Le Poidevin's regress argument

Le Poidevin's argument begins by observing that one who accepts relative identity is committed to true instances the following schema:

(1)

Le Poidevin subsequently argues that an advocate of relative identity cannot consistently affirm (1). Although one who accepts this theory is committed to the denial of absolute identity, (1) involves an appeal to absolute identity, since it implicitly affirms that the very thing (x) which is the same F as y is not the same G as y.

How might an advocate of relative identity theory best respond to this objection? Since this theory is committed to the denial of absolute identity, Le Poidevin suggests that the relative identity theorist will respond by providing us with a suitably relativized version of this identity-statement, i.e. the one which characterizes the relationship between the referents of the first and second instances of ‘x’ (henceforth ‘x₁’ and ‘x₂’). Since the relative identity theorist cannot affirm the absolute identity of x₁ and x₂, the best one can do is to say that x₁ is the same F as x₂. He thus suggests that the relative identity theorist must replace (1) with the following:

(2)

If our goal is to replace affirmations of absolute identity with affirmations of relative identity, then it should be clear that we are no better off than before. For (2) succeeds in avoiding one affirmation of absolute identity only by committing itself to another, namely, the assertion that the very thing (x₂) which is the same F as x₁ is not the same G as y. It is only too clear, moreover, that by substituting a suitably relativized version of this identity-statement, we will be committing ourselves to yet another statement of absolute identity. In this case our goal will be to clarify the relationship between the referents of the first and second instances of ‘x₂’ (henceforth ‘x_2.1’ and ‘x_2.2’). As before, instead of affirming the absolute identity of x_2.1 and x_2.2, the best one can do is to say that x_2.1 is the same F as x_2.2. We must therefore replace (2) with the following:

(3)

Once again, we're still no better off than before. For though we have clarified the relationship between the referents of the first and second instances of ‘x₂’, this statement commits us to yet another statement of absolute identity between the referents of the first and second instances of ‘x_2.2’. For now we are saying that the very thing (x_2.2) which is the same F as x_2.1 is not the same G as y. Since our objective is to replace statements of absolute identity with statements of relative identity, and this goal is clearly not attainable, this account of identity faces an infinite regress of absolute identity relations. And since these are the very relations which relative identity is keen to avoid, this regress thoroughly undermines the credibility of this account.

An extension of Le Poidevin's argument

If Le Poidevin is on the right track here, then relative identity theory is actually worse off than he suggests. In the first place, we should note that (1) is implicitly making not one but two affirmations of absolute identity. In addition to affirming that the very thing (x) which is the same F as y is not the same G as y, (1) also affirms that the very thing (y) which is the same F as x is not the same G as x. So in addition to clarifying the relationship between the referents of the first and second instances of ‘x’, we must also clarify the relationship between the referents of the first and second instances of ‘y’. Replacing these instances of ‘y’ with ‘y₁’ and ‘y₂’ as before, here is our best attempt at a fully relativized version of (2):

(2.1)

Once again, in terms of the theoretical objectives of the relative identity theorist we are clearly no better off than before, since (2.1) does not contain fewer affirmations of absolute identity than (1) or (2). For in addition to affirming that the very thing (x₂) which is the same F as x₁ is not the same G as y₂, (2) also affirms that the very thing (y₂) which is the same F as y₁ is not the same G as x₂.

Now if our objective is to replace statements of absolute identity with statements of relative identity, while neither (2) nor (2.1) is better off in this regard, can we at least say that they are not worse off in this regard? We cannot: (2) and (2.1) are much worse off than (1) in this regard. For the sake of simplicity, let us focus solely on the transition from (1) to (2.1). In the process of eliminating the two affirmations of absolute identity which are implicit in (1), (2.1) is not merely committed to two different affirmations of absolute identity, it is also committed to three additional affirmations of absolute identity. For (2.1) also affirms (i) that the very relation (the same-F relation) which holds between x₁ and y₁ also holds between x₁ and x_2, (ii) that the very relation which holds between x₁ and y₁ also holds between y₁ and y₂, and (iii) that the very relation which holds between x₁ and x₂ also holds between y₁ and y₂. As Colin McGinn has recently observed, what is sometimes spoken of as qualitative or specific identity is really an affirmation of absolute identity at the level of qualities or kinds (McGinn (Reference McGinn2000), 2). Thus, for example, if I say that Rex and Flash are the same breed of dog, then I am saying that the breed to which Rex belongs is identical with the breed to which Flash belongs. This, in turn, suggests that Le Poidevin's regress is not merely a regress of relations between individuals, since there appears to be a parallel regress of identity relations surrounding the properties which are instantiated by these individuals.

