Fregean Senses

The following is a tutorial essay for Philosophy of Logic and Language with James Kirkpatrick. In effect, it is a quick agenda for an hour-long discussion on a guiding question, which for this week was ‘What are the arguments for/against Frege’s theory of sense for proper names?’. A free-standing and much more readable version of some of the material may eventually go up on my Substack.

I. Frege’s Theory of Sense

A proper name is a term like ‘Aristotle’ or ‘the Great Wall of China’, but unlike ‘smokes’ or ‘the actual teacher of Alexander’. Of course, characterising this category merely by ostension is somewhat unsatisfying; but it is enough to get going (and about as well as we can do when the semantics of such terms is what’s under debate).

Frege forwards a theory of sense for the semantics of proper names. It relies on the ideology of sense and referent. The referent of a term is simply its extension; From a contemporary perspective, Frege in effect held that the only propositions are ⊤ (the true) and ⊥ (the false). The extension of an individual is that individual; the extension of a true proposition is ⊤ and a false proposition is ⊥; and the extension of a property of individuals is the function that maps each individual which actually has that property to ⊤ and all other individuals to ⊥; and so on. In effect, everything is modally rigid. for instance, ‘is a person in Oxford in October 2025 with a heart’ and ‘is a person in Oxford in October 2025 with a kidney’ (presumably) happen to each have the same class of individuals as their referent. In an extensional setting, we can easily move between thinking about classes and thinking about their indicator functions. Meanwhile, the sense of a term is—at first pass—the objective guise under which the term presents its referent. This is a feature of the term itself, which does not differ between distinct hearers of the term—by contrast with one’s conception of a term, which may differ in this way.

Frege’s semantic theory is compositional, in that a complex term’s sense is determined by the senses of its constituents. Thus, the referent of a complex term may be computed by first computing its sense from the senses of its constituent terms, and then passing from sense to referent. However, Frege also holds that a complex term’s referent is determined by the referents of its constituents. Thus, the referent of a complex term may alternatively be computed by first passing from the senses of its constituent terms to their referents, and then computing the referent of the complex term from these. That these results must match gives strong constraints on the senses and referents of various terms. For instance, it pushes Frege to accept that the senses and referents of terms are not fixed, but that, for instance, the referent of a sentence shifts what is typically its sense when embedded under an attitude operator like ‘believes that’.

As applied to proper names in particular, then, Frege holds that—like other terms—the primary semantic contribution of a proper name is a sense (a mode of presentation), which may then determine a referent (for a proper name, any individual).

II. Arguments for Frege’s theory of sense

You might think that Frege cases motivate his theory of sense, but it is a bad way to deal with such cases. Such cases are not at all marginal: they crop up everywhere in philosophy (to name three more well-known phenomena that plausibly result from bungling Frege cases: Mary’s Room, the de se, and the is–ought gap). I think Daniel Kodsi is right that the thing to do is to assimilate Frege cases to cases of inconsistency, and to assimilate cases of inconsistency to cases of disagreement (in particular, self-disagreement).

Above, we characterised the sense of a term as the objective guise under which it presents its referent. Of course, this presupposes that each term in fact has a referent. However, some proper names might not refer. Consider ‘Hamlet’, or ‘N’ after we’ve tried to stipulate it as the largest prime (or, for that matter, beforehand). One could try to hold that such names have an empty referent, or that they have zero referent as opposed to no referent. But such desperate moves are unnecessary, if we have something like Frege’s theory of senses: once one understands senses on their own terms, rather than along the reductionist lines sketched above, one can allow for terms with senses but no referent. That is, something fails at the step where sense determines referent, but all goes fine at the step where the term determines its sense.

Indeed, this seems to be what goes on in the case of definite descriptions: ‘the king of France’ determines some sense which picks out as referent whoever is king of France in the circumstance of evaluation. If we are talking about the late 1700s, ‘the king of France’ has the same referent, Louis XVI, as ‘the husband of Marie Antoinette’; but the former has a fairly different sense than the latter. But if we are talking about the present day, where there is no king of France, ‘the king of France’ retains its sense but cannot find a referent. A similar story is plausible for ‘Odysseus’ and ‘N’, above.

To be explicit, then: one reason to buy Frege’s theory of sense for proper names is that it smoothly accounts for empty names, which might otherwise be puzzling. For instance, an obvious alternative to Frege’s theory of sense is that all terms directly pick out their referent. Suppose, then, that the only route to calculating the referent of a complex term is via the referents of its constituent terms. If some term—say, ‘Odysseus’—has no referent, then it is not immediately obvious what the referent of sentences containing that term are supposed to be. Indeed, the sentences plausibly do not refer at all; if they did, this might spell trouble for classical logic, which governs the referents of declarative sentences. But if all one can say about the semantics of a sentence is determined by its referent, then one cannot distinguish semantically between ‘Odysseus is a Greek hero’ and ‘N is a Greek hero’. At least at first pass, this seems problematic. Frege’s theory of senses allows us to talk about the sense of these sentences, and how they differ, despite both sentences lacking a referent.

A second canonical (see, for instance, Kripke 1980) argument for Frege’s theory is that it provides a natural account of how speakers can use terms to talk about their referents: one is competent with a term if one is sufficiently familiar with its sense—which is its primary meaning—and the sense does the work of picking out the referent. Meanwhile, without having senses to make this connection, it is (supposed to be) unclear how some particular proper name comes to have the referent that it in fact does; one must posit ad hoc devices like “naming ceremonies” that hook a name up to its referent as if by magic.

Here is another argument for Frege’s theory of sense (due to Heim 1998). Consider the following sentence.

(1) He hates him.

