11. Sense and Reference

Equality gives rise to challenging questions which are not altogether easy to answer. Is it a relation: A relation between objects, or between names or signs of objects? The reasons which seem to favour this are the following: a = a and a = hare obviously statements of differing cognitive value; a = a holds a priori and, according to Kant, is to be labelled analytic, while statements of the form a = b often contain very valuable extensions of our knowledge and cannot always be established a priori. The discovery that the rising sun is not new every morning, but always the same, was one of the most fertile astronomical discoveries. Even today the identification of a small planet or a comet is not always a matter of course. Now if we were to regard equality as a relation between that which the names ‘a’ and ‘b’ designate, it would seem that a = b could not differ from a = a (i.e. provided a = b is true). A relation would thereby be expressed of a thing to itself, and indeed one in which each thing stands to itself but to no other thing. What is intended to be said by a = b seems to be that the signs or names ‘a’ and ‘b’ designate the same thing, so that those signs themselves would be under discussion; a relation between them would be asserted. But this relation would hold between the names or signs only in so far as they named or designated something. It would be mediated by the connexion of each of the two signs with the same designated thing. But this is arbitrary. Nobody can be forbidden to use any arbitrarily producible event or object as a sign for something. In that case the sentence a = b would no longer refer to the subject matter, but only to its mode of designation: we would express no proper knowledge by its means. But in many cases this is just what we want to do. If the sign ‘a’ is distinguished from the sign ‘b’ only as object (here, by means of its shape), not as sign (i.e. not by the manner in which it designates something), the cognitive value of a = a becomes essentially equal to that of a = b, provided a = b is true, A difference can arise only if the difference between the signs corresponds to a difference in the mode of presentation of that which is designated. Let a, b, c be the lines connecting the vertices of a triangle with the midpoints of the opposite sides. The point of intersection of a and b is then the same as the point of intersection of b and c. So we have different designations for the same point, and these names (‘point of intersection of a and b: ‘point of intersection of b and c’) likewise indicate the mode of presentation; and hence the statement contains actual knowledge.

Accordingly, the reference of the expressions ‘the point of intersection of a and b’ and ‘the point of intersection of band c’ would be the same, but not their senses. The reference of ‘evening star’ would be the same as that of ‘morning star’, but not the sense.

The sense of a proper name is grasped by everybody who is sufficiently familiar with the language or totality of designations to which it belongs; but this serves to illuminate only a single aspect of the reference, supposing it to have one. Comprehensive knowledge of the reference would require us to be able to say immediately whether any given sense belongs to it. To such knowledge we never attain.

The regular connexion between a sign, its sense, and its reference is of such a kind that to the sign there corresponds a definite sense and to that in turn a definite reference, while to a given reference (an object) there does not belong only a single sign. The same sense has different expressions in different languages or even in the same language. To be sure, exceptions to this regular behaviour occur. To every expression belonging to a complete totality of signs, there should certainly correspond a definite sense; but natural languages often do not satisfy this condition, and one must be content if the same word has the same sense in the same context. It may perhaps be granted that every grammatically well-formed expression representting a proper name always has a sense. But this is not to say that to the sense there also corresponds a reference. The words ‘the celestial body most distant from the Earth’ have a sense, but it is very doubtful if they also have a reference. The expression ‘the least rapidly convergent series’ has a sense; but it is known to have no reference, since for every given convergent series, another convergent, but less rapidly convergent, series can be found. In grasping a sense, one is not certainly assured of a reference.

If words are used in the ordinary way, what one intends to speak of is their reference. It can also happen, however, that one wishes to talk about the words themselves or their sense. This happens, for instance, when the words of another are quoted. One’s own words then first designate words of the other speaker, and only the latter have their usual reference. We then have signs of signs. In writing, the words are in this case enclosed in quotation marks. Accordingly, a word standing between quotation marks must not be taken as having its ordinary reference.

1. Sense and Analysis

One of the main characteristics of analytic philosophy is its view of philosophy as consisting essentially in analysis. Analytic philosophers attempt to tackle the traditional problems of philosophy via the analysis of language: a proper conception of language will enable us either to solve or dissolve these problems. The advances Frege made in logic greatly facilitate the task of solving philosophical problems in this manner: so much so that Frege is often hailed as “the founder of analytic philosophy.” But what exactly is analysis? Intuitively, we would want to say that this consists in discovering relationships among senses: the analysis of “X knows that P” as “X has a justified, true belief that P” claims to exhibit an identity in sense between the two expressions. Alternatively, we could view identity in sense as corresponding to the intuitive notion of synonymy: two expressions have the same sense if and only if they are synonymous. The project of analysis would then be construed as the project of discovering philosophically interesting relations of synonymy, or identity in sense, and the employment of these in the solution or dissolution of philosophical problems.

