Saturday, 13 August 2011

On What "Is" Is

Continuing the anti-reductionist theme in the previous post.

Some metaphysicians have wanted to reduce "is" to something else (e.g., Saunders 2003, Ladyman 2005). By "is", I mean "is" as in
6 + 9 is 15
Sir Walter Scott is the author of Waverley.
I don't mean "is" as in
Frank Plumpton Ramsey is the Amy Winehouse of 20th century philosophy
Tuna is the chicken of the sea.
Mathematicians use the symbol "=" for "is", so the question is what is "="? (Maybe, better: what is =?). The most popular answer has usually been: "=" is indiscernibility. More exactly,
$x = y$ iff $x$ and $y$ have all the same properties.
One direction of this (right-to-left) is known as Leibniz's Principle of Identity of Indiscernibles. It can be given a formalization as follows:
(PII) $x = y \Leftrightarrow \forall X(Xx \rightarrow Xy)$
This definition of $=$ is fine with me, but it is a bit conceptually circular (see Savellos 1990 for a useful discussion). For its truth rests on treating being $x$ as a property of $x$. (Known as the haecceity of $x$.) The extension of the property of being $x$ is just the unit set, $\{x\}$. If one's property quantifier ranges over all of these properties (or, equivalently, unit sets), then the above definition comes out true. But if one examines the proof, one needs a comprehension instance using the = predicate. One can find counter-examples if one uses a Henkin model $(\mathcal{A}, S)$ in which = is not first-order definable in $\mathcal{A}$, and whose second-order domain $S$ lacks the required unit sets (see Ketland, 2006, "Structuralism and the Identity of Indiscernibles", Analysis 66.4, pp. 313-4). A simple example of such a structure $\mathcal{A}$ has $dom(\mathcal{A}) = \{0,1\}$ and a single binary relation $R = \{(0,0),(0,1), (1,0), (1,1)\}$. This example is discussed also in Button 2006.

In addition to the Leibnizian definition, there are others, invoking different notions of indiscernibility. Given a formalized language $L$ with just predicates, one can define, for each primitive predicate $P$, a notion of "being indiscernible relative to $P$". If there are finitely many different predicates in $L$, one can take the conjunction of these to give a formula $x \approx y$, which expresses the strongest notion of indiscernibility expressible in $L$. (This approach was first given in Hilbert & Bernays, Grundlagen der Mathematik (1934, Vol 1), and was later discussed on several occasions by Quine.) One can prove various things about this formula. Suppose we are given a language $L$ and $L$-structure $\mathcal{A}$. We can show that the formula $x \approx y$ behaves just like an identity predicate (it satisfies reflexivity and substitutivity); and if $=$ is definable at all in $\mathcal{A}$ by some formula $\phi(x,y)$, then $=$ is definable by $x \approx y$. One may also define two further notions of indiscernibility, which can be shown to be equivalent to $x \approx y$. One is
"$x$ is not weakly discernible from $y$"
(the original idea is due to Quine: see Quine 1976). The other is
"$x$ is polyadically indiscernible from $y$".
And then one can also give examples of structures $\mathcal{A}$ which contain indiscernible but distinct elements (i.e., $=$ is not first-order definable in $\mathcal{A}$). I call such structures non-Quinian and I call structures in which $=$ is definable Quinian. Since $x \approx y$ expresses an equivalence relation on $\mathcal{A}$, one can take the quotient of $\mathcal{A}$ under this relation. In the resulting structure, $=$ is now definable. In reverse, one can take a Quinian structure, and add "indiscernible elements", to get a structure with distinct but indiscernible elements. It's worth stressing that whether identity is definable or not in some interpreted language $L$ depends on the properties definable in $L$.

