## 1. Frege's Life and Influences



According to the *curriculum vitae* that the 26-year old Frege
filed in 1874 with his *Habilitationsschrift*, he was born on
November 8, 1848 in Wismar, a town then in Mecklenburg-Schwerin but
now in Mecklenburg-Vorpommern. His father, Alexander, a headmaster of
a secondary school for girls, and his mother, Auguste (nee
Bialloblotzky), brought him up in the Lutheran faith. Frege attended
the local *Gymnasium* for 15 years, and after graduation in 1869,
entered the University of Jena (see Frege 1874, translation in
McGuinness (ed.) 1984, 92).


At Jena, Frege attended lectures by Ernst Karl Abbe, who
subsequently became Frege's mentor and who had a significant
intellectual and personal influence on Frege's life. Frege
transferred to the University of Gottingen in 1871, and two years
later, in 1873, was awarded a Ph.D. in mathematics, having written a
dissertation under Ernst Schering titled *Uber eine
geometrische Darstellung der imaginaren Gebilde in der Ebene*
("On a Geometrical Representation of Imaginary Forms in the
Plane"). Frege explains the project in his thesis as follows:
"By a geometrical representation of imaginary forms in the plane
we understand accordingly a kind of correlation in virtue of which
every real or imaginary element of the plane has a real, intuitive
element corresponding to it" (Frege 1873, translation in
McGuinness (ed.) 1984, 3). Here, by 'imaginary
forms', Frege is referring to imaginary points, imaginary curves
and lines, etc. Interestingly, one section of the thesis concerns the
representation of complex numbers by magnitudes of angles in the
plane.


In 1874, Frege completed his *Habilitationsschrift*,
entitled *Rechnungsmethoden*, *die sich auf eine Erweiterung
des Grossenbegriffes grunden* ("Methods of
Calculation Based on an Extension of the Concept of Quantity").
Immediately after submitting this thesis, the good offices of
Abbe led Frege to become a *Privatdozent* (Lecturer) at
the University of Jena. Library records from the University of Jena
establish that, over the next 5 years, Frege checked out texts in
mechanics, analysis, geometry, Abelian functions, and elliptical
functions (Kreiser 1984, 
 21).[1] 
 No doubt, many of these texts helped him to
prepare the lectures he is listed as giving by the University of Jena
course bulletin, for these lectures are on topics that often match the
texts, i.e., analytic geometry, elliptical and Abelian functions,
algebraic analysis, functions of complex variables, etc. (Kratzsch
1979).[2]


This course of Frege's reading and lectures during the period of
1874-1879 dovetailed quite naturally with the interests he
displayed in his *Habilitationsschrift*. The 'extension
of the concept of quantity' referred to in the title concerns
the fact that our understanding of quantities (e.g., lengths,
surfaces, etc.) has to be extended in the context of complex
numbers.[3] He
says, right at the beginning of this work:



> 
>  According to the old conception, length appears as something material
> which fills the straight line between its end points and at the same
> time prevents another thing from penetrating into its space by its
> rigidity. In adding quantities, we are therefore forced to place one
> quantity against another. Something similar holds for surfaces and
> solid contents. The introduction of negative quantities made a dent in
> this conception, and imaginary quantities made it completely
> impossible. Now all that matters is the point of origin and the end
> point - the idea of filling the space has been completely
> lost. All that has remained is certain general properties of addition,
> which now emerge as the essential characteristic marks of quantity.
> The concept has thus gradually freed itself from intuition and made
> itself independent. This is quite unobjectionable, especially since
> its earlier intuitive character was at bottom mere appearance.
> Bounded straight lines and planes enclosed by curves can certainly be
> intuited, but what is quantitative about them, what is common to
> lengths and surfaces, escapes our intuition. ... There is
> accordingly a noteworthy difference between geometry and arithmetic in
> the way in which their fundamental principles are grounded. The
> elements of all geometrical constructions are intuitions, and geometry
> refers to intuition as the source of its axioms. Since the object of
> arithmetic does not have an intuitive character, its fundamental
> propositions cannot stem from intuition... (Frege 1874,
> translation in McGuinness (ed.) 1984, 56)
>  


Here we can see the beginning of two lifelong interests of Frege,
namely, (1) in how concepts and definitions developed for one domain
fare when applied in a wider domain, and (2) in the contrast between
legitimate appeals to intuition in geometry and illegitimate appeals
to intuition in the development of pure number theory. Indeed, some
recent scholars have (a) shown how Frege's work in logic was informed
in part by his understanding of the analogies and disanalogies between
geometry and number theory (Wilson 1992), and (b) shown that Frege was
intimately familiar with the division among late 19th century
mathematicians doing complex analysis who split over whether it is
better to use the analytic methods of Weierstrass or the intuitive
geometric methods of Riemann (Tappenden 2006). Weierstrass's 1872
paper, describing a real-valued function that is continuous
everywhere but differentiable 
 nowhere,[4]
 was well known and provided an example of an ungraphable functions
that places limits on intuition. Yet, at the same time, Frege clearly
accepted Riemann's practice and methods derived from taking functions
as fundamental, as opposed to Weierstrass's focus on functions that
can be represented or analyzed in terms of other mathematical objects
(e.g., complex power series).


In 1879, Frege published his first book *Begriffsschrift, eine
der arithmetischen nachgebildete Formelsprache des reinen Denkens*
(*Concept Notation: A formula language of pure thought, modelled
upon that of arithmetic*) and was promoted
to *ausserordentlicher Professor* (Extraordinarius Professor)
at Jena. Although the *Begriffsschrift* constituted a major
advance in logic, it was neither widely understood nor well-received.
Some scholars have suggested that this was due to the facts that the
notation was 2-dimensional instead of linear and that he didn't build
upon the work of others but rather presented something radically new
(e.g., Mendelsohn 2005, 2). Though our discussion below is framed
primarily with an eye towards the system Frege developed in his
two-volume work of 1893/1903 (*Grundgesetze der Arithmetik*),
the principal elements of this later system can be found already in
the *Begriffsschrift* of 1879.


Frege's next really significant work was his second book,
*Die Grundlagen der Arithmetik: eine logisch mathematische
Untersuchung uber den Begriff der Zahl*, published in
1884. Frege begins this work with criticisms of previous attempts to
define the concept of number, and then offers his own
analysis. The *Grundlagen* contains a variety of insights still
discussed today, such as: (a) the claim that a statement of number
(e.g., 'there are eight planets') is a higher-order
assertion about a concept (see Section 2.5 below); (b) his famous
Context Principle ("never ask for the meaning of a word in
isolation, but only in the context of a proposition"), and (c)
the formulation of a principle (now called 'Hume's
Principle' in the secondary literature) that asserts the
equivalence of the claim "the number of *F*s is equal to
the number of *G*s" with the claim that "there is a
one-to-one correspondence between the objects falling under *F*
and the objects falling under *G*" (see Section 2.5 below).
More generally, Frege provides in the *Grundlagen* a
non-technical philosophical justification and outline of the ideas
that he was to develop technically in his two-volume
work *Grundgesetze der Arithmetik* (1893/1903).


