**Summary: **The terms *sense, meaning, denotation, reference *are mostly used
without any critical attempt at defining them. So it frequently happens that *reference *is
used promiscue with *denotation.* The paper shows that at least the approach known as
'transparent intensional logic' is able to offer such definitions which, among other
things, make a fundamental distinction between *denotation* and *reference,* and
make it possible to explicate Frege's *Sinn* in a most inspiring way.As a
consequence of such definitions one basic intuition is supported, viz. that whereas the
sense and the denotation of an expression are given (relatively) *a priori*,
reference cannot be unambiguously determined by the sense and is co-determined by the
state of the world. A logical apparatus is briefly suggested which enables us to exactly
formulate the above intuitions.
**Key words**: sense, meaning, denotation, reference, functional approach,
intensions, constructions.
**1. Semantics vs. Pragmatics**
The topic of the present paper is a* semantic* rather than a *pragmatic*
problem. Unlike Quine I do not replace semantics by pragmatics, and my 'slogan' is *ask
for meaning before asking for use* in contradistinction to the well-known
Quinean-Wittgensteinian slogan (see [Materna 1998]). This means that it is abstract
expressions what we try to analyze, not the concrete events of uttering expressions.
From this viewpoint we must be aware of the fact that the genuinely semantic entities
like *sense, meaning, denotation* are given (relatively) *a priori*; this
viewpoint is, of course, distinct from the viewpoint of theoretical linguistics, since the
fact that a certain expression possesses a certain meaning (in the given language) is from
the latter viewpoint contingent, whereas for semantics the given linguistic convention is
supposed to be already given, so that the given expression *necessarily* possesses
such and such meaning(s), so that we can 'calculate' the meaning independently of
empirical facts. The same result holds for denotation, since the meaning should
unambiguously determine the denotation. One of the results of our analysis consists in the
claim that *reference* - unlike denotation - is not *a priori*.
The area given by this specification can be called *logical analysis of natural
language* (LANL), and it is important especially for philosophical logic, which should
determine the class of *correct arguments*. To adduce an example, the reason why the
following argument is not correct, although it seems to be due to the application of the
Leibniz's principle
The number of (the major) planets is nine.
Necessarily, nine is greater than seven.
*Necessarily, the number of (the major) planets is greater than seven.*
can be found by LANL but not by any descriptive theory such as a general theory of
language.
** **
**
****2. A (not only) terminological mess in semantics**
The origin fo the contemporary semantics and LANL is, at the same time, the origin of
fundamental confusions in this area. I mean Frege's classical [Frege 1892]. Already from
the terminological point of view, Frege's term *Bedeutung* is confusing: in German it
means what should or at least could be translated as *meaning*, but if meaning is
what makes us understand the given expression (this is one of the few points where a
nearly general consensus among semanticists can be expected), then Frege's *Bedeutung*
surely does not fulfil this role, especially when *Bedeutung* of a sentence is 'its'
truth-value. The role of meaning is played by Frege's *Sinn* instead. Therefore the
Fregean logician Alonzo Church (see [Church 1956]) has chosen the term *denotation, *translating
*Bedeutung *into English.
What led Frege to his distinguishing between sense and denotation is well-known. Two
problems he began to solve and failed to do it inspired great a many philosophers and
logicians to writing articles and monographs handling these problems, but mostly the
source of Frege's failure has not been recognized, or at least a wrong therapy has been
chosen. The vagueness of Frege's characteristics of sense and his not consistent and too
coarse-grained use of the denoting relation have been mostly inherited by his followers.
Let aus articulate the mentioned problems:
A. What is *sense*?
B. What does an (empirical) expression denote?
Ad A.: The only formulation used by Frege to characterize sense is *die Art des
Gegebenseins*, viz. *der Bedeutung* ("mode of presentation"), which, on the
one hand is too vague to be construed as being a definition, and, on the other hand,
cannot decide whether sense is structured (this can be implied from his first example with
medians of a triangle), or not (see his more popular example with morning star vs. evening
star).
Ad B.: The second question begins to be a problem in virtue of the fact that the
relation between an expression and the object it denotes (*bezeichnet*) is considered
by Frege to be immediately obvious and essentially the same for empirical and
non-empirical expressions. This Frege's opinion has been called by P.Tichý *Frege's
Thesis* and formulated as follows:
[u]nder the meaning (i.e., *Bedeutung*, P.M.) he does not understand what is
connected with it by linguistic fiat, but rather the object which is presented thereby.
