
varies rather than an expression of language which can vary as to
which determinate number it refers to. And he criticized those
mathematicians who developed `piecemeal' definitions or `creative'
definitions. In the *Grundgesetze* (Band II, Sections 56-67)
Frege criticized the practice of defining a concept on a given range
of objects and later redefining the concept on a wider, more inclusive
range of concepts. Frequently, this `piecemeal' style of definition
led to conflict, since the redefined concept did not always reduce to
the original concept when one restricts the range to the original
class of objects. In that same work (Band II, Sections 139-147),
Frege criticised the mathematical practise of introducing notation to
name (unique) entities without first proving that there exist (unique)
such entities. He pointed out that such `creative definitions' were
simply unjustified.

**The Natural Numbers**. In his seminal work *Die Grundlagen
der Arithmetik*, Frege successfully defined the notion of a
`cardinal number' in terms of the primitive notion of an
*extension* or *set*. The insight behind the definition is
that a statement of cardinal number such as `There are *n*
*F*-things' predicates a higher-order concept of the concept
*F*, namely, that it is a concept under which *n* things
fall. Frege simply defines the (cardinal) number of the concept
*F* (i.e., the number of *F*s) as the extension of the
concept *being a concept equinumerous to F*. On this definition,
the number of planets is identified as the extension of the concept
*being a concept equinumerous to the concept of being a planet*.
In other words, the number of planets is an extension (or set) which
contains all those concepts which, like the concept *being a
planet*, are exemplified by nine objects.

Frege defined the concept of *natural number* by defining, for
every relation *xRy*, the general concept `*x* is an
ancestor of *y* in the R-series' (this new relation is called
`the ancestral of the relation R'). The ancestral of a relation R was
first defined in Frege's *Begriffsschrift*. The intuitive idea
is easily grasped if we consider the relation *x* is the father
of *y*. Suppose that *a* is the father of *b*, that
*b* is the father of *c*, and that *c* is the father of
*d*. Then Frege's definition of `*x* is an ancestor of
*y* in the fatherhood-series' ensured that *a* is an
ancestor of *b*, *c*, and *d*, that *b* is an
ancestor of *c* and *d*, and that *c* is an ancestor
of *d*.

More generally, if given a series of facts of the form *aRb*,
*bRc*, *cRd*, and so on, Frege showed how to define the
relation *x is an ancestor of y in the R-series* (this is the
ancestral of the relation R). To exploit this definition in the case
of natural numbers, Frege had to define both the relation *x
precedes y* and the ancestral of this relation, namely, *x is an
ancestor of y in the predecessor-series*. He first defined the
relational concept *x precedes y* as follows:

In the notation of the second-order predicate calculus, Frege's definition becomes:x precedes yiff there is a conceptFand an objectzsuch that:

zfalls underF,yis the (cardinal) number of the conceptF, andxis the (cardinal) number of the conceptobject other than z falling under F

To see the intuitive idea behind this definition, consider how the definition is satisfied in the case of the number 1 preceding the number 2: there is a concept

- Whitehead falls under the concept
*being an author of Principian Mathematica*, *2*is the (cardinal) number of the concept*being an author of Principia Mathematica*, and*1*is the (cardinal) number of the concept*object other than Whitehead which falls under the concept being an author of Principia Mathematica*

Given this definition of *precedes*, Frege then defined the
ancestral of this relation, namely, *x is an ancestor of y in the
predecessor-series*. So, for example, if 10 precedes 11 and 11
precedes 12, it follows that 10 is an ancestor of 12 in the
predecessor-series. Note, however, that although 10 is an ancestor of
12, 10 does not precede 12, for the notion of *precedes* is that
of *strictly* precedes. Note also that by defining the ancestral
of the precedence relation, Frege had in effect defined *x* <
*y*.

Frege then defined the number 0 as the (cardinal) number of the
concept *being an object not identical with itself*. The idea
here is that nothing fails to be self-identical, so nothing falls
under this concept. The number 0 is therefore identified with the
extension of all concepts which fail to be exemplified.

