All Designators are Rigid

In Naming and Necessity (1973) Kripke introduces the concept of a rigid designator (a term with the same reference across possible worlds) and argues that proper names are rigid designators. He argues that in this way they are different from typical definite descriptions (though he allows that some definite descriptions, e.g., ‘the actual winner of the lottery’, ‘the square of 3’ are rigid designators). Kripke (1973: 48) introduces intuitive tests for rigidity to give us a handle on the notion. ‘x might not have been x’ and ‘something other than x might have been x’ are unambiguously false when ‘x’ is a rigid designator, but not when ‘x’ is non-rigid. Thus, by the tests, ‘Benjamin Franklin’ is rigid and ‘the inventor of bifocals’ is not. Hence according to Kripke, non-rigid designators generate ambiguities when occurring in modal and counterfactual contexts. For example, ‘the inventor of bifocals might not have been the inventor of bifocals’ and ‘if [things had been different in such-and-such ways] the inventor of bifocals would have been richer’ are both ambiguous.

His opponents have either argued that names can be regarded as abbreviations of rigid descriptions (e.g., ‘actualized’ ones) or have tried to deny that names are rigid designators.

The point I shall argue is that no unambiguous descriptions are non-rigid. All unambiguous descriptions are rigid. The appearance of non-rigidity in descriptions is simply an illusion, a manifestation of ambiguity. A consequence is an explanation of the difficulty which has been found in extending the rigid/non-rigid distinction from singular terms to predicates.

This point emerges in passing from a not much noticed discussion from John Burgess (so it is not original with me but not, I think, sufficiently appreciated).^[1] Burgess (2006: 175) notes:

… even when discussing a counterfactual situation in which Bill Gates has given all his wealth to Ivana Trump, it is not unambiguously the case that when we use the description ‘the richest person in the world’ we must be referring to her and not to him. For the description contains an implicit verb, made explicit in ‘the one person to be richer than anyone else in the world’. And this verb is subject to inflection for grammatical mood – ‘to be’ may become the indicative ‘is’ or the conditional ‘would have been’ – in a way that creates a flexibility of reference. Thus ‘If Bill Gates had given all his wealth to Ivana, the richest person in the world would have been female’ is ambiguous between the truth ‘If Bill had given all his wealth to Ivana, the one person who would have been richer than anyone else in the world … would have been female’ and the falsehood ‘If Bill had given all his wealth to Ivana, the one person who is richer than anyone else in the world would have been female’.

The crucial point to note is that since the latter is unambiguously false, according to Kripke’s conception the unambiguous description ‘the one person who is richer than anyone else in the world’ is a rigid designator; so its occurrences in counterfactual statements and categorical statements about the actual world do not have different references. Likewise, the description ‘the one person who would have been richer than anyone else in the world’, occurring in the previous unambiguously true counterfactual conditional Burgess gives, is not a non-rigid designator. It cannot, therefore, be understood as having a reference different in this counterfactual from its reference in categorical statements about the actual world. In fact, it cannot occur in categorical statements about the actual world.^[2] It can occur only in conditional contexts.^[3] So neither unambiguous description in which an explicit verb form occurs, neither ‘the one person who is richer than anyone else in the world’ nor ‘the one person who would have been richer than anyone else in the world’, is a non-rigid designator. All that qualifies as one is the description ‘the richest person in the world’, which is ambiguous. The point generalizes. The only descriptions which manifest non-rigidity are ambiguous ones wherein the ambiguity resides in a verb which must be inflected for grammatical mood to resolve it. Non-rigidity is ambiguity.

There is a complication. Burgess’s example ‘the richest person in the world’ is disambiguated by reading it first as ‘the one person who would have been richer than anyone else’ and secondly as ‘the one person who is richer than anyone else’. The second, indicative, reading is present-tensed. Suppose, however, that we are discussing the last election. Then we may say ‘If such and such, the winner would have been female’, which is ambiguous between ‘If such and such, the person who would have won would have been female’ and ‘If such and such, the person who won would have been female’. But the description ‘the person who won’ is itself ambiguous. This is because it contains the description ‘the person who won’ and the past tense verb present here is ambiguous between an indicative and conditional reading. Compare and contrast, ‘the book I was reading last week might not have been the book I was reading last week’, which has a true reading (it would not have been if I had visited the other library) and ‘the book I am reading today might not have been the book I am reading today’ which does not. The latter is not acceptable English. What is acceptable is, ‘the book I am reading today might not have been the book I was reading today’.^[4] Also consider an example not involving definite descriptions: ‘It might not have been the case that I was now about to be sacked’ (I wouldn’t have been but for a series of coincidences). This is acceptable English. ‘It might not have been the case that I am now about to be sacked’. This is not. Also compare and contrast the ambiguity of ‘If Mary had been sicker on Monday than Jane was on Tuesday … ’ and the lack of ambiguity of ‘If Mary had been sicker on Monday than Jane is today … ’.^[5] So not only a description in which, as in Burgess’s example, a verb is only implicit, but also a description containing an explicit past tense verb may be ambiguous between indicative and conditional readings. But again, once a unique reading is imposed the description cannot be understand as non-rigid.^[6]

