Predicates Rigidly Signifying the “Unnatural”*
Dan López de Sa
Universitat de Barcelona / Université de Genève
dlopez@trivium.gh.ub.es

Preliminary draft – Comments very welcome

I
Philosophers, with few exceptions, think they have a reasonably clear idea of what it is for indexicals and demonstratives, proper names and singular definite descriptions to be rigid. Now in Naming and Necessity (‘N&N’ hereafter), Kripke (1980) nonetheless seems to intend the notion to apply to expressions of different syntactic and semantic categories, and in particular to apply, “suitably elaborated”, to predicates (N&N, 134). And people quite often talk about predicates being rigid and non-rigid (or flexible) in the context of discussion in philosophy of mind, philosophy of science or metaehtics. I think the “suitable elaboration” that underlies this common talk can be captured by the following simple and straightforward proposal:
(*)	A predicate is rigid if and only if the property it contributes to the truth-conditions of the (simple) sentences containing it with respect to counterfactual worlds is the same as the one it makes to them with respect to the actual world; in short: a predicate is rigid if and only if it signifies the same property with respect to all possible worlds.
To illustrate: it seems that ‘is (a drop of) water’ is intuitively rigid whereas ‘has the color Sònia likes best’ is intuitively non-rigid. Does (*) have these desired results? It seems that it does. The property in virtue of which sentences containing ‘is water’ are to be evaluated as true or false with respect to any counterfactual world is, intuitively, the same as the property in virtue of which they are to be so evaluated with respect to the actual world. By contrast, consider a world in which this book remains the color it is but Sònia’s taste for colors changes, due to whatever you want to consider, so that it is no longer true that blue is the color Sònia likes best, but rather yellow is. Now it seems that in order to evaluate the sentence ‘this book has the color Sònia likes best’ with respect to that counterfactual world, the relevant entity is no longer the property of being blue but rather precisely the property of being yellow. So it seems that the property signified by ‘has the color Sònia likes best’ with respect to that world is a different one from that it signifies with respect to the actual world, namely, the property of being yellow rather than the property of being blue. Thus, it seems (*) has the intuitively right result of counting this predicate as non-rigid.

II
Proposals along this line are indeed considered and rejected in the recent literature that focuses on the topic of the notion of rigidity for predicates. A defense of such a proposal would then need to address at least the following kinds of concern. First, something like (*) assumes that the relation of signification between linguistic expressions and entities can be coherently applied to predicates and properties. There are several general doubts that can be raised against this assumption, see inter alia Wright 1998. Important as I think this issue is, I will take in this paper this assumption for granted. Second, some have argued that, on a liberal understanding of what properties are – like Lewisian abundant functions from worlds to extensions – every predicate turns out to be (trivially) rigid according to (*), see inter alia Soames 2002 and Schwartz 2002. For take any (intuitively non-rigid) predicate P. The defender of (*) thinks she can account for the intuitions regarding the non-rigidity of P by being possibly the case that it signifies with respect to the different worlds w’, w’’, ... the different properties Pw’, Pw’’, ... But consider the property P which is had by something in a world w if and only if it has Pw in w. Now, (vagueness aside) any (simple) predication of P would be true with respect to a world iff the relevant object has the property P in that world, so that the supposition that the predicate signifies this property with respect to all worlds seems to yield the right consequences with respect to the truth-values of the relevant (simple) sentences containing the predicate with respect to both actual and counterfactual worlds. But if that was the case, the proposal of (*) would make the predicate a rigid one. This “generalization” problem could be then stated thus: How could it be defended that the predicate P signifies with respect to the different worlds w’, w’’, ... different properties Pw’, Pw’’, ... instead of signifying with respect to all of them the very same property P? I have defended elsewhere that this question can be answered. First, one could reject that ‘properties’ in (*) should be understood in such a liberal sense, so that the claim that in general something like P is a property in the relevant sense is blocked. But furthermore, even granting that liberal reading of ‘property,’ one can appeal to intuitive (contingent) truth of certain identity statement involving, indirectly, the relevant predicate, via its “canonical nominalization” (the expression that results from the predicate by replacing the first verb it contains by its gerund form), which signify the same entity than the corresponding predicate.
	Take for instance the intuitively non-rigid predicate ‘has the color Sònia likes best,’ and consider the following:
	Being blue is having the color Sonia likes best.
I take it that, were we to evaluate this sentence with respect to the actual world, we would intuitively say that it should come out as actually true (even if only contingently so, given that Sonia may prefer yellow. But for that sentence to be true with respect to the actual world, both nominalizations should signify the same property (in the liberal sense) (with respect to the actual world). But ‘being blue’ does not seem to signify (with respect to the actual world) the property (in the liberal sense) of having-the-color-Sònia-likes-best. Therefore, we have a reason to hold that ‘having the color Sònia likes best’ does not signify (with respect to the actual world) it either. So it seems that a proposal like (*) might meet the generalization problem, even without challenging the assumption on the liberal sense of ‘property.’

