André Leclerc (*)
Frege’s Puzzle, Ordinary Proper Names, and
Individual Constants
Resumen: Mi propósito en las siguientes
observaciones es simplemente rememorar las
motivaciones de Frege para introducir el Sinn,
y algunos hechos importantes acerca de su uso
de ‘=’. Pienso que son relevantes al escrutar lo
que ha sido denominado el “puzzle de Frege”.
Primeramente, Frege no utiliza el signo de
identidad ‘=’ exactamente como lo hacemos
nosotros. En segundo lugar, su noción de objeto
no es substantiva (física o mereológica), sino
esquemática. Y en tercer lugar, la noción de
Frege de nombre propio es enteramente distinta
de la nuestra o de la del sentido común. Al
final, lidio con el problema de Glezakos acerca
de la individuación de los nombres. Concluyo,
al igual que Glezakos, que el puzzle de Frege
no es enigmático, mas proporciono razones
un tanto distintas. La teoría de Frege de la
intencionalidad y las reglas que gobiernan el
uso que hacemos de las oraciones esquemáticas
en el proceso de formalización son claves para
entender por qué Frege propuso el problema tal
y como lo hizo. Empero, no creo que la solución
de Frege se sostenga para los nombres propios
ordinarios. Creo que la solución de Frege
funciona mucho mejor cuando los substituyentes
de ‘a’ y ‘b’ son expresiones complejas, como lo
son las descripciones definidas o las oraciones
declarativas completas, debido a que estas
expresiones expresan modos articulados de
presentación, mientras que las expresiones no
complejas, como lo son los nombres propios
ordinarios, no expresan su modo de presentación
en virtud de una convención definida, siendo la
arbitrariedad inevitable.
Palabras claves: Frege. Oraciones de
identidad. Sinn. Nombres propios. Constantes
individuales.
Abstract: My aim in the following
observations is simply to remind Frege’s
motivations for introducing the Sinn, and some
important facts about his use of ‘=’. I think
they are relevant at the time of scrutinizing
what has been called “Frege’s puzzle”. First,
Frege does not use the identity sign ‘=’ exactly
as we do; second, his notion of object is not a
substantive one (physical or mereological), but
a schematic one; and third, Frege’s notion of
proper name is quite different from ours or that
of common sense. At the end, I tackle Glezakos’
problem about the individuation of names. I
conclude, like Glezakos, that Frege’s puzzle
is not that puzzling, but for slightly different
reasons. Frege’s theory of intentionality and the
rules that govern the use we make of schematic
sentences in the process of formalization are keys
to understand why Frege posed the problem the
way he did. However, I do not believe that Frege’s
solution holds for ordinary proper names. I think
that Frege’s solution works much better when
the substituends for ‘a’ and ‘b’ are complex
expressions like definite descriptions or full
declarative sentences, because these expressions
express articulated modes of presentation, while
the incomplex expressions, like ordinary proper
names, do not express their mode of presentation
in virtue of a definite convention, and then
arbitrariety is unavoidable.
Key words: Frege. Identity sentences. Sinn.
Proper names. Individual constants.
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1. Frege sets up his so-called “puzzle” by
using two identity sentences of the language of
first order predicate logic (with identity): “a=a”
and “a=b”. If “a=b” were a relation between two
signs or names, there wouldn’t be any difference
in terms of cognitive value between the two
identity sentences. We can choose any signs we
want, arbitrarily. The shape does not matter. If
it is a relation between the object denoted by ‘a’
and the object denoted by ‘b’, then we loose the
difference between the two sentences given that
“a=b” is true —they both state the same thing:
that an object is identical to itself.
As we know, the language of predicate
logic with identity is extensional, that is, it is a
language in which the only semantic value that
matters for calculations is the extension. All
instances of “a=a”, or so it seems, are truths
knowable a priori; so, “Charles dodgson=Charles
dodgson” is true just in case Charles dodgson is
the same (person) 1 as Charles dodgson. That
language in classical logic works with a nonempty domain. Therefore, the only thing we learn
by understanding “Charles dodgson=Charles
dodgson” is that there is an x such that x is
identical to Charles dodgson. That is not much
informative, indeed. But Frege, correctly, calls
our attention to the fact that instances of “a=b”
behave in a very different way. In arithmetic,
to know (a priori) the truth of an equation like
“(47+63)=(55+55)” I have to do something, to
calculate, and usually, it takes time to discover
truths of the form “a=b”. In some cases, it is an
empirical discovery. “Charles dodgson=Lewis
Carroll” is true just in case Charles dodgson is
the same (person) as Lewis Carroll, but in that
case, you don’t know that simply by inspecting
the sentence. You have to investigate. So, here is
the “puzzle”: given that a sentence of the form
“a=b” is true, given that ‘a’ and ‘b’ denote the
same “object”, then what makes the difference
between “a=a” and “a=b”? In other words, how
a sentence of the form “a=b” can be informative?
