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Gottlob Frege
Gottlob Frege
Friedrich Ludwig Gottlob Frege (b. 1848, d. 1925) was a German mathematician,
logician, and philosopher who worked at the University of Jena. Frege
essentially reconceived the discipline of logic by constructing a formal system
which, in effect, constituted the first ‘predicate calculus� In this formal
system, Frege developed an analysis of quantified statements and formalized the
notion of a ‘proof�in terms that are still accepted today. Frege then
demonstrated that one could use his system to resolve theoretical mathematical
statements in terms of simpler logical and mathematical notions. One of the
axioms that Frege later added to his system, in the attempt to derive
significant parts of mathematics from logic, proved to be inconsistent.
Nevertheless, his definitions (of the predecessor relation and of the
concept of natural number) and methods (for deriving the axioms of
number theory) constituted a significant advance. To ground his views about the
relationship of logic and mathematics, Frege conceived a comprehensive
philosophy of language that many philosophers still find insightful. However,
his lifelong project, of showing that mathematics was reducible to logic, was
not successful.
- 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 founded the modern discipline of logic by developing a superior method
of formally representing the logic of thoughts and inferences. He did this by
developing: (a) a formal system which formed the basis of modern logic, (b) an
elegant analysis of complex sentences and quantifier phrases that showed an
underlying unity to certain classes of inferences, (c) a deep understanding of
proof and definition, (d) a theory of extensions which, though
seriously flawed, offered an intriguing picture of the foundations of
mathematics, (e) an insightful analysis of statements about number (i.e., of
answers to the question ‘How many?�, (f) definitions and proofs of some of the
basic axioms of number theory from a limited set of logically primitive concepts
and axioms, and (g) a conception of logic as a discipline which has some
compelling features. We discuss these developments in the following
subsections.
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. Though this notation was first outlined in his
Begriffsschrift (1879), the most mature statement of Frege's system was
in his 2-volume Grundgesetze der Arithmetik (1893/1903). Frege's
1893/1903 system is best characterized as a logic of terms which, with the help
of a few definitions, grounds the modern predicate calculus. A predicate
calculus is a formal system (a formal language and a method of proof) in which
one can represent valid inferences among predications, i.e., among statements in
which properties are predicated of objects. Frege's earlier 1879 system was more
of a predicate calculus, and as such, was the first of its kind.
In this subsection, we shall examine the most basic elements of Frege's
1893/1903 term logic and predicate calculus. These are the statements involving
function applications and the simple predications which fall out as a special
case.
2.1.1 The Basis of Frege's Term Logic
In Frege's term logic, the complete expressions are all terms, i.e., denoting
expressions. These include: (a) simple names of objects, like â€?â€?and ‘Ï?â€? (b)
complex terms which denote objects, like �2�and � + 1� and (c)
sentences (which are also complex terms). The complex terms in (b) and (c) are
formed with the help of ‘incomplete expressions�which signify functions, such
as the unary squaring function � )2�and the binary addition
function � )+( )� In these functional expressions, � )�is
used as a placeholder for what Frege called the arguments of the
function; the placeholder reveals that the expressions signifying function are,
on Frege's view, incomplete and stand in contrast to complete expressions such
as those in (a), (b), and (c). (Though Frege thought it inappropriate to call
the incomplete expressions that signify functions ‘names� we shall sometimes do
so in what follows, though the reader should be warned that Frege had reasons
for not following this practice.) Thus, a mathematical expression such as
�2�denotes the result of applying the function ( )2
to the number 2 as argument, namely, the number 4. Similarly, the expression �
+ 1�denotes the result of applying the binary function +(( ),( )) to
the numbers 7 and 1 as arguments, in that order.
Even the sentences of Frege's mature logical system are complex terms; they
are terms that denote truth-values. Frege distinguished two
truth-values, The True and The False, which he took to be objects. The basic
sentences of Frege's system are constructed using the expression
� ) = ( )� which signifies a binary function that maps a
pair of objects x and y to The True if x is identical
to y and maps x and y to The False otherwise. A
sentence such as �2 = 4�therefore denotes the truth-value The True,
while the sentence �2 = 6�denotes the truth-value The False.
An important class of these identity statements are statements of the form
‘Æ?x) = yâ€? where Æ’( ) is any unary function (i.e.,
function of a single variable), x is the argument of the function, and
Æ’(x) is the value that the function Æ’( ) has for the argument
x. There can also be identity statements involving binary functions
(i.e., functions of two variables), namely, Æ’(x,y) =
z. And so on, for functions of more than two variables.
If we replace a complete name appearing in a sentence by a placeholder, the
result is an incomplete expression that signifies a special kind of function
which Frege called a concept. Concepts are functions which map
every argument to one of the truth-values. Thus, � )>2�denotes the
concept being greater than 2, which maps every object greater than 2 to
The True and maps every other object to The False. Similarly,
� )2 = 4�denotes the concept that which when squared is
identical to 4. Frege would say that any object that a concept maps to The
True falls under the concept. Thus, the number 2 falls under the
concept that which when squared is identical to 4. In what follows, we
use lower-case expressions like Æ’( ) to talk generally about functions, and
upper-case expressions like F( ) to talk more specifically about
those functions which are concepts.
Frege supposed that a mathematical claim such as � is prime�should be
formally represented as �EM>P(2)� The verb phrase ‘is prime�is thereby
analyzed as denoting the concept P( ) which maps primes to The
True and everything else to The False. Thus, a simple predication like
� is prime�becomes analyzed in Frege's system as a special case of functional
application.
2.1.2 The Predicate Calculus Within Frege's Term Logic
The preceding analysis of simple mathematical predications led Frege to
extend the applicability of this system to the representation of
non-mathematical thoughts and predications. This move formed the basis of the
modern predicate calculus. Frege analyzed a non-mathematical predicate like ‘is
happy�as signifying a function of one variable which maps its arguments to a
truth-value. Thus, ‘is happy�denotes a concept which can be represented in the
formal system as �EM>H( )� H( ) maps those arguments
which are happy to The True, and maps everything else to The False. The sentence
‘John is happy�(�EM>H(j)� is thereby analyzed as: the object
denoted by ‘John�falls under the concept signified by � ) is happy�
Thus, a simple predication is analyzed in terms of falling under a concept,
which in turn, is analyzed in terms of functions which map their arguments to
truth values. By contrast, in the modern predicate calculus, this last step of
analyzing predication in terms of functions is not assumed; predication is seen
as more fundamental than functional application. The sentence ‘John is happy�is
formally represented as �EM>Hj� where this is a basic form of predication
(‘the object j instantiates or exemplifies the property H�.
In the modern predicate calculus, functional application is analyzable in terms
of predication, as we shall soon see.