In the years 1891-1892, Frege gave more thought to the
philosophy of language that would help ground his philosophy of
mathematics. He published three of his most well-known papers,
'Function and Concept' (1891), 'On Sense and
Reference' (1892a), and 'On Concept and Object'
(1892b) in this period. These works were eventually supplemented by
three other papers on the philosophy of language, from a later period,
'The Thought' (1918a), 'Negation' (1918b), and
'Compound Thoughts' (1923). As we shall see in Section
3.3 below, these works have to be considered in light of recent
scholarship providing evidence that Frege heavily borrowed from Stoic
philosophy. Though Frege always motivated and recast (often in
rigorous terms) ideas that he absorbed from others, the documentation
concerning the origins of many of his ideas in the Stoic corpus has to
be taken seriously.



In 1893, he published the first volume of the technical work
previously mentioned, *Grundgesetze der Arithmetik*. In 1896,
he was promoted to *ordentlicher Honorarprofessor* (regular
honorary professor). Six years later (on June 16, 1902), as he was
preparing the proofs of the second volume of
the *Grundgesetze*, he received a letter from Bertrand Russell,
informing him that one could derive a contradiction in the system he
had developed in the first volume. Russell's letter frames the
paradox first in terms of the predicate *P* = 'being a
predicate which cannot be predicated of itself', and then in
terms of the class of all those classes that are not members of
themselves. Frege, in the Appendix to the second volume, rephrased
the paradox in terms of his own system. 



Frege never fully recovered from the fatal flaw discovered in the
foundations of his *Grundgesetze*. His attempts at salvaging
the work by restricting Basic Law V were not successful. However, he
continued teaching at Jena, and from 1903-1917, he published six
papers, including 'What is a Function?' (1904) and
'On the Foundations of Geometry' (Frege 1903b and 1906).
In the latter, Frege criticized Hilbert's understanding and use of the
axiomatic method (see the entry on the
 Frege-Hilbert controversy). From
this time period, we have the lecture notes that Rudolf Carnap took as
a student in two of his courses (see Reck and Awodey 2004). In 1917,
he retired from the University of Jena.


In the last phase of his life, from 1917-1925, Frege
published only the papers mentioned earlier (1918a, 1918b, 1923) and
developed some unpublished fragments of philosophical works.
Unfortunately, his last years saw him become more than just
politically conservative - his diary for a brief period in 1924
show sympathies for fascism and anti-Semitism (see Frege 1924 [1996],
translated by R. Mendelsohn). He died on July 26, 1925, in Bad
Kleinen (now in Mecklenburg-Vorpommern).


## 2. Frege's Logic and Philosophy of Mathematics




Frege provided a foundations for the modern discipline of logic by
developing a more perspicuous method of formally representing the
logic of thoughts and inferences. He did this by developing: (a) a
system allowing one to study inferences formally, (b) an analysis of
complex sentences and quantifier phrases that showed an underlying
unity to certain classes of inferences, (c) an analysis
of *proof* and *definition*, (d) a theory of extensions
which, though seriously flawed, offered an intriguing picture of the
foundations of mathematics, (e) an 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. However, it should be noted that, for those who are
encountering it for the first time, we have simplified the
presentation of Frege's logic in a significant number of ways.
For those already familiar with the notation and who want to see a
more precise presentation, see the entry on
 Frege's logic.


### 2.1 The Basis of Frege's Term Logic and Predicate Calculus




In an attempt to realize Leibniz's ideas for a universal formal
language 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 two systems are best
characterized as term logics, since all of the complete expressions
are denoting terms. Frege analyzed ordinary predication in these
systems, and so they can also be conceived as predicate calculi. 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.




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, all of the terms and well-formed formulas
are denoting expressions. These include: (a) simple names of objects,
like '\(2\)' and '\(\pi\)', (b) complex terms
which denote objects, like '\(2^2\)' and '\(3 +
1\)', 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
'\(2^2\)' denotes the result of applying the
function \((\:)^2\) to the number \(2\) as argument, namely, the
number \(4\). Similarly, the expression '\(7 + 1\)' 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) denoting 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 '\(2^2 = 4\)' therefore denotes the truth-value The
True, while the sentence '\(2^2 = 6\)' denotes The
False.




An important class of these identity statements are statements of the
form '\(f(x) = y\)',
where \(f(\:)\) is any *unary* function
(i.e., function of a single variable), \(x\) is the argument of the
function, and \(f(x)\) is the value of the function for the argument
\(x\). Similarly, \(f(x,y) = z\) is an identity statement involving a
*binary* function of two variables. 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, '\((\:) \gt 2\)' 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 \(f(\:)\) 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 '\(2\) is
prime' should be formally represented as
'\(P(2)\)'. 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 '\(2\) 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 '\(H(\:)\)'. \(H(\:)\) maps those arguments
which are happy to The True, and maps everything else to The
False. The sentence 'John is happy', represented as
'\(H(j)\)', 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
'\(Hj\)', 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 (written: \(Lxy\)) 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 manifested 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; a binary relation is analyzed as a binary function that
maps 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.


### 2.2 Complex Statements and Generality



 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 | Frege-notation | The function which maps The True to The True and maps all other
objects to The False; used to express the thought that the argument of
the function is a true statement. |
| Negation | Frege-notation | The function which maps The True to The False and maps all other
objects to The True |
| Conditional | Frege-notation | The function which maps a pair of objects to The False if the first
(i.e., 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 | Frege-notation | The second-level function which maps a first-level concept
Ph to The True if Ph maps every object
to The True; otherwise it maps Ph 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's 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'sNotation** | **ModernNotation** |
| John is happy | Frege-notation | \(Hj\) |
| It is not the case that John is happy | Frege-notation | \(\neg Hj\) |
| If the sun is shining, then John is happy | Frege-notation | \(Ss \to Hj\) |
| The sun is shining and John is happy | Frege-notation | \(Ss \amp Hj\) |
| Either the sun is shining or John is happy | Frege-notation | \(Ss \lor Hj\) |
| The sun is shining if and only if John is happy | Frege-notation | \(Ss \equiv Hj\) |




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
\(\phi \equiv \psi\) is true whenever \(\phi\) and \(\psi\) are both
true or both false. The only difference is, in the modern sentential
calculus \(\phi\) and \(\psi\) are not construed as terms denoting
truth-values, but rather as sentences having truth conditions. Of
course, Frege could, in his notation, use the sentence '\((\phi
\to \psi) \amp (\psi \to \phi)\)' to assert \(\phi \equiv
\psi\).


