Introduction to Gottlob Frege
(from Contemporary Analytic Philosophy, 2nd Edition, edited by James Baille)
Gottlob Frege was born in Wismar, Germany in 1848. He studied at the
Universities of Jena and Gottingen, where he received his
Ph.D. in mathematics. Frege spent his entire academic career at Jena, first as an unsalaried Privatdozent and eventually as professor of
mathematics. While Frege is now regarded as the greatest logician since Aristotle and a founder of analytic philosophy, this judgment
is retroactive. His achievements were barely noticed by either philosophers or mathematicians when he died in 1925. As Bertrand
Russell remarked of the Begriffssehrift (1879), "In spite of the epoch-making nature of this work, I was, I believe, the first person who
ever read itmore than twenty years after its publication" (Introduction to Mathematical Philosophy, p. 25).
Frege's philosophical theories developed from his work in mathematics.
His original aim was to show how arithmetical
statements could be proved without appeal to empirical facts or intuition (as in Kantian theories) but by reason alone. He advocated
logicism, the thesis that mathematical concepts could be defined in terms of logical concepts, and mathematical truths derived from
logical axioms. Supporting logicism required him to develop a language in which logical properties and relations could be stated
precisely. Since he believed that natural languages were inadequate, being essentially subject to vagueness and ambiguity, he
constructed an artificial language-Begriffsschrift (i.e., Concept Script)-for this task. This new symbolism was designed to contain only
the features relevant to logical inference. The Begriffsschrift presented a propositional calculus in which all logical laws could be
derived from an axiomatic base, together with a predicate calculus capable of formalizing sentences involving multiple generality.
While Frege's own choice of symbolism is no longer used, modern logic is unquestionably based on the framework presented in the
Begriffsschrift and developed in his later works.
This principle originally expressed by Frege was opposed to the then
domnant theory of psychologism, which
regarded the laws of logic as empirical generalizations about the actual psychological processes involved in reasoning.
Such a position resulted from a failure to distinguish the psychological process of thinking (or making a judgment) from
its content (the thought expressed). As Frege put it in TheFoundations of Arithmetic, one should "always separate the
psychological from the logical, the subjective from the objective." While causal laws can be formulated concerning
relationships between psychological processes, thoughts are related by logical laws. The laws of logic are unlike scientific
laws in being essentially normative rather than descriptive. That is, they don't describe how we do infer, but how we should
infer. They state ideals against which our practice can be measured.
Frege argued in The Basic Laws ofArithmetic that psychologism allows
the possibility of creatures governed by
totally different laws of logic who would accept inferences that we would reject, and vice versa. Psychologism bars us
from asking the legitimate question of whose set of laws are the correct ones, since it cannot allow any genuine error or
disagreement between us, merely psychological difference. Frege emphasized that logic has an irreducibly normative
relation to our thoughts, whereas a record of psychological activity remains purely on the descriptive level.
Since Aristotle, logicians had assumed that all sentences had a subject-predicate
structure. Frege regarded this as a
superficial grammatical distinction that need not represent the true logical structure involved. Second, traditional
syllogistic logic dealt only with general sentences asserting relations between properties, namely "all A are B," "no A is B,"
"some A is B," "some A is not B." Frege, on the other hand, took singular sentences as basic. Singular sentences are
constructed from two syntactic types of expression:
Singular terms or names, referring to particular objects.
Concepts, predicate expressions referring to properties that more than one object could have.
Frege treats ordinary names (e.g., "George W. Bush") and definite
descriptions (e.g., "the present president of the
United States") as in the same syntactic category of singular terms, since to be a name is to refer to a particular object.
Frege modeled the name/concept distinction on the mathematical distinction
between argument and function.
Consider the expression "x2 + 1," which expresses a mathematical function. Any number replacing the variable x is called
the argument for that function. The resulting number is called the value of that function for that argument. For example,
the argument 1 gives 12+ 1, namely 2; the argument 4 gives 42+ 1, equal to 17. The variable x doesn't stand for anything
but is merely a way of showing that the expression is essentially incomplete or "unsaturated" and has a value would
yield T only when completed by an argument. Arguments, on the other hand, are complete or saturated independent of
their surrounding functions. They represent objects, so that the numeral "5" represents the number 5, and so on. The essentially unsaturated nature of functions, and their difference to arguments, solved two problems for Frege. First, the fact that the same function can be saturated by a vast
number of different arguments explained how we are capable of understanding and using a seemingly limitless number of
sentences. Second, it explained how something with a truth-value-a sentence-could be constructed from parts that did not.