I think it must asked, however, whether there is even room for this second metaphysical regress, i.e. one which occurs at the level of properties and relations. In the case of (2.1), for example, we are forced to clarify the relationship between the referents of the three instances of ‘F’. While it is clear from (3) that there is no difficulty in formulating a suitably relativized account of the relationship between the referents of the first and second instances of ‘x₂’, I don't see any prospect for applying this strategy to the three occurrences of ‘F’. Although nothing prevents us from disambiguating these terms, as before, as F₁, F₂, and F₃, where are we to go from here? We can hardly say, for example, that F₁ is the same F as F₂, since the referents of these predicates (properties) do not possess the properties which they are. Thus, for example, the property of being a mountain lion is not itself a mountain lion: it is not even an animal, much less a mammal of the appropriate genus. Nor will it help to say that F₁ is the same property as F₂, since this is the same as saying that the property of being F₁ is (absolutely) identical with the property of being F₂. Thus, for example, there is no substantive difference between saying (i) that the property of being a cougar is identical with the property of being a mountain lion, and (ii) that the property of being a cougar is the same property as the property of being a mountain lion. Since this result will also hold for any higher-level properties to which we might appeal, I don't think there is any basis for a regress of identity relations at the level of kinds and properties. And this suggests that relative identity theory is actually worse off than Le Poidevin has indicated, since in affirming (2) (2.1), and (3) one is affirming statements of absolute identity which cannot be avoided.

I might be wrong in thinking that there is no room for a regress at the level of properties and relations. There may well be a logical maneuver here which has escaped my attention. Even so, essentially the same problem remains. If the relative identity theory is committed to the denial of absolute identity, if every affirmation of relative identity which follows the pattern of (1) implicitly affirms multiple statements of absolute identity, and if every attempt at avoiding these statements of absolute identity commits us to subsequent affirmations of absolute identity, then Le Poidevin has not merely shown this theory to be saddled with a vicious infinite regress: he has shown it to be self-contradictory. So if Le Poidevin is on the right track here, this account of identity is not merely one which we should avoid if possible, it is one which we should avoid at all costs. If it is self-contradictory, then any sense that that is useful in theological or other contexts is thoroughly illusory and misguided.

Why Le Poidevin's regress is not inevitable

Le Poidevin's regress argument turns on the fact that statements of relative identity which follow the pattern of (1) involve multiple references to the same individual. Whether this occurs by way of recurring singular terms or variables, the resulting instance of anaphoric cross-reference involves an implicit appeal to absolute identity. And though he does not draw attention to this fact, this implication also holds for multiple instances of the same predicative constant or variable, as we saw with regard to the use of ‘F’ in (2.1). Le Poidevin has thus drawn our attention to a significant issue, namely, whether there is a way of referring to individuals and of describing their properties and relations which is consistent with the thesis that identity is relative. Since it is not difficult to find applications of relative identity which do not confront this issue,Footnote ⁵ the prevailing assumption is evidently that advocates of relative identity are able to engage in these linguistic actions in a manner which is consistent with this theory. What follows is a qualified defence of this assumption. Although Le Poidevin is certainly justified in rejecting the solution which he has proffered on behalf of this theory, he is wrong in thinking that relative identity lacks the resources for avoiding these difficulties.

It is not difficult to avoid the higher-level regress problem. An advocate of relative identity is not obliged to say, with Geach, that there is no such thing as absolute identity, or more precisely, that there are no instances of this relation (Geach (Reference Geach1972), 240–245). For nothing prevents us placing more or less severe restrictions upon the individuals which are allowed to fall under the extension of this relation. In particular, by excluding properties and relations from the extension of this relation we are able consistently to describe an object's properties and relations in a manner which involves absolute identity. We will be able to say consistently, for example, (i) that Fred has the same colour eyes as Mary, and (ii) that Secretariat is the same kind of horse as Man O'War. That is, we will be able to say (i) that the colour of Fred's eyes is (absolutely) identical with the colour of Mary's, and (ii) that the breed to which Secretariat belongs is (absolutely) identical to the breed to which Man O'War belongs. Indeed, this result also holds for relations, including the relation of relative identity. Though Geach has also suggested (Geach (Reference Geach1972), 249) that there are as many kinds of relative identity as there are sortals to which it is attached (so that the ‘___ is the same horse as ___’ relation involves one kind of identity, while ‘____ is the same person as ___’ involves another), we are not compelled to follow him in this regard. Having excluded properties and relations from the extension of this relation, a proponent of relative identity can consistently affirm that the statements (i) ‘George Eliot is the same person as Mary Ann Evans, (ii) ‘Everest is the same mountain as Chomolungma, and (iii) ‘Snippy is the same horse as Lady’ are distinct instances of the same relation (distinct because they involve different relata).