The point that emerges from the passage in Burgess is actually implicit in an earlier discussion of Kripke (1977: 259–60):

If definite descriptions, ιxφ(x), are taken as primitive and assigned reference, then the conventional non-rigid assignment assigns to such a description, with respect to each possible world, the unique object, if any, which would have φ’d in that world …. For example, ‘the number of planets’ denotes eight, speaking of a counterfactual situation where there would have been eight planets …. Another type of definite description, ιxφx, a ‘rigid’ definite description, could be introduced semantically by the following stipulation: let ιxφx denote, with respect to all possible worlds, the unique object that (actually) φ’s (then ‘the number of planets is odd’, as interpreted, expresses a necessary truth). Both kinds of definite descriptions can … be introduced, theoretically, into a single formal language …. Some have suggested that definite descriptions, in English, are ambiguous between the two readings.^[7]

Note that in the semantic stipulation for the rigid description Kripke makes use of a merely parenthetical and optional ‘actually’ and uses a present tense indicative verb form ‘φ’s’. In the stipulation for the non-rigid description the form ‘would have φ’d’ is used and there is an implicit counterfactual conditional ‘would have φ’d if that world had obtained’ (Kripke writes ‘in that world’). The two stipulations are thus:

1. ιxφ(x) is to denote, with respect to any world w, the object which would have φ’d if w had obtained,

2. ιxφx is to denote, with respect to any world w, the object which φ’s.

The second stipulation yields a description which functions like Burgess’s ‘the person who is richer than anyone else in the world’, which is unambiguously rigid. The first stipulation specifies a description which functions like ‘the person who would have been richer than anyone else in the world if Gates had given his fortune to Trump’. So the first form of stipulation does not specify a non-rigid description of the type ‘the x which would have been φ’. A description of this type (‘the x which would have been φ’) can only occur in a conditional context so cannot have a reference which is different according to whether it occurs in conditional or non-conditional contexts. Neither stipulation is therefore a stipulation of a non-rigid meaning for a description (If we say that the description yielded by the first stipulation is the whole ‘the person who … if world w had obtained’ this is a rigid designator.)

But what I take to be the lesson of Burgess’s example and consistent with Kripke’s discussion is that everyday descriptions like ‘the richest person in the world’, ‘the winner’ or the person who won’, in which verbs are only implicit, or past tense but ambiguous between indicative and conditional readings, are themselves ambiguous between the two forms of description and this explains the intuitive appearance that they are non-rigid. Really, no unambiguous descriptions are.^[8]

This may seem to be merely an alternative way of putting Kripke’s point that such descriptions are non-rigid. But the contention is more interesting. It is the prediction that all descriptions in English which will be found to fail Kripke’s intuitive tests for rigidity will also be found to be ambiguous descriptions containing only: an (a) implicit or (b) if explicit, past-tensed, verb which must be inflected for mood to eliminate ambiguity. So ‘the inventor of bifocals’ fails since both ‘the inventor of bifocals might not have been the inventor of bifocals’ and ‘someone other than the inventor of bifocals might have been the inventor of bifocals’ are true.

I think that this way of conceiving of the class of descriptions which come out as non-rigid by Kripke’s tests is illuminating. In particular, it helps with one question which has been much discussed in the literature: whether the Kripkean division between rigid and non-rigid designators can be extended from singular terms to predicates.^[9]

The proposal I have been defending is that in the case of singular terms what we think of, following Kripke’s advice, as non-rigid terms simpliciter are descriptions ambiguous in a certain way, that is, they contain (implicit) verbs which are subject to inflection for grammatical mood.

Now adopting a Fregean framework we can think of predicates as follows. First-level predicates can be understood as verbs denoting concepts satisfied by, or relations among, objects (the referents of singular terms). And verbs denoting concepts satisfied by, or relations among, objects can be thought of as standing for Fregean first-level functions from sequences of objects to truth-values (the special case when the sequences are one-membered is that in which Frege speaks of concepts rather than relations). In the same framework non-rigid parts of speech corresponding to first-level predicates (in natural languages, general terms, adjectives and verbs) would have to be ambiguous and contain (implicit) expressions for second-level relations (just as the ambiguous singular terms contain (implicit) expressions for first-level relations); ones which, in Fregean parlance, are functions from first-level concepts or relations to truth-values (like quantifiers). But do any of our general terms, adjectives and verbs in statements which are translated into the language of first-order logic as first-level predicates behave in this way? Are they ambiguous in this way? If not the distinction between rigid and non-rigid designators, as I suggest we understand it, cannot be drawn at the level of predicates.

That seems to be the case, and if so explains the difficulty philosophers have had in extending Kripke’s discussion beyond singular terms. Non-rigidity in general is an illusion, which only appears in the case of singular terms, and only in the case of those which are ambiguous in the way suggested, due to containing (implicit) inflectable verbs.