## 1. A Basic Characterization of Rigid Designation, its Interest, and the Breadth of its Application


### 1.1 Names, Ordinary Descriptions, and Identity Statements



Before going into refinements, consider a famous application
made by Saul Kripke, who coined the word 'rigid
designator'. Kripke is not the first philosopher to discuss the
idea behind the term he
 coined,[1]
 but his illuminating discussions have made the importance of rigidity
widely appreciated.



Kripke (1980;1971) famously argues that because a rigid designator
designates the same object in all possible worlds, an identity
statement in which both designators are rigid must be necessarily true
if it is true at all, even if the statement is not a priori. His
classic example is the identity statement 'Hesperus is identical
with Phosphorus', which is true, but which was discovered a
posteriori to be true. (Let us understand a "statement" to
be a sentence under an interpretation; and, following Kripke's
notation in 1971 and elsewhere, let us shorten 'Hesperus is
identical with Phosphorus' to 'Hesperus =
Phosphorus' even though Kripke does not do that in 1980.)
'Hesperus' is a name that was given to a heavenly body
seen in the evening, and 'Phosphorus' is a name that was,
unknown to the first users of the name, given to that same heavenly
body seen in the morning. The heavenly body is Venus.



One might initially suppose that since the statement 'Hesperus =
Phosphorus' was only discovered empirically to be true, it must
be contingently true. But, says Kripke, it is necessarily true. The
only respect in which it might have turned out false is not
metaphysical but epistemic: thus, one could as well say of a
geometrical theorem before it is proven that it might or might not
turn out to be true or that it might or might not be provable without
the parallels postulate. But if it is true and is provable without the
parallels postulate, that is a matter of metaphysical necessity. In
the same way, if the statement 'Hesperus = Phosphorus' is
true, that is a matter of metaphysical necessity.



'Hesperus = Phosphorus' is necessarily true if true at all
because 'Hesperus' and 'Phosphorus' are proper
names for the same object. Like other names, Kripke maintains, they
are rigid: each designates just the object it *actually*
designates in all possible worlds in which that object exists, and it
designates nothing else in any possible world. The object that
'Hesperus' and 'Phosphorus' name in all
possible worlds is Venus. Since 'Hesperus' and
'Phosphorus' both name Venus in all possible worlds, and
since Venus = Venus in all possible worlds, 'Hesperus =
Phosphorus' is true in all possible worlds.



A description like 'the brightest non-lunar object in the
evening sky' is, on the other hand, *not* rigid. That
explains why the identity statement



> 
> (H) 'Hesperus = the brightest non-lunar object in the evening
> sky'
> 



is true but not *necessarily* true. While Hesperus is in fact
the brightest object in the evening sky apart from the moon, Hesperus
might have been dimmer: had, say, Hesperus been obscured by cosmic
dust, Mars might have been the object designated by 'the
brightest non-lunar object in the evening sky' rather than
Hesperus. In that case, the above identity statement (H) would have
been false. So the reason that (H) could have been false is that
'the brightest non-lunar object in the evening sky' does
not designate Hesperus rigidly. It designates Hesperus in this world,
which explains why (H) is true, but this description designates Mars
in some other worlds, which explains why (H) could have been false:
(H) would have been false had some other such world been actual.


### 1.2 Standard Clarifications from Kripke



Some potential misunderstandings are well known.



First, a rigid designator is used in a certain way in the actual world. Given *that* meaning, it designates the same object with respect to all possible worlds, regardless of how this term is used, or not used, in those other possible worlds: for
although we identify objects in other worlds by our own names, natives
of some of these worlds use other names (Kripke 1980, p. 77; for
discussion of objections, see Fitch 2004, pp. 103-4). 



Second, although the statement (H) is not necessarily true, it is
nevertheless Kripke's view and the standard view that the object that
is in fact the brightest in the evening sky, Venus (Hesperus),
*is* necessarily identical to Hesperus.



Third and finally, 'Hesperus' is rigid because it picks
out Hesperus in all worlds that *contain* Hesperus. In worlds
not containing Hesperus, the designator fails to name anything other
than Hesperus. There is more than one account of a rigid designator
that conforms to that requirement (many do: Rami 2019). On one basic account, a rigid
designator designates its designatum in every possible world
containing the designatum and in other possible worlds the designator
fails to designate. In places, Kripke suggests that this is his
idea:



> 
> When I use the notion of a rigid designator, I do not imply that the
> object referred to necessarily exists. All I mean is that in any
> possible world where the object in question *does* exist, in
> any situation where the object *would* exist, we use the
> designator in question to designate that object. In a situation where
> the object does not exist, then we should say that the designator has
> no referent and that the object in question so designated does not
> exist (Kripke 1971, p. 146; a disclaimer is reported in Kaplan 1989b,
> p. 570n.8).
> 



In other places, Kripke seems to have in mind another account of
rigidity: one according to which a rigid designator designates its
object in *every* possible world, *whether or not* the
designatum exists in that world. Hence, he says, "If you say,
'suppose Hitler had never been born' then
'Hitler' refers here, still rigidly, to something that
would not exist in the counterfactual situation described"
(Kripke 1980, p. 78).



It may be that no substantive issues ride on which conception of
rigidity is adopted (Stanley 1997a, pp. 557, 566ff.). On the other hand, some philosophers have held that
true statements using a proper name to express that so and so
*might not have existed* are unintelligible unless the relevant
name refers to the object in all worlds, period. That might provide a
substantive reason for favoring the latter type of rigidity (see, e.g., Besson 2009; Gomez-Torrente 2006, p. 250; Plantinga
1985, p. 84; on Plantinga, see also the subsection on "Worlds
and Essences" in the entry on
 actualism),
*obstinate rigidity*, as Salmon calls it (1981, p. 34).

Kripke's quote above hints that obstinate rigidity might afford this
kind of advantage. But Kripke never argues for one position or
another. Despite occasional slips in favor of one or another refined
version of rigidity, he deliberately sidesteps these "delicate
issues" when he gives them his full attention (1980, p. 21, n.
21).



There are stronger and weaker brands of necessity corresponding to the
possible notions of rigidity. Kripke argues that a sentence like
'Hesperus = Phosphorus' is at least "weakly"
necessary (1971, p. 137): true in all those possible worlds in which
Hesperus exists and Phosphorus exists. The statement may enjoy a
stronger necessity, too, which would render it true in all worlds,
 period.[2]
 In this entry, "necessity" is to be understood as weak
necessity (at least).


### 1.3 Rigidity for a Diversity of Grammatical Categories



Rigidity is most straightforwardly applied to proper names of concrete
objects. There is general agreement that 'Hesperus' and
'Biden' refer rigidly. The debate about which
expressions in natural languages are rigid designators is typically
set up with a background assumption that proper names are singular
terms. This assumption is challenged by predicative accounts of names (think of 'every Biden': Fara 2015). Whether such alternatives can "explain why names are rigid" (Lee 2020, p. 253; Schoubye 2018), perhaps in recognizing singular reference or denotation for some uses of names tacitly equivalent to demonstratives or definite descriptions (categories discussed below), remains a matter of controversy.




Rigidity is applied commonly to indexicals, like
'now' or 'you', as well as names. If, when I use
'you' in a context, my utterance means something like
*the person at the door* (Nunberg 1993; Smith 1989 SS5),
then presumably it is not rigid. You may be at the door, but in
another possible world it is another philosopher who comes visiting.
On the other hand, if (as Kaplan proposes: 1989a) 'you'
refers directly to an individual, without amounting to a disguised
description meaning the same as a description like 'the person
at the door', then it would seem to be rigid. So if I say,
pleasantly surprised, "You made good time," we could
discuss reasons for this and conditions under which this might not
have been the case, but in all of our considerations the same
individual, and not anyone else who might have been at my door in good
time or not, is the one in question. The individual in question is the
one who is *in fact* identical to *you* (as any account
could put it). After Kaplan, indexicals have typically been treated as
rigid; but there seem to be nonrigid uses. 



Demonstratives such as 'that' raise similar issues as
indexicals. So also do noun phrases like 'that flower you just
picked' (see King 2001, chap 2, for discussion). As Wolter
(2009, p. 457) observes, "there is not yet consensus"
about which such expressions "allow non-rigid interpretations
and which do not," but it is standard to hold some to be rigid (Nowak 2019).



The paradigm example of *non*-rigid designators, since the work
of of Kripke, has been definite descriptions like
'the brightest non-lunar object in the evening sky'.
However, some definite descriptions, such as 'the successor of
2', are rigid designators; their rigidity seems to arise in
virtue of facts about metaphysical reality, in this case, the
necessity of mathematical facts, rather than facts about the semantic
properties of expressions. Such definite descriptions are therefore
"de facto" rigid designators. The Russellian view that
definite descriptions are quantified expressions is also widely held.
Since it is natural to think of rigidity but not, in view of Russell,
to think of definite descriptions in connection with singularity, the
question dawns of how to think about rigidity with categories of
expressions other than singular terms. This broad topic, which
surfaces in other connections too, has received much discussion in
recent years. 



Another category of terms whose singular status raises questions is
that of natural kind terms like 'gold' and
'water'. Again, these seem widely to be thought singular
at least some of the time. They are frequently counted as rigid. Other
related and again apparently singular terms that are sometimes counted
rigid include 'redness' and 'loudness'. More
controversial are singular terms for properties that are more
artificial: 'bachelorhood', or 'soda pop'. 
Not just singular terms but also general terms, like
'tiger', 'hot', and 'red' are
often recognized as rigid (following Kripke's suggestion: 1980, p.
134). These terms raise complications not present for singular
terms. I discuss kind and property terms toward the end of this entry.



Terms from other grammatical categories, as well, might admit a
rigid/non-rigid distinction (for the broadest account of rigidity available, see Haze 2021). Philosophers have attended relatively
little to the status of verbs and adverbs, in this connection (but
see, e.g., Lopez de Sa 2008). Compare 'to begin
writing', 'to commence writing', and 'to do
what the teacher commanded at time *t*' (where the
command was, "Begin writing"). Evidently, the necessity of
'One is about to begin writing if and only if one is about to
commence writing' is not enjoyed by 'One is about to begin
writing if and only if one is about to do what the teacher commanded
at time *t*'. Arguably, the reason for the modal
differences has to do with the rigidity or non-rigidity, in some
appropriately extended sense, of the contained verbs. Something
similar applies to adverbs. Compare 'She ran quickly' with
'She ran in the manner signaled by her coach' (where the
signal means: *Run quickly!*). 


