Facts and Granularity

We have already noted that (it is natural to say that) the fact that Clark Kent is Superman is distinct from the fact that Clark Kent is Clark Kent. And this seems to reflect, in some way, the fact that our proposition 'Clark Kent is Superman' involves two different concepts, modes of presentation, name-uses or internal meanings, whereas 'Clark Kent is Clark Kent' just involves the one.

As we have argued, these things (let's call them concepts for now) can be individuated at different granularities. This raises the question: should we say that facts can be individuated at different granularities as well?

We will now consider a case which strongly suggests an affirmative answer. We will then consider the further question: does the individuation of facts go hand-in-hand with that of propositions? Consideration of a familiar case will be seen to suggest that it does not: there are contexts where we can naturally distinguish two true propositions which we might count as the same at a coarser granularity, but where it is unnatural to distinguish two facts corresponding to the propositions so distinguished.

Here is the case which suggests that facts can be individuated at different granularities. An untravelled German called Pieter who is, like Kripke's Pierre in France, ignorant of English, uses the name 'Uebermensch' for the hero we know as 'Superman'. He has never heard the term 'Superman', but is privy to the fact that the hero has two guises, and knows that he is called 'Clark Kent' in the non-hero guise. So he assents to 'Clark Kent ist Uebermensch'. He knows, from reading and testimony, quite a bit about the hero, but doesn't know much about his appearance in his hero guise – has never seen a picture, heard or read a detailed description, etc. Let us suppose further that he assumes correctly there must be some name, unknown to him, which is used for the hero in his hero guise, but has no idea what this might be.

In a Pierre-like development, Pieter comes to America and comes into regular contact with Superman. He learns the name 'Superman', and uses it to refer to Superman. But he doesn't realize that this is the hero with two guises, the hero he already knew about in Germany. He just assumes this hero whom he knows as 'Superman' only appears as a hero. He has learnt some English, including the words 'super' and 'man', but just hasn't put two-and-two together.

When Pieter is talking one day with someone privy to the business of Superman having two guises, this person makes some remark, intended to be quite trivial, beginning: 'Even though I realize that Clark Kent is Superman, when ever he wears that suit, I…'. The penny drops. Pieter bursts out with 'Clark Kent is Superman?!', and thinks to himself ('Superman ist Uebermensch!'). The person looks at him, and says 'I didn't realize you weren't aware of that fact'.

I will now explain why I think this case shows that it is natural to individuate facts at different granularities, given different descriptive needs. Before the Pierre-like development, when Pieter was in Germany, we would, on the basis of what we have supposed about him, find it natural to say that he knows who Superman is (although he doesn't know the name he is called by in America, and doesn't know what he looks like), and that he is aware of the fact that Clark Kent is Superman. Nevertheless, as the story develops above, someone has the opportunity to, apparently quite properly, say 'I didn't realize you weren't aware of that fact' right after his outburst, which was 'Clark Kent is Superman?!'. Here, it is natural to say that he became aware of the fact that Clark Kent is Superman, and also of the fact that Superman is the hero he knew as 'Uebermensch'. In this latter connection, he might say: 'I am very surprised at the fact that Superman is Uebermensch – I never even considered the possibility, although I should have worked it out!'.

I submit that the best way of making sense of this situation is to embrace the idea that we carve up facts at different granularities. If we consider Pieter in Germany, without any inkling of what is to take place later, we find it expedient to use a granularity coarser than the one we will end up at, and we take Pieter's sentence 'Clark Kent ist Uebermensch' to state the same fact we state with 'Clark Kent is Superman'. (Although, if we so much as consider speaking of 'the fact that Uebermensch is Superman', we will begin to want to shift to a finer granularity.) Then, once Pieter goes to America, it becomes expedient to shift to a finer granularity and distinguish more facts: there is the fact that Pieter stated in Germany with 'Clark Kent ist Uebermensch', which we might call the fact that Clark Kent is Uebermensch, and there is the fact which now surprises Pieter in America, which we might call the fact that Clark Kent is Superman. It also seems natural to say that Pieter puts knowledge of these two facts together and immediately comes to know a third, which we might call the fact that Superman is Uebermensch.