Although excluding properties and relations from the extension of this relation enables us to avoid the metaphysical regress which obtains at the level of properties and relations, it does not help us to avoid Le Poidevin's official regress, namely, the regress of absolute identity relations which attaches to the objects to which we are ascribing relative identity. Consider, once again, the following proposition:

(2.1)

While an advocate of the relative identity thesis can now consistently affirm that the very relation (the same-F relation) which holds between x₁ and y₁ also holds between x₁ and x₂, he is still in no position to say that the very thing (x₂) which is the same F as x₁ is not the same G as y₂. Although one in this position is not obliged to replace all statements of absolute identity with suitably qualified statements of relative identity, since (2.1) makes it clear that the referents of these terms do fall under the extension of this relation, he is evidently committed to avoiding this one. The question at hand is whether we are able to say that S₁ and S₂ are the same F but different Gs in a manner which does not commit us to statements of absolute identity concerning S₁ and S₂. In what remains of this article I will argue that we can. In particular, I will present two alternatives for avoiding Le Poidevin's regress, alternatives which turn upon competing accounts of singular reference, and upon competing answers to the question of whether we can use singular terms to refer to objects which fall under the extension of this relation.

Although we refer to objects in a variety of ways (e.g. through gestures, diagrams, definite descriptions, and indexical pronouns), at least part of the time we refer to objects by means of proper names. While there is nearly universal agreement that we generally succeed in referring to individuals by means of proper names, there is a persisting debate as to how singular reference is accomplished. Or perhaps I should say that until recently there has been such a debate, since the prevailing view these days is that a proper name does not refer to an object by means of the descriptive content which the speaker associates with this name.Footnote ⁶ On the currently orthodox account of reference, names pick out individuals directly. In Kripke's terminology (1980), proper names are rigid designators: they pick out the same object at all times and in all conceivable counterfactual circumstances (possible worlds) in a manner which has literally nothing to do with the speaker's beliefs about the object in question. Kripke's account of reference is formulated in opposition to the so-called descriptive theory of reference, according to which names refer to objects indirectly, by means of the descriptive content which the speaker associates with them. On a descriptive account, the referent of a given name is the entity which uniquely satisfies this description. If there is no such entity (either because nothing fits this description or because more than one thing fits it), then this name lacks a referent.

Now even if Kripke's case against the descriptive theory is not conclusive, it has nonetheless proved to be historically decisive. It is surely no overstatement to say that the force of his arguments is largely responsible for the recent demise of the descriptive theory of reference and hence for the current ascendency of his favoured alternative. At any rate, the important thing for our purposes is to ascertain the connection between Le Poidevin's regress argument and the now orthodox conception of reference. I will argue that Le Poidevin's regress argument presupposes a direct account of reference, and hence that this regress does not hold for accounts of relative identity which reject this account of reference. In addition, I will argue that this regress is not inevitable for one who embraces the currently orthodox account of reference, and to make good on this claim I shall draw extensively upon the formal logic of relative identity which Peter van Inwagen uses to articulate and defend the mysteries of the Trinity and the Incarnation.

While it might seem artificial to suppose that one account of reference should suffice for all proper names, let us suppose that the names for the objects with which we are now concerned (objects which fall under the extension of the relation of relative identity) denote their referents in one of these two ways. For the sake of argument, let us suppose that proper names function as rigid designators, and hence that names refer to objects directly. At least at first glance, it looks as if accepting this account of reference commits us to denying the relative identity thesis, since on this account referring to a thing involves picking it out as a numerically distinct individual. In a widely discussed article, Alston and Bennett (Reference Alston and Bennett1984) provide a particularly clear argument to this effect:

In the ensuing paragraph Alston and Bennett reiterate this contention in a manner which incorporates Leibniz's principle of indiscernibility:

The relative identity thesis claims, first, that well-formed statements of identity must be made relative to a sortal count noun, and second, that it is possible for an object and an object to be identical relative to one sortal and distinct relative to another. In these passages Alston and Bennett take issue with both of these claims. If we succeed in referring to something by means of ‘a’, and if we also succeed in referring to something by means ‘b’, there is an absolute, fully objective fact of the matter with regard to whether these terms pick out the same thing. In either event, the thesis that a is (absolutely) identical with b is intelligible and fully determinate. So in the absence of a sortal qualifier, the suggestion that a is identical with b is not, as Geach suggests, ‘a vague expression of a half-formed thought’ (Geach (Reference Geach1972), 238). If ‘a’ and ‘b’ do pick out the same thing, then every property had by a is also had by b (and conversely). So if a is b, then we can also say, for any property F which is had by a, that a is the same F as b. Although ‘a=b’ is less informative than ‘’, it is no less determinate. If a≠b, on the other hand, then it is not more informative to be told that , since this is true regardless of whether a and b are both Fs.