In Frege's analysis, the verb phrase ‘loves�signifies a binary function of
two variables: L(( ),( )). This function takes a pair of
arguments x and y and maps them to The True if x
loves y and maps all other pairs of arguments to The False. Although it
is a descendent of Frege's system, the modern predicate calculus analyzes
loves as a two-place relation (Lxy) rather than a function;
some objects stand in the relation and others do not. The difference between
Frege's understanding of predication and the one manifest by the modern
predicate calculus is simply this: in the modern predicate calculus, relations
are taken as basic, and functions are defined as a special case of relation,
namely, those relations R such that for any objects x,
y, and z, if Rxy and Rxz, then y=z.
By contrast, Frege took functions to be more basic than relations. His logic is
based on functional application rather than predication, and so relations become
analyzed as a binary functions which map a pair of arguments to a truth-value.
Thus, a 3-place relation like gives would be analyzed in Frege's logic
as a function that maps arguments x, y, and z to an
appropriate truth-value depending on whether x gives y to
z; the 4-place relation buys would be analyzed as a function
that maps the arguments x, y, z, and u to an
appropriate truth-value depending on whether x buys y from
z for amount u; etc.
So far, we have been discussing Frege's analysis of ‘atomic�statements. To
complete the basic logical representation of thoughts, Frege added notation for
representing more complex statements (such as negated and conditional
statements) and statements of generality (those involving the expressions
‘every�and ‘some�. Though we no longer use his notation for representing
complex and general statements, it is important to see how the notation in
Frege's term logic already contained all the expressive power of the modern
predicate calculus.
There are four special functional expressions which are used in Frege's
system to express complex and general statements:
Intuitive
Significance |
Functional Expression |
The Function It Signifies |
| Statement |
 |
The function which maps The True to The True and maps all other
objects to The False; this is used to indicate that the argument is a
true statement that names The True. |
| Negation |
 |
The function which maps The True to The False and maps all other
objects to The True |
| Conditional |
 |
The function which maps a pair of objects to The False if the first
(i.e., the one named in the bottom branch) is The True and the second
isn't The True, and maps all other pairs of objects to The True |
| Generality |
 |
The second-level function which maps a first-level concept
F( ) to The True if F( ) maps every object
to The True; otherwise it maps F( ) to The
False. |
The best way to understand this notation is by way of some tables, which show
some specific examples of statements and how those are rendered in Frege�
notation and in the modern predicate calculus.
2.2.1 Truth-functional Connectives
The first table shows how Frege's logic can express the truth-functional
connectives such as not, if-then, and, or, and if-and-only-if.
| Example |
Frege's Notation |
Modern Notation |
| John is happy |
 |
Hj |
| It is not the case that John is happy |
 |
¬Hj |
| If the sun is shining, then John is happy |
 |
Ss �Hj |
| The sun is shining and John is happy |
 |
Ss & Hj |
| Either the sun is shining or John is happy |
 |
Ss  Hj |
| The sun is shining if and only if John is happy |
 |
Ss �
Hj |
As one can see, Frege didn't use the primitive connectives ‘and� ‘or� or
‘if and only if� but always used canonical equivalent forms defined in terms of
negations and conditionals. Note the last row of the table �when Frege wants to
assert that two conditions are materially equivalent, he uses the identity sign,
since this says that they denote the same truth-value. In the modern sentential
calculus, the biconditional does something equivalent, for a statement of the
form φ≡Ï?is true whenever φ and ψ are both true or both false. The only
difference is, in the modern sentential calculus φ and ψ are not construed as
terms denoting truth-values, but rather as sentences having truth conditions
(though, in the semantics of the sentential calculus, sentences are assigned
truth-values as their ‘semantic value� and they are considered true/false
according to which truth-value serves as their semantic value).
2.2.2 Quantified Statements
The table below compares statements of generality in Frege-style notation and
in the modern predicate calculus. We say ‘Frege-style" notation because we are
modifying Frege's notation a bit so as to simplify the presentation; we shall
not use the special typeface (Gothic) that Frege used for variables in general
statements, or observe some of the special conventions that he adopted, for
reasons that would distract us from this introduction.
| Example |
Frege-Style Notation |
Modern Notation |
| Everything is mortal |
 |
�xMx |
| Something is mortal |
 |
¬�x¬Mx
i.e.,
�EM>xMx |
| Nothing is mortal |
 |
�x¬Mx
i.e., ¬�EM>xMx |
| Every person is mortal |
 |
�x(Px �Mx) |
| Some person is mortal |
 |
¬�x(Px �¬Mx)
i.e.,
�EM>x(Px & Mx) |
| No person is mortal |
 |
�x(Px �¬Mx)
i.e.,
¬�EM>x(Px & Mx) |
| All and only persons are mortal |
 |
�x(Px �
Mx) |
Note the last line. Here again, Frege uses the identity sign to help state
the material equivalence of two concepts. He can do this because materially
equivalent concepts F and G are such that F maps an
object x to The True whenever G maps x to The True;
i.e., for all arguments x, F and G map x to
the same truth-value.
In the modern predicate calculus, the symbols ‘∀�(‘every� and ‘∃�(‘some�
are called the ‘universal�and ‘existential�quantifier, respectively, and the
variable �EM>x�in the sentence ‘∀xMx�is called a ‘quantified
variable� or ‘variable bound by the quantifier� We will follow this practice
of calling statements involving one of these quantifier phrases ‘quantified
statements� As one can see from the table above, Frege didn't use an
existential quantifier. He was aware that a statement of the form
‘∃x(��could always be defined as ‘¬∀x¬(��
It is important to mention here that the predicate calculus formulable in
Frege's logic is a ‘second-order�predicate calculus. This means it allows
quantification over functions as well as quantification over objects; i.e.,
statements of the form ‘Every function Æ’ is such that …â??and ‘Some function Æ’ is
such that …â??are allowed. Thus, the statement ‘objects a and b
fall under the same concepts�would be written as follows in Frege-style
notation:
whereas in the modern second-order predicate calculus, we write this as:
�F(Fa �Fb)
Readers interested in learning more about Frege's notation can consult Beaney
(1997, Appendix 2), Furth (1967), and Reck & Awodey (2004, 26�4). In what
follows, however, we shall continue to use the notation of the modern predicate
calculus instead of Frege-style notation. In particular, we adopt the following
conventions. (1) We shall often use �EM>Fx�instead of
�EM>F(x)�to represent the fact that x falls under the
concept F; we use �EM>Rxy�instead of
�EM>R(x,y)�to represent the fact that x stands
in the relation R to y; etc. (2) Instead of using expressions
with placeholders, such as � ) = ( )�and
�EM>P( )� to signify functions and concepts, we shall simply use ��
and �EM>P� (3) When replace one of the complete names in a sentence by a
variable, the resulting expression will be called an open sentence or
an open formula. Thus, whereas �<2�is a sentence,
�<x�is an open sentence; and whereas �EM>Hj�is a formal
sentence that might be used to represent ‘John is happy� the expression
�EM>Hx�is an open formula which might be rendered �EM>x is happy�
in natural language. (4) Finally, we shall on occasion employ the Greek symbol φ
as a metavariable ranging over formal sentences, which may or may not be open.