### 1.4 Rigidity for Temporal and other Non-Alethic Modal Talk



Just as rigidity might have bearing on different grammatical
categories, not merely the paradigms to which Kripke applies it,
rigidity might also have bearing on different modalities. For example,
just as 'the brightest non-lunar object in the evening
sky' seems to designate Venus with respect to the actual
*world* but not with respect to other possible worlds in which
Venus is obscured by some other body like Mars, just so the expression
seems to designate Venus with respect to the present *time* but
not with respect to other times at which Venus is obscured by some
other body. By contrast, 'Hesperus' and
'Phosphorus' seem to designate Venus with respect to all
times. (For cautions about the analogy, see Gomez-Torrente
2006, p. 249.) Therefore, again, 'Hesperus = Phosphorus'
seems to be true regardless of the time of utterance; 'Hesperus
= the brightest nonlunar body in the evening sky', on the other
hand, is false when uttered as a passing comet outshines Venus. 



Whether we are talking about *alethic* rigidity (designation of
the same individual with respect to other possible worlds) or
*temporal* rigidity (designation of the same individual with
respect to other times), a commitment to rigidity will be
attended by similar sorts of metaphysical and epistemic
commitments. Here it will do to indicate briefly some
ontological commitments. Just as recognition of alethically rigid designation carries
with it a commitment to some sort of trans*world* identity
(discussed later), so the recognition of temporally rigid designation
carries with it a commitment to some sort of "transtime"
or transtemporal identity. Similar alternatives to rigidity are
available to theorists who oppose rigidity. Hence,
if we adopt the ontological position that individuals are
instantaneous so that we find it natural to think in terms of the
"stages" of you since, although the stages are not
identical, they're similar and ordered in relation to one another
spatiotemporally or causally, then we might naturally take names like
yours to range over different instantaneously-lasting individuals, and
so conclude with Varzi (2003, p. 387), that "the rigidity claim
can't be right and a counterpart-theoretic semantics seems
required."



Familiar, broadly applicable reservations about the reduction of modal
talk to counterparts talk arise in their turn: namely reservations
according to which ordinary speakers' accurate talk of a future and a
past for you uses 'you' and your name
 rigidly since ordinary speakers presuppose or assert that something in the
past and something in the future is *identical* to you.[3] The
counterpart substitute, it will be said, is either a misdiagnosis or a
tacit concession to the falsity of ordinary claims and a masking over
of that falsity by means of a so-called reductive analysis. Quinn, for
example, anticipates the dilemma, maintaining that identity through
time is what is needed to honor claims to an individual's lasting
through a day, and maintaining further that if we conclude that we
*don't* really last over time, and decide anyway that "it
would be convenient to speak" in some counterpart-reductive
fashion in order to be able still to *assent* when ordinary
speakers innocently say 'You were around yesterday and will
probably be around tomorrow', then we end up masking our
disagreement with what ordinary speakers literally claim (1978, p.
348). Metaphysical positions of great consequence are thereby
concealed: 'you'll be around tomorrow' is now, according
to the newly proposed reductive reading, to be accepted on the liberal
terms that tomorrow there is a line of "momentary creatures
endowed with similar properties," none of whom is really you.
But were a reductionist to affirm correctly the usual sentence, now
eviscerated of the usual content, the affirmation should bring you
"no comfort at all" (1978, p. 350; see also p. 347). The
hope is that you yourself are around tomorrow; if others are around
too, that might be good, but it is not enough to satisfy your hope.
(So we would put it if we believed in the reality of different times.
More neutrally, we might put the point as follows. When we conceive of
tomorrow, you're part of what we're conceiving as present, hopefully;
if others are around too, that is irrelevant to this hope.) Since the
sentence 'you'll be around tomorrow' on its original
interpretation expresses comfortingly that your hope is to be
realized, and since the reductive assent does not, the reductive
assent hardly ends up conveying the appropriate opinion about what
ordinary speakers are talking about. 



Kripke presents rigidity first as an *alethic* notion but there
is also a *temporal* interpretation, as we've seen. There are
other interpretations too, which is not surprising if rigidity is
fundamental to all our modal talk: and it seems reasonable to say,
with Follesdal, that it is. "All our talk about change, about
causation, ethics and knowledge and belief, as well as about the other
modalities, presupposes that we can keep our singular terms referring
to the same objects. To the extent that we fail, these notions become
incoherent" (Follesdal 1986, p. 111). 



Furthermore, a term might *rigidly* designate under
*one* interpretation of rigidity but *non*-rigidly designate under a
*different* interpretation of rigidity: thus, 'Hesperus =
Phosphorus' is apparently not *epistemically* necessary,
even though it seems *alethically* and *temporally*
necessary. Accordingly, 'Hesperus' and
'Phosphorus' are plausibly thought not to be epistemically
*rigid*, in which case, as Fitting and Mendelsohn (1998, p. 219)
observe, "it is hard to see whether there are any rigid
designators under the epistemic reading." A counterpart construal (see below) might be offered more plausibly for epistemic possibility than for temporal or alethic possibility, as Kripke himself sometimes suggests (Rabern 2018, pp. 12-13): when we say that Hesperus might for all we know have turned out not to be Phosphorus, we seem to have in mind a possible world in which something else--a counterpart--is present in the morning or evening, not Hesperus (Phosphorus). This entry will
address an alethic interpretation of rigidity for the most part.


## 2. Relations Between Rigidity and Associated Theories


### 2.1 Relations Between Rigidity and Transworld Identity



A rigid designator designates by picking out its designatum not just
with respect to the actual world--as things are--but with
respect to all other possible worlds in which it is present, as well.
That a proper name like your name is rigid by way of picking out its
same designatum (you) with respect to possible worlds or situations
different than those actually obtaining, though that claim is not
without detractors, is a claim that enjoys great appeal. Call the
relevant claim "transworld identity." The claim is
associated with a host of others that might enjoy much *less*
appeal. Many such claims turn out to be separable from rigidity.
Nevertheless, transworld identity remains a substantive commitment of
rigidity, at least as classically understood.


#### 2.1.1 Questionable Controversy



The commitment to transworld identity should not be confused
with a commitment to *realism* about possible worlds. Prominent
champions of rigidity seem favorably inclined to realism (see
Davidson's 2003 collection of Plantinga's essays; see also Kripke
1980, p. 16). But there is no reason an instrumentalist about possible
worlds like Bas Van Fraassen, say, who regards possible-worlds talk as
a useful fiction, cannot appeal to rigidity (as Van Fraassen does:
1984). What matters for the position that a
designator for an individual *x* is a rigid designator, is that
in using that designator to entertain the various counterfactual
situations or worlds *S* in which *x* exists, whether we
do the counterfactual entertaining about *x* by way of
apprehending abstracta, say (if worlds are real objects, abstract
ones), or by way of pretending (if counterfactual talk is a fictional
pretense, say, because worlds are not real) or whatnot, we entertain
situations (or pretend to do so) that, *were* they to obtain,
*would* contain *x*: so we tacitly commit to "the
impossibility that *S* obtain and *x* fail to
exist" (Plantinga 1974, p. 96). Accordingly, suppose I say to
you, "If you'd driven to the park instead of walked, you'd have
missed the refreshing breeze." Here I entertain a nonfactual
possibility associated with your driving to the park. Whom, in that
entertained possibility, do we think of and speak of, as driving and
not walking? The transworld-identity theorist says: you. No one else
"gets into the act" as the driver according to the
transworld-identity theorist (as Lewis would frame that view, without
sympathy: 1986, p. 196).



Realism about worlds, then, is not something rigidity or transworld
identity comes attached to; non-realists have their version.



How about *essentialism*? Whether we want to say that rigidity
or transworld identity is committed on that score might well depend on
the form of essentialism in question. Not many philosophers would
dispute Cartwright's observation that "'Shakespeare'
is rigid only if the *man* Shakespeare could not have existed
without being Shakespeare--or as we might equally well say, only
if it is essential to Shakespeare that he is Shakespeare"
(Cartwright 1998, p. 69). So if Shakespeare could have been Biden
instead of Shakespeare, 'Shakespeare' would not be
rigid.



But even if appeals to the rigidity of a name like
'Shakespeare' (or related appeals to the transworld
identity of Shakespeare) cannot be divorced from essentialist
commitments like the forgoing, such appeals will not commit us to much
at all about what a designatum is essentially *like* with
respect to the various possible worlds in which it exists to serve as
the designatum: whether Shakespeare could have been a brute, say, or
had different parents (matters over which Kripke tellingly disagrees
with, say, Plantinga: Kripke 1980, pp. 110ff; Plantinga 1974, pp.
65ff). 



It is far from clear then, that transworld identity and the rigidity
that depends on it come attached to substantive essentialism of a sort
that settles salient disputes in philosophy about which of your
properties are essential to you. Closely related to the issue of
substantive essences is the issue of substantive *criteria*,
perhaps qualitative criteria. Such concerns, as developed in a metaphysical direction,
will be addressed in connection with counterpart
theory and transworld identity; here let us move in an epistemological direction in order to
address concerns about what we might call transworld
"identi*fication*." 



Kripke addresses the objection that we cannot meaningfully talk about
you, with respect to another possible world, without first having some
qualitative criterion of identity, some qualitatively distinguishing
mark that allows us to pick you out from other objects in the world at
issue, in order to assign your name to the right person, i.e., to you,
as the individual that satisfies the qualitative criterion. This
criterion would appeal to your essence (or *be* "an
 essence": see Plantinga 1985, pp. 85-7; 1974, p. 98; recall, for this example, the minimal requirements of weak necessity),
 in the minimal respect that the criterion must be something that you
and you alone have with respect to any given possible world. As an
objection, the worry is that we know of no such qualitative criterion
so we can *not* meaningfully discuss *you*, with respect
to any merely possible world. 