If we do individuate facts at different granularities, the question arises whether this goes hand-in-hand with the individuation of proposition-meanings. There are really two questions here. It is expedient to operate at different granularities under different circumstances. One thing we can ask is: is it the case that, in any given circumstance, if it is expedient to distinguish two true proposition-meanings, is it also expedient to distinguish two facts, and vice versa? Another thing we can ask is: is it the case that, if it is expedient in some circumstance to distinguish two true propositions meanings, it is expedient in some (possibly distinct) circumstance to distinguish two facts, and vice versa?

It seems to me that the answer to the first question is 'no'. I do not know what to think about the second question, and remain agnostic. I think the 'vice versa' parts hold in both cases; if ever it is expedient in some situation to distinguish two facts (and provided these facts are expressible by propositions at all), it will be expedient to distinguish two corresponding true propositions, and expedient in the very same circumstance at that.

The answer to the first question is 'no', I argue, because, while the vice versa part holds (granted the expressibility of the facts), the first condition doesn't – i.e. it is not the case that, in any given circumstance where it is expedient to distinguish two true propositions, it is expedient to distinguish two corresponding facts. Kripke's original Pierre case gives us an opportunity to see this.

Suppose that London is pretty. (Worries about the subjectivity or indeterminacy of London's prettiness – in short, about there being no fact of the matter, can be easily avoided with a simple alteration of the case.) Now, as we saw in 'Kripke's Puzzle and Semantic Granularity', once Pierre comes to London and forms a second, unconnected conception of it, it becomes expedient to distinguish the proposition he expresses with 'Londres est jolie' from the proposition he would understand by 'London is pretty' – he believes the first, and disbelieves the second. But in this case, as it stands, there is no pressure to distinguish two facts – on the contrary, this is not natural at all. There is just one fact which makes these two propositions true: the fact that London is pretty. Pierre, we might say, is aware of this fact via one mode of presentation (or concept), via one belief-content, but also has another belief-content, involving another mode of presentation, which directly contradicts (or is made false by) this fact.

So, to sum up the preceding discussion: we have reason to think that facts can be carved up at different granularities (the Pieter case), but that their carving up does not go hand-in-hand with that of true propositions (the Pierre case).

In future posts I will discuss skeptical worries about facts, beginning with Strawson's pronouncements on the matter in his famous article 'Truth'.

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'!

On the Problems of Reference and Intentionality II

The following remarks were written when I was wrestling with these problems over an extended period. They might act as a stimulus to someone. Part I is here.

Part II

24. What do I mean when I say 'reference is no ordinary relation' in (22)? Perhaps the point would be better put in terms of reference-ascribing propositions.

25. I have long had the feeling that the desire for a classical analysis of reference ('x refers to y iff ...') comes from within a way of looking at things which involves something which might be called a 'transcendental illusion'.

26. For some reason I want to say: remember, in reference-talk, we are, as always, presupposing a connection between our concepts and their objects. But what this means, if anything, is very unclear.

27. There is the urge to say something like: 'Remember, you can't get outside your own mind'. But what the hell kind of reminder is that?! Who would ever think otherwise? (Satisfaction with such remarks belongs to a relatively low level of philosophizing.)

28. I can imagine someone proposing a two-component theory of reference which makes use of a "disquotational base" together with "equivalence criteria". So when I say 'my concept of John is of John' this comes from the base, and so is similar to 'I have a concept of John, who exists'. But I can also say, for example, this name 'X' seems to be refer to John, this being similar to: This word is used in the same way as my 'John'.