#### 2.2.2 Quantified Statements




The table below compares statements of generality in Frege's notation
and in the modern predicate calculus. Frege used a special typeface
(Gothic) for variables in general statements.




|  |  |  |
| --- | --- | --- |
| **Example** | **FregeNotation** | **ModernNotation** |
| Everything is mortal | Frege-notation | \(\forall xMx\) |
| Something is mortal | Frege-notation | \(\neg \forall x\neg Mx\)
i.e., \(\exists xMx\) |
| Nothing is mortal | Frege-notation | \(\forall x \neg Mx\)  i.e., \(\neg \exists
xMx\) |
| Every person is mortal | Frege-notation | \(\forall x(Px \to Mx)\) |
| Some person is mortal | Frege-notation | \(\neg\forall x(Px \to \neg Mx)\)  i.e.,
\(\exists x(Px \amp Mx)\) |
| No person is mortal | Frege-notation | \(\forall x(Px \to \neg Mx)\) 
i.e., \(\neg \exists x(Px \amp Mx)\) |
| All and only persons are mortal | Frege-notation | \(\forall x(Px \equiv Mx)\) |




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
'\(\forall\)' ('every') and
'\(\exists\)' ('some') are called the
'universal' and 'existential' quantifier,
respectively, and the variable '\(x\)' in the sentence
'\(\forall xMx\)' 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 '\(\exists
x\phi \)' could always be defined as '\(\neg \forall x
\neg \phi\)', where \(\phi\) is any formula.




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 \(f\) is such that ...' and 'Some function
\(f\) is such that ...' are allowed. Thus, the statement
'objects *a* and *b* fall under the same
concepts' would be written as follows in Frege'
notation:



> 
> ![Frege-notation](All-F-a-b.svg)
> 



 and in the modern second-order predicate calculus, we write this
as: 



> 
> \(\forall F (Fa \equiv Fb) \)
> 



Readers interested in learning more about Frege's notation can
consult Beaney (1997, Appendix 2), Furth (1967), Reck & Awodey
(2004, 26-34), and Cook (2013). In what follows, however, we
shall continue to use the notation of the modern predicate calculus
instead of Frege's notation. In particular, we adopt the
following conventions:


1. We shall often use '\(Fx\)' instead of
'\(F(x)\)' to represent the fact that \(x\) falls under
the concept \(F\); we use '\(Rxy\)' instead of
'\(R(x,y)\)' to represent the fact that \(x\) stands in
the relation \(R\) to \(y\); etc.
2. Instead of using expressions with placeholders, such as
'\((\:) = (\:)\)' and '\(P(\:)\)', to signify
functions and concepts, we shall simply use '\(=\)' and
'\(P\)'.
3. When one replaces 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 '\(3 <
2\)' is a sentence, '\(3 < x\)' is an open sentence;
and whereas '\(Hj\)' is a formal sentence that might be
used to represent 'John is happy', the expression
'\(Hx\)' is an open formula which might be rendered
'\(x\) is happy' in natural language.
4. Finally, we shall on occasion employ the Greek symbol \(\phi\) as
a metavariable ranging over formal sentences, which may or may not be
open. Thus, '\(\phi (a)\)' will be used to indicate any
sentence (simple or complex) in which the name '\(a\)'
appears; '\(\phi(a)\)' is *not* to be understood as
Frege-notation for a function \(\phi\) applied to argument \(a\).
Similarly, '\(\phi(x)\)' will be used to indicate an open
sentence in which the variable \(x\) may or may not be free, not a
function of \(x\).



Again, it should be noted that the foregoing has been a simplified
presentation of Frege's logic, for those who are encountering it for
the first time. With these basics in hand, see the entry on 
Frege's logic for a more precise
presentation.


#### 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 two 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 '\(x\)' in
the open sentence '\(x\) loves Mary' as well as the
variable '\(x\)' 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:


* \(Ljm\) [?] \(\exists x (Lxm)\)
* \(Ljm\) [?] \(\exists x (Ljx)\)




The logical axiom which licenses both inferences has the form:



> 
> \(Ra\_1 \ldots a\_i \ldots a\_n \to \exists x(Ra\_1 \ldots x \ldots
> a\_n)\),



 where \(R\) is a relation that can take \(n\) arguments, \(a\_1,
\ldots , a\_n\) are any constants (names), and \(1 \leq i \leq
n\). 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 \(\phi (a)\) is
any statement (formula) in which a constant (name) \(a\) appears, and
\(\phi (x)\) is the result of replacing one or more occurrences of
\(a\) by \(x\), then the following is a logical axiom:



> 
> \(\phi (a) \to \exists x \phi (x)\)
> 



 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, SS47). Basic Law IIa is
formulated as \(\forall x\phi (x) \to \phi (a)\), where \(\phi(a)\) is
the result of substituting \(a\) for one or more variables \(x\) bound
by the quantifier (though see below for a more careful discussion of
this axiom). So the axiom displayed above for the existential
quantifier can be derived from IIa using the rules governing
conditionals, negation, and the definition of \(\exists x\phi\)
discussed above. 




There is one other consequence of Frege's logic of
quantification that should be mentioned. Frege took claims of the form
\(\exists x\phi\) 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
\(\exists xFx\). Some philosophers have thought that this analysis
validates Kant's view that *existence* is not a (real)
predicate.


### 2.3 Proof and Definition


#### 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 ph, is
therefore any finite sequence of statements (with ph 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 may take
as a value.


More importantly, however, Frege was the first to claim that a
properly formed definition had to have two important metatheoretical
properties. Let us call the new, defined symbol introduced in a
definition the *definiendum*, and the term that is used to
define the new term the *definiens*. Then Frege was the first
to suggest that proper definitions have to be both *eliminable*
(a definendum must always be replaceable by its definiens in any
formula in which the former occurs) and *conservative* (a
definition should not make it possible to prove new relationships
among formulas that were formerly unprovable). Concerning one of his
definitions in the *Begriffsschrift* (SS24), Frege
writes: 



> We can do without the notation introduced by this
>  sentence, and hence without the sentence itself as its definition;
>  nothing follows from the sentence that could not also be inferred
>  without it. Our sole purpose in introducing such definitions is to
>  bring about an extrinsic simplification by stipulating an
>  abbreviation.



Frege later 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 objects. 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
practice 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.
Creative definitions fail to be conservative, as this was explained
above. 


