An Attempt to Diagnose the Disagreement over the Relational Explanation of Identity

Here is what I have to say in response to the schmidentity challenge as posed to the sui generis view of identity statements. (See also these two related posts from several years ago.)

OK, so we can grant that you can introduce a 'schmidentity' predicate in the way Kripke describes. We can also grant that this predicate could then get used to do what we do with identity statements. But can we, having granted these things, nonetheless deny that the meaning and function of identity statements is explained with the object-relation story?

I am strongly inclined to do all of this. Why? Because the characteristic function of informative identity statements and their denials - the way they get us to merge and separate mental files, or concepts of individuals - is passed over in this explanation. Going along with the object-relation explanation seems to render this incidental, instead of the main point. That explanation makes it look as though the main function of an 'a is b' statement is also fulfilled by the corresponding 'a is a' statement, which of course it is not.

But, someone may argue, does the object-relation explanation really create this false appearance? And here it would be easy to be dogmatic. There would be something silly about insisting that yes, this sort of explanation really does create this false appearance. After all, my opponent - the philosopher who wants to say that the object-relation story is perfectly adequate, and that there's no problem here, and that anyone who thinks there is is in a muddle - doesn't actually seem to be confused about the fact that 'a is b' statements are often useful in a way that the corresponding 'a is a' statements are not. They would happily admit that. So the difference between us seems to be in whether we are happy to leave this out in our primary explanation, so to speak, of identity statements.

And it is important that I allow that the object-relation explanation of identity statements does show something. It's not as if it is a sheer mistake. It shows that we can so to speak depict identity statements as a special case of relational statements, i.e. statements like 'John loves Mary'. I do not want to deny this, or deny that it is of philosophical interest.

There is something neat or cool about this sort of observation, too. It has a charm to it, similar to the charm possessed by clever hacks (in the sense of computer culture). I think that the philosopher who wants to defend the object-relation story lacks a proper place to put this. They feel the charm, the strikingness, of the explanation, and - not wanting this to elude them - wrongly place it in the "primary explanation" place in their thinking, instead of a place marked something like "striking and potentially instructive thing you can say". So long as we only focus on the "primary explanation" place, it looks like the defender of the object-relation story is missing something and proposing something maddeningly objectionable, but it also looks like the antagonist of the object-relation story is missing something. It is not until we consider other possibilities for the significance of the object-relation story that we are able to give both parties their due.

This, I now think, is a very important point (even though I may not have expressed it very well). I regret that I didn't manage to arrive at this point in my paper on this topic. I also think my predecessors were missing something in this regard.

So, we can grant the possibility of the schmidentity predicate, and the possibility of it coming to be used to do the characteristic work of identity statements, but nonetheless deny that the object-relation story should take pride of place in our explanation of the meaning and function of identity statements. A leftover question here is: should we also deny that the meaning and function of statements made with the 'schmidentity' predicate, if they are being used in the way we use identity statements, is explained by their stipulated semantics? And the answer, I think, is Yes. If they are being used in that way, then the object-relation story should not take pride of place in their explanation. But it is understandable that we should hesitate here, since 'schmidentity' was introduced and defined by means of the object-relation story, and this invites us to look at their use - when they are being used in the characteristic way we use identity statements - as a kind of secondary thing, a happy side-effect.

(I feel like saying something more at this point, which may be more objectionable, about what other use (schm)identity statements may have, apart from their practical use which has to do with merging and separating. A metaphysical use, so to speak. And about what attitude we take to this use, or whether it might be a kind of illusion. And this relates to one of the old posts linked to at the beginning. But I won't do more than make this hint, since these are treacherous waters and I wouldn't want to abuse the goodwill of a differently-minded reader.)

The Schmidentity Challenge (to the Sui Generis View of Identity Statements)

In my (2016) I defended the idea that identity statements are sui generis. More precisely, I defended the idea that identity statements involving proper names (e.g. 'Hesperus is Phosphorus') are not to be explained by the claim that they ascribe a relation which holds between all objects and themselves and in no other case, or for that matter by the claim that they ascribe a relation between names (this latter claim being false). In contrast to my predecessors who railed against the object-relation view, I did not insist that the object-relation claim is false - I decided this was not a very clear thing to insist on, and anyway not really the point - but just that it doesn't explain the meaning and function of identity statements. It may be "something you can say", but it doesn't do that explanatory job. I thought, and still do think, that this is the way forward for the philosopher who feels that there is something fishy about the object-relation view, something which remains even if we succeed in avoiding - most likely by means of senses or similarly-motivated semantic difference-makers - the absurd conclusion that 'Hesperus is Hesperus' and 'Hesperus is Phosphorus' mean the same.