Frege consider as something “obvious”
(offenbar) that the two schematic sentences (more
precisely: all the instances of “a=a” coupled with
all the instances of “a=b”, respecting uniform
substitution) have different cognitive values.
That premise is taken for granted at the outset.
So, ‘a’ and ‘b’ cannot have exactly the same
meaning; therefore, meaning cannot consist
solely in a denotation, since ‘a’ and ‘b’ have
the same. Something more is required.2 What
remains to be done is to find the element that
explains the cognitive difference, and to justify
its introduction. The idea of mode of presentation
or sense (Sinn) comes soon at the very beginning
of Frege’s most famous paper (1892) as a very
natural way to explain that difference. On that
score, I agree with Glezakos: all this is hardly
puzzling.
How puzzling must a problem be in order
to count as a ‘puzzle’? I confess I don’t know.
‘Puzzle’, like any other noun, can be used in a
relatively sloppy way. But if it is a puzzle at all,
Frege has a readily solution and his puzzlement is
not long-lasting. And the solution does not seem
to me as new or revolutionary as it is usually
taken to be. What is new and revolutionary,
however, is the way Frege employs his notion
of Sinn in the development of his philosophical
framework.
2. In Über Sinn und Bedeutung (1892;
hereafter SuB), Frege’s first motivation when he
poses his “puzzle” seems to be the improvement
of his ideal language. The first 1879 version
of it speaks of “content” (Inhalt) without any
further distinction. The second version in The
Basic Laws of Arithmetic (1893) already uses
the sense/denotation distinction. The symbol for
identity is, of course, part of the ideal language in
construction. But there is also an important rule
essential to that language that derives directly
from the principle known as the Indiscernibility
of Identicals. The principle says that if x=y, then
x has all the properties y has and vice versa (in
symbols: x y [(x=y) P (Px Py)]. From this
follows a rule of substitution of co-referential
terms widely used in mathematics and logic:
for any property P, from a=b and P (a) it follows
that P (b) –‘a’ and ‘b’ can be substitute one for
another in P (___) salva veritate (in symbols:
P (a=b)&Pa Pb. That rule works quite well in
extensional contexts, but it fails in intensional
contexts. Frege’s second motivation is to explain
the failure of that important rule in indirect
discourse and to preserve it. Both motivations
have to do with the correct interpretation of
î
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‘=’. With the introduction of another semanticcognitive value, the Sinn, Frege kills two birds
with one stone.
But there is much more. By exploring
the reasons why the Leibnizian rule fails in
indirect discourse, Frege paved the way for a new
philosophical logic. The introduction of sense as
a component of meaning leads to a spectacular
simplification in Frege’s semantics, a decisive
step in the direction of what Alonzo Church
called the “Logic of Sense and denotation”,
which is called today, simply, “Intensional Logic”.
One of the most important theses defended in
SuB is the idea that the denotation of sentences is
a truth-value. The introduction of senses makes
possible the following simplification in Frege’s
system: the denotation of complete sentences
or of a complete interrogative sentence with the
force of a yes/no question is a truth-value. All
true sentences have the same denotation and
all false sentences too, the True and the False,
respectively. different declarative sentences do
not denote, for instance, different “facts”. But it
is as clear as is the summer’s sun that there is all
the (cognitive) difference in the world between
the sense of “A neutron is a particle with zero
charge”, and the sense of “The number of planets
of our solar system is 8”. Both sentences are
names of the True, but this is the only thing they
have in common. In an extensional calculus (like
propositional logic) they can be substitute salva
veritate. But the thoughts or senses they express
are completely different, and a cognitive agent
can believe one and not the other.
3. The identity symbol ‘=’ in the language
of first order predicate logic must be flanked
by symbols for individual constants that stand
for “objects”.3 More precisely, “a=a” and “a=b”
are schemas of identity sentences exhibiting the
logical form of all their respective instances in
a perspicuous way. The constant letter ‘b’ in
the second sentence is there only to indicate
that a different “name” is being used in the
original sentence whose logical form is made
explicit. Most instances considered as examples
by Frege are taken from ordinary language, like
the famous Morning Star/Evening Star example.
However, ordinary language, as we know, is
full of truth-value gaps and intensional contexts.