Frege extended this mathematical model to linguistic expressions. For
example, we can analyze "the capital of France" by taking the
capital of as a function and France as the argument, giving Paris as the value. The same function, when given the argument Portugal, yields
Lisbon as a value. So we can consider the capital of as a function that maps objects onto objects (in this case, countries onto cities). Likewise,
"the Queen of England" takes the function the Queen of , which, when saturated with the argument England, yields Elizabeth Windsor as a
This analysis can be applied to whole indicative sentences. Consider
an expression like "x2 = 36." Clearly, the variable x can be
by a variety of numbers as arguments. When we insert 6, we get the true sentence "62 = 36." When we insert 5, we get the falsehood "52= 36:"
Frege takes The True and The False to be the values of these sentences. He defines a concept as a function that has a truth-value as a value. Frege
considered True and False to be abstract objects referred to by true and false sentences, respectively.
Take a sentence like "Wittgenstein wrote the Tractatus." We can regard
the Tractatus" as a concept which gives the value
True when given the argument "Wittgenstein," but False when given any other argument such as Frege, Sting, and so forth. Alternately, we can
take "Wittgenstein wrote " as the concept and the "Tractatus" as the argument, again yielding the value True. In sum, a sentence is true if the
concept yields the value True for the object represented by the argument. As in the arithmetical examples, True and False are not taken to be
properties of sentences, but abstract objects that are the values of sentences.
Possibly Frege's greatest contribution to the study of logic and language
was his introduction of the quantifiers-the universal quantifier
now represented as (x), and the existential quantifier, (Ex). (In the Begriffsschrift, Frege employed only the universal quantifier, but this and the
existential quantifier are interdefinable.) To explain the use of the quantifiers we need to distinguish first- and second-level functions/concepts. We
have already seen first-level concepts, which take objects as arguments, yielding True and False as values. A second-level concept takes first-level
concepts as arguments, again yielding truth-values.
I will illustrate this using the existential quantifier to analyze statements
involving existence. Consider "ghosts exist." Traditionally, one
would take "ghosts" as the subject, and "exist" as the predicate, so that the sentence ascribes the predicate "existence" to ghosts. But Frege insists
that it isn't about ghosts (i.e., those alleged things), but about the first-order concept " is a ghost." It says that this concept has instances, that is,
that there is some object which, if it were to saturate the concept, would yield True. So the second-order concept " has instances" takes the
firstorder concept "- is a ghost" as an argument. In sum, (Ex)Fx correlates the concept " is an F" with True if and only if some object falls under F.
"(x)" is the universal quantifier. (x) means ` for all values of x'. "(x) Fx" means "all x are F." (x) Fx correlates the concept " is an F"
with the value True if and only if every object in the domain of discourse falls under F (where the domain of discourse is defined contextually, as
with the existential quantifier). Take "everyone is American,' (x) Ax. Suppose I were to utter this in my most recent Philosophy 475 class, taking
that group of students as my domain of discourse. If, every time I replace x with one of the students in the concept Ax and the value True is
obtained, then (x)Ax is true. If even one case gives False, (x)Ax is false. Hence, since this class contained some non-Americans, the sentence was
I now turn to Frege's most famous article, "Sense and Reference." The best way to appreciate the problem that led Frege to distinguish sense and reference is to notice a consequence of his previous theoretical commitments.
Frege was probably the first philosopher to stress the compositionality
thesis: The meaning of a sentence is a function of the
meanings of its constituent parts plus its syntactic structure. Take "Portland is south of Seattle." If you alter the positions of "Portland"
and "Seattle" you get a different sentence-one with a different meaning, truth-condition, and truth-value. These same results happen if
you replace one word with another, for example "San Francisco" for "Seattle."