Thus, ‘Ï?a)â€?will be used to indicate any sentence (simple or complex)
in which the name â€?EM>aâ€?appears; ‘Ï?a)â€?is not to be
understood as Frege-notation for a function φ applied to argument a.
Similarly, ‘Ï?x)â€?will be used to indicate an open sentence in which
the variable x may or may not be free, not a function of
x.
2.2.3 Frege's Logic of Quantification
Frege's functional analysis of predication coupled with his understanding of
generality freed him from the limitations of the ‘subject-predicate�analysis of
ordinary language sentences that formed the basis of Aristotelian logic and it
made it possible for him to develop a more general treatment of inferences
involving ‘every�and ‘some� 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 subject terms whereas the rule
governing the second inference governs reasoning within the predicate, and thus
applies only to the transitive verb complements (i.e., direct objects). In
Aristotelian logic, these inferences have nothing in common.
In Frege's logic, however, 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� That's because the subject John and the direct
object Mary are both considered on a logical par, as arguments of the function
loves. In effect, Frege saw no logical difference between the subject
‘John�and the direct object ‘Mary� What is logically important is that ‘loves�
denotes a function of 2 arguments. No matter whether the quantified expression
‘something�appears as subject (‘Something loves Mary� or within a predicate
(‘John loves something�, it is to be resolved in the same way. In effect, Frege
treated these quantified expressions as variable-binding operators. The
variable-binding operator ‘some x is such that�can bind the variable
�EM>x�in the open sentence �EM>x loves Mary�as well as the
variable �EM>x�in the open sentence ‘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.
Both inferences are instances of a single valid inference rule. To see this
more clearly, here are the formal representations of the above informal
arguments:
- Ljm ��EM>x(Lxm)
- Ljm ��EM>x(Ljx)
The logical axiom which licenses both inferences has the form:
Ra1�EM>ai�EM>
an �
�EM>x(Ra1�EM>x�EM>
an),
where R is a relation that can take n arguments, and
a1,�an are any constants
(names), for any ai such that
1�EM>i�EM>n. This logical axiom tells us that from a simple
predication involving an n-place relation, one can existentially
generalize on any argument, and validly derive a existential statement.
Indeed, this axiom can be made even more general. If φ(a) is any
statement (formula) in which a constant (name) a appears, and
φ(x) is the result of replacing one or more occurrences of a
by x, then the following is a logical axiom:
φ(a) ��EM>xφ(x)
The inferences which start with the premise ‘John loves Mary� displayed
above, both appeal to this axiom for justification. This axiom is actually
derivable as a theorem from Frege's Basic Law IIa (1893, §47). Basic Law IIa
asserts �xφ(x) �φ(a), and the above axiom for the
existential quantifier can be derived from IIa using the rules governing
conditionals, negation, and the definition of �EM>x(� discussed above.
There is one other consequence of Frege's logic of quantification that should
be mentioned. Frege took claims of the form �EM>x(� to be existence
claims. He suggested that existence is not a concept under which
objects fall but rather a second-level concept under which first-level concepts
fall. A concept F falls under this second-level concept just in case
F maps at least one object to The True. So the claim ‘Martians don't
exist�is analyzed as an assertion about the concept martian, namely,
that nothing falls under it. Frege therefore took existence to be that
second-level concept which maps a first-level concept F to The True
just in case �EM>xFx and maps all other concepts to The False. Many
philosophers have thought that this analysis validates Kant's view that
existence is not a (real) predicate.
2.3.1 Proof
Frege's system (i.e., his term logic/predicate calculus) consisted of a
language and an apparatus for proving statements. The latter consisted of a set
of logical axioms (statements considered to be truths of logic) and a set of
rules of inference that lay out the conditions under which certain statements of
the language may be correctly inferred from others. Frege made a point of
showing how every step in a proof of a proposition was justified either in terms
of one of the axioms or in terms of one of the rules of inference or justified
by a theorem or derived rule that had already been proved.
Thus, as part of his formal system, Frege developed a strict understanding of
a ‘proof� In essence, he defined a proof to be any finite sequence of
statements such that each statement in the sequence either is an axiom or
follows from previous members by a valid rule of inference. Thus, a proof of a
theorem of logic, say φ, is therefore any finite sequence of statements (with φ
the final statement in the sequence) such that each member of the sequence: (a)
is one of the logical axioms of the formal system, or (b) follows from previous
members of the sequence by a rule of inference. These are essentially the
definitions that logicians still use today.
2.3.2 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 der Arithmetik, II (1903, 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 (1903, Sections
139-147), Frege criticized 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.
2.4.1 Courses-of-Values and Extensions
Frege's ontology consisted of two fundamentally different types of entities,
namely, functions and objects (1891, 1892b, 1904). Functions are in some sense
‘unsaturated� i.e., they are the kind of thing which take objects as arguments
and map those arguments to a value. This distinguishes them from objects. As
we've seen, the domain of objects included two special objects, namely, the
truth-values The True and The False.
In his work of 1893/1903, Frege attempted to expand the domain of objects by
systematically associating, with each function Æ’, an object which he called
the course-of-values of Æ’. The course-of-values of a function is a
record of the value of the function for each argument. The principle Frege used
to systematize courses-of-values is Basic Law V (1893/§20;):
The course-of-values of the concept Æ’ is identical to the
course-of-values of the concept g if and only if Æ’ and g
agree on the value of every argument (i.e., if and only if for every object
x, Æ’(x) = g(x)).
Frege used the following notation to denote the course-of-values of the
function Æ’:
where the first occurrence of the Greek ε (with the smooth-breathing mark
over it) is a ‘variable-binding operator�which we might read as ‘the
course-of-values of� Using this notation, Frege formally represented Basic Law
V in his system as:
Basic Law V
(Actually, Frege used an identity sign instead of the biconditional as the
main connective of this principle, for reasons described above.)