I defended my negative thesis about the explanation of identity statements against some possible objections in the paper, but one unaddressed challenge I have been thinking about in the years since writing the bulk of the paper (it took a long time to get it published, and I stopped trying for a while) is Kripke's celebrated 'schmidentity' argument. Here it is:

Suppose identity were a relation in English between the names. I shall introduce an artificial relation called 'schmidentity’ (not a word of English) which I now stipulate to hold only between an object and itself. Now then the question whether Cicero is schmidentical with Tully can arise, and if it does arise the same problems will hold for this statement as were thought in the case of our original identity statement to give the belief that this was a relation between the names. If anyone thinks about this seriously, I think he will see that therefore probably his original account of identity was not necessary, and probably not possible, for the problems it was originally meant to solve, that therefore it should be dropped, and identity should just be taken to the relation between a thing and itself. This sort of device can be used for a number of philosophical problems. (Kripke (1980), p. 108.)

As you can see, the schmidentity argument is framed primarily as an argument against the name-relation view of identity statements, which I also argued against. But this argument also threatens my position. As I see it, the challenge is as follows. Kripke's schmidentity predicate is a term which is explicitly introduced - explained, it is natural to say - as ascribing a relation which holds between all objects and themselves and in no other case. So, whatever is true of identity statements, schmidentity statements can be - indeed have been - explained by means of the object-relation stuff which I wanted to say fails to explain the meaning of identity statements. But schmidentity statements could be used to do what we do with identity statements. So then what grounds have we for supposing that identity statements differ semantically from schmidentity statements? Perhaps none. But then if identity statements and schmidentity statements are semantically on a par, and the latter can (are) explained by the object-relation stuff, then so can the former. So now it looks like my position is wrong.

I think this is a serious challenge to my position (about the object-relation claim not being explanatory of identity statements), but I can't help feeling that it misses something and that my position is right in some way. I will try to respond to the challenge in my next post here.

References

Haze, Tristan (2016). On Identity Statements: In Defense of a Sui Generis View. Disputatio 8 (43):269-293.

Kripke, Saul A. (1980). Naming and Necessity. Harvard University Press.

Names and the Publicity of Meaning

One sort of consideration which may seem to augur for Millianism, and against both descriptivism and my view of names, comes from the idea that meanings must be public items, shared by communicators. If the subject matter of semantics is supposed to be the public meanings of linguistic expressions - where this might be conceived as the stuff we must have implicit knowledge of in order to be competent speakers - then it is hard to see what, in any given case, could be essential to using a name correctly, except for using it to denote the right bearer. On the other hand, there does seem to be a technique of using certain empty names like 'Santa Claus' which is more specific than: using it such that it has no bearer. But perhaps we want a minimal conception of semantics on which such specific techniques are regarded as extra-semantic.

Given such a minimal conception of semantics, it will be hard to avoid the conclusion that belief-contents, proposition-meanings and propositions and have more to their identity than their structures and the semantics of their components. (That is, unless we are prepared to bite bullets like: '”Hesperus is Hesperus” means the same as “Hesperus is Phosphorus”'.) And if we accept this, then we must deny that the identity of a proposition can always be reckoned as being determined by its structure plus the meanings of its parts, in the relevant minimal sense of 'meaning'.

We can reinstate compositionality either by moving to a very coarse-grained notion of belief-contents or propositions (and so biting the bullet on 'Hesperus is Hesperus' and 'Hesperus is Phosphorus'), or by moving to a finer-grained conception of the meanings of parts such as names, a conception which will include things beyond public meanings and minimal competence conditions. This latter is, in effect, what I advocate in my view of names as having uses, or being tied to individual concepts, which can differ even though the names do not differ as regards extension.