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We use a lot of empty names, like ‘Excalibur’,
‘Sherlock Holmes’, etc., and fiction is a genuine
and important part of our lives. We do not always
think and speak “seriously and literally”, when the
truth-value of our thoughts and assertions really
matters to us. The existential presuppositions
relatively to what we think and say are not
always fulfilled, and we know that. Furthermore,
we constantly ascribe to each other knowledge,
intentional actions, and a huge variety of mental
states whose contents are specified by the use
of sentences belonging to a public language in
the scope of a verb denoting a mental state. In
Frege’s ideal language (the first version of the
language of first order predicate logic in modern
time), there is no truth-value gap and no place for
indirect discourse. Therein, all names, simple and
complex, have a denotation, and their denotation
is always their usual, customary denotation.
4. The introduction of Sinn as a semantic
value is “new” in SuB and Frege’s system. I
said that Frege’s solution is not overall new
or revolutionary. In fact, it is not hard to find
some equivalent notions in the history of logic.
Actually, the pair <sense; denotation> echoes
other famous distinctions. Port-Royal’s Logique
introduced the pair <compréhension; étendue>;
Leibniz used the pair <intension; extension>,
revived by Carnap centuries later; and Mill
proposed the pair <connotation; denotation>.
The first term of each pair refers to the cognitive
part of language, what we do understand when
we understand sentences. Sinne are what take
us to the denotation when we speak and think
seriously, as it happens most of the time. At least,
it takes us to denotation when there is one. Now,
what about fiction and empty names?
In some versions of free logic, it is
not true that “a=a” can be known a priori.
“Hesperus=Hesperus” looks as a good candidate
for something knowable a priori, but it is
doubtful in cases like “Excalibur=Excalibur”.
From “Excalibur = Excalibur” it certainly does
not follow that there is an x such that x is identical
with Excalibur. After all, the fact that an ordinary
name has a nominatum is not something that can
be known a priori. It is a contingent fact. Positive
free logic says that “Excalibur=Excalibur” is
true just for being an instance of a logical law.
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In a Fregean free logic, the same sentence has
no truth-value. In negative free logic, it is false.4
A false sentence, or a sentence neither true nor
false, is of no use for specifying the content of a
knowledge ascription, since the proposition that
is the content of such an ascription, by stipulation,
must be true. If our concern is ordinary language,
free logic should be taken seriously. But Frege
few excursuses on the topic of empty names and
fiction are open to interpretation and critique.
Some consider incoherent the idea of a mode
of presentation that does not present anything.
Perhaps, mock beliefs are not genuine beliefs, or
beliefs only in a degenerated sense. But why is
it that people knowing a bit of British literature
would reject immediately as wrong the belief
that Sherlock Holmes is a fisherman? The idea
of sentences expressing thoughts with no truthvalue, after all, is not that implausible and should
be treated on a par with the idea of ordinary
proper names with a sense but no denotation.
5. The logical difference between “a=a”
and “a=b” is spelled out in terms of knowledge.
Where “a=a” in classical logic (with a non-empty
domain) always gives rise to analytically true
instances knowable a priori, “a=b” sometimes
represents empirical, scientific discovery. Once
again, Frege’s puzzle amounts to wondering: How
is this possible when “a=b” is true, that is, when
‘a’ and ‘b’ stand for the same denotation? The
answer is simply that the two sentences involve
different cognitive values. This is shown, for
Frege, by the fact that, in normal conditions, it
is not possible for a rational agent not to believe
that a=a, while it wouldn’t be irrational for the
same agent not to believe that a=b, because she
represents the object for which ‘a’ and ‘b’ stand
under different “aspects”, different “modes of
presentation”, different Sinne. As we know, in a
context like “A believes that P”, the substitution
of co-referential terms is not allowed in P because
it might change the belief and run into the risk
of disrespecting the cognitive perspective of the
agent. We do not always feel the need to respect
the cognitive perspective of the agent. Most of
the time, it is pragmatically “negotiable”. But
there are situations in which it is of the utmost
importance to determine exactly under which
aspects an agent represents things or facts. The
president of a country is accused of corruption in
a trial for impeachment. The whole nation wants
to know if he/she knew the facts mentioned in
the accusation. May be the president, in his/her
cognitive perspective, represented the facts under
aspects or modes of presentation making them
something quite inoffensive and usual. May be not.
6. The sense is something we grasp; it
is the cognitive dimension of meaning, what
is understood when we understand linguistic
expressions; and, importantly, it is objective,
that is, the same for all competent speakers in
a determined community. But is it always the
same? “[…] [In] order to think we must use sensesymbol”, 5 says young Frege, and our thoughts
are composed out of Sinne, each one associated
with words by convention. That is the theory.