[Tichý 1992, p.5]
It should be intuitively clear that Frege's Thesis is strongly counterintuitive.
Consider the case of empirical sentences. A consequence of Frege's Thesis is that such
sentences denote truth-values (cf. the well-known 'slingshot argument', e.g., in [Church
1956]). As for the denotation we would have only two sentences, one denoting Truth, the
other Falsity. Not only that, but in the case of empirical sentences which use to change
their truth-value the denotation of such sentences would change with changing facts: the *a
priori* character of denotation would disappear. And the general consequence thereof is
that *denotation is no more unambiguously* *determined by the sense*, which
contradicts Frege's intentions and his characteristics of sense (see above).
The contemporary semantics is subconsciously dissatisfied with the term *denotation*.
Sometimes or probably mostly it simply replaces it with the term *reference*. A
typical example is Linsky's article on referring in [Linsky 1967]; there only the term *refer
*is used where Church would use *denote*, and no distinction between the
medians-example and the morning star vs. evening star-example is seen. We can see,
however, that in the medians-example the point of intersection is unambiguously given by
the sense of the respective expression, whereas the sense of the expression *morning
star* - be it anything - cannot be said to unambiguously determine Venus.
Thus we state the regrettable fact that at least four terms important for
semantics/LANL are used without any rule based on definitions, without any more or less
exact justification: they are
*meaning, sense, denotation, reference.*
One ambiguity could be tolerated: the term *meaning* can be used promiscue with
the term *sense*, because the way the former is used seems to correspond with what
Frege meant introducing the latter. Let us use *meaning* for Frege's *sense*,
therefore. Our question can be formulated as follows:
*What is the distinction - if any - between meaning, denotation, and
reference?*
**
*** *
3. Objects denoted: the elementary case
We will try to show that the conceptual framework proposed and successfully applied
by Pavel Tichý in his *transparent intensional logic* (TIL) is able to disambiguate
the above mentioned chaotic use of terms. Tichý himself did not do it explicitly on the
level of terms (he also sometimes uses *reference* instead of *denotation*) but
his system, explained in many articles and in his last book [Tichý 1988], makes it
possible to correct Frege's Thesis and define meaning (no more an 'obscure entity').
Referring for details to his work (and perhaps to my [Materna 1998]) we will only
describe some fundamental features and notions of TIL. Our first result will consist in
our specification of the denoting relation in the most usual (elementary) case.
Should what is denoted by an expression be independent of empirical facts then no
expression should be construed as denoting distinct objects. This principle contradicts
Frege's Thesis, according to which the expression *the richest man in Europe* denotes
one individual at the time point T and another individual at the time point T'. Also, it
is not at all clear how that expression could *via* its sense unambiguously determine
- even at the same time point - the individual who in fact is the richest man in Europe -
we have to inspect empirical facts to identify such an individual. Thus we can state that
the variability of the denotation - which seems to contradict our principle - is of
twofold character: it is a *temporal* and also a *modal* variability.
To solve this problem we have to apply a principle formulated in [Janssen 1986]:
according to it if we are tempted to say that the denotation (Janssen speaks about
meaning) depends on some circumstances, then let these circumstances be arguments of a
function and say that this very function is the denotation. Now to be able to handle
temporal variability we need time points as elementary entities, and to be able to handle
modal variability we need possible worlds. So the area of objects that can be in the
elementary case denoted should be built up from - among other things - *time points*
and *possible worlds*. This is not sufficient, of course. We need, of course, most
simple material objects called usually *individuals, *and no discourse can be
realized without two simple objects called *truth-values,* say, **T**, **F**.