Frege called the course-of-values of a concept F its
extension. The extension of a concept F records just those
objects which F maps to The True. Thus Basic Law V applies equally well
to the extensions of concepts. Let ‘Ï?x)â€?be an open sentence of any
complexity with the free variable x (the variable x may have
more than one occurrence in φ(x), but for simplicity, assume it has
only one occurrence). Then using the variable-binding operator ε, Frege would
use the expression ‘εÏ?ε)â€?(with a smooth-breathing mark over the first epsilon
and where the second epsilon replaces x in φ(x)) to denote the
extension of the concept φ. Where �EM>n�is the name of an object, Frege
could define ‘object n is an element of the extension of the concept φ�
in the following simple terms: ‘the concept φ maps n to The True�
(i.e., φ(n)). For example, the number 3 is an element of the extension
of the concept odd number greater than 2 if and only if this concept
maps 3 to The True.
Unfortunately, Basic Law V implies a contradiction, and this was pointed out
to Frege by Bertrand Russell just as the second volume of the
Grundgesetze was going to press. Russell recognized that some
extensions are elements of themselves and some are not; the extension of the
concept extension is an element of itself, since that concept would map
its own extension to The True. The extension of the concept spoon is
not an element of itself, because that concept would map its own extension to
The False (since extensions aren't spoons). But now what about the concept
extension which is not an element of itself? Let E represent
this concept and let e name the extension of E. Is e
an element of itself? Well, e is an element of itself if and only if
E maps e to The True (by the definition of ‘element of�given
at the end of the previous paragraph, where e is the extension of the
concept E). But E maps e to The True if and only if
e is an extension which is not an element of itself, i.e., if and only
if e is not an element of itself. We have thus reasoned that e
is an element of itself if and only if it is not, showing the incoherency in
Frege's conception of an extension.
Further discussion of this problem can be found in the entry on Russell's Paradox,
and a more complete explanation of how the paradox arises in Frege's system is
presented in the entry on Frege's logic, theorem,
and foundations for mathematics.
2.4.2 Proposed Foundation for Mathematics
Before he became aware of Russell's paradox, Frege attempted to construct a
logical foundation for mathematics. Using the logical system containing Basic
Law V (1893/1903), he attempted to demonstrate the truth of the philosophical
thesis known as logicism, i.e., the idea not only that mathematical
concepts can be defined in terms of purely logical concepts but also that
mathematical principles can be derived from the laws of logic alone. But given
that the crucial definitions of mathematical concepts were stated in terms of
extensions, the inconsistency in Basic Law V undermined Frege's attempt to
establish the thesis of logicism. Few philosophers today believe that
mathematics can be reduced to logic in the way Frege had in mind. Mathematical
theories such as set theory seem to require some non-logical concepts (such as
set membership) which cannot be defined in terms of logical concepts, at least
when axiomatized by certain powerful non-logical axioms (such as the proper
axioms of Zermelo-Fraenkel 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 nevertheless offered
philosophical logicians an intriguing conceptual framework. The ideas of Bertrand Russell and Alfred North Whitehead
in Principia
Mathematica owe a huge debt to the work found in Frege's
Grundgesetze.
Despite Frege's failure to provide a coherent systematization of the notion
of an extension, we shall make use of the notion in what follows to explain
Frege's theory of numbers and analysis of number statements. Though the
discussion will involve the notion of an extension, we shall not require Basic
Law V; thus, we can use our informal understanding of the notion. In addition,
extensions can be rehabilitated in various ways, either axiomatically as in
modern set theory (which appears to be consistent) or as in various consistent
reconstructions of Frege's system.
In what has come to be regarded as a seminal treatise, Die Grundlagen der
Arithmetik (1884), Frege began work on the idea of deriving some of the
basic principles of arithmetic from what he thought were more fundamental
logical principles and logical concepts. Philosophers today still find that work
insightful. The leading idea is that a statement of number, such as ‘There are
nine planets�and ‘There are two authors of Principia Mathematica� is
really a statement about a concept. Frege realized that one and the same
physical phenomena could be conceptualized in different ways, and that answers
to the question ‘How many?�only make sense once a concept F is
supplied. Thus, one and the same physical entity might be conceptualized as
consisting of 1 army, 5 divisions, 20 regiments, 100 companies, etc., and so the
question ‘How many?�only becomes legitimate once one supplies the concept being
counted, such as army, division, regiment, or
company (1884, §46).
Using this insight, Frege took true statements like ‘There are nine planets�
and ‘There are two authors of Principia Mathematica�to be higher-order
claims about the concepts planet and author of Principia
Mathematica, respectively. In the second case, the higher-order claim
asserts that the first-order concept being an author of Principia
Mathematica falls under the second-order concept being a concept under
which two objects fall. This sounds circular, since it looks like we have
analyzed
There are two authors of Principia
Mathematica,
which involves the concept two, as
The concept being an author of Principia Mathematica
falls under the concept being a concept under which two objects
fall,
which also involves the concept two. But despite appearances, there
is no circularity, since Frege analyzes the second-order concept being a
concept under which two objects fall without appealing to the concept
two. He did this by defining �EM>F is a concept under which two
objects fall� in purely logical terms, as any concept F that satisfies
the following condition:
There are distinct things x and y that fall
under the concept F and anything else that falls under the concept
F is identical to either x or y.
In the notation of the modern predicate calculus, this is formalized as:
�EM>x�EM>y(x�EM>y & Fx
& Fy & �z(Fz �z=x z=y))
Note that the concept being an author of Principia Mathematica
satisfies this condition, since there are distinct objects x and
y, namely, Bertrand Russell and Alfred North Whitehead, who authored
Principia Mathematica and who are such that anything else authoring
Principia Mathematica is identical to one of them. In this way, Frege
analyzed a statement of number (‘there are two authors of Principia
Mathematica� as higher-order logical statements about concepts.
Frege then took his analysis one step further. He noticed that each of the
conditions in the following sequence of conditions defined a class of
‘equinumerous�concepts, where �EM>F�in each case is a variable ranging
over concepts:
| Condition (0): |
Nothing falls under F
¬�EM>xFx |
| Condition (1): |
Exactly one thing falls under F
�EM>x(Fx
& �y(Fy �y=x)) |
| Condition (2): |
Exactly two things fall under F.
�EM>x�EM>y(x�EM>y & Fx &
Fy & �z(Fz �z=x z=y))
|
| Condition (3): |
Exactly three things fall under F.
�EM>x�EM>y�EM>z(x�EM>y &
x�EM>z & y�EM>z & Fx
& Fy & Fz & �w(Fw �
w=x w=y
w=z))
|
| etc. |
|
Notice that if concepts P and Q are both concepts which
satisfy one of these conditions, then there is a one-to-one correspondence
between the objects which fall under P and the objects which fall under
Q. That is, if any of the above conditions accurately describes both
P and Q, then every object falling under P can be
paired with a unique and distinct object falling under Q and, under
this pairing, every object falling under Q gets paired with some unique
and distinct object falling under P. (By the logician's understanding
of the phrase ‘every� this last claim even applies to those concepts P
and Q which satisfy Condition (0).) Frege would call such P
and Q equinumerous concepts (1884, §72). Indeed, for each
condition defined above, the concepts that satisfy the condition are all
pairwise equinumerous to one another.