One way to defend rigid designation in view of the forgoing objection
would be to insist that as successful speakers, we do after all know
of qualitative criteria by which we are able to distinguish
individuals like you by name: some historical figures have in effect
adopted this position, thereby defending rigid
 designation.[4]
 But Kripke would concede that we speakers are not privy to the
content of any qualitative criterion characterizing you. It is hard to
gainsay the concession. We would certainly be hard-pressed to say in
any enlightening way just what qualities characterize you with respect
to the various possible worlds. Kripke would evidently agree that
there is something that it is to amount to you, as opposed to anything
else, and that your essence is what qualifies an
object--you--to be the designatum of your name with respect
to any possible world (he doesn't say that your essence is qualitative
though, and he outright rejects certain versions of that claim). But
Kripke would deny that a speaker would have to *know* this
essence, or to rely upon any other nontrivial criteria distinguishing
you from world to world, in order to refer to you by way of a rigid
designator (Kripke 1980, pp. 15-20; see also Plantinga 1974, pp.
93-98).



Still, we can imagine why others have demurred. After all, if we lack
the ability to say in an informative way just what it is to amount to
you, as opposed to anyone else, with respect to counterfactual
situations, then how *can* we as speakers know enough of what
we're talking about in order to *select you* as the designatum,
as opposed to anyone else, with respect to counterfactual situations?
How can we know that you, no other, are the subject of our thought and
talk? We certainly cannot *define* your name rigidly by means
of essence-invoking criteria if we don't know such criteria.



In response, Kripke (1980, p. 44; see also Plantinga 1974, p. 97)
insists that we may give a satisfactory account of how we designate
you, trans-world, without giving a satisfying account of your essence
or of criteria distinguishing you. Whatever the content of your
essence, we may simply *stipulate* that the bearer of your name
is to be you when we consider other possible worlds in which you do
different things than those that you in fact do. Accordingly, we may
take for granted that *you* are the protagonist of the
different actions at those different worlds, despite our not thinking
about what your essence is.



There is something sensible about this
suggestion. It seems sensible to suppose that ordinary speakers do
something like stipulate: a speaker might naturally insist, "I'm
thinking of *you* driving instead of walking, not someone *like* you." We take the success of
this stipulation for granted, at least for many purposes, in many
contexts.



Even so, the appeal to
stipulation is more like a promissory note than the satisfaction of an
explanatory obligation. The appeal to stipulation puts off for another
occasion any attempt to resolve *how* we succeed at doing what
we take for granted that we manage somehow to do: namely, how we
succeed at referring to the right individual, by means of our
stipulative effort. There has to be some "*reason* the
stipulated situation, when we use a name, contains the object it
does" (Sidelle 1995, p. 99n.4) rather than likely competitors.
It is hardly obvious what that reason would be. To see why, consider
that in order successfully to stipulate that a name is to follow just
you, as a rigid and therefore transworld tracking device, our
stipulative effort has to be able, across worlds, to allow us to
distinguish what is *you* from what is not you but is instead
your *body* (say: assume you are not your body). How is this to
be done without specifying criteria, if you were with your body when
your parents smiled in your direction and baptized you with a rigid
designator, saying "We have decided on a name for the birth
certificate: ...," thereby stipulating that you are to be
called by the name they chose for you? "It is not by
magic," as Jackson (1998, p. 82) reminds us, that your name
"picks out what it does pick out" rigidly--namely
you--despite the competition against you presented by a different
candidate for designation--your copresent body.



So how does it happen that the stipulative effort attaches your name
to you and not competitors? According to a Lockean line of thinking,
the indispensability of *criteria* becomes clear when we
consider that question; stipulation is no substitute for criteria. Unless speakers have essential
criteria in mind to *demarcate* the real referent--you, in
this instance--from alternative candidates like your body, the
speaker would stipulate a rigid designator in vain: his names would
turn out, disappointingly, to be "the Signs of he knows not
what, which is in Truth to be the Signs of nothing"
(Locke, *Essay* III.ii.2, p. 406). On this line of thinking, a
stipulation leaves too much indeterminacy to secure the right
rigidity. To insist that the right designatum is secured without
criteria seems, according to this line of thinking, to insist on
stipulation getting it right by magic. 



So there is more to be said about transworld criteria, even if an
appeal to stipulation is all right so far as it goes. But because the
relevant contributions to the discussion become scattered and
elliptic, a supplementary document is handy for elaboration:



> 
> Stipulating Identity Trans-world, Without Qualitative Criteria for a Designatum to Satisfy
> 


#### 2.1.2 Residual Metaphysical Controversy



It is not obvious that rigidity and its attending transworld identity
run up against anti-realism, nor against any substantive essentialism
or its gainsayers, nor against the intuition that there must be
*some* way to assign reference that avoids (perhaps by way of
qualitative criteria) the charge according to which the relation
between rigid designator and designatum comes by magic. But rigidity
and its attending transworld identity do run into genuine conflict
with certain metaphysical positions about what we're talking about
when we engage in counterfactual talk. The most prominent alternative
to the commitment construes modal discourse about you in terms of *counterparts* of you who are not you and who act in your stead at
other possible worlds. Thus, for David Lewis and fellow counterpart
theorists, what matters to whether you might have driven instead of
walked is indeed whether, in other possible worlds, someone other than
you--but sufficiently similar qualitatively to you--drives
instead of walks. So for Lewis, *you don't* have an important
role, or indeed any role, among the players of the
driving-as-opposed-to-walking worlds: and your *qualities do*.
You are altogether absent from any world we entertain in considering
the driving scenario as a scenario that you *might* have opted
for, and in considering such a scenario we are not entertaining a
thought about you as the protagonist. Let us call this proposal
concerning counterfactual talk "*serious* counterpart
theory" (following Russell 2013, p. 87).



The proposal that it is not you yourself but similar
"counterparts" to you, who are relevant in this way to
what you yourself might have done, is unappealing at face value:
intuitively, as Noonan says, a claim about what you opted not to do
but certainly "might have done hardly seems, at first sight, to
be correctly interpretable as a claim about what *someone else*
(however similar...) does in another possible world"
(SS6 of the entry on
 identity:
 this is a cleaned-up version of the so-called "Humphrey
objection": Kripke 1980, p. 45n.13; 1971, p. 148; cf. Lewis'
replies to the Humphrey objection as originally cast: Lewis 1986,
p. 196).



Indeed, one might go further by way of objections to the
counterpart-construal of modal discourse, as Plantinga and Salmon do.
Plantinga argues at length that the counterpart theorist's mistake is
not really to *misinterpret* 'you might have
driven', by offering an erroneous analysis of what makes it
true; rather, the counterpart theorist's mistake is two-fold: the
first mistake is to *reject* what ordinary speakers are saying
by 'you might have driven', and the second mistake (a
deviation from prudence) is to conceal the disagreement. The
concealment is effected by assigning to the very *same*
sentence used by *ordinary* speakers, 'you might have
driven', an alien interpretation that the (serious) counterpart
theorist can endorse, rather than the interpretation that would in
fact be faithful to what the ordinary speakers are really saying. The
new counterpart-involving interpretation renders reinterpreted
sentences, no longer English, agreeable or offensive to the
metaphysical commitments of a counterpart theorist pretty much when
the respective English sentence does likewise for English speakers. In
this way, the counterpart theorist can affirm and reject the usual
modal *sentences* of English, understanding them differently
though: so the counterpart theorist maintains the deceptive appearance
of agreement with ordinary speakers by means of "a verbal
camouflage," as Salmon agrees (Salmon 1981, p. 236; cf.
Stalnaker 2003, pp. 118ff.).



For her part, the counterparts-invoking theorist might emphasize the
desirability of
 reduction.[5]
 Or she might point to other desiderada: for another likely example,
she might wish to maintain contingency where rigidity commits us to
necessity (Gibbard 1975, p. 188; Lewis 1986, p. 256; Delia Graff Fara
joins many others in counting this as "the main benefit"
of a counterpart construal: 2008, p. 186). One might turn to
counterpart theory for relief from vagueness, since whether an object
*essentially* has any given property can be a vague question;
on the other hand, so too can the "question whether an object
*actually* has a certain property,"as Kripke suggests
(1980, p.115n.57). There are other motives.



The issues at stake between serious counterpart theory and transworld
identity become complex quickly. The tangle of considerations can be
more formally motivated than are the intuitive considerations already
canvassed. Your theory might force unwelcome baroqueness in
expression, when formalized (Linsky and Zalta 1996; Williamson 2000;
2002). Worse, you might find that you cannot even say in the preferred
formal language what you want, intuitively, to say: e.g., "There
could have been things that do not actually exist" (Russell
discusses the history of this example: 2013, p. 485).



If its defenders are right, then counterpart theory can express all
that anyone wants to express--even what transworld identity
theorists want to express (Bacon forthcoming; Russell 2013). If so,
then such qualms about formal resources need not divide
serious-counterpart theorists from transworld identity theorists (and
similarly for transtime identity theorists like Quinn 1978 (Meyer
2013)). But that would not reduce any divisiveness in the foregoing
intuitive metaphysical considerations (these are the considerations
that "must always play a dominant role," too, according to
Kripkean methodology anyway: 1976, p. 411).


### 2.2 Relations Between Rigidity and Associated Theories of Reference



Rigidity is generally discussed in the literature in connection with
several theories about reference that were introduced or reintroduced
or discussed feverishly about the time Kripke called attention to
rigid designation. These theories address either the question of what
it is that designating terms say or express or contribute to the
content of propositions that they are used to articulate, or the
question how it is that designating terms come to say or express or
contribute the content they do. The relationship between rigidity and
these other theories of reference, which is often blurred in the
literature, can be brought into focus in a way that is fairly
uncontroversial, in many instances.


#### 2.2.1 Direct Reference



About the time Kripke named rigidity, Kaplan named the theory of
"direct reference" (in "Demonstratives,"
eventually published as 1989a: see 1989b, p. 571). As the theory is
usually understood, it is the position that the semantic content of a
name or other directly referring expression is nothing more than the
referent: the referent is all that the name contributes to a
proposition expressed by a sentence containing it. So there is no
descriptive information semantically conveyed by a directly referring
expression. If we think of propositions as "structured entities
looking something like the sentences which express them," as
Kaplan invites us to do, we can think of directly referring terms as
terms whose contribution to a proposition lacks the structure that
characterizes the contribution of definite descriptions. In the case
of a definite description, "the constituent of the proposition
will be some sort of complex, constructed from various attributes by
logical composition. But in the case of a singular term which is
directly referential, the constituent of the proposition is just the
object itself" (Kaplan 1989a, p. 494; see also the section
relating possible worlds and structured propositions in the entry on
 structured propositions).