On the direct theory of reference, referring to an object involves picking it out as this object in a manner which singles it out from all others. Since this account of reference is imbued with an absolute conception of identity, it looks as if one who accepts this account is committed to denying the thesis that identity is relative. It is important to note, however, that this implication only applies to contexts in which one is able to refer to objects in this manner. One who is generally committed to this account of reference might acknowledge contexts which rule out the possibility of direct reference, e.g. contexts which include instances of the thesis that identity is relative. In a series of articles defending the mysteries of the Trinity (1988) and the Incarnation (1994), Peter van Inwagen argues that the ‘data’ of divine revelation provide us with just such a context, i.e. one in which distinct persons are the same being (in the case of the Holy Trinity), and one in which distinct beings are the same person (in the case of the Incarnation). As a vehicle for understanding and defending these mysteries, van Inwagen has developed a formal logic of relative identity (henceforth ‘RI-logic’), and since he accepts the direct theory of reference, the language of RI-logic does not include singular terms.

Since ordinary-language sentences pertaining to the mysteries of the Trinity and the Incarnation generally include proper names, using RI-logic to make sense of these mysteries involves translating statements from a form which includes proper names into one which does not. Van Inwagen achieves this end in a manner which is reminiscent of Russell's theory of descriptions, that is, one which involves replacing singular terms with descriptive expressions which a well-informed speaker would likely associate with such names. To see how this works in any sort of detail, we must take a closer look at the linguistic resources of RI-logic.

In common with standard systems of natural deduction, the language of RI-logic includes universal and existential quantifiers, variables for individuals, monadic and polyadic predicates, closed sentences,Footnote ⁷ and the standard sentential connectives. Unlike most systems of predicate logic, however, the language of RI-logic does not include individual constants in addition to individual variables. Nor does it include the symbol for expressing numerical identity (‘=’). Indeed it does not even include the subscripted identity symbol which we have thus far used to express the relation of relative identity (e.g. ‘’). Instead, statements of relative identity are expressed by means of a special class of dyadic predicates.

The language of RI-logic includes two classes of predicates: ordinary predicates and relative-identity predicates (henceforth ‘RI-predicates’). The crucial difference between the two is fairly intuitive: ordinary predicates (such as ‘__ is yellow’ and ‘__ is at least as heavy as __’) express properties and relations which are neutral with regard to questions of number and identity. Ordinary predicates thus cannot involve count nouns, i.e. nouns (such as ‘pencil’ or ‘cat’) which can take an indefinite article and for which one can coherently ask ‘how many?’. Nor can they tacitly involve expressions (such as proper names or demonstrative pronouns) which are tied to specific individuals. Count nouns are restricted to the second class of predicates, e.g. predicates such as ‘__ is the same person as __’ and ‘__ is the same cat as __’. For the sake of convenience, however, van Inwagen allows RI-predicates to take an abbreviated form: the RI-predicate ‘__ is the same person as __’ can also occur as ‘__ is a person’. The rationale for this allowance is straightforward: we cannot say that x is the same person as y without saying (inter alia) that x is a person.

Let us now turn to van Inwagen's method for translating ordinary language sentences into the language of RI-logic. Consider the following propositions:

1. (4) There is one God.

1. (5) God is eternal.

Expressing these propositions in the language of RI-logic requires the following predicates (following van Inwagen, RI-predicates are underlined):

• Bαβ α is the same being as β.

• Dα α is divine.

• Eα α is eternal.

The English word ‘God’ functions as a common noun in (4) and as a singular term in (5). In other words, (4) is telling us that there is exactly one instance of the divine nature, and (5) is telling us that this being is eternal. Using the above dictionary, in the language of RI-logic (4) and (5) are expressed in the following manner:

(4.1)

Footnote ⁸

(5.1)

We are now in a position to appreciate fully van Inwagen's method for translating statements which contain proper names into the language of RI-logic. In each case, this method results in what he refers to as a ‘relative identity surrogate for classical definite descriptions’ (van Inwagen (Reference van Inwagen and Morris1988), 265). Contrast the above propositions with translations guided by Russell's method:

(4.2)

(5.2)

Whereas Russell's method employs statements of (absolute) identity in the consequents of the embedded conditionals, van Inwagen's method employs the RI-predicate ‘___ is the same being as ___’, which is to say that unlike Russell's method, van Inwagen's method for replacing names with descriptions is free from the logic of absolute identity.

Let us now turn to van Inwagen's application of RI-logic to the central mysteries of the Christian faith. Since our understanding of the Incarnation is grounded in a Trinitarian account of the divine nature (we believe that the Second Person of the Holy Trinity assumed our nature), let us begin with his application of RI-logic to doctrine of the Holy Trinity. In connection with this mystery, Christians have traditionally affirmed the following sorts of claims:

1. (6) There are three divine persons.

2. (7) The Father is eternal.

3. (8) The Father is the not the same person as the Son.

4. (9) The Father is the same being as the Son.

Translating (6) into the language of RI-logic requires the following addition to our stock of predicates:

• Pαβ α is the same person as β.

