Frege (1892) held that, when names are embedded under propositional attitude verbs, they refer not to their ordinary referents but to their ordinary senses. For example, consider:

  1. 1.

    If Hesperus is Phosphorus and Hammurabi knew that Hesperus is Hesperus, then Hammurabi knew that Hesperus is Phosphorus.

According to Frege, asserting this sentence would involve something like equivocation: the first occurrence of “Hesperus” would refer to Hesperus, while the second occurrence of “Hesperus” would refer not to Hesperus but to the ordinary sense of “Hesperus”.

It is often thought that, by accepting this doctrine, Fregeans can hold that, although someone who asserted 1 would speak falsely, 1 is still valid, as is the schema:

  • Substitution: If a is b and \(\Phi\), then \(\Phi [b/a]\).

Instances of this schema, such as 1, are obtained by replacing \(\Phi\) with a declarative English sentence \(\varphi\), replacing a and b with proper names n and m, and replacing \(\Phi [b/a]\) with a sentence obtained from \(\varphi\) by replacing an occurrence of n that is not within quotation marks with an occurrence of m.

For example, in what is arguably the locus classicus of contemporary Fregeanism, Kaplan (1968) writes:

Frege’s main idea, as I understand it, was just this. [...A]pparent failures of substitutivity and the like [are] due to confusion about what is denoted by the given [term’s] occurrence. (p. 183)

So we require no special non-extensional logic, no restrictions on Leibniz’ law, on existential generalization, etc., except those attendant upon consideration of a language containing ambiguous expressions. (p. 184)

To illustrate Kaplan’s idea, consider the sentence:

  1. 2.

    If Jane has tenure, then Jane has tenure.

If we equivocate, we can use  2 to speak falsely; for example, by using the different occurrences of “Jane” to refer to different people with that name. But it would clearly be misguided to deny the validity of 2 on this basis. Kaplan’s idea is that it would be similarly misguided for Fregeans to deny the validity of 1 on the basis of the fact that we can use 1 to speak falsely.Footnote 1

Both proponents and detractors of Fregean senses (although not Frege himself) have taken the reconciliation of Substitution with the falsity of sentences like 1 to be the principal virtue of postulating such entities. For example, Kaplan (1968, p. 185) writes that his “own view is that Frege’s explanation, by way of ambiguity, of what appears to be the logically deviant behavior of terms in intermediate contexts [e.g., complement clauses of attitude ascriptions] is so theoretically satisfying that if we have not yet discovered or satisfactorily grasped the peculiar intermediate objects in question, then we should simply continue looking”. And Carnap (1947, p. 136), a detractor of senses, writes:

It seems that Frege was aware of the fact that [Substitution] would lead to a contradiction if the ordinary nominata of names were ascribed also to their oblique occurrences and that the contradiction does not arise if different nominata are ascribed to these occurrences. [...] It is true that Frege does not speak explicitly of the necessity of avoiding a contradiction; he gives other reasons for his distinction between the ordinary nominatum and the oblique nominatum of a name. His reasoning gives the impression that this distinction appeared to him natural in itself, without regard to any possible contradiction. However, I think that to many readers it will scarcely appear very natural and that they, like myself, will see the strongest argument in favor of Frege’s method rather in the fact that it is a way of solving the antinomy.

In this note we will argue that Kaplan and Carnap are mistaken. Whatever its other merits, Frege’s doctrine that names refer to senses when embedded under propositional attitude verbs does not reconcile the validity of Substitution with the threat posed to it by sentences like 1.

Consider the sentence:

  1. 3.

    If Kripke knows that Hesperus is Phosphorus, then Hesperus is Phosphorus.

Assume for the moment that this sentence is valid. (It is, after all, an instance of the schema “If S knows that \(\Phi\), then \(\Phi\)”, perhaps the most basic principle of epistemic logic.) Assume, moreover, that the set of valid sentences is closed under classical propositional logic. So if 1 and 3 are both valid (and we interpret “if ..., then ...” as material implication), then the following sentence must also be valid:

  1. 4.

    If (Kripke knows that Hesperus is Phosphorus) and Hammurabi knew that Hesperus is Hesperus, then Hammurabi knew that Hesperus is Phosphorus.

Now according to Frege, in 4 every word has the same reference in each of its occurrences.Footnote 2 So Kaplan’s strategy for holding that 1 is valid despite expressing something false cannot be applied to 4. By his and Carnap’s lights, Fregeans should deny that 4 is valid, since they should think that it can be used falsely without equivocating. So given our assumptions that 3 is valid and that classical consequences of valid sentences are themselves valid, Fregeans should also deny that 1 is valid.Footnote 3

Some Fregeans might deny that 3 is valid, on the grounds that “knows” is not a ‘logical constant’. But our mode of argument does not essentially rely on 3 being valid. Let schmalidity be that good status, however precisely it is understood, typified by 3 and at which systematic theorizing about knowledge aims. Assume (i) that all valid sentences are schmalid, (ii) that no sentences used falsely without equivocating are schmalid, and (iii) that the set of schmalid sentences is closed under classical propositional logic. Given that 3 is schmalid and 4 can be used falsely without equivocating, (i)–(iii) imply that 1 is not valid.

A common response to this argument has been to grant its conclusion but claim that it misses the point. According to this response, Fregeans should think that there is a theoretically important status that the schema Substitution enjoys despite having non-valid instances like 1. In particular, they should think that weak validity is such a status, where a schema is weakly valid just in case all of its embedding uniform instances are valid, and a sentence is embedding uniform just in case every expression occurring in it is embedded under the same number of attitude verbs in each of its occurrences. Since 1 is not embedding uniform, its invalidity does not threaten the weak validity of Substitution.

We think that this response is half right. Denying that 1 is valid clearly does not bar Fregeans from accepting some version of the idea that true identities license the intersubstitution of their flanking terms, and the claim that Substitution is weakly valid is a natural way for them to make this commitment precise. But this is not because weak validity is a theoretically important status. It isn’t. For consider the schema:

  • Anti-Factivity: If S knows that \(\Phi\), then not-\(\Phi\).

This schema is weakly valid, since it has no embedding uniform instances. But it is clearly not a ‘good schema’ in any interesting sense. The claim that Substitution is weakly valid is interesting because it is equivalent to the claim that Weak Substitution (the schema whose instances are all and only the embedding uniform instances of Substitution) is valid, and Weak Substitution is itself an interesting schema. By contrast, Weak Anti-Factivity (the schema whose instances are all and only the embedding uniform instances of Anti-Factivity) has no instances, and the claim that it is valid is therefore trivial. Fregeans (like everyone else) should consider all of a schema’s substitution instances in assessing its good standing.

We have argued that Fregeans should think that 1 is not a valid sentence of English, and hence that Substitution is not a valid schema. The way in which, according to them, the two occurrences of “Phosphorus” in 1 have different referents is not the kind of equivocation capable of reconciling a false reading of a sentence with that sentence nevertheless being valid.Footnote 4 It is not like the failure of the two occurrences of “Jane” to co-refer on the relevant reading of 2.Footnote 5 This conclusion is not intended as a criticism of Fregean treatments of attitude ascriptions. We hope rather that it will encourage further investigation into the possibilities for systematic theorizing in settings where Substitution is given up.Footnote 6