Both ‘he’ and ‘him’ as used in (1) refer to individuals. What restrictions are there on what these terms can refer to? For a start, any male human (along with some non-human animals) could be the referent of either ‘he’ or ‘him’, and nothing else can. For instance, the ‘he’ might refer to John, and the ‘him’ might refer to John.

However, strikingly, it typically cannot be that both ‘he’ and ‘him’ in (1) refer to John simultaneously! Notice that it is exceedingly difficult to interpret (1) as meaning that John hates himself. However, with the right context, we can overcome this difficulty. For instance, consider (2).

(2) Nobody likes John. His family hates him, his coworkers hate him, and—come to think of it—he hates him, too!

Now, (2) is interpretable with ‘he’ and ‘him’ both referring to John. Similarly, consider (3).

(3) I think that the anonymous caller was actually the candidate herself—she praised her like nobody else would!

What is different about (2) and (3) from ordinary utterances of sentences like (1)? One striking feature is that the uses of ‘he’ and ‘him’ (in (3), ‘she’ and ‘her’) seem to present the referent under two different senses. In (2), ‘he’ has a sense like ‘yet another person who hates John’, while ‘him’ has a sense like ‘this person who everyone hates’. Meanwhile, in (3), ‘she’ has a sense like ‘that anonymous caller’ (on which, indeed, ‘she’ is anaphoric), while ‘her’ has a sense like ‘the candidate herself’ (similarly).

The pattern seems to be that the subject and object in a sentence like (1) cannot share a referent, unless they have distinct senses. Or, more straightforwardly, the restriction is simply that the subject and object cannot share a sense. It is also notable that substituting ‘herself’ for ‘her’ in (3) creates a very different effect.

Importantly, without appealing to the ideology of Fregean senses, it is hard to make such a generalisation. For instance, it cannot be in (3) that the speaker is merely accommodating the audience’s ignorance as to the identity of the subject and object, and for that reason alone saying ‘her’ rather than ‘herself’. For one thing, in (2), such identity is common knowledge, and indeed is only easily interpretable absent other context if one knows that ‘he’ and ‘him’ are both supposed to refer to John. For another, substituting ‘herself’ for ‘her’ in (3) changes the effect drastically.

(4) ? I think that the anonymous caller was actually the candidate herself—she praised herself like nobody else would!

Similarly, it cannot be that what is going in (3) is merely an effect of anaphora on definite descriptions (for instance, one might hypothesise that the anaphor “imports” the entire definite description, and not just its referent). Consider (5), as uttered while pointing first at John and then at his face in a group picture.

(5) Of course, he thinks that the most important member of the band is him.

Here, both ‘he’ and ‘him’ are pronominals, as in the default interpretation of (1); so complicating the theory of anaphora to deal with (3) is insufficient.

Against Frege’s theory of sense for proper names

Frege’s theory of proper names is a pretty good theory. If it were the only theory we had, and no others looked promising, then listing problems for it would not accomplish much: if some theory had no anomalies whatsoever, that would make it anomalous, and perhaps a mark against its strength and informativeness.

Arguments against Frege’s theory of senses, then, are effective only insofar as they promote some alternate theory: if some problem for Frege’s theory of senses is a problem for all theories, it should not make us abandon Frege’s theory.

For instance, one problem for Frege’s theory is that it does not seem to be able to distinguish between the proper name ‘the Great Wall of China’ and the rigid definite description ‘the thing which is actually the largest wall in China’. But this is not a problem if every theory has this issue. Unfortunately for Frege’s theory of senses, not every theory has this issue. For instance, we might characterise proper names as atomic terms of type e (that is, terms that have no subterms and can refer to individuals). Where we take the type system seriously, as higher-order metaphysics does, rather than treating it as a mere book-keeping device. Obviously, this is antithetical to Frege’s theory of senses, which posits that—like definite descriptions—proper names specify some properties, and then pick out whatever satisfies such properties.

One initial disadvantage of Frege’s theory of sense is that it posits a new bit of ideology; if we can theorise without appeal to senses, then that is better. So, undermining the advantages that senses seem to offer is a good way of arguing against Frege’s theory of senses. In particular, if we can supplement the simple characterisation of proper names given just above with independently plausible auxiliary assumptions that account for the data purportedly explained by appeal to senses, then we undermine the need to posit and appeal to senses.

One extant tool we have for replacing senses is interpreted sentences (more generally, interpreted terms). For the most part, what was attractive about senses was that they enabled us to draw distinctions between terms with the same referent. But we can simply use the terms themselves to do that! Interpreted sentences also let us draw similarities between terms that, under Frege’s view, share a sense. For instance, one might hold that the English (6) and the German (7) have something important in common, which is different from (8), which has the same referent (either the true, or perhaps the necessary proposition). An easy explanation for this is that they share a sense.

(6) Gold is an element.

(7) Gold ist ein Element.

(8) Water is H₂O.

However, we can simply note that although (6) and (7) are not identical, they involve coreferential subterms in the same structure; this is something that also sets them apart from (8).

One lingering thing that simply appealing to interpreted sentences does not account for, though, was Heim’s argument. It is an open question whether a plausible alternate account of such data may be found. A sketch of a Gricean story to account for this at the pragmatic level instead: if ‘her’ did refer to Mary, you’d typically have expected the speaker to help you out by saying ‘herself’ instead; since the speaker didn’t, you can safely assume that ‘her’ doesn’t refer to Mary. But this reasoning is naturally undercut when, for instance, saying ‘herself’ would produce a cognitively different result. So we need no semantic restriction on interpreting ‘her’ as a pronominal which co-refers with ‘Mary’, only a pragmatic one. I was introduced to this puzzle by Paul Elbourne; the presentation of this puzzle (pre-emptively closing off alternative escape routes which anti-Fregeans might want to try) benefitted significantly from arguing with Tim Williamson.