Version: v1, Published online: 1998
Retrieved May 11, 2021, from https://www.rep.routledge.com/articles/thematic/trinity/v-1
3. Relative identity
The originator of this approach to the logical problems raised by the doctrine of the Trinity is Peter Geach (1977; Geach and Anscombe 1963), who has developed a theory according to which ‘identity is always relative to a sortal term’, which he has applied to the problems of counting and predication that confront the doctrine of the Trinity. Geach’s work has been continued by Martinich (1978) and van Inwagen (1988). The exposition that follows is a composite of things said by these three authors.
The ‘theory of the relativity of identity’ proceeds from the axiom that there is no such relation as numerical identity simpliciter: there is rather an indefinite number of relations expressed by phrases of the form ‘is the same N as’, where N represents the place of a count-noun. There are, for example, such relations as ‘is the same horse as’ and ‘is the same apple as’, but there is, strictly speaking, no such relation as ‘is the same as simpliciter’ or ‘is numerically identical with’. Identity simpliciter (expressed below by ‘=’) is defined by two characteristics: it is universally reflexive (everything bears identity simpliciter to itself) and it forces absolute indiscernibility (this characteristic is embodied in Leibniz’s Law or the principle of the indiscernibility of identicals: if x=y, then anything whatever that is true of x is also true of y). Relative-identity relations, however, are not in general universally reflexive. (Socrates is not the same horse as Socrates because Socrates is not the same horse as anything; that is to say, Socrates is not a horse.) Relative-identity relations, moreover, cannot be assumed to force absolute indiscernibility – although any given such relation may have this feature. If it were assumed that every relative-identity relation forced absolute indiscernibility, then the logic of relative identities would simply be a fragment of the standard logic of identity simpliciter, and anything that could be said by using relative-identity predicates could be said equally well without them. (If every relative-identity relation forced absolute indiscernibility, then ‘x is the same N as y’ could always be replaced by ‘x is an N and x=y’.)
The logic of relative identities is easily described. Its language is that of first-order predicate logic (without ‘=’ and the description operator, and without singular terms), its two-place predicates being partitioned into two classes (somehow visibly differentiated), the ‘ordinary’ two-place predicates, and the ‘relative-identity’ predicates. Its rules of inference are those of ordinary predicate logic, plus two rules that state, in effect, that relative-identity predicates express symmetrical and transitive relations. Relative-identity logic must do without anything corresponding to Leibniz’s Law, for the reason outlined above. It must also do without singular terms. This is because a singular term is supposed to denote exactly one object (if it does not fail of denotation), and the concept of a singular term therefore involves the notion of identity simpliciter. (If a denotes x and also denotes y, it follows that x=y.) If, however, relative-identity logic is to have any power to represent ordinary, informal reasoning, its users must be able to employ some substitute for singular terms. This can be done through the use of an adaptation of Russell’s Theory of Descriptions. For example, ‘The present pope is bald’ could be read as ‘There is an x such that [x is at present a pope, and, for any y (if y is at present a pope, then y is the same man as x), and x is bald].’ There is, of course, nothing special about the word ‘man’ that dictated its use in this sentence; we might as well have used ‘person’ or ‘animal’ or any of indefinitely many other count-nouns that would apply to anyone who was a pope. The sentence obtained by substituting ‘person’ in the above sentence is not equivalent in relative-identity logic to that sentence; to deduce either from the other, one would need a premise not endorsed by relative-identity logic. For example: ‘For any x and for any y, if x is a man (that is, if x is the same man as something) and if y is a man, then x is the same person as y if and only if x is the same man as y.’ No doubt most people would say that this proposition was true, but it is of the essence of the theory of the relativity of identities not to regard such propositions as truths of logic.
The customary term for ‘what there is one of’ in the Trinity is ‘substance’. (But Geach and Martinich use ‘God’ for ‘what there is one of’ in the Trinity, and van Inwagen uses ‘being’.) The customary terms for ‘what there are three of’ in the Trinity are ‘person’ and ‘hypostasis’. (The relation between the meaning of ‘person’ in Trinitarian theology and ‘person’ in ordinary speech is a matter of dispute.)
All of the propositions of Trinitarian theology that raise logical problems can be represented using two relative-identity predicates (‘is the same substance as’ and ‘is the same person as’), a predicate that expresses the divine nature (‘is a God’ or ‘is divine’), and some predicates that express the relations that individuate the Father, the Son and the Holy Spirit. (The three persons or hypostases have traditionally been held to be individuated by the relations they bear to one another: the Father begets the Son; the Holy Spirit proceeds from the Father and – or through – the Son.) For example, the proposition that there are three divine persons can be expressed as ‘There exist x, y and z, all of which are divine and are such that none of them is the same person as the others, and such that anything divine is the same person as one of them.’ The proposition that there is one God (one divine substance) can be expressed as ‘Something is divine and anything divine is the same substance as it.’ These two sentences are consistent in relative-identity logic. The proposition that God is omnipotent can be expressed as ‘Something is divine and anything divine is the same substance as it and it is omnipotent.’ ‘Reference’ to the Father, the Son and the Holy Spirit can be accomplished by a device similar to the one that was used to ‘refer’ to God in the preceding sentence; in applying this device, use must be made of the predicates that express the relations that individuate the Father, the Son and the Spirit. Van Inwagen has shown (by constructing a model in which the interpretations of these sentences are true and in which ‘is the same person as’ and ‘is the same substance as’ express symmetrical and transitive relations) that the formal analogues of the whole set of logically problematic sentences endorsed by the doctrine of the Trinity are consistent in relative-identity logic. One striking consequence of this result is that the formal analogues of the sentences ‘The Father is the same person as God’, ‘God is the same person as the Son’ and ‘The Father is not the same person as the Son’ are consistent – and this despite the fact that ‘is the same person as’ expresses a transitive relation. (Needless to say, the formal sentences do not have the logical forms suggested by the English sentences they are held to translate.)
The main problem facing this account of the ‘logic’ of the Trinity would seem to be whether it is intelligible. Is it, in the final analysis, intelligible to suppose, for some x and for some y – where x and y are both substances and both persons – that x is the same substance as y, but not the same person as y? Alleged non-theological cases in which x is the same N as y, but not the same M (the statue is the same lump of clay as the vase, but not the same artefact; Dr Jekyll and Mr Hyde were the same man but not the same person; James I of England and James VI of Scotland were the same man but not the same monarch) are all susceptible of lucid and plausible philosophical analyses that do not presuppose that ‘identity is always relative to a sortal term’.
van Inwagen, Peter and Dan Howard-Snyder. Relative identity. Trinity, 1998, doi:10.4324/9780415249126-K105-1. Routledge Encyclopedia of Philosophy, Taylor and Francis, https://www.rep.routledge.com/articles/thematic/trinity/v-1/sections/relative-identity.
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