Substitutions and Models, Part 1: Bolzano, Quine, Tarski, and Boolos

Let ‘\Rightarrow‘ stand for a (not-yet analyzed) relation of logical consequence. According to the substitutional analysis of quantification, \Delta \Rightarrow P iff every substitution scheme \Delta is true under is also one P is true under. According to the model-theoretic analysis, \Delta \Rightarrow P iff every model on which \Delta is true is also a model on which P is true. The substitutional account is generally associated with Bolzano, and the model-theoretic with Tarski. Tarski’s objections to Bolzano’s account are generally thought to tell against the substitutional account and in favor of the model-theoretic. And what goes for logical consequence goes for logical truth — especially as P‘s being a logical truth is generally thought equivalent to it’s being a consequence of the empty set.

Enough of background. Now for the plot: Quine, in Philosophy of Logic, has an argument that — provided the background language is rich enough for elementary arithmetic (say, Peano Arithmetic) — the two accounts of logical truth are equivalent. This is prima facie surprising, and for two reasons: First, Tarski’s objection against Bolzano’s account shows that the two aren’t equivalent in general, and so it’s surprising that they would collapse for languages of this expressive power. Second, certain objections raised by e.g. John Etchemendy (The Concept of Logical Consequence) against Tarski’s official account (or, at least, Etchemendy’s reading of it) seem easily parlayed into an argument that the two are not equivalent for any first-order language.

To thicken the plot, in “On Second-Order Logic,” George Boolos argues that, although Quine is right about the two accounts of logical truth, the result does not extend to logical consequence, even in first-order languages. This is surprising, too: given compactness results, we ought to be able to move from logical truth to logical consequence.

I want to get to the bottom of these issues in a series of four posts. Here’s my plan of attack. In this post, I’ll sketch the two theories and lay out these three prima facie puzzles in more detail. The next post focuses on Quine’s side of things, first solving some of the puzzles and then examining Quine’s argument — and the scope of the premises it requires — in more detail. The third post will be dedicated to understanding Boolos’s argument and seeing how it interacts with the central puzzle to be outlined here. A final post will look at philosophical issues kicked up in the first three.


Here’s this post’s plan of attack. In section one, I’ll lay out the substitutional account in detail and give Tarski’s argument against it. Section two explains the model-theoretic account. Section three gives an argument (derived from Etchemendy’s) that the substitutional account of logical truth predictably delivers different results for first-order languages than the model-theoretic. And in section four, I’ll outline the argument that, if the accounts of logical truth are equivalent, then so are the accounts of logical consequence.

Section 1: The Substitutional Account

Suppose L is an interpreted language. Let a substitution scheme be a function from syntactically simple terms of L to any expressions of L at all that could be substituted without turning sentences into non-sentences. (If there are bound variables, we’ll also need some restrictions to keep them from interfering with each other, but we can ignore that here.) So, for instance, a syntactically simple predicate like `is tired’ (I’m pretending copula are syntactically embedded — just work with me here) could be taken to another simple one like `is hungry’, or to a complex one like `is about to eat every Taco in the tri-state area’. And a connective such as `and’ could be taken to another simple connective, such as `or’, or to a complex connective, such as ‘unless I hear by tomorrow that _______, John is going to think that _________’.

Some substitution schemes will take some simple expressions to themselves. In this case, say that it preserves those expressions.

One final bit of terminology. If P is a sentence and S a substitution scheme, let S(P) be the result of replacing all of P’s simple expressions with their values given by S. And if \Delta is a set of sentences, let S(\Delta) be the set of S(P) for P \in \Delta.

OK, now we’re ready for our definitions of logical consequence (and, by extension, logical truth). If E is a set of expressions, then say that P is an E substitutional consequence of \Delta, written \Delta |\!\!\models_{E} P, iff for every E-preserving substitution scheme S, if all of S(\Delta) are true, then S(P) is true, too.

Now, this doesn’t quite give us a definition of logical consequence; it gives us instead a definition of something like `logical consequence relative to a set of preserved expressions’. The idea, though, is that if we can pick the right expressions, we’ll have our definition of logical consequence: logical consequence is substitutional consequence under substitutions that preserve those. Those are the expressions we call the `logical constants’. Of course, for an arbitrary language, it’s no easy going deciding which expressions are the logical constants. But for, say, first-order languages, the idea is that the logical constants are those that are treated as logically special in our textbooks and so on (you know, quantifiers, truth-functional connectives, etc.)