With the addition of this RI-predicate, (6) is expressed in the following manner:

Footnote ⁹Since (7), (8), and (9) contain the singular terms ‘the Father’ and ‘the Son’, translating these propositions into the language of RI-logic involves replacing these terms with suitably relativized descriptions. It is not difficult to ascertain descriptions which uniquely apply to each of the divine persons, since, as van Inwagen points out, it is a commonplace of Trinitarian theology that each of the divine persons is individuated by his relations to the other two (van Inwagen (Reference van Inwagen and Morris1988), 251). In terms of this tradition, the Father is the one who begets the Son, the Son is the one who is begotten of the Father, and the Spirit is the one who proceeds from the Father through the Son. For the sake of economy, van Inwagen introduces three monadic predicates which are defined in the following manner:

• α begets (∃β)(α begets β).

• α is begotten (∃β)(β begets α).

• α proceeds (∃β)(∃γ)(α proceeds from β through γ).Footnote ¹⁰

In terms of these predicates, then, the Father is the one who begets, the Son is the one who is begotten, and the Holy Spirit is the one who proceeds. So instead of saying, for example, that the Father is eternal, we can say (salva veritate) that the one who begets is eternal. (7) is thus translated in the following manner:

• (7.1) (∃x)[x begets & (y)(y begets→Pyx) & Ex].

Following the same strategy, it is a simple matter to represent the logical form of (8) and (9). For this is to say, in the first place, that the one who begets is not the same person as the one who is begotten, and in the second place, that the one who begets is the same being as the one who is begotten. Here are translations of both statements:

Given the unwieldy nature of these translations, van Inwagen allows for further abbreviations by way of several defined predicates. Here are the defined predicates which are relevant to the propositions which we have translated thus far:

• Gx=_df Dx & (y)(Dy→Byx).

• Fx=_df x begets & (y)(y begets→Pyx).

• Sx=_df x is begotten & (y)(y is begotten→Pyx).Footnote ¹¹

While we are instructed to read these predicates as ‘x is God’', ‘x is the Father’', and ‘x is the Son’', it is important to note that these defined predicates do not express the property of being absolutely identical with the being or person with whom they are associated. Thus, for example, the proposition ‘Gx’ does not say that x is identical with God. Rather, to say that x is G is to say both (i) that x is divine and (ii) that x is the same being as anything (anyone) else that is divine, and this leaves the possibility that x is not the only such reality (person). Finally, by means of these defined predicates we are able to render the following simplified translations:

• (4.2) (∃x)Gx.

• (5.2) (∃x)(Gx & Ex).

• (7.2) (∃x)(Fx & Ex).

• (8.2) (∃x)(∃y)(Fx & Sx & ~Pxy).

• (9.2) (∃x)(∃y)(Fx & Sx & Bxy).

Let us now turn to van Inwagen's application of RI-logic to the mystery of the Incarnation. With the fathers at Chalcedon, van Inwagen affirms that Christ is both fully human and fully divine. For the reasons which we considered at the outset of this article, van Inwagen's account of this mystery is strongly isomorphic with his account of the Holy Trinity. Whereas the Holy Trinity presents us with a plurality of persons who are the same being, the Incarnation presents us with a plurality of beings which are the same person. In particular, this doctrine affirms the existence of two distinct beings: the (fully divine) Son of God and the (fully human) son of the Blessed Virgin. While these cannot possibly be the same being, this doctrine insists that they are the same person. Van Inwagen is thus committed to the following triad:

1. (10) Jesus of Nazareth and Son of God are distinct beings.

2. (11) Jesus of Nazareth and the Son of God are the same person.

3. (12) Jesus Christ is both divine and human.

Translating these statements into the language of RI-logic requires two additional predicates. In the first place, we must add the following ordinary predicate to our working dictionary:

• Mα α is human.

In the second place, we need a description which uniquely identifies Jesus of Nazareth and which can be captured by an ordinary predicate along the same lines as ‘Gx’ and ‘Fx’. Though he intentionally leaves this list open, van Inwagen is confident that anyone who believes in the historical Jesus will be in a position to supply a list of properties which applies solely to him, one, for example, which includes such properties as ‘was born of a virgin’, ‘turned water into wine’, ‘healed lepers’, ‘was crucified’, and ‘was raised from the dead’. If we let the predicate ‘Jx’ stand for the conjunction of these predicates, we can replace statements containing ‘Jesus of Nazareth’ with ‘the being who is J’ in the same way that we used ‘Sx’ to replace ‘the one who is begotten’. Towards this end van Inwagen introduces the following defined predicate:

• Jx=_df Jx & (y)(Jy→Bxy).

With these predicates in hand, we are in a position to translate (10) and (11) in the following manner:

• (10.1) (∃x)(∃y)(Sx & Jy & ~Bxy).

• (11.1) (∃x)(∃y)(Sx & Jy & Pxy).