So it is a new, different semantic value that
Frege introduced in his system. Therefore, senses
supposedly are tied to languages, accessible to
all those who master the conventions and rules
of language. Thus, according to Frege, our
knowledge of senses is a genuine and central
part of our semantic knowledge. The sense of
an expression is denoted by using the sentence
form “The sense of ‘…’”.6 An entire hierarchy
of indirect senses can be constructed in this
way. Just continue the sequence: the sense of
“the sense of ‘… E …’”, etc., where E is any
expression provided with a denotation.7 This way
of denoting the sense of a linguistic expression
ties up strictly the senses to language. However,
it is not a device to specify the sense of an
expression, as Sainsbury seems to suggest; this
is obvious, we’ll see, in case of ordinary proper
names. But old Frege with his doctrine of a
third realm seems to underline the languageindependent character of Sinne. And then it is
hard to separate proper semantic knowledge
from encyclopaedic knowledge.
Frege presents the sense as a semantic
property of an expression, something understood
or grasped by all those who know the language
and are competent speakers. But then it seems
hard to separate sense and linguistic meaning,
especially for complex expressions where the
rules of compositionality for the senses apply
(definite descriptions, full declarative sentences,
complete interrogative sentences). do we always
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grasp the same sense expressed by an expression?
According to Burge (1990/2005), in many
passages, Frege admit that the sense grasped
is not always the one semantically expressed.
He gave as examples the Sinn expressed by
‘number’ (and also ‘inertia’), which became more
precise over a long period of time by the addition
of fractionary numbers, real numbers, complex
numbers, etc. By its very nature, the Sinn cannot
change. What become more precise is not the
Sinn, of course, but our grasping of it. Frege never
faced insistently the issue that became important
in the late sixties and seventies in the philosophy
of science: scientific progress versus change
in meaning, but he anticipated it. As we saw,
semantically, the Sinn expressed cannot change;
it is always the same. But then, what about the
sense of ‘Aristotle’?
My hypothesis is that Frege’s solution to
the puzzle only works for complex names that
express what I shall call articulated modes of
presentation. For incomplex names, like ordinary
proper names, numerals, and demonstratives or
indexicals (for which there is no “completing
Fregean sense” as Perry famously said), the
doctrine of “sense-expression” is totally arbitrary.
There is no obvious convention that picks up
one and only one mode of presentation. You can
know a priori the truth of an identity sentence
like “32=√81” and the modes of presentation
are easy to identify and grasp, because the
expressions involved are complex and some rule
of compositionality applied. But what would be
the mode of presentation expressed by ‘9’ in
virtue of a convention? Would it be the successor
of 8? Why not the last of the ten first integers, or
the predecessor of 10, etc.?
In case of ordinary proper names, the
whole doctrine is vulnerable, hardly consistent,
arbitrary and ontologically expensive. There are
infinitely many senses waiting to be grasped.
For Platonists, this is no problem at all, but an
army of naturalists will disagree. Fortunately,
Frege observes that a complete knowledge of
the denotation is not within our power. Any
object has infinitely many properties and we can
think of an object under infinitely many aspects
or modes of presentation. But our cognitive
capacities are limited; this is shown by the fact
45
that we cannot tell, of any sense given, if it is (or
not) a sense of a determined denotation.
We all know the famous (for some, infamous)
footnote about the sense of ‘Aristotle’. Frege
cannot specify the sense of an ordinary proper
name like ‘Aristotle’ simply by saying “the
sense of ‘Aristotle’”. It would sound ridiculous.
So he suggests that the sense is specified by an
indefinite numbers of definite descriptions, which
are complex names. The sense of ‘Aristotle’
would be specified by one or more sentences like
“the sense of ‘the founder of the Lyceum’”, “the
sense of ‘the most famous Plato’s student’”, etc.
The denotation of ‘Aristotle’ is the individual that
satisfies the condition of being the most famous
Plato’s student, etc. If Frege is right, ordinary
proper names, when they have a denotation,
necessarily have a sense, what we do understand
when we understand the contribution made by a
proper name to the truth conditions of a sentence
uttered in a context. For him, definite descriptions
are proper names too. But any individual has
infinitely many modes of presentation; most of
them will never be communicated, or will be
ignored. Why should we choose one instead
of another? Why Alexander’s teacher rather
than the founder of Lyceum? My encyclopaedic
knowledge about Aristotle is poorer than that
of any specialist, richer than most people I can
meet in the street, and that knowledge is not
(and needs not be) communicated when I use the
name ‘Aristotle’. An ordinary proper name that
denotes must have a sense, but which one? What,
if any, is the semantic rule attached to the name?