Finally, Frege defined:

In other words, a natural number is any member of the predecessor series beginning with 0.x is a natural numberiff either x = 0 or 0 is an ancestor ofxin the predecessor series

Using this definition, Frege derived many important theorems of number
theory. It was recently shown by R. Heck [1993] that, despite the
logical inconsistency in the system of his *Grundgesetze*, Frege
validly derived the Dedekind/Peano Axioms for number theory from a
powerful and consistent principle now known as Hume's Principle ("The
number of Fs is equal to the number of Gs if and only if there is a
one-to-one correspondence between the Fs and the Gs"). Although Frege
used his inconsistent axiom Basic Law V to establish Hume's Principle,
once Hume's Principle was established, Frege validly derived the
Dedekind/Peano axioms without making any further essential appeals to
Basic Law V. Following the lead of George Boolos, philosophers today
call derivation of the Dedekind/Peano Axioms from Hume's Principle
`Frege's Theorem'. The proof of Frege's Theorem was a *tour de force*
which involved some of the most beautiful, subtle, and complex logical
reasoning that had ever been devised. For a comprehensive introduction to the logic of Frege's Theorem, see the entry Frege's logic, theorem, and foundations for arithmetic.

**Frege's Ontology**. In Frege's ontology, functions and objects
were rigorously distinguished as two fundamentally different kinds of
entity. Functions are the kind of thing that take objects as
arguments and map those arguments to a value. Frege did not limit
examples of functions to mathematical functions such as *x* + 3.
He allowed the variable *x* to range over any object, and so
*father of x* is a genuine example of a function---it maps
certain biological offspring to their fathers and maps everything else
to The False. Frege associates with every function, a
*course-of-values*. The course-of-values of a function
explicitly indicates the value of the function for each object that is
supplied as an argument. In addition, Frege believed that there are
two distinguished objects, namely, the truth value The True and the
truth-value The False. Those functions which map objects to truth
values are called *concepts*. For example, not only is the
mathematical function *x* + 3 = 5 a concept (this concept maps the
number 2 to The True and everything else to The False), but so is the
function *x is happy* (which maps anything that is happy to The
True and everything else to The False). Frege defines the
*extension* of a concept to contain just those objects which the
concept maps to The True (as indicated by the course-of-values
associated with the concept).

Frege suggested that *existence* is not a property of objects but
rather of concepts: it is the property a concept has just in case it
has an non-empty extension (i.e., just in case it maps some object to
The True). So the fact that the extension of the concept
*martian* is empty underlies the ordinary claim "Martians don't
exist." Frege therefore took *existence* to be a `second-level'
concept.

**Frege's Puzzle About Identity Statements**. Here are some
examples of identity statements:

117 + 136 = 253.Frege believed that these statements all have the form "

The morning star is identical to the evening star.

Mark Twain is Samuel Clemens.

Bill is Debbie's father.

But Frege noticed that on this account of truth, the truth conditions
for "*a = b* " are no different from the truth conditions for
"*a = a* ". For example, the truth conditions for "Mark Twain =
Mark Twain" are the same as those for "Mark Twain = Samuel Clemens";
not only do the names flanking the identity sign denote the same
object in each case, but the object is the same between the two cases.
The problem is that the cognitive significance (or meaning) of the two
sentences differ. We can learn that "Mark Twain = Mark Twain" is true
simply by inspecting it; but we can't learn the truth of "Mark Twain =
Samuel Clemens" simply by inspecting it. Similarly, whereas you can
learn that "117 + 136 = 117 + 136" and "the morning star is identical
to the morning star" are true simply by inspection, you can't learn
the truth of "117 + 136 = 253" and "the morning star is identical to
the evening star" simply by inspection. In the latter cases, you have
to do some arithmetical work or astronomical investigation to learn
the truth of these identity claims.

So the puzzle Frege discovered is: if we cannot appeal to a difference
in denotation of the terms flanking the identity sign, how do we
explain the difference in cognitive significance between "*a = b*
" and "*a = a* "?