In addition to saying that this divine being (the Son of God) and this human being (Jesus of Nazareth) are the same person, we need to say that this person is Christ and hence that Christ is both human and divine. This, in turn, leads van Inwagen to distinguish between Jesus of Nazareth and Jesus Christ: Jesus of Nazareth is the human being who was born of the Blessed Virgin, while Jesus Christ (or more simply, Christ) is the one who is the same person as this human being and the Son of God (van Inwagen (Reference van Inwagen and Padgett1994), 220). Though we often use these terms synonymously, there are things which are true of Jesus Christ which are not true of Jesus of Nazareth. To see why this matters, consider the following question: How should we understand the conviction that the historical Jesus is both human and divine? We might easily take this belief to be captured by the following proposition:

1. (13) Jesus of Nazareth is both human and divine,

which is easily translated in the following manner:

• (13.1) (∃x)(Jx & Mx & Dx).

Although (13.1) is a credible translation of (13), it is also clearly false. While it is a conceptual truth that an object O is human if O has the properties which are included in ‘J’, it is also a conceptual truth that if O is human, then O is not divine, since the properties involved in being human (e.g. the properties of being contingent, created, and spatially extended) are generally incompatible with the divine nature. When we characterize Jesus as being both human and divine, then, we are speaking about Jesus Christ, namely, the one who is personally identical with the Son of God and the son of the Blessed Virgin. While (13.1) is an adequate translation of the (false) proposition that Jesus of Nazareth is both human and divine, it does not do justice to our belief that Christ is both human and divine.

It turns out that the logical structure of ‘Jesus Christ is F’ is considerably more complicated than that of ‘Jesus of Nazareth is F’. When we say that Christ is F, we are attributing this property to one who is personally identical both with Jesus of Nazareth and with the Son of God. Consider, in this light, the following pair of propositions:

1. (14) Jesus of Nazareth is eternal.

2. (15) Jesus Christ is eternal.

Now compare their forms in the language of RI-logic:

• (14.1) (∃x)(Jx & Ex).

• (15.1) (∃x)((∃y)(Sy & Pxy) & (∃y)(Jy & Pxy) & Ex).

It is fair to say that the logical form of (15.1) bears almost no resemblance to the surface grammar of (15). At the same time, whereas (14.1) is conceptually impossible, as a Catholic Christian I have no reservations concerning (15.1), which predicates eternality of one who is personally identical with the Son of God and Jesus of Nazareth.

Following this pattern of translation, we should have no difficulty in translating the following propositions:

1. (16) Jesus Christ is human.

2. (17) Jesus Christ is divine.

Keeping in mind that attributing a property to the person of Christ involves attributing this property to one who is personally identical with the Son of God and also with Jesus of Nazareth, (16) and (17) are translated into the language of RI-logic as follows:

• (16.1) (∃x)((∃y)(Sy & Pxy) & (∃y)(Jy & Pxy) & Mx).

• (17.1) (∃x)((∃y)(Sy & Pxy) & (∃y)(Jy & Pxy) & Dx).

Since (12) is best understood as saying both (a) that Christ is human, and (b) that Christ is divine (and not that Christ has the impossible property of being both-human-and-divine), this proposition is best translated simply as the conjunction of (16.1) and (17.1):

Before we return to Le Poidevin's regress, I would like to point out that the linguistic resources of RI-logic are only part of what distinguishes van Inwagen's logic of relative identity from (standard) logical systems involving a classical conception of identity. For an adequate sense of this distinction we need to consider briefly the deductive resources of RI-logic. In comparison with other systems of natural deduction, the deductive resources of RI-logic are fairly modest. In the first place, there are only two inference rules for statements involving RI-predicates: one which draws upon the symmetrical nature of these predicates, and one which draws upon the transitive nature of these predicates.Footnote ¹² The first rule allows us to infer that y is the same F as x from the proposition that x is the same F as y. The second rule allows us to infer that x is the same F as z from the propositions (i) that x is the same F as y, and (ii) that y is the same F as z.

In the second place, since there is no identity operator in RI-logic, there is no general principle of indiscernibility. Since we will never encounter the proposition that x=y, we will never be in a position to infer that x is G on the grounds that y is G and x=y. Nor is there a principle of indiscernibility which applies to suitably relativized statements of identity, namely, one which applies to statements involving RI-predicates. That is, in RI-logic it is not the case that for any RI-predicate ‘__ is the same F as __’ and any ordinary predicate Φ,

While there is no general principle of this form, van Inwagen is certainly willing to affirm particular instances of this form, instances which he refers to as dominance statements (van Inwagen (1988), 238). In his terms, an RI-predicate F dominates an ordinary predicate G when F forces indiscernibility with respect to G, that is, when

Thus, for example, van Inwagen affirms such dominance propositions as the following:

1. (18) (x)(y)(Bxy→(Dx↔Dy)).

