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- Frege's Life
- Frege's Advances in Logic
- Frege's Ontology and Philosophy of Language
- Chronological Catalog of Frege's Work
- 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)

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.

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*.

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: 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 conceptx 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

- Whitehead falls under the concept
*being an author of Principia 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 eitherx=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 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.

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

From 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) andn=m, infer S(m)

But Frege, in effect, noticed the 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.

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.

Chronological Catalog of Frege's Work (PDF file=Adobe Acrobat file)

- Beaney, M., 1996,
*Frege: Making Sense*, London: Duckworth - Bell, D., 1979,
*Frege's Theory of Judgment*, Oxford: Clarendon - 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 - Boolos, G., 1998,
*Logic, Logic, and Logic*, Cambridge, MA: Harvard University Press - 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., 1981,
*The Interpretation of Frege's Philosophy*, Cambridge, MA: Harvard University Press - Dummett, M., 1991,
*Frege: Philosophy of Mathematics*, Cambridge, MA: Harvard University Press - Haaparanta, L., and Hintikka, J., (eds.), 1986,
*Frege Synthesized*, Dordrecht: D. Reidel - 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 of 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 - Pelletier, F.J., 2001, "Did Frege Believe Frege's
Principle",
*Journal of Logic, Language, and Information*,**10**/1: 87-114 - Perry, J., 1977, "Frege on Demonstratives",
*Philosophical Review*,**86**(1977): 474-497 - 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 - Ricketts, T., 1986, "Logic and Truth in Frege",
*Proceedings of the Aristotelian Society*, Supplementary Volume 70, pp. 121-140 - 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 - Sluga, H., 1993,
*The Philosophy of Frege*, New York: Garland, four volumes - Wright, C., 1983,
*Frege's Conception of Numbers as Objects*, Aberdeen: Aberdeen University Press

- Die Grundlagen der Arithmetik, (528 KB PDF file), original German text (maintained by Alain Blachair, Académie de Nancy-Metz)
- MacTutor History of Mathematics Archive
- Metaphysics Research Lab Web Page on Frege
- Brian Carver's Web Page on Frege

*First published: September 14, 1995*

*Content last modified: January 30, 2002*