### 2.4 Courses-of-Values, Extensions, and Proposed Mathematical Foundations


#### 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 \(f\), an
object which he called *the course-of-values of* \(f\). 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/SS20;):



> The course-of-values of the concept \(f\) is identical to
> the course-of-values of the concept \(g\) if and only if \(f\) and
> \(g\) agree on the value of every argument (i.e., if and only if for
> every object \(x\), \(f(x) = g(x)\)).




Frege used the a Greek epsilon with a smooth breathing mark above it
as part of the notation for signifying the course-of-values of
the function \(f\):



> 
> \(\stackrel{,}{\varepsilon}\!\!f(\varepsilon)\) 
> 




where the first occurrence of the Greek \(\varepsilon\) (with the
smooth breathing mark above it) is a 'variable-binding
operator' which we might read as 'the course-of-values
of'. To avoid the appearance of variable clash, we may also use
a Greek \(\alpha\) (also with a smooth breathing mark above) as a
variable-binding operator. Using this notation, Frege formally
represented Basic Law V in his system as:



> 
> **Basic Law V**   
>  
> \(
> \stackrel{,}{\varepsilon}\!\!f(\varepsilon) =
> \, \stackrel{,}{\alpha}\!g(\alpha) \equiv \forall x(f(x) = g(x))
> \)
> 




(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
'\(\phi(x)\)' be an open sentence of any complexity with
the free variable \(x\) (the variable \(x\) may have more than one
occurrence in \(\phi(x)\), but for simplicity, assume it has only one
occurrence). Then Frege would use the expression:



> 
> \(\stackrel{,}{\varepsilon}\!\! \phi (\varepsilon)\),
> 



where the second epsilon replaces \(x\) in \(\phi (x)\), to denote the
extension of the concept \(\phi\). Where '\(n\)' is the
name of an object, Frege could define 'object \(n\) is an
element of the extension of the concept \(\phi\)' in the
following simple terms: 'the concept \(\phi\) maps \(n\) to The
True', i.e., \(\phi(n)\). 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 theorem and foundations for arithmetic.


#### 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,
without assuming Basic Law V. It suffices to use our
informal understanding of the notion, though extensions can be
rehabilitated in various ways, either axiomatically as in modern set
theory or as in the various ways logicians have found to weaken
Frege's system. (See T. Parsons 1987, Burgess 1998, Heck 1996, Wehmeier
1999, Ferreira & Wehmeier 2002, Fine 2002, Anderson & Zalta
2004, Ferreira 2005, and Antonelli & May 2005.)


### 2.5 The Analysis of Statements of Number




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 eight
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, SS46).




Using this insight, Frege took true statements like 'There are
eight planets' and 'There are two authors of *Principia
Mathematica*' to be "second level" claims about
the concepts *planet* and *author of Principia Mathematica*,
respectively. In the second case, the second level claim asserts that
the first-level concept *being an author of Principia
Mathematica* falls under the second-level 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 '\(F\) 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:



> 
> \(\exists x \exists y (x\! \neq\! y \amp Fx \amp Fy \amp
> \forall z(Fz \to z\! =\! x \lor z\! =\! y))\)
> 




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
'*F*' in each case is a variable ranging over
concepts:




|  |  |
| --- | --- |
| Condition (0): | Nothing falls under \(F\) 
 \(\neg \exists xFx\) |
| Condition (1): | Exactly one thing falls under \(F\) 
 \(\exists x(Fx \amp \forall y(Fy \to y\! =\! x))\) |
| Condition (2): | Exactly two things fall under *F*.

\(\exists x \exists y (x\! \neq\! y \amp Fx \amp Fy \amp
\forall z(Fz \to z\! =\! x \lor z\! =\! y))\)
 |
| Condition (3): | Exactly three things fall under *F*. 
\(\exists x \exists y \exists z (x\! \neq\! y \amp x\! \neq\! z
\amp y\! \neq\! z \amp Fx \amp Fy \amp Fz \amp\ \)
\(\forall w(Fw \to w\! =\! x \lor w\! =\! y \lor w\! =\! z))\) |
| 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, SS72). 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 \(F\)' to be the extension consisting of all
the concepts that are equinumerous with \(F\) (1884, SS68).
To get started, Frege then defined Zero to be the number of the
concept *being non-self-identical* (1884, SS74). If we use
the notation \(\#F\) to represent *the number of the concept F*,
and use the \(\lambda\)-notation \([\lambda x \: \phi]\) to name the
complex concept *being an object x such that* \(\phi\), Frege's
definition of Zero becomes:



> 
>  \(0 =\_\mathit{df} \#[\lambda x \: x \neq x]\)
>  



Thus, the number 0 becomes defined as the extension of all the
concepts equinumerous to the concept *not being
self-identical*. This extension contains all the concepts that
satisfy Condition (0) above, and so the number of all such concepts is
0. For example, the number of the concept *being a square
circle* is \(0\), since nothing falls under it. Similarly, one
could define the number \(1\) as the extension consisting of all the
concepts that satisfy Condition (1) above, and define the number \(2\)
as the extension of all the concepts that satisfy Condition (2) above,
and so on. But though this would define a sequence of entities which
are numbers, this procedure doesn't actually define the
concept *natural number* (*finite number*). Frege,
however, had a deep idea about how to do this.


### 2.6 Natural Numbers




In order to define the concept of *natural number*, Frege first
defined, for every \(2\)-place relation \(R\), 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 the relation \(R\) was first defined in
Frege's *Begriffsschrift* (1879, SS26, Proposition
76; 1884, SS79). 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'
ensures that \(a\) is an ancestor of \(b\), \(c\), and \(d\) in this
series, that \(b\) is an ancestor of \(c\) and \(d\) in this series,
and that \(c\) is an ancestor of \(d\) in this series.




More generally, if given a series of facts of the form \(aRb\),
\(bRc\), \(cRd\), and so on, Frege showed how to define the relation
\(R^\*\), i.e., *x is an ancestor of y in the R-series*, which
Frege referred to 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,
SS76):



> 
> \(x\) *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 falling
> under* \(F\) *other than* \(z\)
> 
> 
> 




If we again use the notation \(\#F\) to denote the number of \(F\)s
and the \(\lambda\)-notation \([\lambda u \: \phi]\) to name the
complex concept *being an object* \(u\) *such that*
\(\phi\), Frege's definition becomes:



> 
>  \( \mathit{Precedes}(x,y) =\_\mathit{df} 
>  \exists F \exists z (Fz \,\amp\, y\! =\! \#F \,\amp\, x\! =\!
>  \#[\lambda u \: Fu \amp u\! \neq\! z]) \)
> 




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) that falls under the concept
*author of Principia Mathematica other than Whitehead*.