With this notion of equinumerosity, Frege defined ‘the number of the concept
F�(or, more informally, ‘the number of Fs� to be the
extension or set of all concepts that are equinumerous with F (1884,
§68). For example, the number of the concept author of Principia
Mathematica is the extension of all concepts that are equinumerous to that
concept. This number is therefore identified with the class of all concepts
under which two objects fall, as this is defined by Condition (2) above. Frege
specifically identified the number 0 as the number of the concept not being
self-identical (1884, §74). It is a theorem of logic that nothing falls
under this concept. Thus, it is a concept that satisfies Condition (0) above.
Frege thereby identified the number 0 as the class of all concepts under which
nothing falls, since that is the class of concepts equinumerous with the concept
not being self-identical. Essentially, Frege identified the number 1 as
the class of all concepts which satisfy Condition (1). And so forth. But though
this defines a sequence of entities which are numbers, this procedure doesn't
actually define the concept natural number (finite number).
Frege, however, had an even deeper idea about how to do this.
In order to define the concept of natural number, Frege first
defined, for every 2-place relation R, the general concept �EM>x
is an ancestor of y in the R-series� This new relation is
called ‘the ancestral of the relation R� The ancestral of the relation
R was first defined in Frege's Begriffsschrift (1879, §26,
Proposition 76; 1884, §79). 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 �EM>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 (Frege referred to this as:
y follows x in the R-series). 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 (1884, §76):
x precedes y iff there is a concept F and an
object z such that:
- z falls under F,
- y is the (cardinal) number of the concept F, and
- x is the (cardinal) number of the concept object other than
z falling under F
In the notation of the second-order predicate calculus, augmented by the
functional notation �F�to denote the number of Fs and by the
λ-notation ‘[λu φ]�to name the complex concept being a u
such that φ, Frege's definition becomes:
Precedes(x,y)
=df �EM>F�EM>z(Fz &
y=#F & x=#[λu Fu &
u�EM>z])
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 F (e.g., let F = being an author of Principia
Mathematica) and an object z (e.g., let z = Alfred North
Whitehead) such that:
- Whitehead falls under the concept author of Principia
Mathematica,
- 2 is the (cardinal) number of the concept author of Principia
Mathematica, and
- 1 is the (cardinal) number of the concept author of Principia
Mathematica other than Whitehead.
Note that the last conjunct is true because there is exactly 1 object
(namely, Bertrand Russell) which falls under the concept object other than
Whitehead which falls under the concept of being an author of Principia
Mathematica.
Thus, Frege has a definition of precedes which applies to the pairs
<0,1>, <1,2>, <2,3>,� Frege then defined the ancestral of
this relation, namely, x is an ancestor of y in the predecessor-series.
Though the exact definition will not be given here, we note that it has the
following consequence: 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.
Recall that Frege defined the number 0 as the number of the concept not
being self-identical, and that 0 thereby becomes identified with the
extension of all concepts which fail to be exemplified. Using this definition,
Frege defined (1884, §83):
x is a natural number iff either x=0 or 0 is an
ancestor of x in the predecessor-series
In other words, a natural number is any member of the predecessor-series
beginning with 0.
Using this definition as a basis, Frege later derived many important theorems
of number theory. Philosophers only recently appreciated the importance of this
work (C. Parsons 1965, Smiley 1981, Wright 1983, and Boolos 1987, 1990, 1995).
Wright 1983 in particular showed how the Dedekind/Peano axioms for number might
be derived from one of the consistent principles that Frege discussed in 1884,
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�. It was recently shown by R. Heck [1993] that,
despite the logical inconsistency in the system of Frege 1893/1903, Frege
himself validly derived the the Dedekind/Peano axioms from Hume's Principle.
Although Frege used his inconsistent axiom, Basic Law V, to establish Hume's
Principle, once Hume's Principle was established, the subsequent derivations of
the Dedekind/Peano axioms make no further essential appeals to Basic Law V.
Following the lead of George Boolos, philosophers today call the 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.
Before receiving the famous letter from Bertrand Russell informing him of the
inconsistency in his system, Frege thought that he had shown that arithmetic is
reducible to the analytic truths of logic (i.e., statements which are true
solely in virtue of the meanings of the logical words appearing in those
statements). It is recognized today, however, that at best Frege showed that
arithmetic is reducible to second-order logic extended only by Hume's Principle.
Some philosophers think Hume's Principle is analytically true (i.e., true in
virtue of the very meanings of its words), while others resist the claim, and
there is an interesting debate on this issue in the literature. However, for the
purposes of this introductory essay, there are prior questions on which it is
more important to focus, concerning the nature of Frege's logic, namely, ‘Did
Frege's 1879 or 1893/1903 system (excluding Basic Law V) contain any
extralogical resources?� and ‘How did Frege's conception of logic
differ from that of his predecessors, and in particular, Kant's?�For even if
Frege had been right in thinking that arithmetic is reducible to truths of
logic, it is well known that Kant thought that arithmetic consisted of synthetic
(a priori) truths and that it was not reducible to analytic
logical truths. But, of course, Frege's view and Kant's view contradict each
other only if they have the same conception of logic. Do they?
MacFarlane 2002 addresses this question, and points out that their
conceptions differ in various ways:
�the resources Frege recognizes as logical far outstrip those of
Kant's logic (Aristotelian term logic with a simple theory of disjunctive and
hypothetical propositions added on). The most dramatic difference is that
Frege's logic allows us to define concepts using nested quantifiers, while
Kant's is limited to representing inclusion relations.
MacFarlane goes on to point out that Frege's logic also contains higher-order
quantifiers (i.e., quantifiers ranging over concepts), and a logical functor for
forming singular terms from open sentences (i.e., the expression ‘the extension
of�takes the open sentence φ(x) to yield the singular term, ‘the
extension of the concept φ(x)�. MacFarlane notes that if we were to
try to express such resources in Kant's system, we would have to appeal to
non-logical constructions which make sense only with respect to a faculty of
‘intuition� that is, an extralogical source which presents our minds with
(sensible) phenomena about which judgments can be formed. Frege denies Kant's
dictum (A51/B75), ‘Without sensibility, no object would be given to us�
claiming that 0 and 1 are objects but that they ‘can't be given to us in
sensation�(1884, 101). (Frege's view is that our understanding can grasp them
as objects if their definitions can be grounded in analytic propositions
governing extensions of concepts.)