In his previous writings Frege had held that a name's sole contribution
to the meaning of a sentence was referential. The
combination of a referential theory of names plus the compositional ity thesis created a problem that was highlighted in identity
(1) The Morning Star is the Morning Star.
(2) The Morning Star is the Evening Star.
These statements employ the names "the Morning Star" and "the Evening
Star." When these names were introduced in the
ancient world, they were taken to pick out different planetary bodies. However, it was later discovered that they both stood for Venus.
By this stage, Frege had reversed his earlier view to say that such sentences were asserting that identity held between objects,
not names for objects. That is, these statements were about the world, not about linguistic signs themselves. But in that case there are
important differences between (1) and (2), since the latter expresses a posteriori geographical information whereas the former is an
uninformative logical truth, knowable a priori. Second, they clearly express different thoughts, since someone can assent to one and
deny the other without falling into contradiction. Similarly, someone who already knows one and then discovers the other has learned
something new. These latter two points can be raised outside of identity statements:
(3) The Morning Star was hard to locate last Wednesday.Frege's problem is that since `the Morning Star' and `the Evening Star' are coreferential terms, and the sentences are
(4) The Evening Star was hard to locate last Wednesday.
A name's reference is whatever it picks out, so the reference of "the
Morning Star" is Venus. "The Evening Star" has this
same reference. Sense is harder to pin down, but we can distinguish a few main points:
• A sense is a "mode of presentation," a way of conceiving of the thing
• A sense can be grasped by different minds and is therefore objective.
• A term's sense determines its reference in that it is some condition that an object must satisfy to count as its reference. Grasping a sense
involves knowing what condition something would have to satisfy to be the name's reference, without knowing whether it actually does.
• If two expressions have the same sense, then they are coreferential. However, two expressions can be coreferential while differing in
• Sense is compositional, in that the sense of a complex expression is a function of the senses of its constituent parts.
• In natural languages, expressions can have a sense without a reference, for example, "JFK's third wife."
The sense/reference distinction is also applied to whole sentences:
• The reference of a sentence is its truth-value, that is, "The True"
or "The False:"
• The sense of a sentence is the thought it expresses.
• In The Basic Laws of Arithmetic, Frege says that to grasp the sense of a thought is to know its truth-condition, that is, the
mind-independent aspects of reality in virtue of which it is true or false.
If substituting one expression for another can yield a sentence with
a different truth-value, then the expressions are not coreferential. If
substituting one expression for another creates a different thought, then these expressions have different senses. Two thoughts are different if
someone can believe one and not the other without contradiction.
In his late paper "The Thought" (1918), Frege's antipsychologism is
extended to a Platonic view of thoughts (i.e., the senses of sentences)
as objectively existing entities. In line with his principle to "always separate the psychological from the logical, the subjective from the objective,"
he made a sharp distinction between thoughts on the one hand and feelings, images, and mental associations on the other. His main target was
empiricism, which held that words stood for "ideas in the mind," usually considered as mental images. Frege regarded this theory as confusing the
act of thinking with the thought-content itself, since thoughts were essentially sharable whereas ideas were mind-dependent and incommunicable.
That is, only I can have my ideas, and only I can be aware of them.
There are two aspects to the objectivity of thoughts. First, since a
an abstract object, it exists independent of any psychological or neurological process. _; i
The truth (or falsity) of a thought is also objective in that someone's believing p to be
true is neither necessary nor sufficient for p to actually be true. It follows that a sen
tence can be true even if no one ever becomes aware of this fact, or even considers ._
the sentence. Second, different persons can entertain the same thought. Indeed, iden
tity of thought-content is presupposed in the notions of agreement or disagreement.°r~u
For example, when A believes that p, and B disagrees, they are disagreeing about p.
An account of the notion of disagreement requires ascribing the same thought to the
The claim that nonreferring expressions can have a sense raises two
for Frege. Consider a sentence like "JFK's third wife was Belgian." We can under
stand this by knowing what would have to be the case for it to be true. However, given
that reference is compositional, it follows that this sentence itself lacks a referent. In
other words, it has no truth-value, hence there are exceptions to the Law of Excluded
Middle, p v --p. Frege regarded it as a formal weakness in natural languages that these
sentences can be generated.