Thus, Frege has a definition of *precedes* which applies to the
ordered pairs \(\langle 0,1\rangle\), \(\langle 1,2\rangle \),
\(\langle 2,3\rangle\), ... . Frege then defined the
ancestral of this relation, namely, \(x\) *is an ancestor of*
\(y\) *in the predecessor-series*, or
\(\mathit{Precedes}^\*\). Though the exact definition will not be given
here, we note that it has the following consequence: from the facts
that \(10\) precedes \(11\) and \(11\) precedes \(12\), it follows
that \(10\) precedes\(^\*\) \(12\) in the
predecessor-series. Note, however, that although
\(10\) precedes\(^\*\) \(12\), \(10\) does not
precede \(12\), for the notion of *precedes* is that
of *immediately* precedes. Note also that by defining the
ancestral of the precedence relation, Frege had in effect defined
\(x\) < \(y\) relative to the predecessor-series.




Recall that Frege defined the number \(0\) as the number of the concept
*being non-self-identical*, and that \(0\) thereby becomes
identified with the extension of all concepts which fail to be
exemplified. Using this definition, Frege then defined (1884,
SS83) *natural number* as follows:



> 
>  \(x\) is a *number*  =*df* 
>  either \(x\! =\! 0\) or \(0\) is an ancestor of \(x\) in the
>  predecessor-series


which we might represent formally as:



> 
>  \( \mathit{Number}(x) =\_\mathit{df} x\! =\! 0 [?]
>  \mathit{Precedes}^\*(0,x)\)
> 




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 appreciated the importance of
this work only relatively recently (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 \(F\)s is equal
to the number of \(G\)s if and only if there is a one-to-one
correspondence between the \(F\)s and the \(G\)s"). 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 Dedekind/Peano axioms from Hume's
Principle. Although Frege used Basic Law V (which yields an
inconsistency when added to his second-order logic) 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'. For a comprehensive introduction to the subtle and
complex logical reasoning involved in this theorem, see the entry
 Frege's theorem and foundations for
 arithmetic.


### 2.7 Frege's Conception of Logic




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 truths of logic. 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 (see, e.g., Boolos 1997, Wright 1999).


However, for the purposes of this introduction to Frege's work,
there are prior questions on which it is more important to
focus. Whereas Frege thought that the truths of arithmetic are
derivable from analytic truths of logic, Kant thought arithmetic
principles are synthetic, in which case they wouldn't be derivable
from analytic truths. Their different conceptions of logic helps to
explain why these two philosophers came to such different conclusions.
In this section, we therefore turn to the following questions:


1. How did Frege's conception of logic differ from Kant's?
In particular:
	* What resources (or laws) did Kant and Frege both consider to be
	 logical?
	* Did Kant and Frege agree about the content and 
	subject matter of logic?
2. How did Frege's conception of logic differ from that of
later logicians?



An answer to the first question sets the stage for answering the
second. 


#### 2.7.1 How Frege's Conception of Logic Differed from Kant's



One of the most important differences between Kant and Frege concerns
the resources available to logic. Kant's logic is limited to (a)
Aristotelian term logic with a simple theory of disjunctive and
hypothetical propositions, and (b) representing inclusion relations
among concepts (MacFarlane 2002, 26). By contrast, Frege's logic
includes (a) a term-forming operator \(\sm{\varepsilon}\), which
allows one to form the singular term
\(\sm{\varepsilon}f(\varepsilon)\) from the function expression
\(f(\xi)\), and (b) a Rule of Substitution, which allows one to
substitute complex open formulas for free second-order variables in
theorems of logic but also allows one to define, and assert the
existence of, complex concepts, including concepts defined in terms of
quantifiers over concepts. We'll discuss both of these
resources below, but first, the discussion needs some context.



The differences concerning the resources available to logic revolve
around a key issue, namely, whether the additional resources Frege
assigns to logic require an appeal to non-logical constructions,
specifically to a faculty of 'intuition', that is, an
extralogical source which presents our minds with phenomena about
which judgments can be formed. (Recall the discussion above about
Frege's early interest in appeals to intuition.) The debate
over which resources do and do not require an appeal to intuition is
an important one. Frege continued a trend started by Bolzano (1817),
who eliminated the appeal to intuition in the proof of the
Intermediate Value Theorem in the calculus (which in its simplest form
asserts that a continuous function having both positive and negative
values must cross the origin). Bolzano proved this theorem from the
definition of continuity, which had recently been given in terms
similar to the definition of a limit (see Coffa 1991, 27). A Kantian
might 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 appeal to a graph involves
an appeal to intuition, and both Bolzano and Frege saw such appeals to
intuition as potentially introducing logical gaps into a proof. There
are reasons to be suspicious about such appeals: (1) there are
functions which we can't graph or otherwise construct for
presentation to our intuitive faculty, e.g., the function \(f\) which
maps rational numbers to \(0\) and irrational numbers to \(1\), or the
functions noted by Weierstrass, 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 dedicated himself to the idea of eliminating appeals to
intuition in the proofs of the basic propositions of arithmetic. He
explicitly remarked upon this fact in a number of works throughout his
career (1879, Preface/5, Part III/SS23; 1884, SSSS62, 87;
1893, SS0; and 1903, Appendix). Thus, he would deny Kant's
dictum (A51 [B75]), 'Without sensibility, no object would be
given to us', and claim instead that \(0\) and \(1\) are objects
but that they 'can't be given to us in sensation'
(1884, 101). Frege's view is that we can understand or grasp
them as objects if we (a) define them as extensions of concepts, and
(b) show that singular terms of the form
\(\sm{\varepsilon}f(\varepsilon)\) can be axiomatized by an analytic
proposition. (The latter was to be accomplished by Basic Law V, and so
the collapse of this law in light of Russell's paradox
undermined this part of his plan for avoiding appeals to
intuition.)



Moreover, philosophers have questioned whether Frege's Rule of
Substitution (*Grundgesetze I*, 1893, SS48, item 9) also
required an appeal to intuition. The Rule of Substitution allows one
to substitute complex formulas for free second-order variables in
logical theorems to produce new logical theorems. Boolos argued (1985,
336-338) that since Frege's Rule of Substitution is
equivalent to a Comprehension Principle for Concepts, it is
extralogical in character. The Comprehension Principle for Concepts
asserts \(\exists F \forall x(Fx \equiv \phi)\), provided \(\phi\)
doesn't have a free variable \(F\); the proviso blocks the
instance \(\exists F\forall x(Fx \equiv \neg Fx)\), from which one can
quickly derive a contradiction. Thus, the Comprehension Principle for
Concepts asserts the existence of a concept corresponding to every
expressible condition on objects. From Kant's point of view,
such 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 question as to whether his system of second-order logic
(minus Basic Law V), which implies 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.