Direct reference theorists (e.g., Soames 2002, pp. 240, 243), also
called "Millians" after J. S. Mill, insist that
'Hesperus = Phosphorus' and 'Hesperus =
Hesperus' express the same proposition or share the same
content. At first glance, these statements appear to say different
things, but if 'Hesperus' and 'Phosphorus'
contribute no descriptive information (like *is a morning star*
or *is an evening star*) to the proposition expressed by either
sentence, but only the referent, which is the same for each term, then
the sentences have to say the same thing despite first appearances.




It is generally acknowledged that rigidity cannot be
identified with direct reference. That is because some
expressions designate rigidly by means of describing the
designatum: e.g., 'The successor of 2', which rigidly
designates 3. Kripke calls designators like 'The successor of
2' rigid *de facto*, rather than rigid *de jure*:
the description happens to be satisfied by the same object in every
possible world and never anything else. Compare the intuitively
distinct case of *de jure* rigidity in a name, like
'Biden'. Here the intent is to refer to *this
person* in all possible worlds, whatever descriptions may
designate him. Still, not all rigid de jure expressions seem to be directly referential (as some direct reference theorists agree: e.g., Kaplan 1989b, p.
577n.25; cf. Salmon 2003, pp. 486-7; see also Stanley's 1997a
discussion: 570-1). An example of a description that is rigid de jure might be 'The famous
humanist most closely associated, in a, with the Italian
Renaissance', where 'a' is a name for the actual world (on world-indexing, see Plantinga 1974, pp. 62-65), or a similar actualized description.[6] The
designatum must first be picked out in a by whether it meets the
descriptive requirements (with or without the indexing) there in
a: but the purpose of the indexing is to keep the term
referring, with respect to other worlds, to *that same item*
picked out originally in a by virtue of meeting the right
requirements there. So the rigidity seems de jure.


 
Descriptivists claim that names are disguised descriptions: perhaps a rigidified description like the foregoing captures that content of 'Petrarch'. Direct reference theorists deny that names have descriptive content like this. Either way, the rigidity of names and even de jure rigidity of names could plausibly be honored (as opponents of direct reference hold, such as Plantinga 1985, pp. 82-7; Sidelle 1992; 1995; Justice 2003; the same goes for hybid views as well as the polar views discussed: Garcia-Carpintero 2018).


#### 2.2.2 Causal Grounding



Another theory of reference that was named about the time 'rigid
designator' was coined, and that is widely associated with rigid
designation, is the causal theory of reference. All that is relevant
here is one method of term dubbing associated with that theory.
According to a typical causal theorist, many terms are coined in a
"baptismal ceremony," during which the dubber points at an
object in her perceptual field (hence, the object's causal role), and
establishes reference by appeal to this object. The baptismal object
might become the referent, if the term's coiner says something to this
effect: "Term *t* is to be used for *that*
object." The baptismal object might also be a sample of a
substance that becomes the referent, if the term's coiner says
something to this effect: "Term *t* is to be used for the
substance *instantiated* by *that* object." Or the
object might have some other connection to the referent, as might
happen if the term's coiner says something to this effect: "Term
*t* is to be used for the ceremonial function of *that*
ancient artifact," or "Term *t* is to be used for
the leader of *that* wolf pack." Most of these examples
involve a definite description, which is used to "fix the
reference" (Kripke 1980, pp. 54-6, 135), and not as a
synonym: the term is to apply to the designatum even with respect
worlds in which it does not satisfy the description. In all of these
examples, whether the relevant term designates in a given instance
depends on the properties of the object used to ground reference in
the baptismal ceremony. The relevant properties may not be known to
the term's coiner.



Some philosophers seem to think that rigid terms are just those coined
in accordance with a baptism like the
 above.[7]
 But causal grounding is not closely tied to rigidity. Since many
philosophers deny that rigidity applies to designators for kinds (see
below), designators for other entities serve better to illustrate.
First, designators that are rigid can fail to be causally grounded.
'The successor of 2', which is rigid de facto, may be a
case in point. '3' might be an example of a rigid *de
jure* term that is not causally grounded. And there can evidently
be terms for concrete objects that are rigid but not causally
grounded. You coin 'Joy' in the following way:
"'Joy' is to be used for the most joy-filled
individual." Here you use the description to fix the reference,
not as a synonym: whatever individual is most filled with joy is the
designatum, even when we are discussing worlds in which that same
individual is glum. Reference is not secured by way of causal
grounding; you never point to anything in a baptismal
 ceremony.[8]
 Yet the designator is rigid.



So designators that are rigid might fail to be causally grounded.
Designators that are causally grounded might also fail to be rigid.
You find an old painting. After engaging in some convoluted discussion
about legal ownership, you decide to clarify your terms: "Let
the expression 'Originalowner' designate, for any possible
world *w*, the original owner in *w* of *that*
painting" (you point at the painting). You have causally
grounded 'Originalowner' by means of a baptismal ceremony;
but the referent varies from world to world, depending on who first
owned the painting. The term is not rigid.


#### 2.2.3 Descriptivism



The rigidity of names does not rule out descriptivism. As we've seen, disguised descriptions may be world-indexed or otherwise rigidified. The present section discusses strategies for maintaining descriptivism according
to which a name like 'Petrarch' has the same content as ordinary nonrigid descriptions like 'The famous
humanist most closely associated with the Italian
Renaissance'.


##### 2.2.3.1 Names as Wide-Scope Descriptions



One proposal to defend traditional, non-world-indexed descriptivism
has taken the label "widescopism." According to this
proposal, the rigidity of names is or may be treated as a matter of
scope. 'Petrarch', on this view, may be a disguised
ordinary description meaning the same as 'The famous humanist
most closely associated with the Italian Renaissance'. It is
natural to think that 'Petrarch' cannot have this semantic
content, since 'Petrarch might not have been famous' seems
unambiguously true, but 'The famous humanist most closely
associated with the Italian Renaissance might not have been
famous' has a false reading. It can have the same truth
conditions as a scoped sentence something like, 'It might have
been the case that: for some *x*, *x* is the famous
humanist most closely associated with the Italian Renaissance and
*x* has never been famous', which is false. But according
to the proposal in question, the false reading for 'The famous
humanist most closely associated with the Italian Renaissance might
not have been famous' does not show that 'Petrarch'
fails to mean the same as 'the famous humanist most closely
associated with the Italian Renaissance'. The other reading is
the relevant one. According to that reading, the name takes wide
scope, and the truth conditions are the same as those for a sentence
like, 'For some *x*, *x* is the famous humanist
most closely associated with the Italian Renaissance and it might have
been the case that: *x* has never been famous', which is
true.



Kripke (1980 pp. 11-15) points out that there are problems with
this attempt to accommodate rigidity. Some sentences have no modal
operators. When we evaluate these sentences with respect to other
worlds, we do not seem to admit that the designatum changes, as it
would if names were non-rigid descriptions taking the proper scope in
modal contexts. Thus, we can describe a possible world in which
Petrarch dies as an infant. With respect to such a world, would
'Petrarch is never famous' be true? It seems so. But on
the proposal in question, the sentence would say the same as
'The famous humanist most closely associated with the Italian
Renaissance is never famous', which is false with respect to any
world. Also, it seems that we can *make* names appear inside
the scope of a modal operator: "It might have been the case
that: Petrarch never became famous." This would apparently be
impossible to do on the view in question, but we seem to be able to do
it. On the basis of such considerations, Kripke rejects this proposal
for accommodating rigidity within descriptivism.



Kripke's arguments seem to have persuaded most of his audience
(including Salmon 1981, p. 26n.28), and his principal target, Michael
Dummett, has responded by making some modifications (1981, pp.
xvii-xix). Others have followed Dummett with more sophisticated variations (e.g., Sosa 2001; Hunter 2005; cf. Caplan 2005;
Everett 2005; cf. Soames 2005; a good overview is provided by Noonan 2013, chap
13).


##### 2.2.3.2 Assertoric Content and Ingredient Sense



Assuming that Kripke's arguments against widescopism are successful,
it might appear that rigid and nonrigid designators differ in content.
Thus, *any statement *S* containing a proper name cannot
have the same content as a statement *S*' that differs
from *S* just in the respect that a proper name in *S*
is replaced by a non-rigid designator in *S*'*.
Following Stanley (whose precise formulations differ slightly from
this and from each other: 1997a, pp. 568-9; 1997b, p. 135), we
might call this the *Rigidity Thesis*.



But Dummett has still another suggestion that, if successful, would
refute the Rigidity Thesis (or at least one understanding of it).
Consider the similarity between (1)-(4):



(1)
The famous humanist most closely associated with the Italian
Renaissance wrote the epic *Africa*
(2)
The famous humanist *now* most closely associated with the
Italian Renaissance wrote the epic *Africa*
(3)
The famous humanist *here* most closely associated with the
Italian Renaissance wrote the epic *Africa*
(4)
The famous humanist *in* a most closely associated
with the Italian Renaissance wrote the epic *Africa*.


In some intuitive sense, it might seem that these statements share the
same content. Thus, for example, what one believes when one assents to
any of (2)-(4) might seem to be what one believes when one
assents to (1). Even so, (1)-(4) contribute different semantic
values to complex sentences in which they are embedded. What one
believes when assenting to (1') is not what one believes in
assenting, say, to (2'):



(1')
*It will always be the case that* the famous
humanist most closely associated with the Italian Renaissance wrote
the epic *Africa*
(2')
*It will always be the case that* the famous
humanist *now* most closely associated with the Italian
Renaissance wrote the epic *Africa*.


(1') is dubiously true, at best. It might well be that Lorenzo
Valla will overtake Petrarch at some future time, being a famous
humanist more closely associated with the Italian Renaissance than
Petrarch. And Valla did not write the epic *Africa*. On the
other hand, (2') is true, at least assuming, for the sake of an
illustration, that (1) and (2) are true. On that assumption, it does
not matter that Valla might overtake Petrarch later: that will be too
late to affect their respective status as it is *now*.