**Frege's Puzzle About Propositional Attitude Reports**.
Frege is generally credited with identifying the following puzzle
about propositional attitude reports, even though he didn't quite
describe the puzzle in the terms used below. A propositional attitude
is a psychological relation between a person and a proposition.
Belief, desire, intention, discovery, knowledge, etc., are all
psychological relationships between persons, on the one hand, and
propositions, on the other. When we report the propositional
attitudes of others, these reports all have a similar logical form:

If we replace the variable `xbelieves thatp

xdesires thatp

xintends thatp

xdiscovered thatp

xknows thatp

John believes that Mark Twain wroteHuckleberry Finn.

To see the problem posed by the analysis of propositional attitude
reports, consider what appears to be a simple principle of reasoning,
namely, the Principle of Substitution. If a name, say *n*,
appears in a true sentence S, and the identity sentence *n=m*
is true, then the Principle of Substitution tells us that the
substitution of the name *m* for the name *n* in S does
not affect the truth of S. For example, let S be the true sentence
"Mark Twain was an author", let *n* be the name `Mark Twain',
and let *m* be the name `Samuel Clemens'. Then since the
identity sentence "Mark Twain = Samuel Clemens" is true, we can
substitute `Samuel Clemens' for `Mark Twain' without affecting the
truth of the sentence. And indeed, the resulting sentence "Samuel
Clemens was an author" is true. In other words, the following
argument is valid:

Mark Twain was an author.Similarly, the following argument is valid.

Mark Twain = Samuel Clemens.

Therefore, Samuel Clemens was an author.

4 > 3In general, then, the Principle of Substitution seems to take the following form, where S is a sentence,

4 = 8/2

Therefore, 8/2 > 3

S(This principle seems to capture the idea that if we say something true about an object, then even if we change the name by which we refer to that object, we should still be saying something true about that object.n)

n = m

Therefore, S(m)

But Frege, in effect, noticed following counterexample to the Principle of Substitution. Consider the following argument:

John believes that Mark Twain wroteThis argument is not valid. There are circumstances in which the premises are true and the conclusion false. We have already described such circumstances, namely, one in which John learns the name `Mark Twain' by readingHuckleberry Finn.

Mark Twain = Samuel Clemens.

Therefore, John believes that Samuel Clemens wroteHuckleberry Finn.

**Frege's Theory of Sense and Denotation**. To explain these
puzzles, Frege suggested that in addition to having a denotation,
names and descriptions also express a *sense*. The sense of a
expression accounts for its cognitive significance---it is the way by
which one conceives of the denotation of the term. The expressions
`4' and `8/2' have the same denotation but express different senses,
different ways of conceiving the same number. The descriptions `the
morning star' and `the evening star' denote the same planet, namely
Venus, but express different ways of conceiving of Venus and so have
different senses. The name `Pegasus' and the description `the most
powerful Greek god' both have a sense (and their senses are distinct),
but neither has a denotation. However, even though the names `Mark
Twain' and `Samuel Clemens' denote the same individual, they express
different senses. Using the distinction between sense and denotation,
Frege can account for the difference in cognitive significance between
identity statements of the form "*a = a*" and "*a = b*".
The sense of the whole statement, on Frege's view, is a function of
the senses of its component parts. Since the sense of `*a*'
differs from the sense of `*b*', the components of "*a = a*"
and "*a = b*" are different and so the sense of the whole
expression will be different in the two cases. Since the sense of an
expression accounts for its cognitive significance, Frege has an
explanation of the difference in cognitive significance between "*a
= a*" and "*a = b*", and thus a solution to the first puzzle.

Moreover, Frege proposed that when a term (name or description) follows a propositional attitude verb, it no longer denotes what it ordinarily denotes. Instead, Frege claims that in such contexts, a term denotes its ordinary sense. This explains why the Principle of Substitution fails for terms following the propositional attitude verbs in propositional attitude reports. The Principle asserts that truth is preserved when we substitute one name for another having the same denotation. But, according to Frege's theory, the names `Mark Twain' and `Samuel Clemens' denote different senses when they occur in the following sentences:

John believes that Mark Twain wroteIf they don't denote the same object, then there is no reason to think that substitution of one name for another would preserve truth.Huckleberry Finn.