29. We might say: Reference is that relation which is presupposed when anything is spoken of.

30. This ignores some complications (e.g. fiction) but is more appropriate, I think, than:

(1) Existence is that property which is presupposed. and

(2) Identity is that relation which is presupposed

because that distorts the logical form of existence and identity propositions. Reference is more properly a relation (i.e. more properly a property-or-relation, i.e. thing which can be ascribed to things).

31. What do I mean by 'more properly a relation' here? Here is one way to explicate this. Imagine a form of representation where objects are represented with dots or boxes, which are then labelled with names (or otherwise made intrinsically unique). Properties can be indicated by labelled lines which have one point of contact with dots or boxes, two-place relations with labelled directed lines which have two points of contact, etc.

With such a technique, an existence property-representation and an identity relation-representation would serve no purpose. Every object would have exactly one point of contact with an existence line, and exactly two points of contact with a single reflexive identity line.

32. This is, of course, a consequence of the way I am imagining this mode of representation to work. There are of course more sophisticated possibilities, where one uses dots and boxes at a higher semantic level, so to speak - as representations of ideas of objects rather than objects themselves. In that case, one can meaningfully use something like existence and identity lines - but then aren't they more properly seen as ascriptions of the property of designating something and the relation of codesignating respectively?

33. Now, what about reference? Here we can start to see that we can sort reference propositions into classes, depending on how they work and get verified. One distinction we can make is between reference propositions which concern expressions belonging to the same language-system, and those which talk about expressions from another language-system. Let us call the former 'intrasystematic' and the latter 'extrasystematic'.

34. Now we may make a further distinction between disquotational intrasystematic reference propositions, and non-disquotational. (Some of the latter can be used to give all kinds of information, e.g. '”The winner of the race” refers to John' can inform someone that John won the race.)

35. Consider what happens when we use the graphical mode of representation described above, and a new (extralinguistic) object comes to our attention. We draw in a new box or dot (let us say "node" from now on).

But in so doing, we bring into existence a new object - one which stands in the reference relation to the original object. We can represent this fact now too. One way of going would be simply to draw in another node, which represents the last node drawn, and then to connect it to that node with a line indicating the reference relation. But of course this procedure can then be repeated for the new node. This procedure corresponds roughly to giving a name its own name in word language.

36. Another way of going, which corresponds roughly to quotation in word language, would be to introduce a kind of operator on points of contact. An unmarked point of contact is the ordinary case, a marked point of contact indicates that the contacted node is representing itself.

37. Consider the first technique, where we start at the bottom level, and then construct nodes to represent those nodes, etc., as required. This process thus "automatically generates" reference propositions. These automatically generated ones are the disquotational intrasystematic ones.

38. Also, among extrasystematic reference propositions, two kinds of verification criteria can be distinguished. Direct comparison with our systems (looking at arithmetic talk for instance), vs. coordination which "involves the object" more. (The field-linguist would be doing both of these things.)

39. There is a close connection here to the Twin-Earthable/Non-Twin-Earthable distinction. (Perhaps it is that the direct comparison verifications yield propositions about the referents of non-Twin-Earthable expressions, and the “object-involving” ones yield propositions about the referents of Twin-Earthable expressions, but that may be an oversimplification.)

40. Consider Idealism here. On Idealism, perhaps all extrasystematic reference criteria can be reduced to system-coordination. A kind of collapse of Twin-Earthability.

41. For example: I see a man point at a rock and say 'N', and I form the hypothesis that 'N' is a name (which is not part of my system) representing the rock I see in front of me. This is a paradigm case, it might seem, of acting on criteria for extrasystematic reference which cannot be reduced to system coordination. I coordinate a part of his system directly with its object, not with a part of my system.

42. An Idealist may insist that this is not so. Rather (they may say), I am correlating my perceptual representation of the rock with something. But it would seem that if we are to be consistent, we can't really say that the perceptual representation of the rock is correlated by us with the name 'N', since 'N' is not part of our system. Mustn't we now say that we correlate our perceptual representation with our representation of the extrasystematic name 'N'?