TIL is a type-theoretical system based on the above elementary types. Using o (Greek omíkron) for the set {**T**, **F**}, i (iota) for the set of individuals, t
(tau) for real numbers and time points, and w (ómega) for the
set of possible worlds, we define *types of order 1*:
i) o , i , t
, w are *types of order 1.* *
*ii) If a , b _{1},
..., b _{m} are *types of order 1,* then (a b _{1}...b
_{m}), i.e., the set of all partial functions
with arguments (tuples) in b _{1}, ..., b _{m} and values in a , is
a *type of order 1.* *
*iii) Nothing other is a *type of order 1.* ~
It can be shown that this definition covers all (important?, just all?) kinds of object
which can be denoted in the elementary case, i.e., if we ignore the case when an
expression denotes a meaning (of another expression). For we should be able to denote
truth-values, individuals, numbers, classes of objects of any type, relations-in-extension
of various types, propositions, properties of objects of any type, relations-in-intension
of any types, magnitudes, etc., but all of these objects can be associated with a type. So
let a be any type. A class of objects of the type a is an (o a
)-object, since it can be identified with a characteristic function: with any a -object such a class associates **T** if it is a member of it,
and **F **otherwise. An easy analogy holds for relations-in-extension (the general
schema is (o b _{1}...b _{m}) ).
The remaining examples of objects represent *intensions*. What is a proposition?
Setting aside the Russellian "structured propositions" we can accept the contemporary
convention that holds among the possible-world-semanticists, and construe propositions as
functions from possible worlds to (chronologies of) the truth-values. The modal
variability is annihilated by taking propositions to be functions from possible worlds,
the temporal variability disappears as soon as the value of such function is not simply a
truth-value but rather a chronology of truth-values, i.e., a function from time points to
truth-values. According to our definition the type of propositions is ((o t )w ). In
general, *intensions *are functions whose type is ((a t )w ) for a
any type. We will use the abbreviation accepted in TIL, writing a
t w instead of ((a
t )w ). So we have type (o a )t w for properties of a -objects, type (o b _{1}...b
_{m})t w for
relations-in-intension, type t t w for magnitudes, etc.
Especially such objects that Church called *individual concepts* and are denoted
by (empirical) definite descriptions, like our example *the richest man in Europe, *are
i t w
-objects.
Against Frege's Thesis TIL shows that empirical expressions can denote only intensions.
Thus empirical sentences denote propositions rather than truth-values, and rightly so:
imagine that a logical analysis of an empirical sentence would discover the truth-value
'denoted' by that sentence. Then no verification of empirical sentences would be
necessary: instead, any logician would be able (in principle) to 'calculate' the
respective truth-value.
Thus a first distinction can be stated:
*An expression ***denotes** the object (if any) which is unambiguously determined
by its sense.
Therefore:
*An empirical expression denotes an intension, never the value of the intension in
the actual world.*
On the other hand:
*An expression ***refers to** the object which is the value of its denotation in
the actual world-time.
4. Sense (= meaning)
Perhaps the most difficult task is to replace Frege's vague characteristics of
sense by a more or less acceptable but in any case precise definition. Why we cannot
accept Carnap's *intensional isomorphism* neither Cresswell's (and Kaplan's) *tuples*,
is explained in [Tichý 1998, p.8-9] and [Tichý 1994, p.78]. We now only globally
characterize Tichý's *constructions *and try to argue that they are probably the
best starting point to explicating sense.
Let us begin with any example from the area of arithmetics. So consider the expression
7 + 5 = 12
(in honor of Kant): anybody would agree that *expressions *7, 5, 12 denote the *numbers
***7**,** 5**, **12**, respectively. What about + and = ? Even in this point
only some very stubborn nominalists would disagree that + denotes the *function* of
adding (its type being (t t t ) ), and = the identity *relation* (type (o
t t ) ). What does the whole
expression *denote *should also be clear - it is the truth-value **T**. Now what
is the *sense *of the above equality, as the way to this **T** ? Accepting the
useful principle of compositionality we claim that *the sense of that equality is
unambiguously* *determined by the senses of the particular parts of it.* So let us
ask what senses are expressed by these components, i.e., by the expressions 7, 5, 12, +, =
.
Understanding these simple expressions (= knowing their senses) means to be able to *identify
*the respective numbers and functions (even relations are functions, viz.
characteristic functions). Now we can have various ways to such an identification, but to
get such a way for every object presupposes that there are some *primitive ways*
where we have to stop to avoid regressus ad infinitum. Thus our first claim is that *there
are* some *primitive senses. *Not claiming that we need them just when analyzing
our equality we will choose them for the sake of a didactic explanation.
