Various views of rigidity: the problems

The important point to note is that there appear to be (at least) three distinct definitions of rigid designation current in the literature:

RDI: This is the view which Kaplan identifies as being ’the more widely held’. According to this definition, a rigid designator does not, indeed cannot, designate an object at worlds in which that object does not exist. This view is possibly the one held by Carruthers who, in the passage quoted above, observes that rigid designators 'designate the same items in all possible worlds in which they exist’. This does not, of course, preclude the possibility that they also designate entities in (or at) worlds in which those entities do not exist. Carruthers may just be maintaining a neutral position, but he fails to make this clear.

RD2; This is the middle ground, apparently held by Kripke. The definition is neutral with respect to whether a rigid designator does, or does not, designate an object at worlds in which that object does not exist.

RD3: This is the strong position - held by Kaplan. According to this view, a rigid designator has the ’same designation in all possible world’. (Kaplan, 1989a, p 493) irrespective of whether the object thus designated does or does not exist in all of those worlds.

Salmon notes the apparent emergence o f yet a fourth definition:

Dummett (1973), Linsky (1977), and Putnam (1973) each define yet a fourth notion of rigid designator, distinct from each of the three notions given here. They call a designating expression a rigid designator if it designates the same thing with respect to every possible world with respect to which the expression designates anything at all. (Salmon,

1982a, p 33, fh 36).

Salmon comments that it is unlikely that the above-mentioned writers intend to diverge from the definition proposed by Kripke. This may well be so. On the other hand, the confusion over what actually is proposed by Kripke is instructive.

8.3 The problems

It has been suggested by various w r i t e r s t h a t there are distinct difficulties associated with R D I. The most notable o f these takes the following form: If a possible worlds treatment o f modality is adopted, a statement such as that in (24) below may be paraphrased as in (25).

(24) Saul Kripke might not have existed.

(25) There is a possible world in which Saul Kripke does not exist.

The truth or falsity of the propositions expressed by (24) and (25) can then be evaluated by checking the worlds in the model. If there is at least one world in which (or at which) (26) is true, then (24) and (25) will be true also:

(26) Saul Kripke does not exist.

If we assume RD I the problem should now be apparent. At any world in which Saul Kripke does not exist 'Saul Kripke' fails to designate anything. At such a world (26) either cannot be evaluated at all or it is false. For those who are reluctant to adopt a tri-valent logic, Saul Kripke will thus turn out to be a necessary existent, as in every world in the model (26) will be false. For those who are prepared to concede that (26) is unevaluable rather than false (24) will also be unevaluable, whereas intuitively it seems to be true.

Nathan Salmon discusses the problem with respect to (27).

(27) Nathan Salmon is dead.

He writes:

It is assumed that I cease to exist when I die. On the usual theories of the truth-value of a simple subject-predicate sentence containing a non­ denoting subject term we should not expect the sentence displayed above to be true with respect to a future time if the name 'Nathan Salmon' denotes no one with respect to that time. (Salmon, 1982a, p 37).

In other words, if RDI is assumed it becomes impossible to say truthfully o f any individual that he is dead. Kaplan makes a similar observation:

There are worlds in which Quine does not exist. It does not follow that there are worlds with respect to which ’Quine’ does not denote. What follows is that with respect to such a world ’Quine’ denotes something which does not exist in that world. Indeed, Aristotle no longer exists, but ’Aristotle’ continues to denote (him). (Kaplan, 1973, p 503).

If this is correct, then it would seem that Kripke cannot afford to sit on the fence, as he apparently does by adhering to RD2, but must accept Kaplan’s view of rigidity. Unfortunately, the adoption o f RD3, as Kaplan advocates, is not a problem free solution either, but for rather different reasons. The main difficulty is that Kaplan’s view of the rigidity of natural language expressions is subordinated to his view o f the rigidity o f the free variables o f modal logic. Indeed, in order to promote the acceptance of RD3 as the correct definition of rigid designation, he writes;

... it is a striking and important feature o f the possible world semantics for quantified intensional logic, which Kripke did so much to create and popularize, that variables, those paradigms of rigid designation, designate the same individual in all possible worlds whether the individual ’’exists” or not. (Kaplan, 1989b, p 493).

Kaplan’s point is that as the free variables o f modal logic are rigid designators in the sense defined by RD3 then their analogues, the free pronouns of natural language must be rigid designators in the sense defined by RD3 also. As my knowledge o f quantified intensional logic is strictly limited, I will stipulate that with respect to the free variables of that logic Kaplan’s assertion is correct. The problems start when we try to analogise natural language to formal languages.

Now, it is quite clear that in his account of the demonstratives - and other indexical usages - o f natural language Kaplan does assume just such an analogy. That is to say, he really does assume that the free pronouns of natural language

are analogous to the free variables o f logic. As this assumption is o f considerable significance, I shall quote his views at length:

Pronouns in natural language have often been analogized to variables. Pronouns are lexically ambiguous, having both an anaphoric and a demonstrative use. An anaphoric use o f a pronoun is syntactically bound to another phrase occurring elsewhere in the discourse. In meaningful discourse, a pronoun not used anaphorically is used demonstratively. As I saw the matter, a demonstrative use o f a pronoun was simply a syntactically free use. Like a free occurrence of a variable, it requires something extralinguistic, a demonstration ... to assign it a value. Demonstrative and anaphoric occurrences o f pronouns can thus be seen to correspond to free and bound occurrences o f variables. What I want to stress is that the difference between demonstrative and anaphoric uses of pronouns need not be conceptualized primarily in terms of lexical ambiguity; it can also be seen in terms o f the syntactic distinction between free and bound occurrences of terms. I saw the analogy between variables and pronouns as even closer than had been thought. (Kaplan, 1989b, p 572).