Given that (1)-(4) make different semantic contributions to
complex sentences in which they are embedded, must we reject the
earlier suggestion that they share the same content? Here we might use
a tip from Dummett to resist an affirmative answer. We might
acknowledge that (1)-(4) share the same truth conditions, which
amount to at least one bona fide use of 'content'. Thus,
for Dummett, a speaker who could classify sufficiently detailed
possible states of
 affairs[9]
 into those that render any statement like these "correct and
those that render it incorrect, may be said to know the *assertoric
content* of the sentence" (1991, p. 48). And we might also
acknowledge a distinct phenomenon that is responsible for different
contributions (1)-(4) make to larger sentences in which they are
embedded: Dummett calls this distinct phenomenon a sentence's
"ingredient sense." "Ingredient sense is what
semantic theories are concerned to explain" (p. 48).



Armed with the distinction between assertoric content and ingredient
sense, we might reject the Rigidity Thesis. We might maintain that
(1)-(4) have the same assertoric content and hence that they say
the same thing: anyone who asserts one of these (or believes it,
rejects it, and so on), also asserts the others. Where they differ is
in ingredient sense.



There are a number of lines of thinking that converge on the
conclusion that (1)-(4) share the same assertoric content even
though they differ in ingredient sense. As already suggested, some
might find it natural to take (1)-(4) as expressing the same
"semantic content" in some meaningful sense (perhaps the
only meaningful sense): if so, Dummett's distinction indicates how one
might coherently maintain that position. Such a position might be
especially natural in view of "descriptive names," like
Evans' famous 'Julius', which is a rigid designator
for whoever invented the zipper (Evans 1979). It might seem natural to
say that 'Julius is an inventor' has the same content as
'Whoever invented the zipper is an inventor', even though
the former is contingent and the latter necessary. Evans himself
suggests that such pairs of sentences "have the same content,
despite their modal differences" (Evans 1979, p. 187n.10). 



There are other related considerations in favor of saying that
(1)-(4) share the same assertoric content even though they
differ in ingredient sense: here is a sketch of a couple of
considerations from Stanley, who develops Dummett's suggestion in
admirable detail in a series of interesting papers (1997a, 1997b,
2002). First, one might maintain that what one asserts, when one
asserts (1), is simply true or false, and not true now but false
later, true here but false in some other community where Valla is more
highly esteemed than Petrarch, true in a but false in other
possible worlds. Yet what (1) contributes to the likes of
(2)-(4) is not simply true or false: it is rather something like
a function from times, places, or worlds, respectively, to truth
values. That function generates a different truth value depending on
the argument (Stanley 1997a, p. 577; see also King 2003 SS1; for
related claims, see Lewis 1998). Hence, the line of thinking
concludes, what is asserted, when one asserts (1), is assertoric
content and not ingredient sense.



Another line of thinking by which one might argue that (1)-(4)
share the same assertoric content even though they differ in
ingredient sense starts from general considerations about what kind of
animal "content" is. For Stanley, the content of a
statement is closely tied to what it is used to communicate in normal
contexts, where "normal" has to do with the competence of
speakers, their intentions to use words as others do, and so on
(1997b, 136; 2002). In such contexts, he says, (1)-(4) are used
to communicate the same thing. So they have the same content, the same
meaning. Meaning, which facilitates communication in the right
contexts, is constituted by presuppositions on the part of speakers.
In contexts of modal evaluation, two sentences asserting the same
thing can diverge in their contribution to larger sentences or diverge
in truth value with respect to counterfactual worlds under
consideration because "meaning-constituting presuppositions are
irrelevant for modal evaluation. It is the purpose of modal evaluation
to suspend presuppositions," on this way of thinking about
content (Stanley 2002, p. 338; see also 1997b, p. 155).



If the distinction between assertoric and ingredient sense is tenable,
and if (1)-(4) share the same assertoric content even though
they differ in ingredient sense, so that the Rigidity Thesis is false,
then this opens the door to holding that proper names share the same
content as ordinary, non-indexed descriptions after all, even though
names are rigid and ordinary, non-indexed descriptions are not rigid.
In that case, something along the lines of the venerable descriptivism
associated with Frege and Russell (see the subsection on description
theories in the entry on
 reference
 and the section on descriptive theories of proper names in the entry
on
 descriptions),
 which Kripke is widely thought to have devastated, survives. One way
to develop a descriptivism along these lines would be to say that a
name like 'Petrarch' is something like a disguised
description that is shorthand for 'the famous humanist most
closely associated *in* a with the Italian
Renaissance'. Although this description for which
'Petrarch' is shorthand is world-indexed, it shares the
descriptive content of the non-indexed 'the famous humanist most
closely associated with the Italian Renaissance'. On this
suggestion, since 'Petrarch' shares the content of the
rigid description, it also shares the content of the nonrigid
description.



Kripke's reservations about descriptivism go deeper than arguments
from rigidity or indeed arguments from any considerations pertaining
to the metaphysics of modality: there are quite distinct worries about
whether speakers would have to be in possession of the relevant
descriptive information to use 'Petrarch', for example.
Could not someone refer to Petrarch by 'Petrarch' if she
supposed that Valla were the famous humanist most closely associated
with the Italian Renaissance and she associated with
'Petrarch' only the description 'a clergyman who
criticized the Avignon papacy'? If so, a standard argument runs,
the content of 'Petrarch' cannot be the same as that of
'the famous humanist most closely associated with the Italian
Renaissance'. There may be a *few* descriptive names of
the relevant variety in natural language: Dummett adduces 'St.
Joachim', which he takes to have been "introduced as
denoting the father of the Blessed Virgin, whoever that may have
been" (1991, p. 48), and Kripke adduces 'Jack the
Ripper', introduced for the murderer of so and so (1980,
79-80; for doubts, see Devitt 2020, 414-415). But it is unlikely that there are many such descriptive
proper names in natural language. Still, it may be that more
sophisticated descriptivisms appealing to more sophisticated
descriptions could overcome worries like these further,
non-metaphysical worries about typical names (see, e.g., Nelson
2002).


##### 2.2.3.3 Descriptivism, Anti-Descriptivism, and the Importance of Rigidity



None of the foregoing considerations 
 in favor of the fruitfulness of distinguishing assertoric content and
ingredient sense is beyond controversy. One might resist (critics
include Everett 2005, pp. 125-38; King 2007, pp. 168-196;
Shieh 2001, pp. 379-80; Soames 2005, pp. 321-3n.; defenders include Ninian 2012; Stalnaker
2014, pp. 23-24; Yalcin 2014). But the question of whether the
foregoing descriptivist line of thinking is sound, or whether any
descriptivist line is sound, is less important here than the question
of what *rides* on the answer for rigidity.



If the Dummettian line of thinking about assertoric and ingredient sense
 is sound, does rigidity lose its interest? It would appear that the
answer is
 No.[10]
 The fundamental work for rigidity sketched already 
 seems largely independent of these issues.



The fundamental work in question concerns the metaphysics of modality,
for the most part. But assertoric content does not have much to do
with the metaphysics of modality, so the claim that apparently rigid
designators as well as nonrigid designators have a descriptive
assertoric content seems not to threaten the extent or significance of
rigidity: "the propositional content of a sentence in a context
is not its modal content" (Stanley 2002,
 338).[11]



Some of the fundamental work concerns
*epistemology*: e.g., the issue of whether a sentence like
'Hesperus = Phosphorus' is a posteriori as well as
necessarily true. Would the above considerations in favor of distinguishing assertoric from ingredient sense
 undermine Kripke's ideas *here*, at least, even if the
considerations do not undermine Kripke's ideas about modality? That
seems unlikely. On the contrary, the claim that names, though rigid,
share the semantic content of descriptions seems likely to
*vindicate* Kripke's claims about the epistemic status of
statements like the above, if anything. That is because one could say
that 'Hesperus' and 'Phosphorus' differ in
assertoric content (in an extended sense, applied to designators
rather than statements) but they do not differ in ingredient sense or
modal content. Because they do not differ in ingredient sense or modal
content, being rigid designators for the same thing, 'Hesperus =
Phosphorus' is necessarily true. Because they *do* differ
in assertoric content, 'Hesperus = Phosphorus' is a
posteriori; 'Hesperus = Hesperus', by contrast, is a
priori since both occurrences of 'Hesperus' have one and
the same assertoric content. So descriptivism of this sophisticated
sort suggests one means to retain Kripke's epistemic claims.
Anti-descriptivism in the form of direct reference, by contrast, is
incompatible with at least one interpretation of the claim that
'Hesperus = Phosphorus' is "a posteriori",
 because on the theory of direct reference, 'Hesperus' and
'Phosphorus' just share the same content, end of story: so
'Hesperus = Phosphorus' shares the same content as
'Hesperus = Hesperus', which is a priori just in virtue of
content (again, on one interpretation of the claim that the statement
is "a priori").



Rigidity's work is compatible with descriptivism. Although some
philosophers seem to have the impression that the important work that
rigidity performs is to *refute* descriptivism, this seems to
be a mistake. A related impression is that the important work that
*Kripke* performs is to refute descriptivism: the rigidity of
terms and more just falls out of anti-descriptivism, so
anti-descriptivism is what really matters at bottom (see below,
 SS4.2).
 But there are strong reasons to resist this assessment of rigidity's
importance or lack thereof, too.



First, it might be replied that even if names are non-descriptive,
directly referring terms, it might not follow that they are rigid: a
directly referential designator can be nonrigid, at least in principle
(see 
 note 5).
 If this thought is right, then the work that rigidity performs for
names does not follow from anti-descriptivism. More is required.



Further, even if rigidity does follow from anti-descriptivism,
anti-descriptivism does not follow from rigidity: so rigidity's work
is not *limited* to that of direct reference. We have seen that
descriptive designators may be rigid: e.g., world-indexed designators.
But rigidity performs much of the same work on these as on arguably
nondescriptive expressions like names. For example, 'The evening
star in a = the morning star in a' would seem to be
a necessarily true, a posteriori statement whose necessity and
aposteriority cannot be explained in terms of the nondescriptive
nature of the designators flanking the '=' sign.