John believes that Samuel Clemens wroteHuckleberry Finn.

Frege developed the theory of sense and denotation into a thoroughgoing philosophy of language. This philosophy can be explained, at least in outline, by considering a simple sentence such as "John loves Mary". In Frege's view, each word in this sentence is a name and, moreover, the sentence as a whole is a complex name. Each of these names has both a sense and a denotation. Then sense and denotation of the words are basic; but sense and denotation of the sentence as a whole can be described in terms of the sense and denotation of the words and the way in which those words are arranged in the sentence. Let us refer to the denotation and sense of the words as follows:

We now work toward a theoretical description of the denotation of the sentence as a whole. On Frege's view,d[j] refers to the denotation of the name `John'.

d[m] refers to the denotation of the name `Mary'.

d[L] refers to the denotation of the name `loves'.

s[j] refers to the sense of the name `John'.

s[m] refers to the sense of the name `Mary'.

s[L] refers to the sense of the name `loves'.

The sentence "John loves Mary" also expresses a sense. Its sense may
be described as follows. First, **s**[L] (the sense of the name
"loves") is identified as a function. This function maps **s**[m]
(the sense of the name "Mary") to the sense of the predicate `loves
Mary'. Let us refer to the sense of `loves Mary' as **s**[Lm].
Now the function **s**[Lm] maps **s**[j] (the sense of the name
`John') to the sense of the whole sentence. Let us call the sense of
the entire sentence **s**[jLm]. Frege calls the sense of a
sentence a *thought*, and whereas there are only two truth
values, he supposes that there are an infinite number of thoughts.

On Frege's view, therefore, the sentences "4 = 8/2" and "4 = 4" both
name the same truth value, but they express different thoughts. That
is because **s**[4] is different from **s**[8/2]. Similarly,
"Mark Twain = Mark Twain" and "Mark Twain = Samuel Clemens" denote the
same truth value, but express different thoughts (since the sense of
the names differ). Thus, Frege has a general account of the
difference in the cognitive significance between identity statements
of the form "*a* = *a*" and "*a* = *b*".
Furthermore, recall that Frege proposed that terms following
propositional attitude verbs denote not their ordinary denotations but
rather the senses they ordinarily express. In fact, in the following
propositional attitude report, not only do the words `Mark Twain',
`wrote' and `*Huckleberry Finn* ' denote their ordinary senses,
but the entire sentence "Mark Twain wrote *Huckleberry Finn*"
also denotes its ordinary sense (namely, a thought):

John believes that Mark Twain wroteFrege, therefore, would analyze this attitude report as follows: "believes that" denotes a function that maps the denotation of the sentence "Mark Twain wroteHuckleberry Finn.

John believes that Samuel Clemens wroteSince the thought denoted by "Samuel Clemens wroteHuckleberry Finn.

*Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens*(`Concept Notation, a formal language of pure thought, modelled upon that of arithmetic'), Halle a. S., 1879*Die Grundlagen der Arithmetik: eine logisch-mathematische Untersuchung über den Begriff der Zahl*(`The Foundations of Arithmetic: A logico-mathematcial enquiry into the concept of number'), Breslau, 1884*Funktion und Begriff (`Function and Concept'): Vortrag, gehalten in der Sitzung vom 9. Januar 1891 der Jenaischen Gesellschaft für Medizin und Naturwissenschaft*, Jena, 1891- `Über Sinn und Bedeutung' (`On Sense and Denotation'), in
*Zeitschrift für Philosophie und philosophische Kritik*, C (1892): 25-50 - `Über Begriff und Gegenstand' (`On Concept and Object'), in
*Vierteljahresschrift für wissenschaftliche Philosophie*, XVI (1892): 192-205 *Grundgesetze der Arithmetik*(`Basic Laws of Arithmetic'), Jena: Verlag Hermann Pohle, Band I (1893), Band II (1903)- `Was ist eine Funktion?' (`What is a Function?'), in
*Festschrift Ludwig Boltzmann gewidmet zum sechzigsten Geburtstage, 20. Februar 1904*, S. Meyer (ed.), Leipzig, 1904, pp. 656-666 - `Der Gedanke. Eine logische Untersuchung' (`The Thought: A
Logical Enquiry'), in
*Beiträge zur Philosophie des deutschen Idealismus*I (1918): 58-77