'Hesperus and Phosphorus' seems to be a different proposition – seems to mean something different from – 'Hesperus is Hesperus'. And, quite apart from any general thesis about meaning-determination, this difference seems like it has to be laid at the door of the names 'Hesperus' and 'Phosphorus'. And that is what my view of names enables us to do, while remaining invulnerable to Kripkean anti-descriptivist arguments.

Why can't we all go home, then? Well, this kind of solution seems to worry people. It seems like they can see what its virtues would be, but don't feel they can help themselves to it. I suspect that one of the major causes of this reluctance is some kind of conceptual intuition to the effect that meanings – anything worth calling 'a meaning' – have, by definition, to be public and shared by competent communicators. I suspect that another major cause, perhaps even more active, is that people have sensed that on this way of going, there won't be any general story to tell about how to count meanings – i.e. about how to determine whether to say that two expressions, or instances thereof, are synonymous or not.
 
We can appease the first worry to some degree, I think, by allowing that there is a natural conception, which it is not improper to use the word 'meaning' in connection with, according to which meanings, in order to be meanings, must be public and shared. But we can also have a richer, more idiolectic conception, and maintain that this is what we're talking about in connection with names, and the semantic difference between 'Hesperus and Hesperus' and 'Hesperus is Phosphorus'.

Furthermore, these two sorts of conceptions need not be seen as two utterly different things, but as continuous. Both deal with systematic use-patterns of signs, or roles of signs in systems. Taking the 'public and shared' conception as our starting point, we may yet ask: how public and how shared must these use-patterns or system-roles be to count as meanings? We could have a conception on which the patterns or roles must be shared by all who competently speak the language. But how do we individuate languages? Peter Ludlow's recent work on 'the dynamic lexicon' can help prepare the ground for what I am saying here, being consonant with it in important ways.

This appearance of continuity and fluidity is not some nasty imprecision in our philosophy, but a faithful capturing of the facts. People differ from each other – and from themselves over time – in their use of symbols and the way their understandings work, and in most cases, the question whether two symbol-instances align in meaning can be given different answers for different purposes. When we're talking about something we're both familiar with, and our ideas of that thing are similar enough, our talk can be said to align in meaning. But notice that, in speaking just then of ideas being similar enough, I have already hinted that there might be a finer granularity at which our ideas are not type-identical – a finer granularity at which it may be said that we don't mean exactly the same thing. This seems realistic.

Regarding the second worry, my answer is already implicit in the above; I think the most fruitful response is not to try to explain it away, but to embrace it. There is no single way of counting meanings, since we can individuate them and count them differently at different granularities. We are already pushed toward this by considering Kripke's puzzle, and its character as a solution there is only strengthened when we see it has further applications, such as here to this worry about the very idea that names have internal meanings, to questions about the individuation of facts, and elsewhere.

Notes on Identity (and the idea of a 'law of metaphysics' concerning it)

(I) Everything is identical with itself and with no other thing.

Consider what results when we try to express (I) in first-order logic:

(FOL I) (x) x = x  ~ (y)(y = x  ~ y = x)

For a closer articulation, let us introduce a function SelfOf() - the identity function – and a distinctness relation D, yielding

(FOL I 2) (x){ x = SelfOf(x)  ~ (y)(y = x  y D x)}

This shows what we are doing in the case where we use something like (I) to introduce 'identity' and cognates, e.g. making sure it's not taken qualitatively. We are presupposing, using, the notions of self and distinctness/otherness, and showing how identity connects with them; if we had fixed interpretations for 'D' and 'SelfOf', we could use (FOL I 2) to define '='. And we could of course reverse this, presupposing '='. (In the first use, we would naturally emphasise 'self' and 'no other thing', or just 'other', in pronunciation. In the second, we would emphasise the word 'identical'.) We can also presuppose nothing, and take these propositions as just specifying how these notions are to relate to each other.

All these construals can be called grammatical in Wittgenstein's sense. But now, is there some other way of taking (I)? It can certainly look that way in philosophy. ('Law of Metaphysics'.)

It can look as though (I) is ruling out worlds where the identity relation looks like this:

 

or like this:

and ruling in this sort of picture:

 

But no one wants to say that these worlds are real, or even possible. So then can't we – mustn't we - take this ruling in and out as grammatical too?

Somehow, instead of seeing it that way, philosophical thinkers try to give it another, impossible sort of application – or so I want to say. But what is this other application, and how am I to describe it without falling prey to the sort of confused thinking I am trying to correct?

