Frege's schema is inspiring, of course, but it has to be corrected as for the denotational part, and the lacking explication of "sense" has to be added.

Both points have been realized by Pavel Tichý in his numerous articles and in the book "The Foundations of Frege's Logic" [Tichý 1988].

Let us begin with the denotational semantics.

Denotation vs. reference. Using promiscue the term "denotation" and the term "reference", as it is usual, means that a fundamental distiction between empirical and non-empirical expressions is ignored. I do not intend to thoroughly comment the popular claims of the "post-analytical philosophers" who simply follow the superficial attack against the borderline between analytic and synthetic propositions. The power of this attack (started by Quine already in 1952) can be explained by the simple fact that semantic problems have been mixed up with pragmatic problems. Let us therefore accept a following principle which should make this mixing up impossible:

The area of semantics is the area of linguistic convention which associates the expressions of a language with meanings/senses. Within this area the transition of an expression to its sense, but also the transition of the sense to the denotation, is an a priori area: in the case of empirical expressions it means that what an expression denotes must be determined by semantics alone. In other words, the "actual objects" falling under the denotation must be determined not by semantics alone but also by the state of the world.

In possible-world semantics (PWS) this principle can be obeyed as follows:

Empirical expressions denote just intensions, i.e., functions from possible worlds (and time points). To determine the value of an intension in a world/time cannot be the task of semantics: we need experience, i.e., something which cannot be a priori derived from the sense of the given expression.

Thus we can distinguish between denotation and reference: the reference of an empirical expression E in the world W at the time (point) T is the value of the denotation of E in W at T.

To adduce some examples, our (Tichý's) conception leads to following results:

Empirical descriptions denote what Tichý calls offices (and Church individual concepts), i.e., partial functions from possible worlds to (chronologies of) individuals. So

The highest mountain

denotes the function that associates every possible world (and time) with at most one individual. (There are no "possible individuals" , since the universe is the same for all possible worlds.) Thus it is not Mount Everest what is denoted by the above expression, and rightly so: the linguistic convention that equips this expression with its sense is surely independent of the fact that such and such object is the highest mountain, and the way from sense to denotation is still within the area of semantics.

(Therefore, from the premiss

The greatest mountain is in Asia

the conclusion

Mount Everest is in Asia

cannot be logically derived: it does not follow.)

General expressions like table, (to be) yellow, goblin, etc. denote properties (of individuals, of classes, of properties, etc.) rather than sets. This can be very clearly seen when we ask which kind of object is denoted by such fairy-tales-names like goblin: no actual class is mentioned, and rightly so: if we said that the object in question is the empty class (as we would probably say ), then, for example, unicorn would have to denote the same object (viz. the empty class). Properties are again intensions, viz. functions that associate every possible world/time with a class. And the property being a goblin differs from the property being a unicorn : in terms of PWS there are possible worlds where unicorns do and goblins do not exist, and vice versa.

Analogy concerning the distinction between relations-in-extension and relations-in-intension can be easily formulated.

(Declarative) empirical sentences denote propositions, i.e., functions from possible worlds/times to truth-values. Thus Frege's claim that (also the empirical) sentences denote truth-values is wrong; if it were the case, i.e., if an empirical sentence really denoted a truth-value and if denoting were - as it should be - a semantic relation then the truth-value of any empirical sentence could be logically determined: Is there any life in Mars? Why, we need not send some probes to Mars - the semantic analysis of the sentence There are living entities in Mars will do.

The last example concerns the well-known expression The number of (the great) planets (of the Solar system). Should the denotational semantics really determine that this number is 9? As soon as we apply our above principle everything is clear: this expression denotes a magnitude, viz. an intension that associates worlds/times with numbers. To see that in the actual world now the number is 9 we need experience: it is astronomers rather than semanticists who determine this number, i.e., the value of that intension in the actual world now. (By the way, the famous puzzle with the argument

Necessarily, 9 > 7

The number of planets is 9

Therefore: Necessarily, the number of planets > 7

- which has perhaps caused Quine's scepticism as regards the possibility of doing modal logic -

can be easily solved with construing the denotation of the key expression as an intension.)

In all the above examples it is easily seen what the reference of the respective expressions is in the actual world . Thus the highest mountain refers to Mount Everest (but not always), yellow refers to the class of all (actually) yellow things at the given time point, There are living entities in Mars surely refers to a truth-value (True, False? We do not know as yet.), the number of planets refers to 9 (but even in the actual world this number was for a long time zero).

One could object that application of the respective intensions to the actual world would unambiguously determine the references. Alas, since the actual world can be rationally construed only as the set of all (""actual") facts, only an omniscient being could know which of the possible worlds the actual one is.