How do we separate encyclopaedic knowledge
from proper semantic knowledge in cases like
that? How could we establish with precision the
cognitive difference between ‘a’ and ‘b’ in “a=b”
when “a=b” is the formalization of an identity
sentence involving ordinary proper names? On
that score, I take side with direct reference
theorists.
7. As our main topic here is the interpretation
of ‘=’ in Frege,8 we cannot ignore truth-value
names because, in Frege’s system, truth-value
names, like any other simple or compound
names, are allowed to appear on each side of ‘=’.9
And this is perfectly consistent in Frege’s system.
If ‘P’ and ‘Q’ are sentential letters, then “P=Q”
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is a well-formed formula in that system, and
means that the two sentences are names of the
same truth-value. Most of the time, of course,
they will express different thoughts. Today
we use
or between sentences, and ‘=’ only
between constant letters, names in the strict (nonFregean) sense. Names for Frege are signs or
combinations of signs that denote. In his formal
system, all the names denote, and the variables
(Frege’s Roman letters in the Begriffsschrift) are
not names, let alone rigid designators. They do
not denote; they only “indicate”, he says.
When the puzzle is introduced, Frege speaks
of “objects” denoted by the names on each side
of ‘=’, and we know that objects, for him, “stand
opposed to functions.” So physical objects, like
the Moon, count as objects, but also truth-values,
courses-of-values, numbers, and also the possible
denotations of “the sense of ‘… E …’”. But Frege,
surprisingly, seems to accept in The Basic Laws
of Arithmetic, equations involving functional
expressions on both sides of ‘=’, like “
( )”.10 But this must be just a way of saying that
both functions have always the same truth-value
for the same arguments, or that both functions
have the same Wertverlauf (course-of-values).
Frege’s notation for courses-of-values says
exactly this:
Elsewhere, he says
clearly: “[…] the relation of equality, by which
I understand complete coincidence, identity, can
only be thought of as holding for objects, not
concepts” (Frege, 1892a, 175). We can think of a
function –and concepts are functions– in different
ways. These ways of thinking are unsaturated
senses. The course-of-values (an object in Frege’s
system) of two functions can coincide, but the
cognitive significance attached to the predicate
may diverge importantly. The function/concept
denoted by “x is much taller than the average
adults in North America”, and the one denoted
by “x is tall enough to play in the NBA”, may
coincide in truth-value for all arguments, but
certainly not in cognitive significance. Frege
says: “The fundamental logical relation is that of
an object’s falling under a concept: all relations
between concepts can be reduced to this. If an
object falls under a concept, it falls under all
concepts with the same extension […]” (Ibid., 173)
The two complex predicates denote concepts with
the same extension, but they express different
(articulated) modes of presentation.
So, this is an important point we should
bear in mind at the time of scrutinizing Frege’s
(so-called) puzzle. We should be aware of the fact
that he does not use ‘=’ exactly as we do.
8. Individual constants in a regimented
language are a very simplified version of proper
names, but they are far from retaining all the
complexity of ordinary proper names. Individual
constants retain a very small part of it. They
stand for individuals, and that is part of their
contribution to the truth conditions of the sentences
in which they appear. Some would say this is
their only contribution. Regimented, perspicuous
languages are extremely useful when it comes to
establishing the validity of an argument or the
logical form and truth conditions of sentences of
ordinary language when they masquerade their
logical form. But the use of individual constants
is very limited in a regimented language; in
natural languages, proper names do not always
serve the same and unique purpose of standing
for a referent. We call someone a ‘Hercules’ just
because he looks very strong, and a ‘Casanova’ a
successful man with a disposition to womanize.
This is what we call ‘antonomasia’ in Rhetoric.
Ordinary proper names like ‘Ramses III’ or
‘Elisabeth II’ tell us something more than other
ordinary names. They situate the bearer in time
by telling us, for instance, that Elizabeth II comes
after another famous queen with the same name.
Frege introduced his famous puzzle by using the
common notation for identity sentences in logic,
but after considering a geometrical example,
he turned to ordinary language to introduce his
notion of Sinn. As he used the term ‘Eigenname’
in a very extensive way, definite descriptions
count as proper names, and also predicates
and complete sentences. Russell and the neoRussellians would not follow Frege’s practice of
flanking two quantified terms on both sides of
‘=’. But ‘The Morning Star’ is a proper name?
