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Equality of Senses
One of the fundamental challenges for every theory of senses is the notion of equality of senses.
In our setting the notion of sense is naturally relativized to (the knowledge of) an agent A . A very naive attempt would be to introduce a notion of equality of senses relativized to an agent A , identifying the sense expressed by two terms if and only
if A can prove the equality t = s. This would allow to separate the denotation from
the senses of two terms denoting the same object in cases where A does not have
the proof of the corresponding equality at hand. But it would compromise Frege's original idea, as the senses of “morning star” and “evening star” should clearly stay different even if somebody knows that both denote Venus.
Still, we may obtain an interesting notion of equality of senses if we allow for the closure of the mode of presentation under some equalities. This can be illustrated best by use of the geometric example: we said that Intsec(a, b) and Intsec(b, c) should be considered as different modes of presentation of the point p. It seems to be, however, that Intsec(a, b) and Intsec(b, a) do not give us different modes of presentation of the same point. In technical terms, this means that the mode of presentation is not changed when we invoke the symmetry of the relation Intsec.
It is not our aim to specify concrete criteria concerning which (type of) equations should be taken into account for the equality of senses. In contrast, we think that equality of senses should not only be relativized to an agent (or an agent's knowledge) but that it could also be graduated and that it depends on the chosen axiomatic context.
The rôle of the axiomatic context can be exemplified by the natural numbers: if they are introduced as a commutative semigroup, commutativity is, of course, “build in”
and t + s should not have a sense different from that of s + t . If, however, the natural
numbers are introduced by use of the Peano Axioms, the commutativity of addition
requires a rather non-trivial proof by induction, and, in this context, one might say that the sense of t + s differs from the one of s + t , as the required recursion over
(only) one of the summands to calculate the value may lead to substantially different computations.
This last example shows that, for our notion of mode of presentation, the underlying axiomatic setting forms an integral part of the sense of a term.15
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