The difficulty is that Frege’s solution to the problem of infornativeness appears to rule out this intuitive description of the analytic project. Recall that the problem was to account for the fact that whereas

(23) The Morning Star is the Evening Star

is potentially informative,

(24) The Evening Star is the Evening Star

is not. For Frege, (23) is potentially informative because although “The Morning Star” and “The Evening Star” both refer to the same object, they have different senses. This contrasts with (24), whose uninformativeness is explained by the fact that the two signs flanking the “is” of identity have the same sense. Thus, the informativeness of some identity statements is explained in terms of the signs on either side of the identity sign having the same reference but expressing different senses; while the uninformativeness of other identity statements is explained in terms of the signs on either side of the identity sign having the same sense as well as the same reference. The problem with this is that given the assumption that analysis concerns relations between senses, it also entails that there can be no such thing as an informative analysis.^[1] To see this, suppose that analytic philosophers have succeeded in showing by analysis that knowledge is justified true belief.

Then consider (25) as follows:

(25) Knowledge is justified true belief.

Suppose that analysis really is concerned with discovering relations in the realm of sense, and that (25) is a good analysis. Then consider a philosopher who understands (25). Since he grasps the sense of (25) he must also grasp the senses of its constituents. Since “Knowledge” and “Justified true belief” ex hypothesi have the same sense, it follows from^[2] that anyone who grasps (25) must know that they have the same sense. But sense determines reference,^[3] so since the philosopher knows that they have same sense, he must also know that they have the same reference. From Miller (2007: 11) “The semantic value of a complex expression is determined by the semantic of its parts,” it follows that he must know the truth-value of (25). So, any philosopher who understands (25) must know its truth-value. But this is just to say that there can be no such thing as an informative and interesting analysis. And if there can be no such thing as an interesting and informative analysis, what becomes of the project of analytic philosophy? The dilemma for Frege here is clear. Either he gives up his explanation of why (23) and (24) differ in their potential informativeness, or he renders himself unable to account for the possibility of informative analyses, like that encapsulated in (25). The only way of avoiding this dilemma would seem to be to reject the intuitive conception of analysis as a project concerning relations of sense. This may be possible, but it would further deepen the mystery as to what sense actually is.

2. Reference and Contexts

It should be recalled that Frege’s solution to the problem of substitution into belief contexts is summed up as in (26) in which “Mark Twain” refers, not to the celebrated American author, but rather to the sense of “Mark Twain,” as it appears outside of belief contexts:

(26) John believes that Mark Twain is Samuel Clemens

As we saw, Frege expresses this point in the following way: in belief contexts expressions refer to their customary senses, or, the indirect reference of an expression is its customary sense.

Now consider the following example of a perfectly well-understood sentence in (27):

(27) James believes that John believes that Mark Twain is Samuel Clemens.

It is natural to suppose that just as the occurrence of “Mark Twain” in an indirect context such as (26) refers to the sense it possesses in a direct context such as (28), the occurrence of “Mark Twain” in a doubly indirect context such as (27) will refer to the sense which it possesses in a singly indirect context such as (26).

(28) Mark Twain was an American

So, on Frege’s account, the name “Mark Twain,” as it appears in (27) refers to the sense that it possesses in (26), what Frege calls its indirect sense.

But what is this indirect sense? All we know is the reference of “Mark Twain” as it appears in (26), namely, its customary sense. But this won’t allow us to work out the sense of “Mark Twain” as it appears in (26), since the sense-reference relation is many-one: the same referent can be associated with indefinitely many senses (as Russell put it, “there is no route back from the reference to the sense”). The upshot of this is that we seem to be unable to say what the sense of “Mark Twain” in (26) is, and therefore unable to say what the reference of “Mark Twain” is in (27). Given that the reference of an expression is what determines the contribution it makes to the truthvalues of sentences in which it appears, we should be unable to appraise sentences involving doubly indirect contexts such as (26) for truth or falsity. But, of course, we are perfectly capable of making such appraisals: so Frege’s account of indirect sense and indirect reference is thrown into doubt.

