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Morning Star Versus Evening Star Revisited
Frege's example of difference of senses in “morning star” and “evening star” became a classic. It is intuitively clear that there are two different senses, although there is only one reference.
Possible worlds semantics does not cope well with this example. Taking Kripke's  famous distinction of rigid and non-rigid (use of) terms into account, one can consider “morning star” and “evening star” as definite descriptions10 which should be non-rigid. But “Venus”, as a proper name, is supposed to be rigid. Now, however, in the worlds in which “morning star” and “evening star” are supposed to be different, we would have “two copies” of Venus, let's call them VenusM and VenusE . Leaving aside the question which of them should be the Venus, the problem is that for these two Veneres the astronomical laws have to fail—otherwise they would coincide again.11 Is it really the case that—to understand the difference of the sense of morning star and evening star—we would have to consider worlds with different astronomical laws? In our view, the difference in the sense of morning star and evening star should not depend on the astronomical laws at all—it depends, to go back to Frege, only in the mode of their presentation.
In our account, we would take (appropriate) terms tM and tE representing “morning star” and “evening star” in a sufficiently formalized astronomical theory as definite descriptions which both could serve as defining a planet. It is now a new task to prove the equality tM = tE by use of astronomical laws (together with the empirical astronomical observations which are formalized as statements involving tM and tE ).
We may say that the fact that the denotations of tM and tE are equal follows from the identity of (tM )M ≡ (tE )M ≡ Venus in the real world, while the equality of the modes of description tM = tE follows from the proof in our astronomical theory. The
need of performing this proof explains the epistemic difference between identities and equalities.12
We here consider only equalities between terms, which may refer to mathematical objects or to objects of our real world.
As said, in first-order logic, equality is axiomatized as a universal congruence relation, thus directly linked to extensionality (the congruence axioms include the compatibility with all functions and relations).
Working in an epistemic context, however, one may note that not all (true) equalities might be known by an agent13 A . Thus, the equalities known by A may not be complete with respect to the identities which hold in the intended model of A 's knowledge. This incompleteness has to be understood with respect to the combination of interpretation and identity as described in Sect. 2: for two terms t and s,
(t )M ≡ (s)M may hold, but A doesn't know t = s.
The incompleteness can arise from two different sources. On the one hand, an agent may have an “underaxiomatized” representation of the world. On the other hand, agents are not supposed to be logically omniscient, and will miss (fail to know) those equations which they haven't yet proved.
The first case may apply in the morning star/evening star example, when the agent does not know the astronomical laws to derive the fact that both terms refer to the same object.14
The second case may apply to the geometric example, if the agent didn't perform the mathematical proof of the equality of the two intersections.
In both cases, the equalities the agent knows are incomplete with respect to the identities which hold in the appropriate model. Now, the equality relation = of A
(considered as the set of equalities known by A ) may serve to express some intensionality with respect to the outer extensionality, given by ≡ (or all true equalities).
If we analyze A 's knowledge we should allow the substitution of two terms only if A 's knowledge comprises the corresponding equality—the underlying identity in the model is irrelevant. With only these identities in mind, we may observe the intensional phenomena in A 's knowledge.
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