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Basically the title question: How can I tell if a term is a rigid designator?

So far all I have is that proper names are (or can be, since there is a difference between "being named Barack Obama" and "being Barack Obama") rigid designators. For other things, indicatives for example, I'm not sure.

Is there some test, or set of tests I can apply to a term to decide if it designates rigidly?

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By definition, due to Kripke, a rigid designator is a term that refers to the same object in all possible worlds. Rigid designators are philosophically interesting because "an identity statement in which the identity sign is flanked by two rigid designators must be necessarily true if it is true at all, even if the statement is not a priori". For example, according to Kripke, "Hesperus is Phosphorus" (the evening star is the morning star) is analytic a posteriori, something Kant would not have thought possible.

According to Hughes, “that proper names are rigid, and that identity statements involving only proper names are accordingly necessarily true or necessarily false... [is] as close to uncontroversial as any interesting views in analytic philosophy”, although there is a minority view disputing even that. Beyond that what is or is not a rigid designator is subject to controversy, because truth of many philosophical claims depends of whether some of the terms are assumed to be rigid designators. Kripke himself extends rigid designation to "natural kinds", such as 'apis mellifera' (European honeybee). It is not immediately clear what "object" this species name refers to, one suggestion is "an abstract object" a la Plato, another is to extend designation to kinds, as Kripke does, so that it rigidly refers to "honeybee kind", as opposed to bumblebee or any other "kind", in all possible worlds. Other examples of natural kinds are 'water', 'heat', or 'redness'. One can then say that the corresponding natural properties, 'honeybee-ish', 'watery', 'hot', and 'red', are also rigid.

Aside from technical objections, like distinguishing between natural and artificial kinds ("wine", "bachelor") philosophers pointed out that the theory is "committed to substantive metaphysical positions about abstract objects like kinds or properties, concerning which a theory of language ought to remain neutral". Indeed, Kripke produced an argument against materialism about the mind, which involves rigid designation for 'water'. A compromise is the Cook's theory of rigid application, where no properties are rigid, and for kinds "what makes the application of a general term rigid is that any item to which the term applies, in any possible world, is part of the extension of that term in all worlds in which it exists".

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