Home Philosophy Advances in Proof-Theoretic Semantics

## Morning Star Versus Evening Star RevisitedFrege'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 [9] 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 Venus In our account, we would take (appropriate) terms We may say that the fact that the denotations of need of ## EqualityWe 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 Working in an epistemic context, however, one may note that not all (true) equalities might be known by an agent13
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 (considered as the set of equalities known by If we analyze |

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