That, anyway, is the idea. But in “On the Concept of Logical Consequence,” Tarski argued against the view more-or-less as follows. Suppose that L is really impoverished. Suppose, for instance, that along with logical vocabulary, it only has one predicates and two names (each of which means what they do in English):

Names Predicates
Washington was president.
Lincoln

Now consider the argument:

  1. Lincoln was president.
  2. Therefore, Washington was president.

Intuitively, this shouldn’t count as a valid argument. But — given the extreme impoverishment of the language — it does. Any substitution scheme that preserves the logical terms won’t have any way of substituting in something for (1) that makes it true while substituting in something for (2) that makes it false. The problem, essentially, is that the only predicate the language has doesn’t divide between Lincoln and Washington, and the language doesn’t let us talk about anything else.

Consider, for instance, the scheme that sends `Lincoln’ to `Washington’, `Washington’ to `Lincoln’, and `was president’ to `was not president’. This makes the conclusion false — but also makes the premise false! And no other substitution scheme does the job either.

The problem is apparent: it’s simply the impoverishment of the language that’s to blame. If we had the predicate `had a beard’ in the language, the argument would count as invalid: the scheme that sends `was president’ to `had a beard’ (and kept the names the same) would make the premise true and the conclusion false. But since our language doesn’t have such a predicate, the problem remain.

Notice the nature of the problem. The complaint isn’t that the substitutional account delivers the wrong verdict about any arguments we ever actually encounter. We speak a very rich language (English, at least for readers of this blog, and perhaps others as well). The problem is that the account delivers the wrong verdict about arguments in some possible language. So if we’re going to let this worry move us to a different account of logical consequence, notice that we must be implicitly endorsing a principle like,

UNIVERSAL: If an account of logical consequence is correct, it must deliver the right results for any possible language.

For those who accept UNIVERSAL, the natural “fix” behind the basic Bolzanian idea is to move to the model-theoretic account of logical consequence.

Section 2: The Model-Theoretic Account

The basic idea behind the model-theoretic account is that there are a range of things, models, with a certain structure, and a relation of `truth in’ defined between these models and sentences of L. If the language is first-order, we can be a bit more precise: a model is an ordered pair \langle D, I\rangle of a domain D and an interpretation function I. D is just any old set of things. I is a function from names of L to objects in D and predicates of L to subsets of D (or subsets of D^{n}, for n-placed predicates) that provides the `extensions’ of those predicates on the model — provides the things in the domain that, according to the model, are the way the predicate says they are. We also need a definition for `truth in’ — thanks to the necessity of dealing with quantifiers and variables, this is a fairly complex recursive one in terms of satisfaction, but I’m not going to make hay about the details here, since it’s pretty easy to see which sentences should be true on a model given that we know what terms’ denotations and predicates’ extensions are.

For instance, a model for our simple language from above would give us a set of things D, and then tell us which of those things got assigned `Lincoln’, which got assigned `Washington’, and which were in the extension of `was president’. Then a sentence like, say, `Lincoln was president’ would be true iff the thing assigned to `Lincoln’ on the model was in the extension assigned to `was president’ on that model, and a sentence like `Something was not president’ would be true iff there were things in the domain that weren’t in the extension of `was president’.

A model is called a model of a sentence iff the sentence is true on it, and a model of a set of sentences iff all the sentences in the set are true on it. If every model of \Delta is also a model of P, we write \Delta \models P, and say that P is a model-theoretic consequence of \Delta.

The model-theoretic account of consequence insists that logical consequence is model-theoretic consequence. It’s fairly easy to see that the model-theoretic account won’t suffer from the problem that plagued the substitutional. Consider again the argument:

  1. Lincoln was president.
  2. Therefore, Washington was president.

For it to be valid, every model of 1 must be a model of 2. But there are models of 1 that aren’t models of 2. Just take a model with a domain of two things, say a and b which assigns `Lincoln’ to the first, `Washington’ to the second, and assigns the set \{a\} to the predicate `was president’.

Section 3: Inequivalence

The foregoing is enough to show that the two accounts are inequivalent; that is, it’s enough to show that the following does not hold for arbitrary languages L:

(EQ) \Delta |\!\!\models P \textrm{ iff } \Delta \models P

We might think that’s not a huge deal: perhaps for suitably expressive languages EQ does hold. (This, in effect, is what Quine will argue.)