The debate over which resources require an appeal to intuition and which do
not is an important one, since Frege dedicated himself to the idea of
eliminating appeals to intuition in the proofs of the basic propositions of
arithmetic. Frege saw himself very much in the spirit of Bolzano (1817), who
eliminated the appeal to intuition in the proof of the intermediate value
theorem in the calculus by proving this theorem from the definition of
continuity, which had recently been defined in terms of the definition of a
limit (see Coffa 1991, 27). A Kantian might very well simply draw a graph of a
continuous function which takes values above and below the origin, and thereby
‘demonstrate�that such a function must cross the origin. But both Bolzano and
Frege saw such appeals to intuition as potentially introducing logical gaps into
proofs. There are good reasons to be suspicious about such appeals: (1) there
are examples of functions which we can't graph or otherwise construct for
presentation to our intuitive faculty—consider the function ƒ which maps
rational numbers to 0 and irrational numbers to 1, or consider those functions
which are everywhere continuous but nowhere differentiable; (2) once we take
certain intuitive notions and formalize them in terms of explicit definitions,
the formal definition might imply counterintuitive results; and (3) the rules of
inference from statements to constructions and back are not always clear. Frege
explicitly remarked upon the fact that he labored to avoid constructions and
appeals to intuition in the proofs of basic propositions of arithmetic (1879,
Preface/5, Part III/§23; 1884, § 62, 87; 1893, §0; and 1903, Appendix).
This brings us to one of the most important differences between the Frege's
logic and Kant's. Frege's second-order logic included a Rule of Substitution
(Grundgesetze I, 1893, §48, item 9), which allows one to substitute
complex open formulas into logical theorems to produce new logical theorems.
This rule is equivalent to a very powerful existence condition governing
concepts known as the Comprehension Principle for Concepts. This principle
asserts the existence of a concept corresponding to every open formula of the
form φ(x) with free variable x, no matter how complex φ is.
From Kant's point of view, existence claims were thought to be synthetic and in
need of justification by the faculty of intuition. So, although it was one of
Frege's goals to avoid appeals to the faculty of intuition, there is a real
question as to whether his system, which involves an inference rule equivalent
to a principle asserting the existence of a wide range of concepts, really is
limited in its scope to purely logical laws of an analytic nature.
One final important difference between Frege's conception of logic and Kant's
concerns the question of whether logic has any content unique to itself. As
MacFarlane 2002 points out, one of Kant's most central views about logic is that
its axioms and theorems are purely formal in nature, i.e., abstracted from all
semantic content and concerned only with the forms of judgments, which are
applicable across all the physical and mathematical sciences (1781/1787,
A55/B79, A56/B80, A70/B95). By contrast, Frege took logic to have its own unique
subject matter, which included not only facts about concepts (concerning
negation, subsumption, etc.), identity, etc. (Frege 1906, 428), but also facts
about ancestrals of relations and natural numbers (1879, 1893). Logic is not
purely formal, from Frege's point of view, but rather can provide substantive
knowledge of objects and concepts.
Despite these fundamental differences in their conceptions of logic, Kant and
Frege may have agreed that the most important defining characteristic of logic
is its generality, i.e., the fact that it provides norms (rules, prescriptions)
that are constitutive of thought. This rapprochement between Kant and Frege is
developed in some detail in MacFarlane 2002. The reader will find there reasons
for thinking that Kant and Frege may have shared enough of a common conception
about logic for us to believe that equivocation doesn't undermine the apparent
inconsistency between their views on the reducibility of arithmetic to logic. It
is by no means settled as to how we should think of the relationship between
arithmetic and logic, since logicians have not yet come to agreement about the
proper conception of logic. Many modern logicians have a conception of logic
that is yet different from both Kant and Frege. It is one which evolves out of
the ideas that (1) certain concepts and laws remain invariant under permutations
of the domain of quantification, and (2) that logic ought not to dictate the
size of the domain of quantification. But this conception has not yet been
articulated in a widely accepted way, and so elements common to Frege's and
Kant's conception may yet play a role in our understanding of what logic is.
(For an excellent discussion of Frege's conception of logic, see Goldfarb
2001.)
While pursuing his investigations into mathematics and logic (and quite
possibly, in order to ground those investigations), Frege was led to develop a
philosophy of language. His philosophy of language has had just as much, if not
more, impact than his contributions to logic and mathematics. Frege's seminal
paper in this field ‘Über Sinn und Bedeutung�(‘On Sense and Reference� 1892a)
is now a classic. In this paper, Frege considered two puzzles about language and
noticed, in each case, that one cannot account for the meaningfulness or logical
behavior of certain sentences simply on the basis of the denotations of the
terms (names and descriptions) in the sentence. One puzzle concerned identity
statements and the other concerned sentences with relative clauses such as
propositional attitude reports. To solve these puzzles, Frege suggested that the
terms of a language have both a sense and a denotation, i.e., that at least two
semantic relations are required to explain the significance or meaning of the
terms of a language. This idea has inspired research in the field for over a
century.
3.1.1 Frege's Puzzle About Identity Statements
Here are some examples of identity statements:
117+136 = 253.
The morning star is identical to the evening
star.
Mark Twain is Samuel Clemens.
Bill is Debbie's
father.
Frege believed that these statements all have the form �EM>a=b� where
�EM>a�and �EM>b�are either names or descriptions that
denote individuals. He naturally assumed that a sentence of the form
�EM>a=b�is true if and only the object a just is
(identical to) the object b. For example, the sentence �17+136 = 253�
is true if and only if the number 117+136 just is the number 253. And the
statement ‘Mark Twain is Samuel Clemens�is true if and only if the person Mark
Twain just is the person Samuel Clemens.
But Frege noticed (1892) that this account of truth can't be all there is to
the meaning of identity statements. The statement �EM>a=a�has a cognitive
significance (or meaning) that must be different from the cognitive significance
of �EM>a=b� 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 -- you have to examine the world to see whether the two
persons are the same. Similarly, whereas you can learn that �17+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 �17+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. Now the problem becomes clear: the meaning of
�EM>a=a�clearly differs from the meaning of �EM>a=b� but given the
account of the truth described in the previous paragraph, these two identity
statements appear to have the same meaning whenever they are true! For example,
‘Mark Twain=Mark Twain�is true just in case: the person Mark Twain is identical
with the person Mark Twain. And ‘Mark Twain=Samuel Clemens�is true just in
case: the person Mark Twain is identical with the person Samuel Clemens. But
given that Mark Twain just is Samuel Clemens, these two cases are the same case,
and that doesn't explain the difference in meaning between the two identity
sentences. And something similar applies to all the other examples of identity
statements having the forms �EM>a=a�and �EM>a=b�
So the puzzle Frege discovered is: how do we account for the difference in
cognitive significance between �EM>a=b�and �EM>a=a�when they are
true?