This is mathematical logic (for more technical results about defining identity and indiscerniblity notions, see Ketland 2006 and Ketland 2011, "Identity and Indiscernibility", Review of Symbolic Logic 4, and A. Caulton and J. Butterfield 2011, "On Kinds of Indiscernibility in Logic and Metaphysics", BJPS). But on the underlying philosophical issue, why should we have to define, or reduce, "=" anyway? Why not take "=" to be a primitive notion? Why not say,
Is is just what it is, and not another thing?
This is, more or less, Frege's suggestion in his 1891 review of Husserl and endorsed in Savellos 1990. I put it, in 2006, like this:
Must the identity relation on positions be defined in terms of the other distinguished relations? Or might the identity relation for positions be taken as primitive? For my part, I see no compelling reason why the identity relation, in general, should not be thought of as primitive. The reasons sometimes given for not taking identity as primitive seem to me to be anti-realist, reductionist or verificationist in spirit. The contrary view, that identity is primitive and indefinable, was advocated by Gottlob Frege, in his 1891 review of Edmund Husserl’s Philosophie der Arithmetik: ‘since any definition is an identity, identity itself cannot be defined’ (see Geach & Black 1980, p. 80). In an illuminating article, Elias Savellos has similarly argued that ‘identity must be viewed as an indefinable, primitive notion’ because ‘any attempt to define identity is bound to be circular, since the intelligible understanding of the notion of identity must make recourse to the intelligible understanding of identity itself’ (1990, p. 476). (Ketland 2006, pp. 305-6.)

References

 Button, Tim. 2006. "Realistic structuralism’s identity crisis: a hybrid solution". Analysis 66.
 Caulton, Adam and Butterfield, Jeremy. 2011: "On Kinds of Indiscernibility in Logic and Metaphysics", BJPS (to appear).
 Frege, Gottlob. 1891: Review of E. Husserl, Philosophie der Arithmetik (1891).
 Hilbert, David and Bernays, Paul. 1934: Grundlagen der Mathematik, Vol 1.
 Ketland, Jeffrey. 2006: "Structuralism and the Identity of Indiscernibles", Analysis 66.
 -- . 2011: "Identity and Indiscernibility". Review of Symbolic Logic 4.
 Ladyman, James. 2005: "Mathematical structuralism and the Identity of Indiscernibles". Analysis 65.
 Quine, Willard V. 1976. "Grades of Indiscriminability", J. Philosophy 73.
 Saunders, Simon. 2003: "Physics and Leibniz’s Principles". In Symmetries in Physics: Philosophical Reflections, edited by K. Brading and E. Castellani. CUP.
 Savellos, Elias 1990: "On defining identity". Notre Dame Journal of Formal Logic 31.

Update: I've updated this post.

1. Hi Jeff,

I'm not sure someone who wants to define x = y' in terms of AX(Xx iff Xy)' will find the existence of the Henkin models you describe very problematic. Of course, they might say, if we consider non-standard models of AX' things will go wrong, but that's also true of ='. All' means all, and not some other thing!

Am I missing something here?

Best,
Sam

2. Hi Sam, agree entirely!
But some in the PII physics literature don't like haecceities - they're not legit physical properties. So, the countermodel gives a very simple example of what happens when the range of the 2nd order quantifier doesn't include all unit sets.
Cheers,
Jeff

3. (This is the second time in a short while that I've posted a link to some of my work in a comment here, but I can't help it. It's bound to stop soon, since I haven't actually done that much work.)

A couple of years ago I wrote a paper in which I argue that identity statements should be regarded as being fundamentally different from other relational statements. Not only was I urging that 'the identity relation is primitive', but that identity statements are of their own logical form, irreducible to the others. (Some things Strawson said in his book Subject and Predicate in Logic and Grammar are very similar to what I was urging. More recently, Recanati's work.)

This sort of formulation no longer does it for me, and I've come to abandon the idea of arguing against the orthodox view that identity is a relation between objects - but still I think that this doesn't do enough to mark the special nature of identity statements. This in turn affects (or infects) our thinking about modality.

On Identity Statements: Against the ascriptional views

4. Hi Jeff,

I wonder what you think about the following two responses:

(1) It's not at all clear what the existence of non-standard models tells us. For instance, there are non-standard countable models of ZF, but that doesn't undermine my belief that there are uncountably many sets. Similarly, I might say that the Henkin models don't (and shouldn't) undermine my belief that AX.....' works as a reduction of identity. To undermine that belief, what one would want is an example of clearly distinct x and y which (actually) share all of the same properties (in the sense of properties' that Phi Phys people have in mind).

(2) Couldn't we simply claim that there has been a confusion over the term property' -- that someone proposing the reduction really meant AX' to be interpreted as Axx' or even as ranging over Fregean concepts?

Thanks again,
Sam

Ps, I assume that their notion of property rules out unordered pairs as well (otherwise the reduction would work in all three membered domains)?