Suppose we limit our consideration to *names*. By ignoring
non-names, we can ignore *descriptive* designators that are
rigid, provided that names are nondescriptive, directly referring
designators: suppose again that they are. Still, it might be wise to
distinguish sharply the significance of rigidity from that of the
theory of direct reference. That is because rigidity's importance does
not hang on the outcome of controversies surrounding the theory of
direct reference (a controversial theory, as proponents concede:
Salmon 2003, p. 475; Soames 2005, p. 3). Where direct reference is
called into question, it should not be thought that rigidity is
thereby called into question. The fate of rigidity's significance is
distinct from the fate of direct reference's
 significance.[12]



Finally, even if direct reference is to be taken for granted and
rigidity for names does follow from direct reference, one might not
draw the conclusion that rigidity loses interest. Here is an analogy.
Few would say that if the theory of special relativity, with all of
its claims about the relativity of space and time, follows from the
theory of general relativity, then special relativity loses interest
because in that case, general relativity is what really matters:
special relativity and much more just follow from it. There is
evidently something wrong with this attempt to devalue the theory of
special relativity. It would be better to say that part of what is
interesting about general relativity concerns its implications for the
relativity of space and time, as explained by the theory of special
relativity: special relativity retains its interest as part of a
larger, more fundamental theory. In the same way, rigidity might be
said to lose no interest even if it is explained by something more
fundamental.



It might be tempting, then, to think that if descriptivism survives,
then rigidity is useless because its work, which is to refute
descriptivism, is left unperformed. It might also be tempting to think
that if descriptivism is defeated but by considerations that are more
fundamental than rigidity and that give rise to rigidity, then
rigidity is superfluous because its work is performed but by other
phenomena. The foregoing paragraphs offer reasons for thinking that
these tempting lines of thought are erroneous. If these reasons are
sound, then unless there are other, quite unheralded ties between
rigidity and descriptivism or its opposition, rigidity's interest
would seem to be largely independent of the fate of these rival
theories about semantic content.


## 3. Philosophical Work for Rigidity.



As already indicated, rigidity does interesting epistemological and
metaphysical work. Sections (3.1)-(3.5) below discuss various
areas of work.


### 3.1 The Necessary A Posteriori



Because of the rigid designation of the names it contains,
'Hesperus = Phosphorus' is necessarily true. Since
empirical work on the part of astronomers was required to determine
that the statement is true, we seem to have an illustration of
necessary, a posteriori truth. Thus, rigid designation seems to play a
crucial role in securing necessity with surprising epistemological
significance. Prior to discussion about rigid designation, the
necessary a posteriori was generally thought to be an empty category.
Rigid designation, along with other referential mechanisms discussed
in connection with it, has
changed all that, going "against the assumption of centuries of philosophers," as Linda Zagzebski relates (Zagzebski 2017, p. 213). Thus, Putnam observes that rigidity has
"startling consequences for the theory of necessary truth"
(1975, p. 232).



> 
> Since Kant there has been a big split between philosophers who thought
> that all necessary truths were analytic and philosophers who thought
> that some necessary truths were synthetic a priori. But none of these
> philosophers thought that a (metaphysically) necessary truth could
> fail to be a priori (Putnam 1975, p. 233).
> 



There is no question that a major reason for excitement about rigidity
is that it underwrites the necessary a posteriori (see also Schwartz 2002, pp.
270ff.). However, in light of the theory of direct reference, this
apparent coup may have been overstated or misunderstood. As we
observed, direct reference theorists insist that 'Hesperus =
Phosphorus' and 'Hesperus = Hesperus' express the
same proposition or share the same content even though these
statements have the appearance of saying different things. For direct
reference theorists, to say that Hesperus = Phosphorus is to say no
more and no less than to say that Hesperus = Hesperus. Since it is a
priori that Hesperus = Hesperus, it is a priori that Hesperus =
Phosphorus. For this reason, the claim that rigidity presents
propositions that are necessarily true and a posteriori might be
disputed (Soames 2002, pp. 240, 243; Salmon 1986, pp. 133-42;
Fitch 1976; for further discussion, see Fitch 2004, pp. 110-13;
Hughes 2004, pp.
 84-108).[13]



So it is controversial, though common, to say that rigidity assures
that the proposition *Hesperus = Phosphorus* is necessarily
true and a posteriori. On the other hand, it may be less problematic
or less controversial to say that rigidity assures that the
*sentence* or *statement* (interpreted sentence)
'Hesperus = Phosphorus' is necessarily true even though we
can find this out only a posteriori (Fitch 2004, pp. 110-13;
Hughes 2004, pp. 106-7; cf. Kripke 1980, pp. 20-1). If it
is tenable to say that rigidity assures that some *sentences*
or *statements* like this are necessarily true though a
posteriori, further questions immediately impose themselves. Here is a
salient question: Does the aposteriority that characterizes the
necessarily true statements retain its epistemological significance
and surprise, given that it is associated with statements rather than
propositions? The question carries some importance in view of the
popularity of Millianism; yet despite all of the debate about the
correctness of Millianism, the question has been neglected.



Here is one challenge to the proposal that the relevant statements are
necessarily true and a posteriori in any interesting respect. The
proposal apparently amounts to this: the sentence 'Hesperus =
Phosphorus' is necessarily true as it is used by English
speakers and it is an a posteriori matter to determine that it is. But
if this is all that the a posteriori necessity of 'Hesperus =
Phosphorus' amounts to, then all necessary statements are a
posteriori in the relevant way. So, the necessary a posteriori loses
its interest. To find out how English speakers use any sentence at
all, one needs an empirical look at the world. Even 'The circle
cannot be squared', which is generally regarded as a priori,
seems to come out a posteriori on the account. It is an a posteriori
matter to determine that 'The circle cannot be squared' is
necessarily true *as used by English speakers*. The reader is
invited to think over whether the intended position lacks interest in
this or other ways (there is room in the literature for
contributions: for discussion, see LaPorte 2020, p. 326). 


### 3.2 The Contingent A Priori



According to Kripke, a proper understanding of rigidity also reveals
that some contingent statements are a priori. Kripke considers the
standard meter stick, which has been used to define the length of one
meter. Someone decreed that the expression 'one meter' is
to be used for the unit of length of that stick, perhaps at a certain
time in the stick's existence. The stick, *S*, might not have
been the length that it was at the time 'one meter' was
coined: had it been heated or cooled it might have been longer or
shorter. As things are, it was not heated or cooled. So the sentence
'*One meter* = the length of *S* (at time
*t*0)' is true in the actual world but it is
false in some possible worlds: it is contingently true. Yet *S*
has been used to *define* 'one meter', in the sense
that it fixes the reference (see above), so the definers of 'one
meter' know a priori the truth of the sentence 'The length
*one meter* is instantiated by *S* (at time
*t*0)'. The reason that the sentence is
contingently true is that 'one meter' is a rigid
designator for the length one meter. 'The length of *S*
(at time *t*0)' is, on the other hand, a
non-rigid designator for one meter; in this world it designates one
meter but in other worlds, those in which *S* is heated or
cooled, 'the length of *S* (at time
*t*0)' designates other lengths.



The contingent a priori has not been as widely discussed as the
necessary a posteriori. The introduction of the contingent a priori is
perhaps less surprising than the introduction of the necessary a
posteriori, in view of the plausible contingency of a priori
statements that philosophers all along recognized, like the plausible
contingency of the statement, as uttered in a context, 'I
exist'. Further, Kripke's examples of the contingent a priori
are perhaps more controversial than his examples of the necessary a
posteriori. Many dispute that the examples really are examples of the
contingent a priori (Donnellan 1977; Hughes 2004, pp. 84-107;
Plantinga 1974, pp. 8-9n.; Salmon 1987-1988; cf. Kripke 2019, Other Internet Resources for a defense; Oppy 1994; Salmon 2020 distinguishes an interesting notion related to the contingent a priori that Kripke has allegedly identified). And even if they are contingent and a priori, many feel they
lack much significance. They are not "scary," as Donnellan
puts it (also Fitch 2004, p. 121). On the other hand, confusions are
like loose threads: they sometimes work their way to apparently remote
material. Confusion about the contingent a priori can and arguably
does lead to confusion and apparent paradox in unsuspected areas of
philosophy (an alleged example is discussed in LaPorte 2003). Kripke
(1980, pp. 14-15) recalls that considerations about the
contingent a priori led in his mind to a number of clarifications
concerning designation, which again is hardly surprising given the
interconnectedness of different issues in a fabric.


### 3.3 Theoretical Identities



The statements of necessity associated with rigid designation are
interesting from an epistemological standpoint, as just indicated.
They are also interesting from a metaphysical standpoint.



Rigid designation seems needed on the part of both designators if
there are to be true "theoretical identity statements":
statements in which a designator designates by way of expressing
explicitly in some manner one of the designatum's theoretically
interesting essences. The argument is not often put explicitly. It is
helpful to outline explicitly the line of reasoning in order to bring
out more clearly what stands or falls with rigidity.



Not all identity statements widely accepted as necessary specify a
theoretically interesting essence in the relevant respect.
'Hesperus = Phosphorus' or 'Cicero = Tully' do
not; each sentence contains only names for an object. But take, on the
other hand, 'Cicero = the organism descended from sperm
*s* and egg *e*', where '*s*'
and '*e*' are names for the right sperm and egg.
Many regard this sentence as necessarily true and revealing of a
theoretical essence of Cicero: roughly, a theoretically interesting
property possessed by Cicero in all possible worlds and never
possessed by anyone else (bear in mind, for this example, limited demands of weak necessity).
 In order for any identity statement to be true of necessity, both
designators must be rigid if either is. And if 'the organism
descended from sperm *s* and egg *e*' indeed
expresses an essence of Cicero, it must designate just Cicero in all
possible worlds, and hence rigidly: so, given that the statement at
issue is necessarily true, 'Cicero' must be rigid, too.
Alternatively, if 'Cicero' is rigid, as is widely
acknowledged, then in order for the statement at issue to be
necessarily true, 'the organism descended from sperm *s*
and egg *e*' must be rigid and so express an essence of
Cicero.



The foregoing paragraphs address a line of reasoning according to
which rigid designation is needed for certain statements to come out
necessarily true. Usually, rigidity is discussed with a view not to
its being *necessary* for the completion of a job but rather
with a view to its being *sufficient* for the completion of a
job. The rigid designation of both designators flanking an identity
sign in a true theoretical identity statement is sufficient to
establish the necessity of the statement. If we acknowledge that
'the organism descended from sperm *s* and egg
*e*' is rigid, and we acknowledge that
'Cicero' is rigid, then we are committed to the necessity,
in case of truth, of 'Cicero = the organism descended from sperm
*s* and egg *e*'. The reasoning is parallel to
that given for the case of 'Hesperus = Phosphorus'
above.