43. But still, this does not give the Idealist the distinction I want to have between external-object-involving verifications of extrasystematic reference-propositions and verifications of extrasystematic reference propositions which involve "merely internal" comparison of systems. For in the internal case they also have to talk about our representations of some extrasystematic expression.

44. Why is this interesting? Not 'in case Idealism is true'!

45. When '”A” refers to B' is not disquotational, it seems that for practical purposes it means the same as the statement that 'A' and 'B' codesignate, and can therefore be understood as being verified by correlation of (aspects of) the role of two signs.

46. x refers to y iff x codesignates with 'y' iff x's referent = y.

47. But not all reference propositions which look like disquotational ones are such. For example: '”N” refers to N in German too'. Or when setting up a new language, as in formal Peano arithmetic: '"0" refers to 0'.

48. We should take note of the singularity of a truly disquotational reference-thought.

49. In philosophy, we think: 'N' refers to N. Then we think: how?

50. Now, that first thought is a thought to the effect that a certain symbol stands in the reference relation to a certain object. That much is clearly true to say.

51. Compare the thought, as had by an English speaker, that 'Deutschland' refers to Germany. This too can be truly said to be a thought to the effect that a certain symbol stands in the reference relation to a certain object. Likewise the thought that 'John' refers to the man I met yesterday.

52. But clearly the first, disquotational thought is a very different beast from the latter ones. What worries me is the effect of their assimilation under the rubric: reporting a reference relation between symbol and object. Reference propositions are propositions which report such relations (connections), reference thoughts are thoughts that such relations hold.

53. We might want to say that the disquotational reference propositions and thoughts are a subset of all reference propositions and thoughts, characterized by the fact that the symbol which the thought is about is also used to represent the related object. And now we may ignore this subset and concentrate instead on the non-disquotational subset.

54. There is a problem with this formulation, however. For example, suppose we are told that someone used a certain expression which itself is named E, but one does not know which expression E is. One may learn something about E, namely that it refers to Venus. Suppose E is actually 'Venus', the same word used in the same sort of language system. In that case, on my formulation above, the thought that E refers to Venus would be classed as disquotational, even when the thinker doesn't know that E is the name 'Venus'.

53. So it seems what we really wanted is: the disquotational subset is characterized by the fact that the same symbol is used twice over – to specify the referer, and the referent. But even this doesn't quite work, for '”The bearer of 'N'” refers to N' fulfills that condition.

54. The difference between disquotational and other reference statements is reminsicent of the difference between trivial (repetitive) and informative identities. However, the problem is in a sense inverted: the naive relational view of identity statements makes all instances look trivial, including the nontrivial ones, whereas the naive relational view of reference statements makes all instances look nontrivial, including the trivial ones. (This use of 'trivial' may not be fully warranted, but suffices for making the point.)

55. It is instructive to compare ' "N" refers to N ' with 'The word "N" has reference, and it is used in the way it is used.' Or even just: 'The word "N" has reference, and it refers to the object which does in fact refer to'.

56. What looks like (and in some sense is) a tautologous appendage changes the modal profile radically.

57. It certainly seems that having purely disquotational reference propositions in the mix can blind us to how the rest work, owing to the special direct way the former are verified. But would it be right to discount these as degenerate?

58. 'London' refers to London. This fact could have an interesting historical explanation. (Contrast 'London is London'.)

Part III coming soon.

On the Problems of Reference and Intentionality

The following remarks were written when I was wrestling with these problems over an extended period. They might act as a stimulus to someone.

Part I

1. What determines reference?

2. Not, in general, descriptive or conceptual content associated with the name. Although the usual counterexamples to descriptivism do not apply to a range of names - for example, names of mathematical objects.

3. Not, in general, a causal connection between the object and the name. Both because the kind of connection in question is unclear, and because there is clearly no such thing in many cases - e.g. mathematical objects. But there is no clear counterexample in a range of cases: names of spatiotemporal individuals which someone in the linguistic community at some time has been perceptually acquainted with, for example.

