According to our assumption, the thought that Hesperus is shining (that Hesperus is Hesperus) and the thought that Phosphorus is shining (that Phosphorus is Phosphorus) stand for the same state of affairs.
For example, the thought that Hesperus shines stands for the state of affairs of Hesperus's shining.
It is plausible to suppose that the question over whether statements like 'Hesperus = Phosphorus', or perhaps 'If Hesperus exists then Hesperus = Phosphorus', are strongly necessary or merely weakly necessary depends on whether 'Hesperus' and 'Phosphorus' are "strongly" rigid (obstinate) or merely "weakly" rigid in this sense.
A theory committed to Hesperus would automatically be committed to Phosphorus, assuming the necessity of identity, even if the theory explicitly claims that Hesperus exists and that Phosphorus doesn't.
Given the plausible assumption that the proposition expressed by a sentence is a function of the semantic values of its constituent expressions together with their mode of combination, it seems inevitable that what the latter sentence says is just what the former sentence says assuming, with the naïve theory, that 'Hesperus' and 'Phosphorus' possess the same semantic value.
In the opening paragraph of "On Sense and Reference" (Frege 1892), Frege argues that a theory that identifies the semantic value of a name with its referent the naïve theory cannot differentiate between the contents of 'Hesperus = Hesperus' and 'Hesperus = Phosphorus'.
One can use 'is' to express identity, as in 'Hesperus is the planet Venus'; but in 'Hesperus is bright'is'indicatesates predication.
On this view, 'Hesperus' is semantically associated with a complex predicate say, for illustration, a predicate of the form 'E x) & S x)', suggesting 'evening star'.
On referentialist principles, the two statements express the same proposition, the one that is true in all worlds in which Venus (the planet that 'Hesperus' and 'Phosphorus' are two names for) is visible in the eastern sky.
However, we may want impossible worlds where exactly one of these fails even though Hesperus is necessarily identical with Phosphorus, for instance, to model the cognitive state of an agent who believes one proposition, but not the other.
In (3), the oblique reference of 'Hesperus' is its ordinary sense.