- `Anwendungen der Begriffsschrift',
*Jenaische Zeitschrift ür Naturwissenschaft***XIII**(1979), Supplement II, pp. 29-33 - `Über die wissenschaftliche Berechtigung einer
Begriffsschrift' (`On the Scientific Justification of Concept
Notation'),
*Zeitschrift für Philosophie und philosophische Kritik*,**LXXXI**(1882): 48-56 - `Über den Zweck der Begriffsschrift' (`On the Purpose of
Concept Notation'),
*Jenaische Zeitschrift für Naturwissenschaft*,**XVI**(1883), Supplement, pp. 1-10 - `Erwiderung',
*Deutsche Literaturzeitung*,**VI**/28 (1885): column 1030 (a brief reply to Cantor's review of the*Grundlagen*) - `Über das Trägheitsgesetz' (`On the Law of Inertia'),
*Zeitschrift für Philosophie und philosophische Kritik*,**XCVIII**(1891): 145-161 - Review of E. Husserl's
*Philosophie der Arithmetik*, Vol. I, in*Zeitschrift für Philosophie und philosophische Kritik*,**CIII**(1894): 313-332 - `Kritische Beleuchtung einiger Punkte in E. Schröders
*Vorlesungen über die Algebra der Logik*, in*Archiv für systematische Philosophie*,**I**(1895): 433-456 - Letter to the Editor,
*Rivista di Matematica*,**VI**(1896-9): 53-59 - `Über die Grundlagen der Geometrie',
*Jahresbericht der Deutschen Mathematiker-Vereinigung***XV**(1903), Part I pp. 293-309, Part II pp. 368-375 - Notes to P. E. B. Jourdain, `The Development of the Theories of
Mathematical Logic and the Principles of Mathematics: Gottlob
Frege',
*The Quarterly Journal of Pure and Applied Mathematics***XLIII**(1912): 237-69 - `Der Gedanke. Eine Logische Untersuchung',
*Beiträge zur Philosophie des deutschen Idealismus***I**(1918): 58-77 - `Die Verneinung. Eine Logische Untersuchung',
*Beiträge zur Philosophie des deutschen Idealismus***I**(1918): 143-157 - `Logische Untersuchungen. Dritter Teil: Gedankengefüge',
*Beiträge zur Philosophie des deutschen Idealismus***III**(1923): 36-51