If we now put aside their differences about logic's resources
and the appeal to intuition, there are other ways in which the Kantian
and Fregean conceptions of logic differ. Both MacFarlane (2002) and
Linnebo (2003) point out that one of Kant's 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) and are applicable across all the
physical and mathematical sciences (1781 [1787], A55 [B79], A56 [B80],
A70 [B95]; and 1800, 15). And Kant takes the laws of logic to be
normative and prescriptive (something one can get wrong), and not just
descriptive (1800, 16); they provide *constitutive norms* of
thought (MacFarlane 2002, 35; Tolley 2008). Indeed, Linnebo takes two
theses, that logic is formal and provides laws that are constitutive
norms of thought, to be distinctive of Kant's conception
(Linnebo 2003, 240).



By contrast, Frege rejects the idea that logic is a purely formal
enterprise (MacFarlane 2002, 29; Linnebo 2003, 243). He took logic to
have its own unique subject matter, which included not only facts
about concepts (concerning negation, subsumption, etc.) and identity,
but also facts about relations (e.g., their properties and
ancestrals). Frege (1906, 428 [1984, 338]) says:



> 
> Just as the concept point belongs to geometry, so logic, too, has its
> own concepts and relations; and it is only in virtue of this that it
> can have a content. Toward what is thus proper to it, its relation is
> not at all formal. No science is completely formal; but even
> gravitational mechanics is formal to a certain degree, in so far as
> optical and chemical properties are all the same to it. ... To
> logic, for example, there belong the following: negation, identity,
> subsumption, subordination of concepts.



And, of course, as we've seen, Frege supposed that there is a domain
of special logical objects (*courses of values*), among which
he defined, and - until confronted by Russell's paradox -
took himself to have proved facts about, *extensions*
and *natural numbers* (1884, 1893/1903). Logic, then, is not
purely formal, from Frege's point of view, but rather can provide
substantive knowledge of concepts and objects.



There is some question, however, as to the extent to which Frege took
logic to provide constitutive norms of thought. Linnebo suggests that
Frege eventually rejected this idea. Though he offers various
arguments for thinking that Frege moved away from the constitutivity
thesis, his [Linnebo's] main argument concerns the fact that Frege
wanted to position Basic Law V as a logical claim, but that Basic Law
V doesn't seem to be a constitutive norm of thought (Linnebo 2003,
247). This is a persuasive reason, though it does make one wonder
what Frege could have meant, in the second volume
of *Grundgesetze* (SS147), when he said, concerning Basic
Law V:



> 
> If there are logical objects at all ... then there must also be a
> means of apprehending, or recognizing them. This ... is performed
> ... by the fundamental law of logic that permits the
> transformation of an equality holding generally
> \([\forall x(f(x)\! =\! g(x))]\) into an
> equation
> \([\sm{\varepsilon}f(\varepsilon) = \sm{\alpha}g(\alpha)]\).
> [Author's note: the equations in brackets were added for the sake of
> clarity.] 



Given Frege's commitment to logical objects as part of the
content of logic, the above passage suggests that he might have
regarded the law which transformed an equality holding generally into
an equation as a constitutive norm of thought.



But many Frege scholars are convinced that Frege took the laws of
logic to provide constitutive norms of thought (MacFarlane 2002,
Taschek 2008, Steinberger 2017). MacFarlane, in particular, argues
that Kant and Frege may have agreed that one of the most important
characteristics of logic is its generality, and that this generality
consists in the fact that it provides normative rules and
prescriptions. He notes that "[t]he generality of logic, for
Frege as for Kant, is a *normative* generality: logic is
general in the sense that it provides constitutive norms for
thought *as such*, regardless of its subject matter"
(2002, 35). So, though they may differ as to which principles are
logical, there may be at least one point of reconciliation concerning
how Kant and Frege conceived of logic. 


#### 2.7.2 How Frege's Conception Differed from Later Logicians



Given the constraints of the present entry, we shall not attempt to
discuss this question in any detail; instead, we present only the
barest of outlines. After all, modern logicians and philosophers of
logic have not yet come to agreement about the proper conception of
logic. Many have a conception of logic that is yet different from
both Kant's and Frege's, one that was, to some extent,
anticipated by Bolzano, namely, that logical concepts and laws remain
invariant under reinterpretation of the non-logical constants or under
permutations of the domain of quantification. But since this modern
conception is still a matter of debate, it may be that elements of
Frege's conception will yet play a role in our understanding of
what logic is.


It is important to recognize just how much Frege took himself to be
focusing on the *content*, as opposed to the *form*, of
thoughts. His concern to more precisely represent the content of
thoughts is stated explicitly in an 1882 lecture before Jena's
Society for Medicine and Natural Science, where he distinguished his
1879 system from Boole's logic by saying:



> I was not trying to present an abstract logic in formulas;
> I was trying to express contents in an exacter and more perspicuous
> manner than is possible in words, by using written symbols. I was
> trying, in fact, to create a "*lingua
> characteristica*" in the Leibnizian sense, not a mere
> "*calculus ratiocinator*"--not that I do not
> recognize such a deductive calculus as a necessary constituent of a
> *Begriffsschrift*. [Frege 1882, V.H. Dudman (trans.)
> 1968] 



So Frege was not just trying to develop an abstract reasoning system
for the precise derivations of theorems from axioms (see van
Heijenoort 1967 for discussion). Frege was at least as interested in
formalizing the *content* of reasoning as he was in formulating
the rules for deriving a given thought from some group of
thoughts. Frege would *not* have regarded the logical axioms of
his formal systems as axiom *schemata*, i.e., as a
metalinguistic sentence patterns whose instances (i.e., the sentences
of the object language that match the pattern) are axioms (see
Goldfarb 2001 for discussion). Nor would he have agreed that the
logical axioms of his system were uninterpreted sentences. His unease
with the modern conception of an uninterpreted formal system was
expressed in his reaction to Hilbert's *Foundations of
Geometry* (Frege 1906, 384 [1984, 315]):



> 
> The word 'interpretation' is objectionable, for when
> properly expressed, a thought leaves no room for different
> interpretations. We have seen that ambiguity simply has to be rejected
> ...



Instead, Frege thought that his logical axioms are (a) fundamental
truths governing negation, conditionalization, quantification,
identity, and description, and (b) principles from which other such
fundamental truths could be derived. Indeed, even though Frege
sometimes introduces methods for abbreviating these truths, he takes
great pains to insist that these abbreviations are to be understood in
terms of the full content being expressed. For example, he summarizes
the law of universal instantiation (Basic Law IIa) in 1893/SS47
using a formula that we would write nowadays as:



> 
> \(\forall xFx \to Fy\)
> 



But closer inspection of SS20, where this principle is first
discussed, and of SS17, where he introduces some notational
conventions used in SS20, makes it clear that the above is
shorthand for:



> 
> \(\forall F \forall y(\forall xFx \to Fy)\)
> 



This latter is a sentence; it is not a schema, nor an open formula
with a free variable, nor an uninterpreted sentence. Rather it
asserts something that Frege takes to be fundamental law, namely: for
any property \(F\) and for any object \(y\), if everything
falls under \(F\), then \(y\) falls under \(F\).