2. (19) (x)(y)(Bxy→(Mx↔My)).Footnote ¹³

The first tells us that ontic identity (sameness of being) dominates divinity, and the second tells us that ontic identity dominates humanity. And while he refers to these propositions as conceptual truths, they do not function as logical truths in RI-logic. Logically speaking, one does not get these propositions ‘for free’. As substantive metaphysical theses, their truth is not ‘underwritten by the formal logic of relative identity’ (van Inwagen (Reference van Inwagen and Morris1988), 238).

To understand why RI-logic is not ‘a mere notational variant of a fragment of the classical logic of identity’ (van Inwagen (Reference van Inwagen and Padgett1994), 203 f.), we need to consider two dominance propositions which van Inwagen does not accept, and indeed which he is compelled to deny:

1. (20) (x)(y)(Bxy→Pxy).

2. (21) (x)(y)(Pxy→Bxy).

The first proposition claims that ontic identity dominates personal identity, while the second affirms the converse of this claim. Given van Inwagen's accounts of the Trinity and the Incarnation, it is easy to see why RI-logic cannot include these propositions. We must deny the first claim if we believe that the Father and the Son are distinct persons who are the same being, and we must deny the second if we believe that the Son of God and Jesus of Nazareth are distinct beings who are the same person. Van Inwagen freely admits that while we believe (on the authority of divine revelation) that these things are true, we shall perhaps never understand how they are true. The mystery of the Holy Trinity tells us that ontic identity does not dominate personal identity. How can this be, when ontic identity clearly does dominate humanity and divinity? We cannot say. The mystery of the Incarnation tells us that personal identity does not dominate ontic identity. But how is it possible for a plurality of beings to be the same person? We haven't the faintest idea. While RI-logic does not enable us to see how these things can be, it does provide us with a credible basis for deflecting the charge that they cannot be. I should add, moreover, that van Inwagen's defence of these mysteries is more than simply an appeal to the logic of relative identity, since in both cases he includes an informal proof of the consistency of the theological propositions which are expressed in the language of RI-logic.Footnote ¹⁴

We are now in a position to return to Le Poidevin's regress. Recall that this regress begins with the observation that an advocate of relative identity is committed to true instances of the following schema:

(1)

Since this regress argument is explicitly directed against those who use relative identity in defence of the mystery of the Incarnation, let us begin with an instance of this schema which pertains to this doctrine. By replacing ‘F’ and ‘G’ with sortal concepts for Person and Being, here's a closed and suitably qualified version of (1):

(22)

According to Le Poidevin, one who employs relative identity in defence of the Incarnation is committed to there being true instances of this proposition. It is not difficult to identify a plausible candidate, particularly if our goal is to represent the logical form of the following:

1. (23) Jesus of Nazareth is the same person as the Son of God, but not the same being as the Son of God.

Since the substitution instances of a quantified expression are generated by removing its quantifiers and by uniformly replacing the variables which are bound by these quantifiers with singular terms, for a translation of (23) which is an instance of (22), all we need is singular terms for the proposed relata. Letting ‘a’ stand for Jesus of Nazareth, and letting ‘b’ stand for the Son of God, here is the sort of statement which Le Poidevin's regress objection has in mind:

(23.1)

As before, it is not difficult to appreciate the force of Le Poidevin's regress: this proposition implicitly affirms that the very individual (a) which is the same person as b is not the same being as b. At this point it should be clear why this difficulty arises: this instance of cross-reference involves an implicit appeal to absolute identity because its constitutive acts of singular reference are themselves imbued with an absolute conception of identity. That is, (23.1) involves an implicit appeal to absolute identity because it employs singular terms, terms which are understood to be picking out a discrete individual in distinction from all others. For only then can this statement implicitly claim that ‘the very thing’ (a) which is the same person as b is not the same being as b.

For some time we have been working under the assumption that singular terms refer to objects in the way that Le Poidevin's regress argument requires, that is, that names refer to individuals directly and hence not in a manner which is mediated by the descriptive content which the speaker associates with this name. But suppose for a minute that this account of reference is mistaken. Suppose, instead, that names are associated with sets of weighted descriptions such that a name's referent in a given context is the object which best satisfies this description. Since there is no guarantee that a recurring name will denote ‘the very same’ object, particularly when it is being used in different contexts, one who accepts this account of reference is not committed to the thesis that ‘the very thing’ (a) which is the same person as b is not the same being as b. That is, since this account of reference is not imbued with a classical conception of identity, on this account we can use recurring instances of a singular term in a manner which does not commit us to statements of absolute identity. It is thus no accident that the accounts of relative identity which are defended by Deutsch and Zemach involve an explicit rejection of the direct theory of reference.Footnote ¹⁵ Since Le Poidevin's regress argument depends upon a direct theory of reference and these accounts of relative identity reject this account of reference, we may surmise that these accounts of relative identity are immune from Le Poidevin's regress. Is this regress inevitable for an advocate of relative identity who accepts the direct theory of reference? In what remains of this article I will argue that it is not.