5. Hi Sam, agree on both your point.

I think the second-order Leibnizian reduction is ok - it works, so long as the range of "AX" is right. That is, = is strict indiscernibility. But there is a kind of circularity involved, because the properties required - haecceities - presuppose identity: in lambda notation, the property of being $x$ is:
$\lambda_y(y = x)$.

In the phil physics literature, there are examples where people allege that PII is violated (usually by considering quantum indiscernibility - i.e., symmetrized n-particle states). E.g., see
Cortes, Alberto. 1976: "Leibniz's Principle of the Indentity of Indiscernibles: A False Principle", Philosophy of Science 43.

http://philpapers.org/rec/CORLPO

To this, I always say "PII isn't really violated: rather, it's just a case where the property quantifiers are restricted".

So, then the question comes down to: is the property of being $x$ a physically legitimate property? I would say , "Yes". I think that Simon Saunders's view is that being $x§ isn't a genuine physical property. Cheers, Jeff PS - does LaTeX view right in your web browser? 6. Hi Tristan, thanks for sending the link to your paper. I had a brief read through. One question you have there concerns what is so special about identity? Well, the binary relation of identity on a set D is the set {(x,x): x in D}. So, it's simply the diagonal of the set D. Why is it special? Why is it a logical relation? Tarski gave a standard answer to this. So long as |D| > 1, the identity relation on D is one of exactly 4 binary relations on D that is *invariant under any permutation of D*. These are (i) the empty relation (i.e., the emptyset). (ii) the universal relation (i.e., the complement of (i)). (iii) the identity relation (i.e., the diagonal). (iv) the distinctness relation (i.e., complement of (iii)). Suppose E is a binary atomic predicate in L, and E(t1, t2) is an atomic sentence, where t1 and t2 are L-terms. Suppose I is an interpretation of L. Then the standard truth definition for atomic formulas gives us: (a) E(t1, t2) is true in I iff (I(t1), I(t2)) is in I(E). where I(t) is the denotation of t in I, and I(E) is the relation that E denotes in I. Suppose next that I(E) is the identity relation. Then: (b) E(t1, t2) is true in I iff I(t1) = I(t2). And this is Frege's view in his famous 1892 article: the sentence "Phosphorus = Hesperus" is true iff the denotation of "Phosphorus" is the denotation of "Hesperus". We freely use the notion of identity in the metatheory. So, we do not impose any demand of reduction. Cheers, Jeff 7. Hi Jeff, Thanks! So, the dialectic is something like this (?): If one means to interpret$\forall X$as a quantifier over legit physical properties, then there are examples of distinct objects which have all the same properties (or so someone like Cortes thinks -- thanks for this reference!). On the other hand, if we are more liberal with our interpretation of$\forall X$, we find that in order to prove the reduction successful, we need to appeal to instance of the comprehension schema which include$=$, and this is somehow circular. I suppose it would be nice to know more about what the rules are here (i.e. what a non-circular proof might look like -- whether the constraint of a non-circular proof is at all fair once made clear). For instance, here's a simple (silly) proof which does not appeal to instances of comprehension including$=$. If$x\not=y$, then$\exists z(x\in z \wedge y\not\in z)$. By second-order comprehension on$v \in z$we get:$\exists X(Xx \wedge \neg Xy)$as required! Of course, I assume you won't be happy with this either, but it would be nice to know why. Best, Sam Ps, yes, but I hadn't thought about typing in LaTeX myself until now! 8. Hi Sam, Yes, that is pretty much the dialectic on this stuff over the last thirty years or so. Your proof is cool, sure. (Do I give the impression of complaining at lot? Cue Morgenbesser joke: "Why is there something rather than nothing?" "Even if there was nothing, you'd still be complaining".) The proof works within a (first-order) theory where identity is already explicitly defined, by the axiom of extensionality,$\forall z(x \in z \leftrightarrow y \in z) \rightarrow x = y$. A similar proof would start with the theory of a strict linear order,$<$, and then we have a theorem,$(\neg (x < y) \wedge \neg (y < x)) \rightarrow x = y)$. Suppose$a \neq b$. Then either$a < b$or$b < a$. By comprehension,$\exists X \forall x(Xx \leftrightarrow (x < b \vee b < x))$. So, by the above,$Xa$. But$\neg Xb$, since$<$is irreflexive. So,$\exists X (Xa \wedge \neg Xb)$. Qed. Maybe I'm being a bit unclear here, though. We want to consider a theory lacking an explicit definition of identity and then introduce a definition of identity somehow. So, suppose$T$in$L$is a first-order theory where we don't assume$=$as a primitive in$L$. Can we introduce a definition of identity which somehow doesn't presuppose$=$? Cheers, Jeff 9. Let me post this rather technical and boring comment just to give a clearer idea. Suppose$L$is a second-order language, with identity as a primitive (i.e., including the deductive rules for =). Suppose$T$is a theory in$L$. Lemma: Suppose$T$contains Comprehension and an explicit definition of$=$.Then$T$proves PII. 1.