What makes these mistakes, or confusions, hard to correct is their slippery, elusive character. (This no doubt has to do with the way so many of our key terms slip and slide so naturally between slightly different uses.)

It is not as if my opponent thinks their Law of Metaphysics is empirical, or anything like that. Rather, they have a spurious category for it, I want to say – or, at the very least, they spuriously categorize it.

They fail to make a category at the right level, so to speak – fail to regard identity statements as being sufficiently special in their fundamental workings. Instead, they are assimilated to other relational statements, giving rise to the conception of a special sort of fact (which we have some kind of special access to, perhaps).

A purported picturing of the world as being one way rather than another. A purported division in some space of representations. Also, perhaps some vague conception of an analogue of empirical verification – as it were, rational perception (cf. Plato, Goedel, the notion of an “eye of Reason”). The sort of thing no one would ever feel like positing for 'Bachelors are unmarried' (except perhaps in a heroic effort to be consistent).

How is it that this comes about here, and not, say, with 'Bachelors are unmarried'? I believe it arises from the mixture of two modes of representation. This should become clear in a moment.

* * *

It is odd the way (I) can inspire phrases like 'Law of Metaphysics' while 'Everything exists' is much less likely to. Of course, for the latter to be false, there would have to be things which do not exist, which is obviously contradictory. OK, but isn't that also true of the following natural description of “what it would take for (I) to be false”?: there would have to be things which do not bear the identity relation to themselves, or distinct objects between which it holds.

It is as though the two conflicting things here are less similar, a bit further apart in language, than in the existence case.

What if I had said 'there would have to be things which aren't themselves, or things which are other things'? That sounds more flatly contradictory; no one, unless doubly perverse, would formulate a “Law of Metaphysics” running 'Things are themselves'. (Butler's Dictum is not generally presented as a Law of Metaphysics.) The flavour changes markedly when we bring in 'identical', talk of 'bearing the identity relation' etc. That is where we begin to get the mixture of modes of representation.

A natural picture to illustrate, or put by, 'Everything is what it is' would be:

Things are represented as dots and are shown “just being themselves”. Whereas the natural picture to put by (I) is: 

 

And here we get the feeling of a contrast, of substance, of something being ruled out. Namely stuff like:

 

The mixture of two modes of representation here consists in the fact that each dot in the picture is taken to represent a different object, and yet lines are drawn indicating the identity relation – lines which could only be of use if two dots sometimes represented one object.

* * *

With identity statements, merely specifying the relation in question (identity) and the pair of objects involved in the ascription (assuming the names involved refer) fails to specify 'what is said' in any natural sense, however minimal. This problem cannot be avoided either by expelling repetitive identities from language – there can be multiple different non-repetitive identities concerning one and the same object.

If you find this strange or unacceptable, I suggest you have in the back of your mind a conception of relations which identity does not really fall under. Perhaps you are picturing something like the dots above, and imagining relations as encoding further information on top of that structure. This may be a good conception, in which case you should stop classifying identity as a relation, stop classifying identity statements along with propositions like 'John loves Mary'. This, rather than, e.g., moving to an unnatural conception of 'what is said' – which is, I think, what hard-line Millians such as Scott Soames do.

This move in semantics to an unnatural conception of what is said, then, may have an origin related to that of the conception of a Law of Metaphysics discussed critically above. Also, they can be made to support each other: if it's a 'substantial metaphysical fact' that everything is identical to itself, then a repetitive identity is an instance of this, and so perhaps it inherits some metaphysical substantiality for itself, in which case perhaps there is an informative, substantive extensional core to extensionally equivalent identity statements – something which they all say. Conversely, if we take this last thing for granted: what sort of thing does a repetitive identity say which an empirically informative counterpart also says? It had better not be something trivial, since the latter doesn't seem to say anything trivial, in any sense of 'say'. And so what sort of thing is this non-trivial thing which repetitive identities say? Perhaps we could call it 'an instance of a Law of Metaphysics'!

Identity Expressed with One-Place Predication

Introduction

Frege's famous paper On Sense and Reference begins with the question of whether identity is a relation. Frege then goes immediately on to ask whether it is a relation between objects, or their names. The latter question then sees most of the action.

