Frege's Puzzle

When we assert an identity, what state of affairs in the world must obtain for our assertion to be true? What, in other words, are the metaphysics behind identity statements?

On the other hand, when we assert an identity, what can we be said to know about the world in truthfully making the assertion? What, in other words, is the epistemology of identity statements?

For example, suppose you did not know who was teaching Intro to Logic this semester. You might ask a friend, who would correctly assert that,

Piker is the professor.

This seems humdrum and unproblematic, to be sure. 'Piker is the professor' is true iff the state of affairs of Piker being the professor of Intro to Logic obtains, and we know that that state of affairs obtains when we have evidence that, other things being equal, Piker has consented to offer it and has proceeded to do so.

Consider, however, the assertion that

1 + 1 = 2

What state of affairs must obtain in light of that (true) assertion, and how do we know it to be true? That is, echoing Moore in his "A Defence of Common Sense", we accept that it is true while finding ourselves utterly at a loss how to analyze it.

Worries about the nature of mathematical entities and our access to them prompts the antecedent puzzle of the nature of identity itself. Frege's genius was to grasp that this puzzle could be understood in terms of the language of identity itself.

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? In my Begriffsschrift I assumed the latter. The reasons which seem to favour this are the following: a = a and a = b are 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.

Let us unpack this using a concrete example.

In antiquity it was not understood that the star that sometimes appeared in the early evening sky just above the horizon, Hesperus, was the same star, Phosphorus, that sometimes appeared early in the morning. Hesperus and Phosphorus were simply the planet Venus, as it turned out.

Now, let us assume three principles:

A. The meaning of a term is give by its reference. That is, it is enough to understand the meaning of a term to grasp what the term designates. A term's reference exhausts its meaning.

B. The meaning of a sentence is a function of the meanings of its parts. For example, it suffices to understand the sentence, 'Hesperus is bright', that one understand the term 'Hesperus' and the predicate 'is bright'.

C. If any two statements have the same meaning, then they convey the same information inasmuch as they are synonymous. That is, statements which mean the same thing have the same cognitive value, as Frege puts it, because they express the same basic proposition.

Frege's Puzzle may now be framed as three individually true yet jointly false propositions:

1. 'Hesperus = Hesperus' and 'Hesperus = Phosphorus' have the same meaning [by (A) and (B) above].

2. 'Hesperus = Hesperus' and 'Hesperus = Phosphorus' have the same cognitive value [by (C) and (1) above].

3. 'Hesperus = Hesperus' and 'Hesperus = Phosphorus' do not have the same cognitive value [because 'Hesperus = Phosphorus' is informative, while 'Hesperus = Hesperus' is not.].

That is, 'Hesperus = Hesperus' and 'Hesperus = Phosphorus' differ in cognitive value because 'Hesperus = Hesperus' is trivial, a priori, and necessary (in order, it is unsurprising, knowable without experience of the world, and true no matter how the world is arranged), where 'Hesperus = Phosphorus' is non-trivial, a posteriori, and contingent (in order, it is surprising, knowable only by having experience of the world, and the world could have been so arranged that Hesperus and Phosphorus turned out to be distinct. In the simplest terms possible, we would never be surprised to hear that 'Hesperus = Hesperus'; we were very surprised, however, to learn that 'Hesperus = Phosphorus'.

Thus, taken individually, (1), (2), and (3) seem true as best we can tell, yet they are just as clearly inconsistent with one another, so they cannot all be true. Something has got to give.

Consider, to use another example, that the surprise Mary Jane lacks upon being informed that Spiderman is Spiderman is abundant when she learns that Spiderman is Peter Parker. Yet we would be at least as surprised by her divergent reactions if the names 'Spiderman' and 'Peter Parker' contributed no more than the individual they co-designate to the meanings of statements in which they occur. The natural--though not uncontested--conclusion to draw is that 'Spiderman' and 'Peter Parker' have distinct cognitive significance for Mary Jane, a distinction rooted, no doubt, in the very different ways Peter Parker has presented to her at different times, wearing different garb, while taking very different social roles.

One puzzle is how to explain the distinct cognitive significance co-referring names might have, since it seems, at least in the case of Mary Jane, that 'Spiderman' and 'Peter Parker' contribute more than just an individual to the meanings of the claims she makes using them. Frege's famous solution to this puzzle is reject (A) above and distinguish the reference 'Peter Parker' has from the sense it expresses: 'The mild-mannered, somewhat geeky college student and long-time friend of Mary Jane' might more clearly express the sense of 'Peter Parker' if the sense expressed by 'Peter Parker' captures how Peter Parker has presented himself to Mary Jane as Peter Parker, whereas 'the red-suited web-slinging super-hero and love-interest of Mary Jane' might more clearly express the sense of 'Spiderman'. Thus the senses expressed by different names might diverge even when those names designate the same individual.

Senses are not, however, mere epiphenomena of the names that express them. They are functional. Mary Jane is surprised to learn that Spiderman is Peter Parker because she naturally assumed that the wildly divergent senses expressed by 'Spiderman' and 'Peter Parker' determined distinct references. That is, names express modes of presentation or senses we use to determine, and communicate to others, reference. The reference of a term is thus mediated by its sense or mode of presentation. As Frege goes on to say,

The regular connexion between a sign, its sense, and what it means is of such a kind that to the sign there corresponds a definite sense and to that in turn a definite thing meant, while to a given thing meant (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 figuring as a proper name always has a sense. But this is not to say that to the sense there also corresponds a thing meant. The words "the celestial body most distant from the Earth" have a sense, but it is very doubtful if there is also a thing they mean. The expression "the least rapidly convergent series" has a sense but demonstrably there is nothing it means, 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 meaning [referring to] anything.

Frege here makes at least six claims with respect to expressions, senses, and references in natural language:

1. Every expression having a sense has the same sense in the same context of use.
2. Every expression having a sense and having a reference has the same reference in the same context of use.
3. For every reference there may be multiple senses.
4. For every sense there may be multiple expressions.
5. Expressions need not have a sense.
6. Senses need not have a reference.