$T \vdash \forall x\forall y(x = y \leftrightarrow \phi(x,y))$2.$T \vdash \forall y \exists X \forall x(Xx \leftrightarrow \phi(x,y))$3.$T \vdash \forall x \phi(x,x)$4.$T \vdash \forall x \forall y(\phi(x,y) \rightarrow \phi(y,x))$5.$T \vdash \phi(a,a)$6.$T \vdash \phi(b,a) \rightarrow \phi(a,b)$7.$T, a \neq b \vdash \neg \phi(a,b)$8.$T, a \neq b \vdash \neg \phi(b,a)$9.$T \vdash \exists X \forall x(Xx \leftrightarrow \phi(x,a))$10.$T \vdash \forall x(Ax \leftrightarrow \phi(x,a))$11.$T \vdash Ab \leftrightarrow \phi(b,a)$12.$T \vdash Aa \leftrightarrow \phi(a,a)$13.$T \vdash Aa$14.$T, a \neq b \vdash \neg Ab $15.$T, a \neq b, \vdash Aa \wedge \neg Ab$16.$T \vdash a \neq b \rightarrow (Aa \wedge \neg Ab)$17.$T \vdash a \neq b \rightarrow \exists X(Xa \wedge \neg Xb)$18.$T \vdash x \neq y \rightarrow \exists X(Xx \wedge \neg Xy)$19.$T \vdash \forall x \forall y(\forall X(Xx \rightarrow Xy) \rightarrow x = y)$10. Hi Sam, At line 9 above in the "meta-derivation", the (parametrized) Comprehenson instance asserts the existence of the haecceity of$a$. I then call this property$A$at line 10. Then, using the properties of$=$,$T$proves both$Aa$and (assuming$a \neq b$,$\neg Ab$. But this works because$=$is explicitly defined in$T$by$\phi(x,y)$to begin with! So, line 9 may as well be:$T \vdash \exists X \forall x(Xx \leftrightarrow x = a)$. Cheers, Jeff 11. So, continuing, ... if$T$is a second-order theory containing an explicitly definition of$=$(by$\phi(x,y)$say), we get, 1.$T \vdash \forall x\forall y(x = y \leftrightarrow \phi(x,y))$(defn of$=$) 2.$T \vdash \forall y \exists X \forall x(Xx \leftrightarrow \phi(x,y))$(instance of Comp) 3.$T \vdash \forall y \exists X \forall x(Xx \leftrightarrow x = y)$4.$T \vdash \exists X \forall x(Xx \leftrightarrow x = a)$5.$T \vdash \forall x(Ax \leftrightarrow x = a)$6.$T \vdash Aa$7.$T, a \neq b \vdash \neg Ab $8.$T, a \neq b, \vdash Aa \wedge \neg Ab$9.$T \vdash a \neq b \rightarrow (Aa \wedge \neg Ab)$10.$T \vdash a \neq b \rightarrow \exists X(Xa \wedge \neg Xb)$11.$T \vdash x \neq y \rightarrow \exists X(Xx \wedge \neg Xy)$12.$T \vdash \forall x \forall y(\forall X(Xx \rightarrow Xy) \rightarrow x = y)$And the proof uses an instance of comprehension at line 3, containing$=$. Cheers, Jeff 12. Hi Jeff, Thanks for responding. You're right that the main question in that paper was 'What's special about identity?'. If I re-write it, maybe as a book chapter, I'll make that clearer. That's a cool point about there being just four kinds of permutation-invariant relation. That's definitely part of what's special about identity. However, I think one has to be careful to separate the name-view of identity (Geach's view, perhaps Frege's view in the Begriffschrift) from the more minimal claim that 'a = b' is true iff 'a' and 'b' codenote. That last claim, interpreted in a certain way, everyone would presumably agree with. But talk of truth-conditions is ambiguous: here, we're talking about the conditions in which the statement has a certain meaning and says something true, but there is another notion of truth-conditions central to philosophy, where we just take what the actual statement actually says, and talk about the conditions in which what it says is true is true. In the second sense, I think it's wrong that Hesperus = Phosphorus iff 'Hesperus' and 'Phosphorus' codenote, for the name 'Hesperus' for instance might have denoted some other object. 13. Hi Tristan, "In the second sense, I think it's wrong that Hesperus = Phosphorus iff 'Hesperus' and 'Phosphorus' codenote, for the name 'Hesperus' for instance might have denoted some other object." This makes an assumption about the modal status of semantic axioms, and I'm sceptical about it. The view I'm sympathetic to is this. First, denotation (of a string of letters) is language relative. For it is meaningless to say, of the finite sequence ("c", "a", "t") that it denotes anything, simpliciter. Rather the finite sequence ("c", "a", "t") denotes-in-English the set of cats. Uninterpreted strings lack semantic properties. So, we need to include this language relativity when properly writing down sentences of the semantic meta-theory for an interpreted object language L. For example, (1) "snow is white" is true-in-English iff snow is white. (2) "Hesperus" denotes-in-English Hesperus (This is even more apparent when the object language is not homophonically translated into the metalanguage.) Next, given this language relativity, one can argue that languages are very finely individuated. The slightest syntactic, semantic or pragmatic change leads to a new language. If that's right, then semantic axioms expresses (in the metalanguage) necessary truths! A fuller argument is this: there is no possible world where "Hesperus" denotes-in-English the moon. There is a possible world where "Hesperus" denotes-n-L the moon, but L is not English. The same goes for what a string "says" or "expresses". If S is a string, then what it says or expresses is always relative to the language L it belongs to. So, to consider your example, (3) Hesperus = Phosphorus iff "Hesperus" denotes-in-English what "Phosphorus" denotes-in-English On the view I just described, (3) expresses (in the metalanguage) a necessary truth. Clearly we do have some L such that "Hesperus" denotes-in-L Alpha Centauri and "Phosphorus" denotes-in-L Betelgeuse. But L isn't English. (There is a complication here: there is, strictu dictu, no such thing as English. Rather, there are countless idiolects, all of which are so similar, lexically, syntactically, semantically, pragmatically, that we idealize them as a single overarching entity. But - particularly in the present discussion - this can be very misleading.) On this view, languages do not modally (or even temporally) vary. An alternative view would try to introduce some modal notion of "counterpart of a language". 