*The Frege Reader*, M. Beaney (ed.), Oxford: Blackwell, 1997*The Foundations of Arithmetic. A logic-mathematical enquiry into the concept of number*, trans. by J. L. Austin, Oxford: Blackwell, second revised edition, 1974*Translations from the Philosophical Writings of Gottlob Frege*, ed. and trans. by P. Geach and M. Black, Oxford: Blackwell, second revised edition, 1970*The Basic Laws of Arithmetic*, (Exposition of the System: Volume I), ed. and trans. by M. Furth, Berkeley: U. of California Press, 1964 (this contains the introductory fragment from Volume I of Frege's*Grundgesetze*. and the Appendix to Volume II)- `The Thought: A Logical Enquiry', trans. by A. and M. Quinton,
*Mind*,**LXV**(1956): 289-311 - `Compound Thoughts', trans. by R. Stoothoff,
*Mind***LXXII**(1963): 1-17 *The Foundations of Geometry*, trans. by M. E. Szabo,*The Philosophical Review*,**LXIX**(1960), pp. 3-17- `About the Law of Inertia', trans. by R. Rand,
*Synthese*,**X11**(1961): 350-363 - `On the Scientific Justification of a Concept-script', trans. by
J. M. Bartlett,
*Mind*,**LXXIII**(1964), pp. 155-60 - `Begriffsschrift, a formula language, modelled upon that of
arithmetic, for pure thought', trans. by S. Bauer-Mengelberg, in
J. van Heijenoort (ed.),
*From Frege to Gödel, a source book in mathematical logic, 1879-1931*, Cambridge, MA: Harvard University Press, 1967 - `On the Purpose of the Begriffsschrift', trans. by
V. H. Dudman,
*The Australasian Journal of Philosophy*,**XLVI**(1968), pp. 89-97 *Conceptual Notaton and Related Articles*, trans. and ed. by Terrell Ward Bynum, Oxford: Clarendon, 1972*On Foundatons of Geometry and Formal Theories of Arthmetic*, trans. E.-H. W. Kluge, New Haven: Yale University Press, 1971.*Posthumous Writings*, ed. by H. Hermes, F. Kambartel, and F. Kaulbach, trans. by P. Long and R. White, Chicago: U. of Chicago Press, 1979*Philosophical and Mathematical Correspondence*, ed. by G. Gabriel, H. Hermes, F. Kambartel, C. Thiel, and A. Veraart, trans. by H. Kaal, Chicago: U. of Chicago Press, 1980*Collected Papers on Mathematics, Logic, and Philosophy*, ed. by B. McGuinness, trans. by Black, Dudman, Geach, Kaal, Kluge, McGuinness, and Stoothoff, Oxford: Basil Blackwell, 1984- `On Herr Peano's
*Begriffsschrift*and My Own',*Australasian Journal of Philosophy*,**47**(1969): 1--14

- Boolos, G., 1986, "Saving Frege From Contradiction",
*Proceedings of the Aristotelian Society*,**87**(1986/87): 137-151 - Boolos, G., 1987, "The Consistency of Frege's
*Foundations of Arithmetic*", in J. Thomson (ed.),*On Being and Saying*, Cambridge, MA: The MIT Press, pp. 3-20 - Currie, G., 1982,
*Frege: An Introduction to His Philosophy*, Brighton, Sussex: Harvester Press - Demopoulos, W., (ed.), 1995,
*Frege's Philosophy of Mathematics*, Cambridge, MA: Harvard - Dummett, M., 1973,
*Frege: Philosophy of Language*, London: Duckworth - Dummett, M., 1991,
*Frege: Philosophy of Mathematics*, Cambridge, MA: Harvard University Press - Heck, R., 1993, "The Development of Arithmetic in Frege's
*Grundgesetze der Arithmetik*",*Journal of Symbolic Logic*,**58**/2 (June): 579-601 - Klemke, E. D. (ed.), 1968,
*Essays on Frege*, Urbana, IL: University of Illinois Press - Parsons, T., 1981, "Frege's Hierarchies of Indirect Senses and the
Paradox of Analysis",
*Midwest Studies in Philosophy: VI*, Minneapolis: University fo Minnesota Press, pp. 37-57 - Parsons, T., 1987, "On the Consistency of the First-Order Portion of
Frege's Logical System",
*Notre Dame Journal of Formal Logic***28**/1 (January): 161-168 - Parsons, T., 1982, "Fregean Theories of Fictional Objects",
*Topoi***1**: 81-87 - Resnik, M., 1980,
*Frege and the Philosophy of Mathematics*, Ithaca, NY: Cornell University Press - Ricketts, T., 1997, "Truth-Values and Courses-of-Value in Frege's
*Grundgesetze*", in*Early Analytic Philosophy*, W. Tait (ed.), Chicago: Open Court, pp. 187-211 - Salmon, N., 1986,
*Frege's Puzzle*, Cambridge, MA: MIT Press - Schirn, M., (ed.), 1996,
*Frege: Importance and Legacy*, Berlin: de Gruyter - Sluga, H., 1980,
*Gottlob Frege*, London: Routledge and Kegan Paul - Wright, C., 1983,
*Frege's Conception of Numbers as Objects*, Aberdeen: Aberdeen University Press

- MacTutor History of Mathematics Archive
- Metaphysics Research Lab Web Page on Frege
- Brian Carver's Web Page on Frege

Edward N. Zalta

*First published: September 14, 1995*

*Content last modified: March 6, 1998*