Since he accepts the direct theory of reference, van Inwagen insists that a consistent logic of relative identity be purged of singular terms. This, in turn, requires him to represent the logical form of (23) in a manner which does not include such terms. As before, the best way to appreciate the significance of van Inwagen's method for eliminating proper names is to compare it with Russell's method. Using the predicates defined above (including the RI-predicates for personal and ontic identity), here is the translation of (23) which is suggested by Russell's theory of descriptions:

Now compare (23.2) with van Inwagen's fully relativized translation of (23), first in its unabbreviated form and then in its abbreviated form:

The crucial difference between these methods for eliminating singular terms should now be clear: where Russell's method for eliminating singular terms employs an absolute conception of identity, van Inwagen's method employs a relativized conception of identity. For the sake of simplicity let us focus solely on (23.4). By translating (23) into the language of RI-logic we are able to refer to and describe these realities in a manner which does not involve an absolute conception of identity. Note that, like (23.1), (23.4) is a conjunction. Note, in addition, that as a closed and suitably qualified version of (1), (22) contains variables which are bound by the same quantifier. Thus, as an instance of (22), (23.1) employs singular terms which occur in both conjuncts. In contrast, the conjuncts of (23.4) contain only variables, but these conjuncts do not contain variables which are bound by the same quantifier. It is worth pointing out, however, that this last feature of (23.4) is not crucial to the success of van Inwagen's approach. For nothing prevents us from affirming a more compact version of this proposition:

• (23.5) (∃x)(∃y)(Jx & Sy & Pxy & ~Bxy).

Note that unlike (23.4), this proposition involves distinct RI-predicates which attach to variables which are bound by the same quantifier. Whereas (23.4) tells us both (a) that Jesus of Nazareth and the Son of God are the same person, and (b) that Jesus of Nazareth and the Son of God are the same being, (23.5) tells us, in the same breath, that Jesus of Nazareth and the Son of God are both distinct beings and the same person. Although we cannot consistently say that there is an x which is both human and divine, we can say that there is an x which is both personally identical with the Son of God and ontically distinct from the Son of God. For we have not assumed (and indeed we have denied) the thesis that personal identity dominates ontic identity.

I would like to conclude with four observations concerning the following propositions:

(1)

(22)

1. (23) Jesus of Nazareth is the same person as the Son of God, but not the same being as the Son of God.

(23.1)

• (23.5) (∃x)(∃y)(Jx & Sy & Pxy & ~Bxy).

First, although (23.1) and (23.5) are both translations of (23), (23.1) is an instance of (22) while (23.5) is not. Since (23.5) contains only individual variables, it is not a substitution instance of any quantified sentence.

Second, recall that Le Poidevin's regress argument begins with the observation that one who affirms relative identity is committed to true instances of (1). Since one who follows van Inwagen in affirming (23.5) is affirming this proposition in the place of (23.1), one who affirms (23.5) is not committed to the thesis that there are true instances of (1) or (22). Indeed, one who affirms (23.5) on these grounds should deny that there are true instances of (1) or (22). With RI-logic firmly in place, therefore, Le Poidevin's regress argument does not even get off the ground.

Third, one who thinks that (23.5) is true cannot say that this proposition is true in virtue of its substitution instances. Since RI-logic does not contain singular terms, this proposition does not have substitution instances. At the same time, its truth conditions are clear enough: it is true just in case its constituent descriptions succeed in referring to realities which are related in the specified manners. That is, this proposition is true if and only if Jesus of Nazareth and the Son of God are both distinct beings and the same person. While we are clearly not in a position to demonstrate the truth of this proposition, the question at hand is not what we are able to show, but rather what we are in a position to believe consistently.

Finally, it is worth pointing out that, like (1), (22), and (23.1), (23.5) involves multiple instances of anaphoric cross-reference. In particular, (23.5) contains recurring variables bound by the same quantifier which attach to multiple RI-predicates. It thus tells us, for example, that the x which is the same person as y is not the same being as y. At least at first glance, it looks as if this proposition is subject to the same worries as (1), which tells us that the very thing (x) which is the same F as y is not the same G as y. Does this mean, in short, that (23.5) is itself subject to Le Poidevin's regress, since it is implicitly affirming a statement of non-relative identity? It does not. My first inclination is to say that since these instances of cross-reference arise out of instances of singular reference and description which are free from the logic of absolute identity, it stands to reason that the resulting instances of cross-reference are themselves free from the logic of absolute identity. And yet it must be said that the statement of identity which arises out of this instance of cross-reference certainly appears to be a statement of absolute identity, since it isn't qualified by a sortal concept. Even so, there is no threat of a regress here, since we can always revert to (23.4), which does not contain recurring variables which attach to multiple RI-predicates. I thus conclude that van Inwagen's defence of the Incarnation is immune from Le Poidevin's regress argument, and more generally that it is possible for an advocate of relative identity who accepts the prevailing account of reference to avoid this regress.Footnote ¹⁶