Sense. Unless an expression denotes a sense (of another expression) the objects denoted are set-theoretical objects. This holds also of intensions, for they are functions from possible worlds, and as such they are sets of ordered tuples. Any such function can be reached in inifinitely many

ways. This can be seen already when we observe a truth function. Whether we apply the truth function ,implication' to a pair of truth-values, or the truth function ,disjunction' to the application of the truth function ,negation' to a truth-value, and a second truth-value, the resulting mapping is the same, and you cannot distinguish in this mapping implications, negations, disjunctions, etc. The ,target' is simple, the procedure identifying the target is, in general, complex and contains some parts which are not contained in the target. (As for simplicity of set-theoretical objects, see the excellent paper [Tichý 1995].) The Fregean sense cannot be simple, since it is a "mode of presentation" (die Art des Gegebenseins), and therefore an abstract procedure. The best definition of such abstract procedures is, as far I know, Tichý's definition of constructions. (See [Tichý 1988].) This definition is based on ramified hierarchy of types (an essential modification of Russell's), where the basic (atomic) types are o (the set {T , F} of truth-values, i (the universe, i.e., the set of individuals), t (the set of time points/real numbers), and w (the logical space, i.e., the set of possible worlds), and the types of order 1 are these atomic types together with sets of functions (ab₁...b_m) with arguments in b₁,...,b_m and values in a. To define the higher order types we have to define constructions. The most important ones are:

i) variables, construed as abstract procedures that construct objects dependently on valuation (they are said to v-construct objects). Infinityely many variables are at our disposal for every type. The letters x,y,z,...,p,q,...,f,g,... etc. are names of variables rather than variables themselves.

ii) Trivialization, ⁰X, where X is any object or construction; trivialization constructs just the object X.

iii) Composition, [XX₁...X_m] , where X is a (v-)construction of a function of type (ab₁...b_m), and X_i v-construct an object of type b_i . This kind of construction v-constructs the value (if any) of the function v-constructed by X . Since functions are construed as partial functions, composition can be v-improper, i.e., not to construct anything. (Compare application of dividing to a pair of numbers where the second is zero.)

iv) Closure, [lx₁...x_mX] , where x₁,...,x_m are pairwise distinct variables, and X is a construction. This kind of construction v-constructs a function (mapping) the arguments of which are all the possible valuations of x₁,...,x_m and values are determined by X when the variables following l take the respective values. (See l-calculi.)

Now on the basis of this definition constructions of order n can be defined, and the set of all these constructions (say, *_n) is a type of order n + 1.

So the sense of an expression can be construed as a construction of the respective order.

Example: Let us consider the sentence The number of planets is greater than 7. First of all, a type-theoretical analysis must me made. We get: The number of/ (t(oi)), planet/ (((oi)t)w) (abbreviated (oi)_tw), greater than/ (ott). Since the sentence is obviously an empirical sentence it must denote a proposition, i.e., an o_tw-object. Thus we have

[lw[lt [⁰Gr [⁰N ⁰Pl_wt] ⁰7]]]

where the respective procedure abstracts over possible worlds (the variable w) and time points (the variable t) and takes the truth-value T in such worlds/times where the number of the elements of the class of those individuals that are planets in those worlds/times is greater than 7.

Contextualism vs. transparency. The Fregean solution of the semantics of expressions in s.c. "indirect contexts" is a contextualistic one. This means that the following principle is accepted:

Indirect contexts concern what the expression in such a context denotes. What a concept denotes is, however, dependent on the context.

The transparent conception claims that every expression denotes the same object (and expresses the same sense) in every context. The problem is that some contexts are sensitive to the denotation ( the case of modalities), some other contexts are sensitive to the sense (the case of propositional and notional attitudes).

All this can be argued for in terms of the exact apparatus of transparent intensional logic, to be found especially in the above mentioned book.

References

[Frege 1892] G.Frege: Über Sinn und Bedeutung. Zeitschrift für Philosophie und philosophische Kritik 100, 25-50

[Church 1956] A.Church: Introduction to Mthematical Logic. Princeton

[Tichý 1988] P.Tichý: The Foundations of Frege's Logic. De Gruyter

[Tichý 1995] P.Tichý: Constructions as Subject Matter of Mathematics. In: W.Depauli-Schimanovich, E.Kohler, Fr.Stadler, eds: The Foundational Debate. Kluwer A.P., 175-186

This work was supported by the Research Support Scheme of the Open Society Support Foundation, grant No.: 166/1998.

Edited: 1998