If I am not wrong, the old Fregean distinction between sense and reference can be read in terms of a distinction between intension and extension: The star of the night and the star of the morning are descriptions that share the same extension in the actual world but differ in "sinn", in the sense that they do not pick out the same items in alternative possible worlds.

Yet, as Frege underlines, Phosphorus and Hesperus share the same reference but not the same sense. And, under a rigid designator theory of names, thay also share the same reference in every possible world. But then we must admit that it is not correct to identify intensions with "Sinn". If this is not the right way to capture that concept, how should we analyze the notion of fregean sense? Artwork

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    We can think of meaning not as a single value attached to a term, but as a set of possible funhouse mirrors the term can be reflected in, distorting and changing it in different ways. One of the mirrors gives the sense, another mirror gives the reference, another mirror gives the connotation, another mirror gives the hidden subtext, and different people who have different thoughts about the term may have different mirrors.
    – causative
    Aug 9, 2021 at 1:30
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    One can use technical modal apparatus for definite descriptions just as for Kripkean rigid designators, but the notions of possibility will be different. Rigid designation applies to names under Kripke's so-called metaphysical possibility. However, the intent in distinguishing Phosphorus and Hesperus is epistemic, it concerns what is known, and, taken as definite descriptions, they do not share reference in all epistemically possible words. For all a person knows, it is possible that the star in the evening is distinct from the star in the morning, so the intensions are different.
    – Conifold
    Aug 9, 2021 at 5:58
  • So the notion of metaphysical necessity does not conform to the analysis of senses as intensions, but the notion of epistemic possibility does conform to that analysis, am I right?
    – PwNzDust
    Aug 9, 2021 at 20:43

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I don't think Frege's theory of sense and reference is compatible with the rigid designator theory of reference (or, at least, it would take some sophisticated theoretical footwork to render them compatible). Frege is generally considered a classical descriptivist; the rigid designation theory of reference developed as an alternative to descriptivism. I haven't read Kripke's Naming and Necessity in a very long time, but I still consider that the canonical text for the rigid designator theory of reference and its implications in the philosophy of language. As for how to understand Frege's notions of "sense" and "reference," "sense" defines the concept by certain attributes and "reference" picks out the objects that instantiate that concept. You can't have a reference without a sense, but you can have a sense without a reference (if I recall correctly). I think that aligns pretty well with contemporary uses of the terms "intension" and "extension."

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