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Put antother way

Proposition 1: for anything that exists in this world I can find/construct, even if just by assertion, a rigid designator that applies to it.

Proposition 2: like proposition 1 but over all possible worlds.

What are the status of these propositions?

I think that in the context of reading Kripke, a definition of "anything" in this would be "any object/entity that we can discriminate/delineate via natural language". This is based on the position that if we can talk about it in a sensible way, then we could apply a proper name to it, and thus it would have a rigid designator.

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I took Kripke to be saying that we can rigidly designate something, and we can rigidly designate anything precisely when we pick that thing out across all possible worlds.

See http://plato.stanford.edu/entries/rigid-designators/

A rigid designator designates the same object in all possible worlds in which that object exists and never designates anything else.

Though an important qualification is that a name that rigidly designates from the standpoint of our world may not be a name that would rigidly designate from a different world, but it does for us.

Proposition 1 seems to follow tautologically from the definition. Proposition 2 seems to suffer from two important ambiguities:

  1. It is not clear that an event of rigid designation has occurred.
  2. The meaning of the phrase "over all possible worlds" is unclear vis-a-vis the act of rigid designation.

I could probably write more ... but it's been 8 years since I've read Kripke, and I'm not quite sure where you're going with this.

Or to word it another way in answer to your title question, everything could have a rigid designator if anyone bothered to assign one to it, which often happens in the form of a name that is given to pick it out in all possible worlds from the privileged location of this world.

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  • Item 2 was to clairfy that if an object does not exist in the real world, but is stipulated to exist in some possible world, it is still sensible to have a rigid designator to it.
    – Dave
    Jun 21 '15 at 18:57
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    There is a trade off. We are free to assign rigid designators to anything but they may fail to refer in any world other than the one from which they are assigned, and the theory trivializes. Kripke's critics contend that this is exactly what happens with anything other than proper names. Since identity of an object is largely determined by its relations the idea that there can be the "same" object in different worlds is peculiar even for proper names, but at least one can argue that for them it is a good enough idealization.
    – Conifold
    Jun 21 '15 at 20:37

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