'Cicero = the organism descended from sperm *s* and egg
*e*' shows that statements about individuals can, in
effect, be theoretical identity statements. The most commonly
discussed theoretical identity statements concern kinds. They include
'Gold = the element with atomic number 79' and
'Water = H2O' (Kripke 1980, pp. 138-40,
148). Similar reasoning might apply. Here the reasoning is again
nearly always left implicit; one must fill in gaps to present a
general line of thought that seems to accord with a widely held, if
not widely articulated, tradition. For that tradition, in order for it
to be the case that 'Water = H2O' is
necessarily true, both designators must be rigid if either is. So if
the statement is necessarily true and either 'water' is
rigid by virtue of being a name for a substance or else
'H2O' is rigid by virtue of expressing an
essence, *both* designators need to be rigid. And again, the
rigidity of designators in a true theoretical identity statement is
*sufficient* for the statement's necessity. If
'H2O' is a rigid designator of a substance, and
'water' is a rigid designator of a substance, and
'Water = H2O' is true, then that statement is
necessarily true.



Theoretical identity statements are of interest across the gamut of
philosophy. Various writers appeal to rigid designation to support
views about the essences not only of kinds from chemistry or biology
but also kinds from philosophical subdisciplines apparently far
removed from empirical science: ethics and aesthetics, for example
(see, e.g., Carney 1982; Funkhouser 2014, pp. 4, 164; Gampel 1997;
Leddy 1987; Noonan 2020). The reader interested in pursuing any of these traditions
should beware that attempts to apply rigidity in the foregoing
subdisciplines, especially early attempts, sometimes confuse rigidity
with other phenomena recognized by popular theories of reference: see
clarifications in section 2.2. 



By far, the most attention concerning theoretical identity statements
has focused on the area of philosophy of mind. That is next on the
agenda.


### 3.4 Mind



Kripke appeals to rigid designation in arguing against the
 identity theory of mind,
 which is really a constellation of theories. A broad division between
identity theories can be drawn between type identity theories and
token identity theories: here is a standard Kripkean argument against
token identity theories, which are often thought to be more compelling
than type identity theories. Kripke is typically understood to put
forward something like this argument. 



> 
> *The Argument Against Token Identity*: Suppose that every
> particular mental event is identical to some particular brain event.
> Then we can take some pain and name it '*P*', and
> we can do the same for the corresponding brain event, calling it
> '*C*' (for "c-fiber firing," assuming
> for the sake of argument that that is the corresponding brain event).
> According to the identity theory, *P* = *C*. But in
> reality, *P* [?] *C*. *C* could have existed
> without *P*: that very c-fiber firing could have obtained even
> while there was no corresponding *P*. Your c-fibers could have
> been blasting away even as you felt nothing at all. In the same way,
> *P* could have existed without *C*. You could have felt
> that same dull, throbbing sensation in your forehead without the
> corresponding c-fibers having ever acted up. Now, merely showing that
> *C* could have existed without *P* and that *P*
> could have existed without *C* does not yet belie
> '*P* = *C*'. For this statement might be
> true necessarily or contingently. Since *P* and *C*
> could each exist without the other, the statement cannot be
> necessarily true. But there is still the possibility that the
> statement is contingently true. There are contingently true
> statements, as we have seen: e.g., 'Hesperus = the brightest
> non-lunar object in the evening sky'. However, it cannot be that
> '*P* = *C*' is contingently true, as
> 'Hesperus = the brightest non-lunar object in the evening
> sky' is. 'Hesperus = the brightest non-lunar object in the
> evening sky' is contingently true because 'Hesperus'
> is a rigid designator and 'the brightest non-lunar object in the
> evening sky' is non-rigid. Because 'Hesperus' and
> 'the brightest non-lunar object in the evening sky'
> designate the same object at the actual world, 'Hesperus = the
> brightest non-lunar object in the evening sky' is true; because
> 'Hesperus' and 'the brightest non-lunar object in
> the evening sky' designate different objects at some other
> worlds, 'Hesperus = the brightest non-lunar object in the
> evening sky' is false at those other worlds. That is why
> 'Hesperus = the brightest non-lunar object in the evening
> sky' is true but contingently so. '*P* =
> *C*', on the other hand, contains two rigid designators.
> '*P*' is a name for that very pain, and so rigidly
> designates it. '*C*' is a name for that very brain
> event, and so rigidly designates it. Because '*P*'
> and '*C*' are both rigid designators,
> '*P* = *C*' has to be true of necessity if
> it is true at all. Since it is not true of necessity, it is not true
> at all.
> 



Some premises in this bare-bones argument are in obvious need of
support, like the modal intuition that *P* and *C* could
have existed each in the absence of the other. Support has been
provided and rejected by different writers. Many philosophers who
reject anti-materialist arguments along these lines nevertheless make
use of rigid designation as a resource in order to frame
*materialist* views that recognize a parallel between
'*P* = *C*' and 'Water =
H2O'.



Kripke's views concerning the philosophy of mind could hardly be
called popular. But, as the introduction to an edited volume covering
the topic says, "Even authors who disagree with Kripke's
fundamental picture tend to present their arguments against an
implicitly or explicitly Kripkean backdrop" (Gendler and Hawthorne 2002, p. 26).


## 4. Objections to Rigidity



Various objections to the coherence or usefulness of rigidity have
been put forward by specialists in the philosophy of language, though
the objections do not seem to have done much to damage the widespread
appeal of rigidity, which is commonly taken for granted in mainstream
literature from various areas of philosophy. (Whether your typical
philosopher is *entitled* to embrace rigidity is another
matter: so there is a case for saying, on grounds independent from the
foregoing objections, that rigidity and the necessity accompanying it
stand or fall with a robust version of the analytic-synthetic
distinction (see 
 note 4).
 If that case is solid, then either popular Quinean antagonism toward
analyticity should prompt the rejection of rigidity or else the appeal
of rigidity should prompt the rejection of the familiar antagonism
toward analyticity.) What resistance to rigidity there is may be
addressed in general terms first, in order to save for its own
subsection
 (SS4.2)
 treatment of objections specifically directed at the application of
rigidity to terms for properties, which are especially contended.


### 4.1 Objections to Rigidity in General



There are arguments against the coherence or applicability of
rigidity for concrete-object designators (for
discussion, see Fitch 2004, pp. 102-10), but here we find relative
calm. As Hughes says, the position "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" (2004, p. vii). 



Still, in saying that rigidity is widely favored, we should be
prepared to qualify in view of those I've called "serious
counterpart theorists," who must adopt a revisionist
understanding of what it is for a designator to designate
"rigidly" (or alternatively, they can just frame their
position as a rejection of rigidity: e.g., Varzi 2003, already
quoted). Usually, counterpart theorists are motivated by objections to
some of the necessity that rigidity secures, according to the usual
understanding. It appears to some (e.g., Gibbard 1975) that not all
identity statements containing just names are necessarily true.
Consider this scenario: you mold one chunk of clay into the top half
of a statue, and another chunk of clay into the bottom half. You stick
the halves together, thus bringing into existence both a statue, call
it 'David', and a large lump of clay, call it
'Lumpl'. Suppose, next, that you explode the statue: both
David and Lumpl go out of existence. It is tempting to say that
'David = Lumpl' is true, but not necessarily true. It
could have been that: David exists and Lumpl exists but 'David
[?] Lumpl' is true. This would have been the case had Lumpl
survived David: e.g., had you squeezed the clay statue into a ball,
thereby changing the shape of the relevant lump of clay, instead of
exploding it, thereby bringing an end to the lump of clay.



If both 'David' and 'Lumpl' are *rigid
designators*, then 'David = Lumpl' is necessarily true
if it is true in any possible world at all. So, if it is even possible
that David [?] Lumpl, then David [?] Lumpl. David and Lumpl are
distinct, however intimately related, because they have different
modal properties: Lumpl could instantiate ballhood, while David could
not. The rigidity of 'David' and 'Lumpl'
evidently requires as much. In order to say that 'David =
Lumpl' is contingently true, Gibbard gives up the thesis that
'David' and 'Lumpl' are rigid designators in
the usual respect. He maintains that designators are
"rigid," very loosely speaking, only with respect to a
sortal: 'David' is statue-rigid, always referring to the
same statue, while 'Lumpl' is lump-rigid, always referring
to the same lump. Accordingly, 'Possibly, David is no
statue' is false, while 'Possibly, Lumpl is no
statue' is true. What about the *object* that is, as a
matter of fact, both a statue and a lump? For Gibbard, objects have no
modal properties and it makes no sense to say that an object in some
other possible world is identical to an object in the actual world.
This is counterintuitive, but again Gibbard does offer a non-standard
semantics to accommodate modal talk.



David Lewis (1986, chap 4) offers a similar strategy for maintaining
that 'Possibly, David is no statue' is false, while
'Possibly, Lumpl is no statue' is true. For Lewis, what
makes a sentence like the above true or false is whether counterparts
in other possible worlds associated with 'David' or
'Lumpl' are all statues. The use, in discourse, of
different names like 'David' and 'Lumpl' can
create different contexts, thus altering the counterparts that come
into play in determining the truth of sentences containing the names.
The sentences 'Possibly, David is no statue' and
'Possibly, Lumpl is no statue' can vary in truth value
because the counterparts relevant to the truth value of the former
sentence are all statues but the counterparts relevant to the truth
value of the latter sentence are not.



Orthodox rigidity proponents who distinguish between the referents of
'David' and 'Lumpl' say that there is more
than one object at the same place and time where the statue stands.
The intimate relationship between David and Lumpl is usually
understood as one of *constitution*: David is constituted by
Lumpl. Constitution has been championed by various philosophers to
address a variety of related issues (see, e.g., Baker 2000 on personal
identity).



Another possibility, which avoids commitment to constitution, is to
maintain an eliminativism about statues or clay lumps. One might
maintain that there is no real entity to name "David," in
the way that the story above requires; there is only Lumpl in a
statuesque form. Accordingly, the problem of accounting for the modal
status of a sentence to which 'David' contributes, like
'David = Lumpl', never arises. (For an appeal to
eliminativism to handle various candidates for constitution, see Fitch
2001, pp. 382-3, 391n.5; Hershenov 2005; Merricks 2001.) In
order to work as a general alternative approach, eliminativism would
have to work for all of a wide variety of would-be cases of
constitution.