It should be noted that the use of the past tense in the above extract is misleading. In this passage Kaplan reviews and reaffirms the conclusions he reached in Demonstratives. He is not setting the stage for a reversal o f opinion. This is evidenced by his next comment:

I believe that the case o f the free pronoun, the demonstrative, can take a lesson from the case o f the free variable.

What is striking about the above passage is Kaplan's apparent indifference as to whether a pronoun such as 'he', say, is best defined as being lexically ambiguous or as a single lexeme with multiple functions. However, I shall argue in chapter two that when certain characterisations of direct referentiality and de jure rigidity

are taken into consideration, the lexical status of pronouns as being either homonymously ambiguous or univocal across functions may be seen to be an issue o f paramount importance. Furthermore, I shall argue that if pronouns are not ambiguous - and I hope to show that they are not - then it will follow that they cannot be directly referential or de jure rigid either. If this is correct, then they cannot be equated with the variables o f quantified intensional logic.

If all this can be demonstrated, it will affect Kaplan's claims concerning rigidity quite profoundly. In the passage quoted above, Kaplan claims that variables 'are paradigms o f rigid designation'. Now, if being a paradigm is to mean anything, then this claim may be reduced to:

If anything is a rigid designator then a (free) variable is.

Or less idiomatically:

If anything is a rigid designator then free variables are rigid designators.

This may be expressed as the simple modus ponens formula: P -> Q,

where P = 3x (rigid designator (x)),

and Q = Vy (free variable (y )-> rigid designator (y)).

As was seen above, Kaplan further claims that pronouns may be analogised to variables and that the free pronoun 'can take a lesson' from the free variable. If this claim is to be taken seriously, then it must surely mean that the free pronoun is to natural language what the free variable is to quantified intensional logic. Kaplan's intention must, therefore, be to claim that, with respect to natural language, a free pronoun is a prototypical rigid designator. That is to say:

If anything is a rigid designator then free pronouns are rigid designators.

This also may be more formally expressed as R ^ S,

where R = 3x (rigid designator (x)),

and S = Vy (free pronoun (y) rigid designator (y)).

Now, if it can be shown that free pronouns (i.e. demonstratives or other indexical usages) are not rigid designators - that is to say, if it can be shown that ~S - then by modus tollens it follows that ~ R. This may be presented syllogistically:

R - > S ~ S

/. ~ R

This may be read as:

If there is something that is a rigid designator then free pronouns are rigid designators.

Free pronouns are not rigid designators.

Therefore there is not something that is a rigid designator.

Less formally:

If free pronouns are not rigid designators then nothing is.

I am inclined to think that this is correct, and that if it can be shown that free pronouns really are not rigid designators, then the whole edifice of rigid

designation would be radically undermined, and indeed nothing in natural language would designate rigidly.

However, such a conclusion at this point would be precipitate. It could be that the major premise in the syllogism is not sustainable. That is to say, it might be argued that if free pronouns are not rigid designators, they quite clearly cannot be prototypically rigid. This would leave the way open for the suggestion that even though free pronouns may be shown not to be rigid, nonetheless there may be another category o f terms - proper names say - which are.

Denying the major premise but hanging onto the thesis o f rigidity seems to me to be a gallant but ill-advised - possibly even a doomed - move. If the major premise is denied, then with respect to natural language the proponents of the thesis of rigidity will have lost the support of quantified intensional logic, since the analogy between variables and pronouns will have been shown to be unsustainable. That is to say, if pronouns cannot be analogised to variables, then the formulae of quantified intensional logic which demonstrate the rigidity of free variables will tell us nothing whatever about the semantics of natural language.

Therefore, if the analogy between variables and pronouns disintegrates, Kaplan's strongest argument in favour of RD3 - i.e. that this analogy requires that definition RD3 be adopted - will disintegrate also. I suggest that the effect of this would be to add to the uncertainty over whether rigid designation can, in fact, be coherently defined as a function of natural language. I do not dispute the fact that it may be possible to construct formal languages which presuppose rigid designation, and in which free variables are indeed rigid designators. However, such rigid designation is rooted in the arcane metaphysics o f possible worlds, and it is not clear to me that stipulating the nature and content o f such worlds tells us anything at all about the workings of natural language.

Although it has yet to be demonstrated that the pronouns of natural language cannot be rigid designators, it must now be clear that an analysis o f their functions

and those of other pro-terms - the ’indexicals’ o f natural language - is of fundamental importance to the thesis o f rigidity as it now stands. Therefore, although proper names constituted my original motivation for challenging that thesis, it is by means o f an investigation into the functions of the pro-terms of natural language that I shall cany that challenge forward.

chapter two