May be it should be counted as one; ‘Phosphorus’
surely is. ‘The Eiffel Tour’ looks like a definite
description, but it is surely a proper name. It
would be silly to pretend that there are possible
worlds, in which “the Eiffel Tower” could have
denoted another structure, say, the CN tower
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in Toronto. ‘The author of Waverley’ is not an
ordinary proper name, but it contains one in modo
obliquo. What counts as a proper name depends
largely on context. It is a pragmatic business.
At the end of Kubrick’s movie, Spartacus, a
Roman centurion announces that the slave who
will identify Spartacus’ body, dead or alive, will
have his life spared. When Antoninus, Spartacus’
best friend, sees that he was about to surrender,
he gets up and says, “I am Spartacus!” followed
immediately by the other defeated slaves. They
all get up and say “I am Spartacus!” ‘Spartacus’
is a proper name. Only one person could truly
say, “I am Spartacus”. Are Antoninus and his
friends lying to the centurion? Well, that is not
the impression we have. A lie is something a liar
always tries to conceal; in that case, they do not
even try to conceal anything. They know that
the centurion would never believe they are all
called ‘Spartacus’. So, they are not lying. At that
point, the name “Spartacus” became the symbol
of something to be proud of. Is it still a proper
name? Yes, but it became something more, and
that “something more” is not captured by an
individual constant in a process of formalization.
Proper names of famous people are special too.
They are salient. If I say “Balzac was a French
writer” I will be understood immediately as
speaking of Honoré de Balzac, even though my
intention was speaking about Jean-Louis Guez
de Balzac (a French writer of the XvII Century),
simply because the first is much more famous
and comes first to mind. Some ordinary proper
names are used to designate a huge set of events
organized in a certain way or with some more
or less unifying characteristics, with no simple
bearer, like ‘Second World War’ or ‘Renaissance’.
For someone using a schematic notion of object,
like Frege, this is not a problem. The denotation of
‘Second World War’ is not an interval of time, but
complex sequences of events running in parallel
with some structuring elements (treatises and
agreements amongst nations, for instance). The
set of ordinary proper names is not something
well unified. This is why the formalization of
ordinary language sentences is a process that
regularly involves decisions guided by some
previous metaphysical conceptions.
47
9. What is a word? How do we individuate
words? In Latin, ‘bellum’ (war) has the same
form in the nominative, the vocative and the
accusative. Is it the same word? Is the shape so
important? Is the qualitative identity of the tokens
enough? As the function is different in the three
cases, the answer should be “no”. An English
translation in each case will render a different
periphrasis, with different prepositions. So, the
shape is not enough.
What is a name? This question is more
specific. It is a bit depressing for me to write
‘André Leclerc’ in the Google search engine and
to behold the result… So many people with the
same name! But is it the same name? Glezakos
(2009, 202-207) raises that important question,
following a suggestion made by Kaplan. Suppose
there are three André Leclerc: one living in
Brazil, one in Quebec, and another in France.
Let’s follow the old rule —correctly criticized by
Wittgenstein and Austin—, but only for ordinary
proper names of non-fictional individuals: unum
nomen, unum nominatum. We could distinguish
or individuate the three names, by using a
subscripted letter like this: AndréB, AndréQ, and
AndréF (or simply distinct numerals instead of
letters). The individuation of the names, in these
cases, presupposes the capacity to distinguish the
bearers of the names. Then what will we do about
fictional names, like Excalibur and Pegasus?
What does not exist has no real qualities or
properties. So what can we do to discriminate
and differentiate Excalibur from Pegasus? Well,
just to give a too short explanation, let us say that
these names are always introduced in narratives
that provide us with enough characteristics to
recognize the figments that are the bearers of
these names when they make an appearance,
for example, in a movie. This is why we can
say that the Sherlock Holmes of “A Scandal in
Bohemia” is the same as the one in “The Hound
of the Baskervilles”, even if the actors are not the
same. It is enough to suppose that the character
is the same.
The situation described above with the name
‘André’ is extremely common when we consider
ordinary proper names. How many people called
André, Marco, Steve, Patricia, etc. Names sharing
the same shape are in fact distinct from each other
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ANdRé LECLERC
because we know somehow that their respective
denotations are distinct too. How do we know
that? By applying cognitive resources, concepts
(in a non-Fregean sense) or modes of presentation,
enabling us to distinguish and classify what is in
fact distinct. Ordinary proper names are always
used against a background of information and
knowledge. First of all, we need to know in
which community (or sub-community) we are in
order to identify the “relevant” André or david,
or Marco. The cognitive resources involved in
the epistemic process of distinguishing different
bearers of the “same name” (or names with the
same shape) are not necessarily those associated
by a semantic rule to the proper name (if there
is such stable association at all). Many proper
names are introduced through descriptions, in
narratives, by demonstration, etc. For me, and
for a long time, ‘Kurt Gödel’ was the name of
the man who demonstrated that arithmetic is
incomplete. Today, after seeing photographs, I
know he was also the man wearing funny round
spectacles, and with a strange lock in his hair, as
he got older.
