Identity and necessity

January 7, 2017

This week I have been pondering the question of whether the God of the Philosophers (a being who is omniscient, omnibenevolent, omnipotent etc.) is the same being as the God of the scriptures (Eleanor Stump thinks he is, for example), and in particular the question of whether, if he is identical, he is necessarily identical. The philosopher Saul Kripke is famous for upholding the so-called ‘necessity of identity’ thesis, and for many years – I first studied his magisterial and influential Naming and Necessity in 1979 – I thought I understood what his argument was. Now I am less sure.

My puzzle is that there are various ways we can get to the thesis of the Necessity of Identity, yet Kripke apparently accepts none of these. The thesis was first proposed (as far as we know) by modal logic pioneer Ruth Barcan Marcus in 1947 (‘Identity of Individuals in a Strict Functional Calculus of Second Order’, JSL 1947 12-15), although her paper was nearly rejected after Quine, the reviewer, found her methods ‘laborious and often rather obvious, while she seems to avoid the more difficult and interesting questions’. Quine later published a less laborious demonstration of the thesis in 1953 (‘Three Grades of Modal Involvement’ JSL 1953, 168-169), which involves just two assumptions, namely the Principle of Identity, that necessarily a=a, and Substitutivity, that a=b and Fa implies Fb. From these two it clearly follows that if a = b, then necessarily a = b. (Hint: let ‘F’ be ‘necessarily a = —’, start with Fa, and substitute ‘b’ for ‘a’).

That is all clear and good. The problem is that Kripke doesn’t want to assume Substitutivity. He questions the universal substitutivity of proper names (N&N p.20), and as is well known he agrees with Frege that the identity of Hesperus the evening star and Phosphorus the morning star had to await discovery by a scientist (Pythagoras) and is thus not knowable from first principles. So ‘It is true from first principles that Hesperus is Phosphorus’ is false, yet ‘It is true from first principles that Hesperus is Hesperus’ is true! If we can change the truth value of the statement simply by substituting a name for the same planet, how can Substitutivity be true?

Nor does he want to assume that ‘Hesperus is Phosphorus’ is true in virtue of its meaning. He says (N&N p.20) that some critics of his doctrines, (‘and some sympathizers’) have taken him to be implying

that a sentence with ‘Cicero’ in it expresses the same ‘proposition’ as the corresponding one with ‘Tully’, that to believe the proposition expressed by the one is to believe the proposition expressed by the other, or that they are equivalent for all semantic purposes. Russell does seem to have held such a view for ‘logically proper names’, and it seems congenial to a purely ‘Millian’ picture of naming, where only the referent of the name contributes to what is expressed. But I (and for all I know, even Mill) never intended to go so far. My view that the English sentence ‘Hesperus is Phosphorus’ could sometimes be used to raise an empirical issue while ‘Hesperus is Hesperus’ could not shows that I do not treat the sentences as completely interchangeable.

This blocks a second route to the necessity of identity. Clearly if ‘a=a’ means the same thing as ‘a=b’, and if ‘a=a’ is necessarily true in virtue of its meaning, then ‘a=b’ is also necessary, since it expresses exactly the same proposition. But Kripke does not endorse such equivalence of meaning. Why then does he believe that the necessity of identity is a universal principle?

It is surprisingly difficult to find any positive argument in his work. Most of his well-known arguments are negative ones, demonstrating that apparent exceptions to the necessity principle are not exceptions at all. For example, we can suppose a situation in which some planet other than Hesperus was called ‘Hesperus’. But that would not be a situation in which Hesperus itself was not Phosphorus (N&N p.108). The only positive argument I could find is this:

If names are rigid designators, then there can be no question about identities being necessary, because ‘a’ and ‘b’ will be rigid designators of a certain man or thing x. Then even in every possible world, ‘a’ and ‘b’ will both refer to this same object x, and to no other, and so there will be no situation in which a might not have been b. That would have to be a situation in which the object which we are also now calling ‘x’ would not have been identical with itself. Then one could not possibly have a situation in which Cicero would not have been Tully or Hesperus would not have been Phosphorus. (‘Identity and Necessity’ p. 154, there is a similar argument in N&N p.104).

Let’s unpack this. Kripke’s notion of a rigid designator is clear enough, and is an important contribution to the philosophy of language. A rigid designator is one which designates the same object in every possible world. Or if you don’t like ‘possible world’ talk, a term which designates the same in a proposition prefixed by a modal operator like ‘it is necessary that’ or ‘it is possible that’ as when not so prefixed. Then his argument looks like this:

1. Let ‘a’ rigidly designate a and ‘b’ rigidly designate b
2. Suppose a=b
3. Then there is a single thing x, such that x=a and x = b
4. ‘a’ designates x and ‘b’ designates x
5. If ‘a’ designates x rigidly, ‘a’ designates x in every possible world, likewise ‘b’
6. If ‘a’ and ‘b’ designate x in some possible world w, and not a=b, then not x=x
7. Therefore a=b in w
8. But w was any possible world. Therefore, necessarily a=b.

Steps 1 and 2 are suppositions, step 3 follows from 2 by the nature of identity. Step 4 I will discuss shortly. Step 5 follows from the definition of rigid designator, step 6 probably requires further assumptions, but looks OK. Step 7 follows by contradiction, and step 8, the conclusion, by the principle that if we can prove p for any arbitrary w, then p holds for every w.

Let’s return to step 4, as I promised. We agree that ‘a’ designates a. And that a=x, by assumption. Why on earth would it follow that ‘a’ designates x? Well, let F be ‘‘a’ designates —’, so ‘Fa’ says that ‘a’ designates a. And let x=a. How do we get from Fa and x=a to Fx? By our old friend Substitution, no less. Yet Kripke claims to reject the universal applicability of Substitution. He could argue that it simply fails to hold in this case, but the thing about being a logical principle is that if it fails in even one case, it has to fail in every case, unless we can find a sufficient reason why it fails in that case, but then of course the principle has to include that reason. Heavy stuff.

Ed Buckner
Ed Buckner studied and taught philosophy at the University of Bristol in England. He has a number of publications in the area of both analytic philosophy and medieval logic and philosophy. He is the author, with Jack Zupko, of Duns Scotus on Time & Existence, a translation of an early work by the philosopher-theologian Duns Scotus, with a comprehensive and detailed commentary. Now mostly retired, he curates the Logic Museum, a collection of primary sources in the history of logic.

2 thoughts on “Identity and necessity”

  1. About your original question at the top of this post, is this a question about words or about things? Seems to me it must be the former: e.g. does “Deus” (used by Anselm) co-refer with the term “ho theos” used by Paul?

    Taken as theoretical entities, one might think Deus and ho theos must be different, as the definitions differ – but of course two people can theorize about the same being, and one gets it more correct than the other.

    I guess I’m trying to get a grip on the religious or theological import of all of this…

  2. “How do we get from Fa and x=a to Fx? By our old friend Substitution, no less.”

    Let’s see… substitution is that necessarily, if two terms co-refer, then substituting one for the other always preserves the truth of the original sentence. Hasn’t this been shown false by people like Cartwright? e.g. “Superman is so-called because of his powers” (true) and “Clark Kent is so-called because of his powers.” (false)

    But intuitively, necessarily, if “a” refers to the object a, and x *just is* a, then “a” also refers to x. Maybe I’m assuming here that names primarily or only refer, that they’re not encoded descriptions…

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