3.1.2 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:
x believes that p
x desires that
p
x intends that p
x discovered
that p
x knows that p
If we replace the variable �EM>x �by the name of a person and replace
the variable �EM>p �with a sentence that describes the propositional
object of their attitude, we get specific attitude reports. So by replacing
�EM>x �by ‘John�and �EM>p �by ‘Mark Twain wrote Huckleberry
Finn�in the first example, the result would be the following specific
belief report:
John believes that Mark Twain wrote Huckleberry
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.
Mark Twain=Samuel Clemens.
Therefore, Samuel Clemens was an author.
Similarly, the following argument is valid.
4 > 3
4=8/2
Therefore, 8/2 > 3
In general, then, the Principle of Substitution seems to take the following
form, where S is a sentence, n and m are names, and
S(n) differs from S(m) only by the fact that at least one
occurrence of m replaces n:
From S(n) and n=m, infer
S(m)
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.
But Frege, in effect, noticed the following counterexample to the Principle
of Substitution. Consider the following argument:
John believes that Mark Twain wrote Huckleberry Finn.
Mark Twain=Samuel Clemens.
Therefore, John believes that Samuel
Clemens wrote Huckleberry Finn.
This 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 reading
Huckleberry Finn but learns the name ‘Samuel Clemens�in the context of
learning about 19th century American authors (without learning that the name
‘Mark Twain�was a pseudonym for Samuel Clemens). John may not believe
that Samuel Clemens wrote Huckleberry Finn. The premises of the above
argument, therefore, do not logically entail the conclusion. So the Principle of
Substitution appears to break down in the context of propositional attitude
reports. The puzzle, then, is to say what causes the principle to fail in these
contexts. Why aren't we still saying something true about the man in question if
all we have done is changed the name by which we refer to him?
To explain these puzzles, Frege suggested that in addition to having a
denotation, names and descriptions also express a sense. The sense of
an expression accounts for its cognitive significance—it is the way by which one
conceives of the denotation of the term. The expressions ��and �/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
�EM>a=a�and �EM>a=b� Since the sense of �EM>a�differs from
the sense of �EM>b� the components of the sense of �EM>a=a�and the
sense of �EM>a=b�are different, guaranteeing that 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 �EM>a=a�and �EM>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 wrote Huckleberry Finn.
John believes that Samuel Clemens wrote Huckleberry
Finn.
If they don't denote the same object, then there is no reason to think that
substitution of one name for another would preserve truth.
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, the
words ‘John�and ‘Mary�in this sentence are names, the expression ‘loves�
signifies a function, and, moreover, the sentence as a whole is a complex name.
Each of these expressions has both a sense and a denotation. The sense and
denotation of the names are basic; but sense and denotation of the sentence as a
whole can be described in terms of the sense and denotation of the names and the
way in which those words are arranged in the sentence alongside the expression
‘loves� Let us refer to the denotation and sense of the words as follows:
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 expression ‘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 expression
‘loves�
We now work toward a theoretical description of the denotation of the
sentence as a whole. On Frege's view, d[j] and
d[m] are the real individuals John and Mary, respectively.
d[L] is a function that maps d[m] (i.e., Mary)
to a function which serves as the denotation of the predicate ‘loves Mary� Let
us refer to that function as d[Lm]. Now the function
d[Lm] maps d[j] (i.e., John) to the denotation
of the sentence ‘John loves Mary� Let us refer to the denotation of the
sentence as d[jLm]. Frege identifies the denotation of a
sentence as one of the two truth values. Because d[Lm] maps
objects to truth values, it is a concept. Thus, d[jLm] is the
truth value The True if John falls under the concept d[Lm];
otherwise it is the truth value The False. So, on Frege's view, the sentence
‘John loves Mary�names a truth value.
The sentence ‘John loves Mary�also expresses a sense. Its sense may be
described as follows. Although Frege doesn't appear to have explicitly said so,
his work suggests that s[L] (the sense of the expression
‘loves� is a function. This function would map 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 again, Frege's work
seems to imply that we should regard s[Lm] as a function which
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.
With this description of language, Frege can give a general account of the
difference in the cognitive significance between identity statements of the form
�EM>a=a�and �EM>a=b� The cognitive significance
is not accounted for at the level of denotation. On Frege's view, the sentences
�=8/2�and �=4�both denote the same truth value. The function
( )=( ) maps 4 and 8/2 to The True, i.e., maps 4 and 4 to The True. So
d[4=8/2] is identical to d[4=4]; they are both
The True. However, the two sentences in question express different thoughts.
That is because s[4] is different from s[8/2].
So the thought s[4=8/2] is distinct from the thought
s[4=4]. Similarly, ‘Mark Twain=Mark Twain�and ‘Mark
Twain=Samuel Clemens�denote the same truth value. However, given that
s[Mark Twain] is distinct from s[Samuel
Clemens], Frege would claim that the thought s[Mark Twain=Mark
Twain] is distinct from the thought s[Mark Twain=Samuel
Clemens].
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 �EM>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 wrote Huckleberry
Finn.
Frege, therefore, would analyze this attitude report as follows: ‘believes
that�denotes a function that maps the denotation of the sentence ‘Mark Twain
wrote Huckleberry Finn�to a concept. In this case, however, the
denotation of the sentence ‘Mark Twain wrote Huckleberry Finn�is not a
truth value but rather a thought. The thought it denotes is different from the
thought denoted by ‘Samuel Clemens wrote Huckleberry Finn�in the
following propositional attitude report:
John believes that Samuel Clemens wrote Huckleberry
Finn.
Since the thought denoted by ‘Samuel Clemens wrote Huckleberry Finn�
in this context differs from the thought denoted by ‘Mark Twain wrote
Huckleberry Finn�in the same context, the concept denoted by ‘believes
that Mark Twain wrote Huckleberry Finn�is a different concept from the
one denoted by ‘believes that Samuel Clemens wrote Huckleberry Finn�
One may consistently suppose that the concept denoted by the former predicate
maps John to The True whereas the the concept denoted by the latter predicate
does not. Frege's analysis therefore preserves our intuition that John can
believe that Mark Twain wrote Huckleberry Finn without believing that
Samuel Clemens did. It also preserves the Principle of Substitution---the fact
that one cannot substitute ‘Samuel Clemens�for ‘Mark Twain�when these names
occur after propositional attitude verbs does not constitute evidence against
the Principle. For if Frege is right, names do not have their usual denotation
when they occur in these contexts.
Chronological Catalog of Frege's Work (PDF)
Works by Frege Cited in this Entry
| 1879 |
Begriffsschrift, eine der arithmetischen nachgebildete
Formelsprache des reinen Denkens, Halle a. S.: Louis Nebert.