Now, let’s go back to “a=a”. What does it
mean exactly to say, “[…] the ability to recognize
that the name is the same seems to involve the
ability to recognize that the referent is the same”?
(Glezakos, 2009, 205, italics in the text) The
shortest answer is that Frege himself stipulates it.
We do not start with logical forms. We formalize
what we say and think in accordance with what
we understand in the first place. Generally,
substitution must be uniform in logic; so the two
ʻa’ must replace the same name. To recognize
and classify anything, in perception or thought,
we must apply concepts (still in a non-Fregean
sense). There is no recognition without that.
Frege’s theory of intentionality does not allow the
possibility of our thoughts and assertions being
about something without the intermediation
of something like modes of presentation or
aspects. The same holds for Husserl, and today
for Searle and Crane. There wouldn’t be any
mental or linguistic reference without that. When
he introduces the Sinn, Frege just points at
something that must be already there according
to his own principles. Furthermore, Frege makes
it clear that if a term has a denotation, it must
have a sense. It would be incoherent to pretend
that names on both sides of ‘=’ could lack a sense.
So I am inclined to agree with Glezakos when
she says: “[…] what emerges is that […] [Frege’s]
“puzzle” and his solution are in fact of a piece.”
Frege’s solution involves his whole framework.
But could it be otherwise?
I do not believe that ordinary proper names
express a sense in virtue of conventions widely
held in linguistic communities. Therefore Frege’s
solution is not working well when ‘a’ and ‘b’
stand for ordinary (incomplex) proper names.
But it works much better when the expression is
complex (definite descriptions or full declarative
sentences), when the expression expresses an
articulated mode of presentation. I also agree with
Glezakos that the puzzle is not very puzzling, may
be for slightly different reasons. However, if this
is so, we should reflect one moment on what leads
so many people to see in the first paragraphs of
SuB the presentation of a puzzle.
Perhaps, the diagnosis that accounts for
the illusion of a (problematic) puzzle is this: we
just focus too much on the schematic sentences
in the first paragraph of SuB, and forget about
the infinitely many substitution instances. By
themselves, the schematic sentences do not
represent anything. Schematic sentences like
“a=a” and “a=b”, either are the result of a process
of formalization, or are used to indicate in
abstracto a logical form as a tool to make explicit
some properties like reflexivity, transitivity and
symmetry. They just retain what is common to
infinitely many instances sharing the same form.
In the formalization process, we substitute names
for constants and the substitution, of course,
must be uniform. When we compare the logical
behaviour of “a=a” and “a=b”, the constant letter
‘a’ must replace the same name. Otherwise, we
open the way for the fallacy of equivocation.
Cases like that of ‘Paderewski’ are to be analysed
before any formalization. The way we formalize
depends upon the way we understand what we
formalize in the first place. Formalization may
help to reveal different possible readings. If
someone believes that Paderewski is a musician
and that the politician with the “same name”
is another person, then that person believes
(wrongly) something of the form “a b”, and
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FREGE’S PUzzLE, ORdINARY PROPER NAMES, ANd INdIvIdUAL CONSTANTS
certainly not something of the form 'a a'. We
have two names in this case. What distinguishes
them? It could be perceptual or psychological
modes of presentation applied to the bearers,
or linguistic modes of presentation used in two
different narratives introducing the names, but
not necessarily modes of presentation associate
by convention to the name ‘Paderewski’ (if
there are such conventions at all). As I said
earlier, if the sense of a proper name is really
something semantic in character, it must be tied
up to the name in virtue of a convention, a social
regularity. It must be there in all circumstances
of use. But this is highly doubtful in case of all
ordinary proper names. Part of the problem with
the so-called “Frege’s puzzle” is that the doctrine
of sense-expression (that says what it is for an
expression to express its sense) works well only
for complex names and not for incomplex ones.
But for us, today, ordinary proper names seem
to be the best candidates the ‘a’ and ‘b’ in the
initial formulas. Be that as it may, I think that
Frege’s puzzle, as posed by Frege himself, is not
very puzzling, and that when the ‘a’ and ‘b’ are
substitutes for ordinary proper names, Frege’s
solution fails.
We use regimented languages to formalize
what we say and think, and we formalize in
accordance with our previous, situated
understanding. In many cases, a little hermeneutic
work is necessary before using the resources
of a formal language. And the basis of any
hermeneutic work is always the spontaneous
linguistic understanding we exercise all the time
in familiar circumstances.
