A proposition can occur in discourse with or without being asserted (Geach, “Assertion”). For example, if I assert “If P then Q”, Q occurs without being asserted (though, of course, I certainly could assert it). However, we have to distinguish between two sets of cases. First, there are cases where a proposition occurs in an assertion without itself being asserted. Second, there are cases where it appears as though a proposition occurs without being asserted when, in fact, what occurs is merely a name.
The first sorts of cases occur when logical operations are applied to form new propositions such as “P v ~P” or “If P, Then Q”. These propositions do not add any descriptive content over and above what is described by the simple, unasserted propositions that they contain. Instead, they simply depend for their truth/falsehood upon the logical relations that hold between the states of affairs that the simple propositions describe.
The second sorts of cases occur when a simple proposition seems to occur unasserted within a proposition that, unlike propositions of the first sort, *does* describe some further bit of reality. For example, “Q” seems to occur within “A believes that Q” even though the latter proposition apparently describes a further relation between “Q” and a subject. However, despite appearances, I think these can only be understood as cases where “Q” is a name that refers to some proposition that, correctly or incorrectly, describes reality. Among other things, this is why “A believes ‘Hesperus is a star’” does not entail “A believes ‘Phosphorus is a star’”. It also explains why “A believes ‘Unicorns have four legs’” does not entail “Unicorns have four legs” (since the former proposition does not describe a relation between A and whatever some proposition describes).
If the above claims are true, a problem arises for inferences of the following form:
1. John believes that ‘P’
2. P
3. John believes truly that ‘P’ (or “John’s thought is true” or “‘P’ is true”, etc.)
As discussed previously, ‘P’ occurs in the first premise as a name. Furthermore, John’s belief is true iff the proposition the ‘P’ names is true. It may seem that this is precisely what the second premise affirms. However, despite appearances, the first and second premises have nothing in common with one another. In the first premise, ‘P’ is a name. In the second, “P” is a proposition (affirming the name ‘P’ makes no more sense than affirming the name ‘John’). Affirming “P” does not license us to infer that ‘P’ is true any more than it licenses us to infer that some arbitrary proposition named ‘A’ is true. As a matter of fact, ‘P’ names the very proposition that we affirm, so the conclusion is true, but there’s no logical reason why ‘P’ could not name a completely different proposition. As a matter of logical inference, concluding that John’s belief is true is totally unjustified.
In order to avoid this problem, one might hold that genuine propositions can occur in intensional contexts without being asserted. However, one must then explain why such propositions have their unusual features that differ from those that result from ordinary logical operations. Alternatively, one might reject inferences of the sort presented above and explain their apparent legitimacy on other grounds. Finally, one might deny that intensional propositions describe relations between subjects and propositions at all. In this case, one must give an alternative account of what this form of discourse amounts to and why the relevant inferences are justified. In my opinion, this final option holds the most promise.
“That the propositional sign is a fact is concealed by the ordinary form of expression, written or printed. (For in the printed proposition, for example, the sign of a proposition does not appear essentially different from a word. Thus it was possible for Frege to call the proposition a compounded name.)” - Ludwig Wittgenstein