4. What determines reference? - What gives this question its power?

5. The question seems to be: given that someone is referring to a particular object, what determines which particular object that is?

6. This is to be distinguished from: What is it which characterizes all cases of representation of, or reference to, an object? I.e. what do they all have in common which distinguishes them from other phenomena, apart from 'involving representation of an object'? But it is close to: Given some particular object O, what characterizes all cases of representation of, or reference to, O?

7. The thing we just can't accept is that we have nothing definitive to say in answer to the question: what determines reference?

8. It is extremely remarkable and important that people have tried hard to say something definitive here. It not a matter of course that anyone is moved in this way by that question. (Many people innocent of philosophy would find the whole thing quite unnatural and meaningless.)

9. I would like to talk about the feeling that, when the investigation goes empirical and scientific, we have left the realm of essence, so to speak. Is that right, anyway? Could one not maintain that by increasing our knowledge of the contingent facts - how reference actually gets going - we may be better able to penetrate to the essence? I.e. to what it takes in principle for reference to get going?

10. Well, historical development itself cannot be part of the essence. It is clear enough that there is no logical impossibility in a whole lot of organisms being spontaneously generated out of nowhere and beginning to speak of things.

11. When we think of the representation of external, physical objects, some kind of correspondence or covariance theory seems attractive. One thinks of primitive representation (or proto-representation) which is of this kind.

12. However, the merest thought of arithmetic, and Wittgenstein's builders, at once makes all this look inessential.

13. There is no reason why one cannot erect a concept which applies, for example, only to the representation of physical phenomena. But if one is inclined to say that we refer also to abstracta, and to wonder how reference in general is possible, then an investigation of the first concept alone will never satisfy.

14. It is clear enough that the idea of referring, which is primitively connected with the ideas of looking-at and pointing - the picture of representation and represented - is widely useful, and widely used.

15. Now, it looks very much as though one might, having noticed this, ask for an explanation of why it is. I.e., what makes the picture of representation and represented apply in this wide range of cases. Or better: what is it about all these cases which makes the picture of representation and represented apply to them?

16. This traces the problem down quite deeply. At this point we see that none of the common 'theories of reference' is going to help us here.

17. No answer comes to this question. This seems to have something to do with the fact that, once you are talking about a potential object of reference, you have already prejudged, so to speak, that the picture of representation and represented applies. Wherever we think of an object, there already we have the material for a higher-order thought about our representation of that object.

18. 'The problem of the essence of the representation of things leads, in a certain sense, to the problem of the essence of things.' The problem of the essence of things is obviously no ordinary one! 

19. To explain this tracing again: whenever we find ourselves in conditions where we can refer to an object, we thereby find ourselves in conditions where we can apply the picture of reference.

20. But, this is not the same as saying: wherever the conditions for objecthood are fulfilled, thereby are the conditions for a representation. For this is refuted by imagining a world in which there is just one object and nothing else. (This sort of distinction is important all through logical philosophy. It appears to be neglected, for instance, when people try to analyze existence and identity statements as designation and codesignation statements respectively.)

21. Two senses of truth conditions, two conceptions of what I will call fulfilment worlds for representations:

(1) All conditions/worlds in which what the representation actually represents to be the case is the case.

(2) The narrower set of all conditions/worlds in which the representation exists, represents what it actually represents, and where what it represents is the case.

22. What I was saying above now comes to: all the fulfilment worlds in the second sense for 'x exists' are automatically fulfilment worlds in the first sense for '"x" refers to x'. Or: once 'x exists' is fulfilled in the second sense, '"x" refers to x' is fulfilled in the first. (This holds not just for 'x exists' but for any proposition which contains 'x' in an extensional context.)

22. By dwelling on this we can start to see that reference is no ordinary relation.

23. As long as one is prepared to ignore the distinction between (1) and (2), or to openly switch from one to the other mid-analysis, then one can analyze existence and identity both in terms of reference. (Early Frege, Geach.)

Part II.

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.)