Notice that the unintuitive idea that more than one object can exist
in the same place at the same time is not necessarily removed if we
accept contingent identity. Gibbard, for example, acknowledges that
Lumpl and David are distinct objects in any world *w* in which
David but not Lumpl comes to an end by being squeezed into a ball; but
despite the failure of identity to hold between David and Lumpl in
*w*, the two share one spatiotemporal location *before*
David is destroyed because the material of the clay lump that is Lumpl
is precisely the material of David.



A good collection of papers on this topic is Rea (1997), and a
good introduction is his "Introduction."


### 4.2 Objections to the Application of Rigidity to Terms for Kinds and Properties



Both Kripke (1980) and Putnam (1975) famously extend the notion of
rigidity to terms for *natural kinds*. Here controversy has
ensued. Let us begin with a singular term as an example; general terms
introduce additional complications (for discussion, see Linsky 1984;
Soames 2002; Salmon 2003; 2005).



What is rigidly designated by a kind term like '*Apis
mellifera*'? Apparently it can not rigidly designate a
scattered object comprised of all particular honeybees, say, since
those honeybees that presently go about their work might not have
existed. And others that do not exist might have existed instead (but
controversy and complications attend this observation: see, e.g.,
Gomez-Torrente 2006; Zouhar 2009, p. 211). To be rigid, a term
has to designate the same thing in all possible worlds.



Some philosophers have proposed that '*Apis
mellifera*' rigidly designates an abstract object, a kind or
property. One might say that '*Apis mellifera*'
designates the honeybee *kind* in every possible world: it
never designates any other kind, like the bumblebee kind, say. That
might be said to make the term rigid. Something similar could be said
for 'water', 'heat', and
'redness'. But a number of objections have been
issued.



One broad sort of objection, if it should be called that, is that
rigidity does, in some respect, the work attributed to it; but
rigidity is superfluous to the task of explaining that work because
the work is the result of other, more fundamental phenomena (for
different approaches along these lines, see Jylkka 2008, pp.
69-72; Schwartz 2002; Soames 2002, e.g., pp. 310-11). For all this objection tells, rigidity might remain interesting as a special case of more fundamental phenomena. More damaging for property rigidity would be its rejection in favor of competing mechanisms alleged to do comparable work (Haukioja 2012; Nimtz 2019).



An objection more specifically directed toward efforts at extending
rigidity to expressions for just kinds and properties, is that such
efforts would also have to extend the status of rigidity to terms for
*artificial* kinds or properties (see, e.g., Gluer and Pagin
2011; Haukioja 2012; Nimtz 2019; Schwartz 2002; 2018; 2020; whether this vast application of rigidity should be thought problematic might be questioned: see e.g., Salmon 2005). The singular terms 'soda
pop', 'bachelorhood', 'livestock' and so
on, would also seem to designate an abstract object rigidly (*soda
pop, bachelorhood, livestock*, respectively), if terms for natural
kinds and natural properties do so. So the foregoing proposal for the
rigidity of kind and property terms might be thought to be too
indiscriminate in what it allows to be rigid--maybe even so
indiscriminate that it mistakes sameness of *meaning* across
possible worlds, which *every* meaningful expression enjoys, for
sameness of *designation*, which only rigid designators are
supposed to have (Schwartz 2002, p. 272; 2020, p. 263n.). 



The basic objection here to our counting certain *singular* terms
to be rigid designators for properties might also tell against
counting certain *general* terms to be rigid designators. The
thesis under fire is that according to which certain general terms
rigidly designate properties, by virtue of designating the right
property in every possible world. So according to this thesis,
'watery', like its singular variant 'water',
is rigid by virtue of designating water with respect to every possible
world: and something similar can be said for 'honeybee',
'hot', and 'green'. Again, the relevant
objection would be that if the foregoing general terms were rigid for
that reason, then something similar could be said for general terms
that cut more artificially, like 'bachelor'; and that is
supposed to be a problem. 



A second group of objections to extending rigid status to expressions
for properties specifically (just tacitly include kinds with
properties--kinds are often counted as a sort of property)
centers around the idea that the proposal seems committed to
substantive metaphysical positions about abstract objects like kinds
or properties, concerning which a theory of language ought to remain
neutral (Devitt 2005; Haukioja 2006; Marti 1998, p. 487). For
example, the proposal seems committed to rejecting nominalism. Some
defenders of property terms' rigidity are happy so to commit (Linsky
1984). Other defenders of property terms' rigidity seek ways of
accommodating nominalism (Marti 1998). 



Even if we are happy to commit to properties, problems remain. Just as
a paucity of properties can create problems (nominalism), so can a
plenum of properties. We've already seen that expressions for
artificial kinds and properties like 'livestock' might
create troubles. There's a different sort of problem raised by
still-more artificial properties: a worry arises that allegedly
non-rigid designators for properties always end up *rigidly*
designating some gruesome relational property or other in the plenum.
Consider 'the biological species typically farmed for
honey', which we might initially be inclined to take for a
nonrigid designator of Apis mellifera. "*In virtue of
what*," as Devitt would ask (2005, p. 154; see also 2009, p.
240), does such an expression "nonrigidly designate that,"
i.e., Apis mellifera, rather than rigidly designate the property
*being the biological species typically farmed for honey*? This
problem has been much discussed. Whether indeed there is a problem
here might depend on whether expressions for properties are singular
terms or whether they are predicative. Arguably the problem is more
tractable for singular terms than for thoroughly predicative terms,
which seem especially unaccommodating. 



For related reasons, most workers reject any rigid-nonrigid
distinction for purely predicative expressions. That would include
'green' and 'honeybee', on a
nominalist's approach according to which there is no property to
designate and on some realist approaches, too, such as a Fregean
approach according to which properties exist but cannot be named.
Other still-more widely accepted candidates for purely predicative
status would be the predicates 'is green', and 'is a
honeybee'. 



'Green' and 'honeybee' are weaker candidates
for purely predicative status than are predicates containing them,
because according to a now-popular view, 'green' or
'honeybee' is "singular" in a *broad*
sense. That popular view recognizes a "'singular
occurrence' of a general term" like 'green'
(Inan 2008, p.227), so that those holding the view do not count
'green' to be "purely predicative" in the
strict respect of the foregoing paragraph, even though they might for
all that count the containing expression '... is green' as
purely predicative (e.g., Soames counts
'green'/'... is green' as singular/predicative
respectively), and such theorists might accordingly apply the
rigid-nonrigid distinction to the singular 'green'
but not to '... is green' (Soames 2010, pp. 90-91; 2006, p.
712; see also Robertson 2009, p. 136n.23). 



How could 'green', a general term, have singular status
too? According to the most commonly cited (but not the only)
suggestion along these lines, although general terms are not
*first*-order singular terms--on the contrary, they're
general terms precisely by virtue of being *first*-order
general terms--they are *higher*-order singular terms
(Linsky 1984; Salmon 2003; 2005). 



A few workers maintain both that there are purely predicative general
terms and that some of them, like 'honeybee', are rigid.
According to a now-familiar suggestion along these lines, what makes
the application of a general term F rigid is that any item *x* to
which the term applies, in any possible world, is part of the
extension of F in *all* worlds in which *x* exists, *x*
being an *F* essentially (Cook 1980; Devitt 2005; Devitt and
Sterelny 1999, pp. 85-6; Gomez-Torrente 2006). This
notion of rigid application might hold some attraction for someone
bothered by the objections just mentioned against the view that terms
rigidly designate properties. It would appear that terms for natural
kinds but not artificial ones come out rigid: thus,
'honeybee' is rigid because any particular honeybee is
essentially a honeybee and could not have been something else like a
dragonfly or a fir tree instead. But 'livestock' does not
apply rigidly according to this account because animals that are
livestock are not essentially so. Neither does '(is a member of
the-/the-) species typically farmed for honey' apply rigidly
according to this account. Furthermore, the account appears to be open
to nominalism (about properties): nominalists recognize the legitimate
*application* of general terms. 



However, even if rigid application evades the above difficulties, it
has its limitations. One commonly noted problem is this: consider
identity statements like 'Water = H2O', or 'Red
light = light with wavelength 680 nm'. Rigidity is apparently
supposed to guarantee the necessity, in case of truth, of identity
statements like these. Typically, fans of rigid application rewrite
identity statements to have this form: 'Anything is (an)
*F* if and only if it is (an) *F*''. But many
philosophers, including proponents of rigid application, have pointed
out that it could be the case for two general terms F and F'
that, (i) necessarily, every *F* is essentially *F* and
every *F*' is essentially *F*', so that F and
F' are rigid appliers and (ii) in the actual world,
'Anything is F if and only if it is F'' is true,
even though (iii) God might have created an *F* that is not an
*F*'. In that event, 'Anything is *F* if and
only if it is *F*'' comes out true with respect to
the actual world but not necessarily true (for an attempt to get
around this difficulty, see Gomez-Torrente 2006).



Apart from the foregoing minority suggestion of tying essentialism to
rigidity, there has not been much done by way of articulating any
purely predicative version of rigidity; again, prospects for any such
account are generally thought to be grim. There is however at least
one rival account of predicative rigidity to compete with the
foregoing essentialism-centered account of predicative rigidity. The
proposal follows a lead from Fregeans who recognize "definite
ascriptions" (Heintz 1973) for properties corresponding to
definite descriptions for entities they are willing to name. Thus,
Geach indicates that he would recognize a proper answer to the
question "what is the color of Antarctica?" to be
"clouds, snow, etc.," or that an English statement like
the following is unobjectionable: "Honeybees are *the species
typically farmed for honey*." In sentences like the
foregoing, seemingly singular talk about an abstract entity, such as a
"color" or a "species," would have to be
interpreted as talk about *concrete objects* instead: clouds,
bees, etc. (see, e.g., Anscombe and Geach 1961, pp. 155-6). Let
us suppose, for the sake of argument, that expressions like
'white' and 'the color of Antarctica' are
purely predicative, either because there are no properties to
designate or because they cannot be designated by a singular
expression: still, the *contrast* between a typical one-word
predicate like 'white' and the foregoing definite
ascription remains of interest for its bearing on the modal and
epistemological status of biconditionals, in the ways relevant to
rigidity, according to this line of thinking: compare 'Something
is the color of Antarctica if and only if it is white' with
'Something is albescent if and only if it is white'
(understand 'albescent' as a synonym for
'white'). That contrast suggests that we might articulate
an account of rigidity that does not presuppose singular designation
of any sort (Haze 2021; LaPorte 2013, chapter 5).