Notes
1.
2.
3.
Here I set aside the issue raised by d. Wiggins
(1980/2001), the question whether there is a
dependency of identity sentences on sortals, and
whether identity is relative or not.
For a reconstruction of the puzzle along these
lines, see (Sainsbury & Tye, 2012, 2-3).
By ‘object’ here I do not have in mind necessarily
a substantive notion of object, like that of a
physical object. Actually, Frege’s notion of
object wasn’t a substantive one. See (Frege, 1893,
49
35-36): “Objects stand opposed to functions.
Accordingly, I count as objects everything that
is not a function, for example, numbers, truthvalues, and the courses of values […]”. But
physical objects, like the Moon, are objects too.
4. On identity sentences and free logic, see (Crane,
2013, chap. 3). See also (Lambert, 2004).
5. (Frege, 1882, 155). By ‘sense-symbol’ the
translator means here that signs can be perceived
(seen or heard).
6. Frege made that suggestion in SuB. See (Sainsbury,
2002, 4).
7. See (Parson, 1981, 37-58). I said “provided with
a denotation” because in most system we need
an axiom of foundation, that is, there must be a
denotation in the initial segment of the series.
8. Interestingly, Nathan Salmon defends that the
puzzle has virtually nothing to do with identity.
The puzzle is more general in nature; according
to Salmon, it’s a problem about the way pieces
of information are encoded in sentences. See
(Salmon, 1983 12-13).
9. (Frege, 1893, 36, 37, 40, 69, et passim). Of
course, this is not what we do today. We do not
put ‘complex names’ on both sides of ‘=’, not
even definite descriptions when they are clearly
quantified terms.
10. See (Frege, 1893, 36, 40, et passim). The variable
is metalinguistic.
References
Beaney, M. (Ed.) (1997). The Frege Reader. Oxford:
Basil Blackwell.
Burge, T. (1990). Frege on Sense and Linguistic
Meaning. In T. Burge Truth, Thought, Reason.
Essays on Frege. Oxford: Clarendon Press, 2005.
Crane, T. (2013). The Objects of Thought. Oxford:
Oxford University Press.
Frege, G. (1892). On Sinn and Bedeutung. In M.
Beaney (Ed.) (1997), 151-171.
. (1892a). Comments on Sinn and Bedeutung.
In Beaney (Ed.) (1997), 172-180.
. (1879). Begriffsschrift. In J. van Heijenoort
(Ed.) (1967) From Frege to Gödel. A Source Book
in Mathematical Logic. Cambridge: Harvard
University Press.
. (1893). The Basic Laws of Arithmetic.
Translated by M. Furth. Berkeley: University of
California Press.
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. (1882). On the scientific justification of
concept-script. Translated by J. M. Bartlett. Mind.
LXXIII (290), 155-160.
. (1918) Thought. In M. Beaney (Ed.) (1997).
Glezakos, S. (2009). Can Frege pose Frege’s Puzzle?
In J. Almog & P. Leonardi (Eds.) (2009) The
Philosophy of David Kaplan. Oxford: Oxford
University Press.
Lambert, K. (2004). Free Logic. Selected Essays.
Cambridge: Cambridge University Press.
Parson, T. (1981). Frege’s Hierarchy of Indirect Senses
and the Paradox of Analysis. In Midwest Studies
in Philosophy. 6 (1), 37-58.
Salmon, N. (1983). Frege’s Puzzle. Cambridge Mass:
MIT Press Brandford Books.
Sainsbury, M. (2002). Departing from Frege. London:
Routledge.
Sainsbury, M. & Tye, M. (2012). Seven Puzzle of
Thought. Oxford: Oxford University Press.
Wiggins, d. (2001). Sameness and Substance Renewed.
Cambridge: Cambridge University Press.
(*) André Leclerc (Universidade Federal do
Ceará, CNPq) is a Canadian working in Brazil
since 1995. He is currently associate professor at
the Federal University of Ceará. He is a fellow
researcher of the Brazilian Council for Scientific
and Technological development (CNPq). He was
president of the Brazilian Society for Analytic
Philosophy (2012-2014), and Treasurer of the
Latin-American Association for Analytic
Philosophy. He publishes mainly in Philosophy
of Mind and Philosophy of Language.
Received: Monday, September 8, 2014.
Approved: Monday, September 22, 2014.
Rev. Filosofía Univ. Costa Rica, 53 (136 Extraordinary), 41-50, May-August 2014 / ISSN: 0034-8252
André Leclerc