14. Hi Jeff, Thanks so much for this! (And no, you don't give the impression of complaining!) So, I wasn't assuming that extensionality held in the set theory. Indeed, I assumed that I was working with some facts about sets with urelements, and so even if I had assumed extensionality, it wouldn't give us an explicit definition of identity in general (only identity for sets). The point I had in mind was that there are lots of facts about sets concerning identity. For instance, it is a fact about sets that if$x\not= y$, then there is some set which separates them (we need not assume that the set in question is a singleton). And the question is which of these facts would it be circular\non-circular to appeal to in proving that the second-order reduction is adequate. Another example of facts that the reductionist could appeal to might (?) be the following. It is a fact about abstract semantics that if$x\not= y$, then we could introduce a name$n$, which names$x$but does not name$y$: formally:$\Diamond^L \exists n[Name(n,x) \wedge \neg Name(n,y)]$. Again, we can then apply the relevant instance of second-order comprehension to get our conclusion. Here, again, it doesn't look like we need or need to presuppose an explicit definition of identity. Hope that makes sense! Best, Sam 15. Hi Sam, Thanks - that makes perfect sense. Oops! You're right. I'm replying too quick and trying to get the proof of PII sorted out - the principle you use isn't extensionality. The proof you give uses "if$x$and$y$are distinct, they belong to distinct sets" and not "if$x$and$y$are distinct, they have distinct elements". I.e.,$\forall z (x \in z \rightarrow y \in z) \rightarrow x = y$. But surely that does function as a definition of identity (in set theory, with urelements, even)? I think the same would hold for the modal semantic approach you mention too, "if$x$and$y$are distinct then$x$and$y$could be distinctly named". Again, that seems to be a definition of identity. (For me, a fact can be a definition.) If so, such definitions give us a formula$\phi(x,y)$such that$x = y \leftrightarrow \phi(x,y)$. Then we can prove PII, as above. Cheers, Jeff 16. Hi again Tristan, I think the distinction you make at the end of your comment corresponds to the distinction between 2 kinds of meta-statement from a semantic metatheory for an interpreted language, (1) "Phosphorus = Hesperus" is true-in-English iff the denotation-in-English of "Phosphorus" is the denotation-in-English of "Hesperus". (2) "Phosphorus = Hesperus" expresses-in-English the proposition that Phosphorus = Hesperus. The first, (1), uses the clause (for atomic sentences) from the extensional semantic (inductive) definition of truth for English; while (2) should be a theorem of some intensional semantic meta-theory for English - stating what propositions are expressed (in English) by what strings. I agree that (1) doesn't state the propositional content of "Phosphorus = Hesperus". It does, however, allow us to unwind the truth condition, eventually to (3) "Phosphorus = Hesperus" is true-in-English iff Phosphorus = Hesperus. (assuming axioms for what "Phosphorus" and "Hesperus" denote-in-English.) The view I'm sympathetic is that semantic claims like (1), (2) and (3) express necessities. Cheers, Jeff 17. Sam, quick PS on my comment on your last comment, which mentions separability. In the RSL paper, Definition 2.6, I consider defining indiscernibility in terms of sets & relations "separating" objects$a$and$b$. For a set$X$, it's the usual topological notion ($a \in X \wedge b \notin X$). For an$n$-ary relation$R$(with$n>1$), it's defined using permutations of arguments (as in the clauses of the Hilbert-Bernays-Quine indiscernibility formula). E.g., if$R$is a ternary relation and$\underline{d} =(d_1,d_2)$is an ordered pair then, then;$(R,\underline{d})$separates$a$and$b$iff$(Rad_1d_2 \leftrightarrow \neg Rbd_1d_2) \vee (Rd_1ad_2 \leftrightarrow \neg Rd_1bd_2) \vee (Rd_1d_2a \leftrightarrow \neg Rd_1d_2b)$. Then:$a$and$b$are separable iff some set$X$, or some$(R, \underline{d})$separate$a$and$b$. (All relative to some structure, of course.) At the end of the article, Section 4.3, I relate the simplest kind of separability (i.e.,$T_0$) of a topological space$S=(D,U)$to definability of identity in the corresponding first-order structure$\mathcal{M}_S$. Under a certain side condition,$\mathcal{M}_S$is Quinian iff$S$is$T_0\$.