III
Now, even granting that the notion of signification can be coherently applied to predicates and properties, and that (*) does not have as a consequence that every predicate turns out (trivially) to be rigid, there is third kind of concern that defenders of (*) should face – and this is the focus of the present paper. This can be put in the following terms: rigid predicates should turn out to be exactly those predicates signifying natural properties, properties whose essences do not involve facts about our responses in front of instantiations thereof. This is, I take it, the ultimate content of the recent reply of Schwartz (2002), which would be my main target here, to Laporte (2000), who defends something along the lines of (*).
Clearly there is an important difference between natural kind terms like ‘gold’ and nominal kind terms like ‘bachelor’ – and isn’t this difference based on the rigidity of the one and non-rigidity of the other? (Schwartz 2002, 266)
I won’t dispute that there is such an important difference, of course, but I don’t see any reason for the claim that this is based on the rigidity of the one and non-rigidity of the other. One nice way of illustrating the point is by means of colors. There is a long and deep dispute as to whether colors are primary, fully objective properties or rather secondary, dispositional or response-dependent ones. Take the predicate ‘is red.’ Most people would agree that there should be some descriptive material conventionally associated with it that plays at least a reference-fixing role, being responsible of the apriori truth of some biconditional along the lines of:
x is red iff x is disposed to produce experiences as of red in normal color perceivers under normal illuminating conditions.
So actual normal color perceivers have a general disposition to respond in certain ways to instances of a redness under certain conditions.  There should be a deep explanation of this general disposition, which would involve a real property common to all relevant instances. Now the debate previously alluded to concerns the nature of this property. But almost all participants in this debate, on both sides, tend to agree in that, whatever the nature of it turns out to be, ‘is red’ rigidly signify this property, contrary to what intuitively seems to happen with ‘has the colors Sònia likes best,’ so that the previous biconditional is contingent. Here is Crispin Wright’s:
We are … obliged to pay attention to the following intuition about colour: that had the typical visual equipment of human beings been very different, or had the lighting (by day) on the earth typically been of a quite different character — perhaps resembling the illumination generated by sodium street lighting — that need have made no difference to the colours things actually are. The extensions of ‘red’ and ‘green’ would not have been different if all human beings had been colour blind, and would not change if they were to become so” (Wright 1992, 113)
So it seems that ‘is red’ could be rigid, even on the assumption that it is a secondary property. But if it is possible to have a predicate rigidly signifying a secondary quality, why couldn’t one rigidly signify an artifact property, like ‘is a knife,’ or a social property, like ‘is a bachelor,’ or an evaluative property, like ‘is funny’? The claim that rigid predicates should coincide with natural kind predicates, by itself, seems completely ungrounded.