Translated as Concept Script, a formal language of pure thought
modelled upon that of arithmetic, by S. Bauer-Mengelberg in J.
vanHeijenoort (ed.), From Frege to Gödel: A Source Book in
Mathematical Logic, 1879-1931, Cambridge, MA: Harvard University
Press, 1967. |
| 1884 |
Die Grundlagen der Arithmetik: eine logisch-mathematische
Untersuchung über den Begriff der Zahl, Breslau: W. Koebner.
Translated as The Foundations of Arithmetic: A logico-mathematical
enquiry into the concept of number, by J.L. Austin, Oxford:
Blackwell, second revised edition, 1974. |
| 1891 |
‘Funktion und Begriff� Vortrag, gehalten in der Sitzung vom 9. Januar
1891 der Jenaischen Gesellschaft für Medizin und Naturwissenschaft, Jena:
Hermann Pohle. Translated as ‘Function and Concept�by P. Geach in
Translations from the Philosophical Writings of Gottlob Frege, P.
Geach and M. Black (eds. and trans.), Oxford: Blackwell, third edition,
1980. |
| 1892a |
‘Über Sinn und Bedeutung� in Zeitschrift für Philosophie und
philosophische Kritik, 100: 25-50. Translated as ‘On
Sense and Reference�by M. Black in Translations from the
Philosophical Writings of Gottlob Frege, P. Geach and M. Black (eds.
and trans.), Oxford: Blackwell, third edition, 1980. |
| 1892b |
‘Über Begriff und Gegenstand� in Vierteljahresschrift für
wissenschaftliche Philosophie, 16: 192-205.
Translated as ‘Concept and Object�by P. Geach in Translations from
the Philosophical Writings of Gottlob Frege, P. Geach and M. Black
(eds. and trans.), Oxford: Blackwell, third edition, 1980. |
| 1893 |
Grundgesetze der Arithmetik, Jena: Verlag Hermann Pohle, Band
I. Partial translation as The Basic Laws of Arithmetic by M.
Furth, Berkeley: U. of California Press, 1964. |
| 1903 |
Grundgesetze der Arithmetik, Jena: Verlag Hermann Pohle, Band
II. |
| 1904 |
‘Was ist eine Funktion?� in Festschrift Ludwig Boltzmann gewidmet
zum sechzigsten Geburtstage, 20. Februar 1904, S. Meyer (ed.),
Leipzig: Barth, 1904, pp. 656-666. Translated as ‘What is a Function?�by
P. Geach in Translations from the Philosophical Writings of Gottlob
Frege, P. Geach and M. Black (eds. and trans.), Oxford: Blackwell,
third edition, 1980. |
| 1906 |
‘Über die Grundlagen der Geometrie�(Second Series), Jahresbericht
der Deutschen Mathematiker-Vereinigung 15, pp.
293-309 (Part I), 377-403 (Part II), 423-430 (Part III). Translation as
‘On the Foundations of Geometry (Second Series)�by E.-H. W. Kluge, in
On the Foundatons of Geometry and Formal Theories of Arthmetic,
New Haven: Yale University Press, 1971. |
- Beaney, M., 1996, Frege: Making Sense, London: Duckworth.
- Beaney, M., 1997, The Frege Reader, Oxford: Blackwell
- Bell, D., 1979, Frege's Theory of Judgment, Oxford: Clarendon.
- Bolzano, B., 1817, ‘Rein analytischer Beweis des Lehrsatzes� in Early
Mathematical Works (1781�848), L. Novy (ed.), Institute of Czechoslovak
and General History CSAS, Prague, 1981.
- 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., 1990, ‘The Standard of Equality of Numbers� in G. Boolos
(ed.), Meaning and Method: Essays in Honor of Hilary Putnam,
Cambridge: Cambridge University Press, 261-77.
- Boolos, G., 1995, ‘Frege's Theorem and the Peano Postulates� The
Bulletin of Symbolic Logic 1, 317-26.
- Boolos, G., 1998, Logic, Logic, and Logic, Cambridge, MA: Harvard
University Press.
- Coffa, J.A., 1991, The Semantic Tradition from Kant to Carnap, L.
Wessels (ed.), Cambridge: Cambridge 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.
- Furth, M., 1967, ‘Editor's Introduction� in G. Frege, The Basic Laws
of Arithmetic, M. Furth (translator and editor), Berkeley: University of
California Press, pp. v-lvii
- Goldfarb, W., 2001, ‘Frege's Conception of Logic� in J. Floyd and S.
Shieh (eds.), Future Pasts: The Analytic Tradition in Twentieth-Century
Philosophy, Oxford: Oxford University Press, 25-41.
- 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.
- Hodges, W., 2001, ‘Formal Features of Compositionality� Journal of
Logic, Language and Information, 10: 7-28.
- Kant, I., 1781, Kritik der reinen Vernunft, Riga: Johann
Friedrich Hartknoch, 1st edition (A), 1781; 2nd edition (B), 1787. Translated
as Critique of Pure Reason by P. Guyer and A. Wood, Cambridge:
Cambridge University Press, 1998.
- Klemke, E. D. (ed.), 1968, Essays on Frege, Urbana, IL:
University of Illinois Press.
- MacFarlane, J., 2002, ‘Frege, Kant, and the Logic in Logicism�
Philosophical Review, 111/1 (January): 25-66.
- Parsons, C., 1965, ‘Frege's Theory of Number� in M. Black (ed.),
Philosophy in America, Ithaca: Cornell, 180-203.
- 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.
- Reck, E., and Awodey, S. (trans./eds.), 2004, Frege's Lectures on
Logic: Carnap's Student Notes, 1910-1914, Chicago and La Salle, IL: Open
Court.
- 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.
- Ricketts, T., forthcoming, Cambridge Companion to Frege,
Cambridge: Cambridge University Press.
- 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. (ed.), 1993, The Philosophy of Frege, New York:
Garland, four volumes.
- Smiley, T., 1981, ‘Frege and Russell� Epistemologica 4: 53-8.
- 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
- Gottlob
Frege, Jena, und die Geburt der modernen Logik (Werner Stelzner, Jena)
- Frege,
Gottlob, by Kevin Klement (U. Massachusetts/Amherst), in the Internet
Encyclopedia of Philosophy.
- Begriffsschrift in LaTeX, documentation for installing and
using begriff.sty, a LaTeX package for typesetting Frege's concept script in
LaTeX.
Frege, Gottlob: logic,
theorem, and foundations for arithmetic | logic: classical |
logic: intensional | logicism | mathematics, philosophy of | neologicism | Principia
Mathematica | quantification | reference | Russell, Bertrand | Russell's paradox
| sense/reference distinction
Acknowledgements
I would like to thank Kai Wehmeier, whose careful eye
as a logician and Frege scholar caught several passages where I had bent the
truth past the breaking point.
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