Is there any way Frege can get round this problem? Why does he need to distinguish between the sense of “Mark Twain” in an indirect context such as (26) and a direct context such as (28) in the first place? In other words, why can’t he just identify the indirect sense of “Mark Twain,” the sense it possesses in (26), with its customary sense, the sense it possesses in (28)? Given that we know the sense of “Mark Twain” as it appears in (28) (or at least we are supposing that we do, for the sake of argument), this would solve the problem about the reference of “Mark Twain” as it appears in the doubly indirect sentence (27). As Dummett^[4] points out, however, this option is not straightforwardly available to Frege because of his adherence to thesis (see footnote 17 above), that sense determines semantic value. It follows from thesis (see also footnote 17) that if an expression has a certain reference in one context, but a different reference in another, then that expression must have different senses in each of the two contexts. Now, given that “Mark Twain” as it appears in (28) refers to the celebrated American author, and as it appears in (26) refers to its customary sense, it follows that

“Mark Twain” has a different reference in each of these, and hence must express a different sense in each. In other words, the sense of “Mark Twain” in (26) is different from the sense it possesses in (28). The identification of indirect sense with customary sense is frustrated, so that we are left in the dark as to what the indirect sense, and hence the doubly indirect reference of “Mark Twain,” actually is.

Dummett (1973) suggests that the whole difficulty arises from interpreting thesis (see footnote 17) as meaning that the semantic value of an expression must be determined by its sense alone. Dummett claims that this is a mistake, based on a misleading tendency to speak about the semantic values of expressions in isolation from the senten tial contexts in which they occur.

Indeed, Frege explicitly counsels against this tendency in the second of the three fundamental principles set out in the introduction to The Foundations of Arithmetic:^[5]

Never to ask for the reference of a word in isolation, but only in the context of a proposition.

Dummett interprets this as a claim that only a particular occurrence of an expression in a sentence has a semantic value, and that this semantic value is determined jointly by the sense of the expression together with the kind of context in which it occurs. Then, as Dummett puts it in his (1973).

The sense of a word may thus be such as to determine it to stand for one thing in one kind of context, and for a different thing in some other kind of context. We may therefore regard an expression occurring in an indirect context as having the same sense as in a direct context, though a different reference.

Frege can thus after all equate the indirect and customary senses of “Mark Twain,” putting the differences in reference in (26) and (28) down to the fact that the expression is appearing in different types of context in each.

How plausible is this solution to the difficulty? There are at least two points that one can make here. First, Dummett’s suggestion that the difficulty can be solved by equating the indirect and customary senses of “Mark Twain” appears to be ad hoc. All Dummett has shown is that there is nothing to prevent Frege from identifying indirect and customary sense. But in order to convincingly get Frege out of trouble here, we surely require some positive grounds for identifying customary and indirect sense, grounds that go beyond the observation that we avoid the relevant difficulty if we make the identification. Second, this point is made all the more pressing by the fact that there are actually some reasons against making the identification. Suppose that we do equate customary and indirect sense, and consider two expressions which arguably have the same customary sense, say, “chiropodist” and “foot-doctor.” Then, it follows that “chiropodist” and “foot-doctor” have the same indirect sense, and therefore the same reference in doubly indirect contexts.

Now it follows, from thesis,^[6] that the move from (29) to (30) should not result in a change in truth-value. But of course it is easy enough to think of cases in which (29) is true yet (30) is false.

(29) John believes that Frank believes that all chiropodists are chiropodists.
(30) John believes that Frank believes that all chiropodists are foot-doctors.

So the identification of customary and indirect sense threatens a clash with one of Frege’s fundamental theses about semantic value. The upshot is that Frege’s solution to the problem of substitution into belief contexts is not entirely satisfactory. And the definition of semantic value in Frege: Philosophy of Language should be repeated here as:

The semantic value of any expression is that the feature of it which determines whether sentences in which it occurs are true or false.

(Dummett (1973: 91)

Footnotes and references:

[1]:

According to Alexander Miller (2007). Philosophy of Language, this is sometimes known as the “Paradox of analysis,” pp. 47.

[2]:

“If someone grasps the senses of two expressions, and the expressions actually have the same sense, then he/she must know that the two expressions have the same sense.” (Alexander Miller 2007: 30).

[3]:

“The sense of an expression is that ingredient of its meaning which determines its semantic value.” (Ibid., pp. 28).

[4]:

M. Dummett. 1973. Frege: Philosophy of Language. London: Duckworth. See also Miller (2007)

[5]:

G. Frege. 1953. The Foundations of Arithmetic. Evanston, IL: Northwestern University Press.

[6]:

“Substitution of a constituent of a sentence with another which has the same semantic value will leave the semantic value (i.e., truth-value) of the sentence unchanged.” (Miller 2007: 12)