IV
If I understand him right, Schwartz’s tries to ground this claim by appealing to a desideratum that a notion of rigidity for predicates should satisfy: namely, that it plays an analogous role to explaining why certain statements are necessary if true than the one played by rigidity in the case of singular terms. Now, on the face of it, the rigidity of the predicates, in the sense of (*), seems to play an analogous role vis-à-vis the necessary if true character of identity statements involving the nominalization of those predicates. Take:
(1)	Being water is being made of H2O-molecules
(2)	Being water is being the substance instances of which fall from the sky in rain and fill the lakes and rivers.
Arguably the predicates ‘is water’ and ‘is made of H2O-molecules’, but not ‘is the substance instances of which fall from the sky in rain and fill the lakes and rivers,’ are rigid according to (*) as intuitively required. The question was whether this fact plays a role in accounting how it is that (1), but not (2), is necessary if true, analogous to the one played by the fact that ‘Hesperus’ and ‘Phosphorus’, but not ‘the morning star’ are rigid with respect to the corresponding :
(1’)	Hesperus is Phosphorus
(2’)	Hesperus is the morning star
Remember that the latter could be, for present purposes, summarized as follows: Suppose ‘Hesperus is Phosphorus’ is actually true and suppose ‘Hesperus’ and ‘Phosphorus’ are rigid singular terms. Now consider an arbitrary world with respect to which ‘Hesperus’ signifies an object. By the rigidity of ‘Hesperus’ this object is the same as the one ‘Hesperus’ signifies with respect the actual world, which, by the actual truth of the identity statement, is the object that ‘Phosphorus’ signifies with respect to the actual world, and which, (#’) by the rigidity of ‘Phosphorus’, is the object that ‘Phosphorus’ signifies with respect the considered possible world. Hence ‘Hesperus is Phosphorus’ is true with respect to this arbitrary possible world, and hence ‘(if Hesperus exists) Hesperus is Phosphorus’ is necessarily true. Nothing similar could be argued in the case of (2’), as the essential step (#’) illustrates.
	But now essentially the same argument (and hence the same role for the assumptions concerning rigidity) applies to (1) as opposed to (2). Suppose ‘being water is being made of H2O-molecules’ is actually true and suppose ‘is water’ and ‘is made of H2O-molecules’ are rigid predicates (in the sense of (*)). Now consider an arbitrary world with respect to which ‘being water’ signifies a property instantiated in it. By the rigidity of ‘being water’ this property is the same as the one ‘being water’ signifies with respect to the actual world, which, by the actual truth of the identity statement, is the property that ‘being made of H2O-molecules’ signifies with respect to the actual world, and which, (#’) by the rigidity of ‘being made of H2O-molecules,’ is the same property as ‘being made of H2O-molecules’ signifies with respect to the possible world considered. Hence ‘being water is being made of H2O-molecules’ is true with respect to this arbitrary possible world, and hence ‘(if there is water,) being water is being made of H2O-molecules’ is necessarily true. Nothing similar could be argued in the case of (2’), as the essential step (#’) illustrates.