This is a confusing issue. What is a relation, anyway? What does it mean to hold that identity statements ascribe a relation, as opposed to doing something else? Might there not be various ways of categorizing things, perhaps involing, or giving rise to, slightly different senses of 'relation'?

It is not my aim here to give an overall discussion of the main philosophical problems surrounding identity statements. My purpose is to show how easily we can modify first-order logic with identity (FOL=) so that identity statements are treated as one-place predications rather than two-place relational predications. Comparing the result with natural language identity statements such as 'Hesperus is Phosphorus' makes the occurence of 'is' look more like a copula ("the 'is' of predication") rather than a relation symbol (some special "'is' of identity"). Sentences like 'Hesperus is identical to Phosphorus' then look, by contrast, more comparable to the familiar '=' form in logic - that is, more like they contain a relation-symbol.

I had thought of this possibility before, at least in part, but it came forcefully to mind recently when I was reading Delia Graff Fara's draft paper, 'Names as Predicates'. The theory put forward there is sophisticated, but my basic thought was: if, as Fara argues, 'Hesperus is Phosphorus' is not an identity, but a statement attributing to Hesperus the property of being Phosphorus, then what do count as identity statements? Statements involving variables? But they can also be treated as one-place predications. Instead of saying these are not identity statements, why not let them be the paradigms of identity statements, and just say that identity statements can be construed as one-place predications? (For Fara, I think, an example of a genuine identity statement would be 'Hesperus is identical to Phosphorus' - cf. the paragraph above. That is, Fara makes it a requirement of identity-statementhood that the statement have a two-place relational syntax, whereas I don't wish to. This is a fairly unimportant terminological difference as far as I can see.) 

The Strategy

We make three modifications to the ordinary syntax and semantics of first-order logic with identity:

- Instead of having a special symbol '=' in our stock of two-place predicates, we add two pointy bracket symbols '<' and '>' to the vocabulary.

- Add the following clause to the recursive specification of the well-formed formulae: For all terms T, '<T>' is a one-place predicate. ('T' here is a syntactic variable, specifically a term placeholder.)

- Instead of mapping '=' to a set of repetitive ordered pairs - one for each object in the domain, containing that object twice, i.e. "the identity relation" construed extensionally - we add the following rule to the semantics: For any term T which has a referent, let the sole member of <T>'s extension be T's referent.

(Note on quantified formulae: this works most clearly with the style of semantics where one considers models which contain a new constant in place of the variables bound by the quantifier, but it also works with variable-assignment semantics, if we class assignments to variables as referents.)

Now, in place of, e.g., 'a = b', we write '<b>a'. In place of '∃x (x = x)', we write '∃x(<x>x)', etc.

Remarks

This way of doing things is interesting in that we can, in an important sense, say everything we said with '=', while using a language that doesn't suggest any talk about identity as a relation which holds between all objects and themselves. The illumination this affords is, I think, the sort of thing Wittgenstein was talking about when he wrote the following:

Each time I say that, instead of such and such a representation, you could also use this other one, we take a further step towards the goal of grasping the essence of what is represented. (Philosophical Remarks, sect. 1.)

I am also reminded, in an obscure way, of this unforgettable passage in Russell's Logical Atomism lectures:

There is a good deal of importance to philosophy in the theory of symbolism, a good deal more than at one time I thought. I think the importance is almost entirely negative, i.e. the importance lies in the fact that unless you are fairly self-conscious about symbols, unless you're fairly aware of the relation of the symbol to what it symbolizes, you will find yourself attributing to the thing properties which only belong to the symbol. That, of course, is especially likely in very abstract subjects such as philosophical logic, because the subject-matter that you are supposed to be thinking about is so exceedingly difficult and elusive that any person who has ever tried to think about it knows you do not think about it except perhaps once in six months for half a minute. (Logical Atomism Lectures, Logic and Knowledge, p. 185.)

On Identity Statements

I've just posted a longer article on my homepage, On Identity Statements: Against the ascriptional views.

Apart from minor revisions, it is about 18 months old now. I would not write it in the same way now, but I still hold the views expressed there. Comments welcome, here or by email (my email address is on my homepage and on the 'About/contribute' page here).

UPDATE 21/06/2016: I have removed the link, as a descendant of this paper called 'On Identity Statements: In Defense of a Sui Generis View' has finally been accepted for publication.