Pavel Materna
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.


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), 0X constructs this very object without any change.

ii) Composition is a construction symbolized by [XX1...Xm], where X constructs (maybe dependently on valuation - this possibility will be presupposed in the following) a function (type (a b 1...b m) ) and Xi constructs an object of the type b i. It constructs the value (if any) of that function on the arguments constructed by Xi for 1L iL m.

iii) Closure is a construction symbolized by [l x1...xmX], where x1...xm 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+ 07 05] 012]

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) 0M(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 0H(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 0M and 0H. 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, Xwt instead of [[Xw]t], we have


as the construction of an individual if Y constructs a class of individuals. Could Y be 0M ? 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 [0Hwt0Mwt]

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.


[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