Frege's understanding of, and attitude towards, the formulas of
his formal language goes a long way towards explaining why his
position, in the now famous debate with Hilbert about the status of
the axioms in a formal system, is not an unreasonable one. At first
glance, it looks as if Frege has mistakenly challenged Hilbert's
method of relative interpretability, whereby one can prove the
consistency and independence of axiom systems by re-intepreting, and
thereby reducing, them to systems assumed to be
consistent. Frege's objections about what exactly has been
established by these relative consistency proofs may seem misguided to
a modern ear. But since Frege and Hilbert understood the notions
of *consistency* and *independence* differently, they
didn't always directly engage with the other's ideas.
Blanchette nicely shows, both in the entry on the
 Frege-Hilbert controversy and in her
book (2012, Ch. 5), that if the notions of consistency and
independence are understood Hilbert's way, then Hilbert's
methods do establish what he says they do, but that if these notions
are understood Frege's way, they don't. The reader should pursue
these works for a more detailed explanation and nuanced discussion of
the disagreement.



One can appreciate how Frege and Hilbert might have failed to engage
with one another by considering a simple analogy. Consider the
inference from "\(x\) had a nightmare" to "\(x\) had
a dream" and ask the question, is the latter a logical
consequence of the former? If one examines the inference *purely
formally*, as Hilbert might, then the sentences have the form
'\(Fx\)' and '\(Gx\)' and the question
becomes, does \(Fx\) logically imply \(Gx\)? The answer for Hilbert
would be 'No', because one can *interpret*
'\(F\)' and '\(G\)' in such a way that the
inference fails, e.g., just assign the standard meaning of
'dream' to '\(F\)' and assign the standard
meaning of 'nightmare' to '\(G\)', i.e.,
interpret '\(F\)' as the property *having a dream*
and interpret '\(G\)' as the property *having a
nightmare*. This shows that, from this purely formal point of
view, "\(x\) had a nightmare" doesn't logically
imply "\(x\) had a dream". (From this purely formal
viewpoint, one additionally needs the premise \(\forall y(Fy \to Gy)\)
to infer \(Gx\) from \(Fx\).)



Although Frege had a formal system in which the open sentence \(Fx\)
doesn't logically imply \(Gx\), he takes logical consequence to
be a relation among *thoughts*. He wouldn't answer the
question, of whether one thought is a logical consequence of another,
solely by looking at the form of the sentences that express them, but
rather by looking at the content of those sentences. Given the meaning
(content) that 'nightmare' and 'dream' in fact
have, "\(x\) had a dream" is a logical consequence of
"\(x\) had a nightmare", for *having a nightmare*,
i.e., *having a bad dream*, logically implies *having a
dream*. So, for Frege, this would clearly be a case of logical
consequence.



This analogy might help one to see how Frege and Hilbert might differ
in their approach to questions of consistency and interpretation. On
Frege's view, the consistency of a group of axioms depends on
content, and if the form of these axioms, under logical analysis,
sufficiently captures their content, this consistency will be
inherited by their formal representations as well. Of course, in
proving consistency, Hilbert was concerned primarily to determine
whether an axiom system entailed a contradiction having the form
\(\phi \amp \neg \phi\). So, given this *formal* goal,
Hilbert's methods are useful and immune to criticism.



But this brings us to one final issue that is crucial to Frege's
conception of logic, namely, the extent to which his formal
representations capture the content of the claims being
analyzed. This issue is relevant because Frege's primary tool for
analyzing the content of a mathematical or philosophical claim is by
way of representing the content in a system that *axiomatizes*
the fundamental concepts that are needed for the analysis. This issue
is the subject of the first half of Blanchette 2012. To see what is at
stake, we vary the example from the one used in Blanchette 2012
(24). Frege would represent the arithmetical law:



> 
> No (natural) number precedes zero
> 


in the first instance, as:



> 
>  \( \neg \exists x(\mathit{Number}(x) \amp \mathit{Precedes}(x,0)) \)
> 


Then if we substitute Frege's definitions of
\(\mathit{Number}(x)\), \(\mathit{Precedes}(x,y)\), and \(0\), as
described in Sections 2.5 and 2.6 above, his representation of the
arithmetical law becomes:



> 
>  \(\neg \exists x((x\! =\! 0 \lor \mathit{Precedes}^\*(0,x)) \amp \\ \
>  \ \ \exists F \exists z(Fz \amp 0\! =\! \#F \amp x\! =\!
>  \#[\lambda u \: Fu \amp u \! \neq\! z])) \)
> 



Though the formal representation could be taken further, if we expand
the definitions of \(\mathit{Precedes}^\*\), \(\#F\), and \(\#[\lambda
u \: Fu \amp u \!\neq\! z]\), enough has been said to pose the
question: why think that by deriving the formal representation from
more fundamental principles, Frege has derived the arithmetic law that
no natural number precedes Zero? This question is tackled in some
detail in in the early part of Blanchette 2012, which investigates
Frege's understanding of conceptual analysis. Her answer
(Chapter 4) is that the formal representation of the arithmetic law
has to be (self-evidently) logically equivalent to a good analysis of
the original. If it is, then notwithstanding Frege's failed
reduction of numbers to extensions, a derivation of the formal
representation from more general logical laws of the kind represented
in his system would have in fact achieved the goal of reducing
arithmetic to logic. The reader is directed to her work for discussion
of this important point.


## 3. Frege's Philosophy of Language




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. However, Bobzien (2021) offers compelling
documentation suggesting that significant elements of Frege's
philosophy of language were adapted from Stoic logic. In this
section, therefore, we first rehearse a key element of Frege's
philosophy of language, namely, the distinction between *sense*
and *denotation*. To motivate this distinction, we go through
the two key puzzles Frege attempted to solve (Section 3.1) and then
examine how the distinction between the sense and denotation of terms
(and sentences) solve these puzzles (Section 3.2). Finally, we attempt
to put Frege's philosophy of language in context (Section 3.3), in
light of new research arguing for the significant influence that
(discussions of) Stoic texts had on his ideas.


### 3.1 Frege's Puzzles



Frege's seminal paper in the philosophy of language is
'Uber Sinn und Bedeutung' ('On Sense and
Reference', 1892a). 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 subordinate 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 and we discuss it in what follows.
(See Heck and May 2006 for further discussion of Frege's
contribution to the philosophy of language.)