But given that we have some primitive senses at our disposition we have to answer a
second question : *In which manner do the primitive senses combine so that they result
in determining the non-primitive sense of the whole expression?*
Many semanticists seem not to see this problem. They say that this 'synthesis' is
determined by the grammar of the given language. (see, e.g., [Sluga 1986]). But then we
can say with Tichý in [Tichý 1988, p.36-37]:
If the term '(2 × 2) - 3' is not diagrammatic of anything, in other words, if the
numbers and functions mentioned in this term do not themselves combine into any whole,
then the term is the only thing which holds them together. The numbers and functions hang
from it like Christmas decorations from a branch. The term, the linguistic expression,
thus becomes more than a way of *referring* to independently specifiable subject
matter: it becomes *constitutive* of it.
Independently of this excellent formulation the French computer scientist J.-Y. Girard
formulates a very similar thought, saying [Girard 1990] about the equality 27 × 37 = 999:
[t]he denotational aspect ... is undoubtedly correct, but it muisses the essential
point:
There is a finite *computation* process which shows that the denotations are
equal. It is an abuse... to say that 27 × 37 *equals *999, since if the two things
were *the same* then we would never feel the need to state their equality. Concretely
we say a *question*, 27 × 37, and get an *answer*, 999. The two expressions
have different *senses* and we must *do* something (make a proof or a
calculation...) to show that these two *senses *have the same *denotation*.
The last sentence uses the terms *sense *and *denotation* not in a standard
way, it should be reformulated as follows: "... to show that the two expressions differ
in senses but have the same denotation", but the idea is clear enough and is in full
harmony with the idea of the preceding quotation.
Thus we have suggested a motivation for the choice of Tichý's constructions the
definition of which is contained in [Tichý 1988]. Here only main points:
i) The primitive constructions are *variables* (as not linguistic expressions but
'incomplerte constructions' constructing objects dependently on valuation) and *trivialization*:
where X is any objects (or even construction), ^{0}X constructs this very object
without any change.
ii) *Composition* is a construction symbolized by [XX_{1}...X_{m}],
where X constructs (maybe dependently on valuation - this possibility will be presupposed
in the following) a function (type (a b
_{1}...b _{m}) ) and X_{i }constructs
an object of the type b _{i}. It constructs the
value (if any) of that function on the arguments constructed by X_{i} for 1L *iL m*.
iii) *Closure* is a construction symbolized by [l *x*_{1}*...x*_{m}X],
where *x*_{1}*...x*_{m} are pairwise distinct varaibles
constructing objets of types b _{1},...,b _{m}, respectively (types not being necessarily
distinct) and X is a construction constructing members of a type a
. It constructs a function in the way well-known from the l
-calculi.
(We have omitted two other constructions from Tichý's book, they are not important
here.)
Now our Kantian equation gets the following construction as its sense :
(7. 5, 12 / t , + / (t t t ), = / (o t t ) )
[^{0}= [^{0}+ ^{0}7 ^{0}5] ^{0}12]
The way the constructions are defined makes it possible to derive the resulting
denotation. The sense given by the above construction is, of course, distinct from the
above chain of characters: the respective composition does not contain brackets etc. - the
above chain of characters only fixes in a standardized way what the *abstract procedure*
called composition does, it fixes particular 'steps' of that procedure.
To show some example from the area of senses of empirical expressions, let us analyze
the expression *the highest mountain.*
We already know that what is denoted by this expression is an individual role, an i t w -object
and that Mount Everest, not being mentioned in this expression, is not the right
candidate. (it is 'only' the reference of that expression.) Now looking for the way in
which the i t w
-object is determined, i.e., looking for the sense of our expression, we have to determine
the primitive senses of *the highest* and *mountain*. Type-theoretically, *mountain*
is clear: it denotes a property of individuals, so an (o i )t w -object;
let its primitive sense be (for the sake of simplicity) ^{0}M(ountain). *The
highest* is type-theoretically not as simple, but we can determine the type of the
denoted object after some brief consideration. Our first claim will be that the expression
is an empirical one. Thus its type will be a t w for some type a
. But a has to be a type of a function: this function, if
applied to a class of individuals, selects that individual (if any) which is the highest
one in that class. So a is obviously a function of the type (i (o i )). The
whole type is thus (i (o i ))t w _{.}
The primitive sense of *the highest *will be ^{0}H(ighest).