V
I don’t think Schwartz would disagree so far. But he points to the fact that a proposal like (*) has as a consequence not only natural kind predicates like ‘is water’ and ‘is made of H2O-molecules’ are rigid but also (arguably) ‘is a knife’, ‘is a bachelor’ or ‘is funny.’ It is the rigidity of these predicates which would not support the required explanatoriness and would thereby challenge (*):
[W]e do not need the semantics of rigidity to explain the necessity of ‘soda = pop’ or ‘bachelors are unmarried males’ or ‘hunters are people who hunt’. These types of statements were well known to be necessary on the basis of meanings long ago Saul Kripke was born. Furthermore, the supposed rigidity of terms like ‘hunter’, ‘bachelor’, and ‘soda’ does not support necessarily true but a posteriori identities like ‘Hesperus = Phosphorus’ or ‘water = H2O’. If there are no necessary a posteriori propositions with ‘bachelor’, ‘hunter’, ‘soda’, then their supposed rigidity offers nothing new or interesting at all and extending rigidity to them illuminates nothing. (Schwarzt 2002, 271)
Claims about interest or illuminatingness or lack thereof are usually hard to assess. But in any case the second point of Schwartz here (I will consider the first one in the next and final section) seems to me to be wrong: ‘is a bachelor’ and the like do support necessarily true but a posteriori identities. The easiest way is just by rigidifying on a non-rigid predicate signifying the same property with respect to the actual world. Thus:
	Being a bachelor is having the marital status CW actually has
seems to be completely a posteriori. And the argument of its necessary if true character is exactly analogous as before: Suppose ‘Being a bachelor is having the marital status CW actually has’ is actually true and suppose ‘is a bachelor’ and ‘has the marital status CW actually has’ are rigid predicates (in the sense of (*)). Now consider an arbitrary world with respect to which ‘being a bachelor’ signifies a property instantiated in it. By the rigidity of ‘being a bachelor’ this property is the same as the one ‘being a bachelor’ signifies with respect to the actual world, which, by the actual truth of the identity statement, is the property that ‘having the marital status CW actually has’ signifies with respect to the actual world, and which, by the rigidity of ‘having the martial status CW actually has,’ is the same property as ‘having the martial status CW actually has’ signifies with respect to the possible world considered. Hence ‘Being a bachelor is having the marital status CW actually has’ is true with respect to this arbitrary possible world, and hence necessarily true. 
	The same is true, I submit, of all the other relevant cases: ‘Being funny is having the most salient feature actually had by The Simpsons’ etc. That suffices in my view for rejecting Schwartz’s second point in the preceding quote. But it is worth emphasizing that the recourse to rigidifiers might be dispensable. Some predicates for artifacts, legal institutions or psychological states and processes might rigidly signify properties less than perfectly natural specifiable, by the relevant experts, by means of other rigid predicates. Those would give rise to a posteriori necessary identifications, which might even state the (less than perfectly natural) essences of the properties in question.

VI
This last consideration might rise a worry concerning what has been said so far, which might be related to the first point in Schwartz’s quote above.  For consider:
	Being water is being made of H2O-molecules
	Being a television is being something with function F
These seem to be not only necessary a posteriori identifications, but such that arguably state the essence of the properties in question. By contrast, the proposal of (*) has it that predicates like ‘is a bachelor’ are also rigid, whereas what would give the essence of the property in question, along the lines of
	Being a bachelor is being an unmarried man
 is necessary, but not a posteriori but a priori. And I grant that it is also the case that predicates of the like of ‘is funny:’ they (rigidly, according to (*)) signify properties whose essences are not “hidden” in the way that the essences of natural and some less than perfectly natural properties are.
	(Observe that even in those cases in which the identifications are apriori, the explanation of their necessary if true character would also appeal to the rigidity of the relevant expressions, as it is also the case with singular terms:
	Hesperus is the actual morning star.)
 The last thought against (*) to consider would then be the following: rigidity for predicates should explain why certain aposteriori identifications are true where they reveal facts about the “hidden” essences of the properties signified. But that, of course, looks close enough to the original, ungrounded, claim that rigid predicates should coincide with natural (or natural enough) kind predicates as to amount to a restatement of it rather than to a consideration which grounds it.  

References:
Kripke, Saul, 1980: Naming and Necessity, Oxford, Basil Blackwell
LaPorte, Joseph, 2000: ‘Rigidity and Kind’, Philosophical Studies, 97/3, 293-316
Soames, Scott 2002: Beyond Rigidity: The Unfinished Semantic Agenda of Naming and Necessity, Oxford University Press
Schwartz, Stephen P. 2002: ‘Kinds, General Terms, and Rigidity: A Reply to Laporte’, Philosophical Studies 109, 265-77
Wright, Crispin 1992: Truth and Objectivity, Cambridge, MA: Harvard University Press
Wright, Crispin 1998: ‘Why Frege Does Not Deserve His Grain of Salt: a Note on the Paradox of "The Concept Horse" and the Ascription of Bedeutungen to Predicates’, Grazer Philosophische Studien 55, New Essays on the Philosophy of Michael Dummett , eds J.Brandl and P.Sullivan (Vienna: Rodopi),  239-63