The second problem arises over the consistency of the notion of sense.
grants that expressions can have sense but no reference. However, he also describes
sense as a "mode of presentation" of its referent. But if an expression has no referent,
then there's nothing to be a mode of presentation of.
Returning to our original example, Frege concludes that sentences (1)
(2) have the same references, but different senses [as do (3) and (4)]. Their senses dif
fer because they express different thoughts. Their component parts, "the Morning
Star" and "the Evening Star," can be shown to differ in sense simply because their
substitution yields a different thought.
However, the theory hits a problem when faced with nonextensional
contexts such as those involving propositional attitudes (e.g., S believes that p, S fears
(5) Bob believes that the Morning Star was hard to locate last Wednesday.
(6) Bob believes that the Evening Star was hard to locate last Wednesday.
As before, we get the latter sentence from the former by substituting corefer
ential terms. However, it is clearly possible for one of these sentence to be true and
the other false. But by the principle of compositionality, shouldn't the two sentences
be guaranteed to have the same truth-value?
The problem of belief-contexts is one case of nonextensional contexts
where standard rules of substitutibility fail). Others include
• Quotational contexts, for example, "bachelor" has eight letters, but "unmarried male"
• Indirect speech, for example, "Bob said that Lewis Carroll wrote the Alice books."
• Modal contexts, "Necessarily George W. Bush is George W. Bush" is true but
"Necessarily George W. Bush is the current United States president" is false, since
Gore might have won.
Frege's way out involves his theory of indirect reference. The basic
idea is that in non-extensional contexts a term's ordinary sense (i.e.,
its Sinn in
extensional contexts) becomes its reference. In other words, while the same combination of words, "the Morning Star was hard to locate last
Wednesday" occur in both (3) and (5), in the former its reference is either True or False (depending on its visibility at that time) whereas in the
latter it refers to Bob's thought that the Morning Star was hard to locate last Wednesday. Likewise, the expression "the Evening Star was hard to
locate last Wednesday," as occurring in (6), refers to a different thought entertained by Bob. On the level of names, in (3) and (4) "the Morning
Star" and "the Evening Star" have different senses yet the same reference, whereas in (5) and (6) they thereby have different referents,
corresponding to distinct ways of conceiving of Venus.
In closing, consider whether the theory of indirect reference can cope with
an example like this:
(7) Bob believes that the Morning Star is made of cheese, but it is not.
Prima facie, the referent of "the Morning Star" and "it" is the same. Indeed it is hard to understand the sentence if this isn't so. But Frege seems bound to deny it since "the Morning Star" occurs in an intensional context whereas "it" does not.
Frege's work is gathered in the following books:
The Frege Reader, ed. Michael Beaney (Oxford: Blackwell, 1997). This
is the best selection of Frege's writings currently available, collecting
all his important papers, along
with extracts from his books, his correspondence, and posthumously published writings.
The Basic Laws of Arithmetic, traps. Montgomery Furth (Berkeley: University of California Press, 1965).
Collected Papers on Mathematics, Logic and Philosophy, ed. Brian McGuinness (Oxford: Basil Blackwell, 1984).
Conceptual Notation and Related Articles, traps. T. W. Bynum (New York: Oxford University Press, 1972).
The Foundations of Arithmetic, traps. J.L. Austin (Oxford: Basil Blackwell, 1953).
Translations from the Philosophical Writings of Gottlob Frege, 3d ed.,
ed. P .T. Geach and Max Black
(Oxford: Basil Blackwell, 1980).
The leading commentator on Frege is undoubtedly Michael Dummett. However,
his books are long and his style difficult. Still, anyone with a serious
interest in Frege
ought to study them, beginning with Frege: Philosophy of Language (Cambridge, Mass.: Harvard University Press, 1973), followed by The Interpretation of Frege's
Philosophy (Cambridge, Mass.: Harvard University Press, 1981).
Frege, Anthony Kenny, (Oxford: Blackwell, 2000), is a short introduction,
usefully organized in chronological sequence rather than by subject matter.
Thus, each of
Frege's works is given a selfcontained section.