18. Hi Jeff,

Thanks, that was really helpful! I'm much clearer now on what the constraints are, i.e. what count's as a circular proof in this context. Need to think more about it!

Also, thanks for the reference to your RSL paper, will certainly read when I get chance.

Best,
Sam

19. Hi Jeff,

OK, you've brought home to me that my claim - that the co-reference analysis of identity, read in a certain way, is false - vaguely involved a certain way of way of individuating languages and "content".

And to be sure, this way is less fine-grained than the one you lean towards.

At this point, I'm wondering whether there is any clear disagreement. I'm very pluralistic about different ways of individuating contents and interpreted expressions - I think your way is coherent, but I'm inclined to think that ordinary uses of the relevant expressions tend to invoke a less fine-grained conception.

So, part of what I'm wondering is whether you have some sort of claim in the back of your mind for your way - e.g. that it is the only reasonable way of going, is useful for X or Y, fits best with usage (or some region thereof), etc. Any thoughts?

There's a way of individuating languages I'm interested in which is pretty fine-grained, but stops short of your conception in not requiring - loosely speaking - identity of reference/extension for identity of language, but still requiring identity of sense (in some sense!). (This hangs together with my semantic externalism, and I use it to make sense of the necessary a posteriori.)

On that approach to individuating expressions-in-languages, what I said initially still seems right to me.

20. Hi Tristan,

Right, all turns on how fine-grained the individuation of languages is. So, we might call the view I'm sympathetic to Ultra-Fine Individuation:

L is identical to L'
iff L and L' are identical in syntax, phonology, semantics and pragmatics.

I can't, however, give any definite argument for this highly theoretical position about language individuation! The argument is to consider our intuitions about small changes (including modal ones) in L.

String/expression individuation is much less fine-grained than language individuation though, because the same string can belong to different languages. So, a string doesn't have any particular meaning/referent intrinsically attached, as it were.