The sense of the whole expression has to be such a construction which would construct
the individual role that an individual has to play to be the highest mountain, so the type
of the constructed object (and thus of the denotation of our expression) will be i t w .
Not having at our disposal up to now the theory which would apply the above frame to a
particular language we have to replace precise rules by intuitive considerations, which,
however, are of key importance.
If we choose *w* as the (say, the first one) variable ranging over possible
worlds, and *t *as the variable ranging over time points, we can see that the
construction we look for will be
[l *wl t* X]
for X being a construction constructing (maybe dependently on valuation) individuals.
To find such a construction we have to use the constructions ^{0}M and ^{0}H.
The first suggestion is: observe that the value of H in the given world-time is a function
from classes of individuals to individuals. Thus applying H to (first) a possible world
and (then) to a time point we get simply a function from classes of individuals to
individuals. Writing, in general, X_{wt} instead of [[X*w*]*t*],
we have
[^{0}H_{wt}Y]
as the construction of an individual if Y constructs a class of individuals. Could Y be
^{0}M ? Surely not, since M is not a class but a property. But applying M to
world-times represented by thevariables *w, t* we get a class (a definite class after
*w* and *t* are evaluated). So we have now (omitting the outermost brackets):
l *wl t* [^{0}H_{wt}^{0}M_{wt}]
and can check that this construction serves as the sense of our expression to determine
the intended denotation: According to the above definitions or characterizations we can
see that the above construction constructs the function whose value in the world *W *is
a chronology which at the time point T returns that individual (if any) which is (in W at
T) the highest one in the class of those individuals that are in W at T mountains. But
this is exactly what we mean by that expression. We cannot mean Mt Everest by it, since it
is not mentioned in it; indeed we do not believe that this expression is a well formed
expression of English because of the necessity to have some other name for Mt Everest
--this expression is fully meaningful even at the time when nobody knew which mountain is
the highest one. Of course, if we knew a priori which of the possible worlds is the actual
one we would be able to 'calculate' Mt Everest from the sense of the expression *the
highest mountain*.
** **
**
****5. Conclusion**
The above conceptual frame makes it possible to distinguish semantic relations which
are frequently confused together. To sum up the results in a brief form we can state:
We can distinguish between the relations
A. *expression - sense (=meaning), expression - denotation, sense - denotation *on
the one hand, and
**B. ***expression - reference, sense - reference, denotation - reference *on the
other hand.
We claim (and have adduced some arguments above) that
*The relations sub ***A**. are (relatively) **a priori**, whereas the relations
sub **B** are necessarily mediated by experience, and are, therefore, **not a priori.**
**
** **
** **
**One of the 'byproducts' of our conception is our ability to logically distinguish
between *synonymy *and a mere logical equivalence. Only those expressions which share
their *sense* can be called synonymous. The (logical) *equivalence* of the
expressions E and E' means only that E and E' share their *denotation.* **
****References**
[Church 1956] Church, Alonzo: *Introduction to Mathematical Logic I. *Princeton
[Frege 1892] Frege, Gottlob: Über Sinn und Bedeutung. *Zeitschrift für Philosophie
und philsophische Kritik 100, 25-50* *
*[Girard 1990] Girard, J-Y.: *Proofs and Types. *Cambridge UP
[Janssen 1986] Janssen,T.M.V.:* Foundations and Applications of Montague Grammar.
Part I. *Amsterdam
[Linsky 1967] Linsky, Leonard: Referring. In: Edwards,P.,ed.: *The Encyclopedia of
Philosophy 7, 95-99. The Macmillan Co & The Free Press, New York* *
*[Materna 1998] Materna, Pavel: *Concepts and Objects. *Acta Philosophica
Fennica 63, Helsinki
[Tichý 1988] Tichý, Pavel: *The Foundations of Frege's Logic. *De Gruyter
[Tichý 1992] Tichý, Pavel: *Sinn & Bedeutung* Reconsidered. *From The
Logical Point of View, 1992/2, 1-10 * *
*[Tichý 1994] Tichý, Pavel: The Analysis of Natural Language. *From the Logical
Point of View 1994/2, 42-80.*
* *
**This paper has been supported by the grant No 401/99/0006 of the Grant Agency of
Czech Republic**
**
**Edited: 2000 |