Home Philosophy Advances in Proof-Theoretic Semantics
Frege did not define the notion of mode of presentation, but he did give two quite illustrating examples of it.
The first one is taken from geometry: a particular point may be presented as intersection of two lines a and b or as intersection of the lines b and c. Though the intersections take place at the very same point, we would say that the two modes of presentation differ: one refers to the lines a and b, the other to the lines b and c.
Assuming a suitable axiom system for geometry, this system will provide terms which serve as definitions for the intersections expressed by, say, Intsec(a, b) and Intsec(b, c). Assuming that both terms refer to the same point p of the plane, it requires some reasoning in the given axiomatic framework to derive the equality
Intsec(a, b) = Intsec(b, c). This equality is epistemically different from a simple reflexive equality, like Intsec(a, b) = Intsec(a, b).
We propose to use Intsec(a, b) to obtain a mode of presentation of p, and
Intsec(b, c) to obtain another mode of presentation of the same point.7 We may say that a term t of our formal language expresses (to use Frege's wording) a mode of presentation if it may be used as a mathematical expression to define a newly introduced constant A. We do not say that the term is the mode of presentation, as—with Frege—the latter is surely not a syntactic object (this would be the mode of designation, [5, p. 157]). The way the mode of presentation should be located between the purely syntactical level and the semantical level will be discussed in more detail below. But let us note, that our mode of presentation is clearly different from any form of reference in model-theoretic terms.
Let us now turn to the more prominent example given by Frege. By “morning star”, Venus is presented as the star8 visible in the morning, by “evening star” as visible in the evening. Thus, the sense of “morning star” differs from that of “evening star”, although both refer to the same object. We may use “the star visible in the morning” and “the star visible in the evening” as the expressions which give us the mode of presentation, using the same argument as above: these expressions may serve as terms t defining a constant A (“morning star”, “evening star”, or even “Venus”).9
Thus, we may extend our working definition of mode of presentation given above for mathematical terms to terms in general, saying that a term t may express a mode of presentation if it can be used as definiens in a clause like “Let A be t .” Later we shall see how proofs enter.