#### 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
'\(a=b\)', where '\(a\)' and
'\(b\)' are either names or descriptions that
*denote* individuals. He naturally assumed that a sentence of
the form '\(a=b\)' is true if and only if the object \(a\)
just is (identical to) the object \(b\). For example, the sentence
'\(117+136 = 253\)' 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 (1892a) that this account of truth can't be
all there is to the meaning of identity statements. The statement
'\(a=a\)' has a cognitive significance (or meaning) that
must be different from the cognitive significance of
'\(a=b\)'. 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
'\(117+136 = 117+136\)' 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. Now the problem becomes clear: the meaning of
'\(a=a\)' clearly differs from the meaning of
'\(a=b\)', 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
'\(a=a\)' and '\(a=b\)'.




So the puzzle Frege discovered is: how do we account for the
difference in cognitive significance between '\(a=b\)' and
'\(a=a\)' 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 '\(x\)' by the name of a person
and replace the variable '\(p\)' with a sentence that
describes the propositional object of their attitude, we get specific
attitude reports. So by replacing '\(x\)' by
'John' and '\(p\)' 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 Identity Substitution (this is not to be
confused with the Rule of Substitution discussed earlier). If a name,
say \(n\), appears in a true sentence \(S\), and the identity sentence
\(n=m\) is true, then the Principle of Identity 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.



> 
>  \(4 \gt 3\)
>   
> 
>  \(4 = 8/2\)
>   
> 
> Therefore, \(8/2 \gt 3\)
> 




In general, then, the Principle of Identity 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 Identity 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 Identity
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?


### 3.2 Frege's Theory of Sense and Denotation




To explain these puzzles, Frege suggested (1892a) that in addition to
having a denotation, names and descriptions also express a
 *sense*.[5] 
 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 '\(4\)' 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. (See May 2006b for
a nice discussion of the question of whether Frege believed that the
sense of a name varies from person to person.) 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 those of the form
'\(a=b\)'. Since the sense of '\(a\)' differs
from the sense of '\(b\)', the components of the sense of
'\(a=a\)' and the sense of '\(a=b\)' are
different. Frege can claim that the sense of the whole expression is
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
Identity 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' (though
see Heck & May 2011 for discussion on whether the sense of a
predicate maps the sense of a name to a thought). 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 the function ( )
*loves Mary*. This latter function serves as the denotation of
the predicate 'loves Mary' and we can use the notation
\(d[Lm]\) to refer to it semantically. 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.[6]




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
 \(s[jLm]\).[7]
 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 '\(a=a\)' and
'\(a=b\)'. The cognitive significance is not accounted for
at the level of denotation. On Frege's view, the sentences
'\(4=8/2\)' and '\(4=4\)' 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 '*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 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 Identity 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.


### 3.3 Frege's Philosophy of Language in Context



In an important paper, Bobzien (2021) develops an inductively strong
argument to the conclusion that Frege adopted ideas by the Stoics, as
the latter were described in Carl Prantl's four volume *History
of Western Logic* (*Geschichte der Logik im Abendland*,
1855-1870) and in Diogenes Laertius' *Lives of the
Philosophers* (*Vitae Philosophorum*, 2nd century CE). The
Stoic ideas described in these two works originate with Zeno of
Citium, Cleanthes, Chrysippus of Soli, Diogenes of Babylon, etc., and
so both works are 'secondary' literature. The relevant
reports in Diogenes' *Lives* (which are largely quoted in
Prantl) are among the closest we have to the source material for Stoic
writings on language and logic, while Section VI of Volume 1 of
Prantl's *Geschichte* is a 95-page summary of Stoic
views. Bobzien notes how widely read Prantl's work was, and how
Frege's training left him well-qualified to understand the many quoted
passages in Greek and Latin in his work. She then compiles evidence to
the conclusion that Frege borrowed heavily (and without attribution)
from these works. This evidence is cumulative, for there is no smoking
gun. We can't point to any dog-eared copies with marginalia in
Frege's own hand, or any copies of works by Prantl and Diogenes
that can be traced to Frege's library, or any records of Frege
checking out the works by Prantl and Diogenes from the Jena University
library.[8]



But in the 50-page Section III of her paper, Bobzien constructs
numerous tables of comparison that place passages from the Stoic and
Fregean corpora side-by-side and interlaces them with observations and
careful scholarship that connect the most salient (transliterated)
Greek phrases with their German counterparts. The overall effect is a
kind of *res ipsa loquitur* - the thing speaks for itself
- and places the ball squarely in the court of those who would
attempt to deny the identities and similarities. The evidence doesn't
challenge the insight and ingenuity that Frege brought to bear when
he *formalized* the language and logic of mathematics (and
certain constructions of natural language) in terms of the 19th
century understanding of functions, and so the formal representations
presented earlier in this entry were something that was *not*
borrowed from the Stoics. And though Frege independently motivated the
ideas he absorbed from the Stoics, Bobzien shows how many aspects of
Frege's underlying approach to language is directly founded on
ideas from fifteen hundred years ago and so are not as new as
originally thought. 



It remains to get a sense of the range of the elements in common, some
of which were pointed out by others in previous
 research.[9]
At the most general level, Bobzien compares the ways that the Stoics
and Frege relate semantic 'content' (Greek *lekta*
v. Fregean *Sinn*) to the linguistic expression of that
content. Both distinguish the 'incomplete'
(*ellipe* v. *ungesattigt*) content of
predicates from the complete content of sentences, and there are
parallels in what the Stoics and Frege have to say about complete
'assertible' contents (*axiomata*
v. *Gedanken*), especially in connection with the fact that
these contents are the primary bearers of truth and falsity. Bobzien
then compares the two philosophies with regard to other kinds of
complete contents, such as those expressed by commands, sentence
questions, expressions of emotion, word questions, and indexicals. In
each case, Frege's descriptions of the content presented by
these linguistic expressions closely follow those of the Stoics.
Another group of similarities arises in her comparison of the
doctrines regarding complex contents produced with propositional
connectives, such as those expressed by sentences with negations
(including contradictories, double negations, etc.), conjunctions,
disjunctions, conditionals, and sentences with 'because'.
A comparison of the two approaches to quantification and universality
concludes the paper.



By producing so many passages in parallel between the Stoic and
Fregean corpora, the details and evidence accumulate, thereby becoming
increasingly persuasive. Any student of Frege who wants to understand
the full range of his views on language will have to come to grips
with how those views are related to Stoic philosophy. Though Bobzien duly
notes the others who had previously remarked on some of the
similarities (see again 
 footnote 9), her paper of
2021 is a good place to start, since the evidence assembled there is
so comprehensive.