But here’s a simple argument, inspired by one John Etchemendy gives against (his reading of) Tarski in The Concept of Logical Consequence. The (adapted) argument shows that for any first-order language, EQ fails.* It goes a little something like this.

Let L be a first-order language. Presumably, its quantifier ranges over more than one object. Now consider the `counting sentence’

(E2) \exists x\exists y(x \ne y)

This sentence is true. Is it a logical truth? Or — more to the point — do we have |\!\!\models (E2)? Do we have \models (E2)?

Well, note first off that \not\models (E2). There are always models with just one thing in the domain; on the model-theoretic conception, then, this does not count as a logical truth.

But the substitutional account disagrees. All of the terms in (E2) are logical terms, so any of the substitutions we care about for evaluating consequence will be ones that keep (E2) the same. (That is: they’re substitutions S where S(E2) = (E2)!) Since (E2) is in fact true, and the substitutions don’t turn it into any different sentence, it remains true under all substitutions — and so |\!\!\models (E2).

Maybe we can fiddle with this. Here’s one way: allow that `preservation’ of quantifiers can include trading in simple quantifiers for complex `restricted’ ones — allow \exists x, to be taken, say, to \exists x(Fx \wedge \underline{\ \ \ \ \ })‘. (If we want to preserve duality — that is, if we want \exists x P to be equivalent to \neg\forall x \neg P — we’ll need to insist that substitution schemes have to `mesh’ in a certain way with how they treat their existential and universal quantifiers.) Once we revise our account, we no longer have |\!\!\models (E2). If `F‘ is a predicate that in fact applies to only one thing, the following counts as a false substitution instance:

(E2*) \exists x(Fx \wedge (\exists y(Fy \wedge (x \ne y)))

But this modification brings model-theoretic and substitutional consequence into agreement on (E2) only by making them disagree about the argument form:

  1. Fa
  2. Therefore, \exists x Fx

This is straightforwardly valid on the model-theoretic account. But on the (new, modified) substitutional account, the following is a fair substitution instance:

  1. Fa
  2. Therefore, \exists x(\neg Fx \wedge Fx)

(We get it by the substitution scheme that takes \exists x to \exists x(\neg Fx \wedge \underline{\ \ \ \ \ }).) And this disagreement about valid arguments will turn into a disagreement about logical truth, since the model-theoretic account will, and the substitutional account will not, reckon the conditional with 1 as antecedent and 2 as consequent a logical truth.

It is possible to further modify the substitutional account to make it march in step with the model-theoretic one, at least given that L’s quantifiers in fact range over infinitely many objects. The modifications basically impose large scale constraints on the substitution instances — constraints that, for instance, if you substitute \exists x(Fx \wedge \underline{\ \ \ \ \ }) for \exists x, you have to also substitute names of things that satisfy F for names, and predicate expressions that are satisfied only by F-things for predicates. However motivated these restrictions might be (and they don’t seem very), they don’t help the prima facie puzzle. For the prima facie puzzle is that the two accounts disagree, but Quine is going to argue that they do agree, contra what’s already happened. And Quine’s argument isn’t predicated on any fancy-schmancy jazzing up of the old substitutional account; it’s predicated on the boring substitutional account of the sort described back in Section 1. So what gives?

Section 4: Consequence and Truth

The puzzle in Section 3 admits a simple solution, but I’ll wait until the next post to give it. Here, I want to present a deeper puzzle. At the end of the next post, we ought (if Quine does his work right) to be convinced that, for expressive enough languages, the following restricted version of EQ holds:

(EQ-R) |\!\!\models P \textrm{ iff } \models P

In the final post, Boolos is going to give us a general argument that, even for these more expressive languages (especially for them, in fact), EQ fails. All the languages in question are first-order. And there is a fairly straightforward argument that EQ holds for a given first-order language iff EQ-R holds for that language.

The argument stems from three properties of first-order logic. The first is compactness. I’ll give a generic version of it here:

(COMPACTNESS) \Delta \Rightarrow P iff there is a finite subset \Gamma \subseteq \Delta where \Gamma \Rightarrow P.

(In other words, no valid first-order arguments require infinitely many premises in order to be valid.) The second theorem is a kind of generalized deduction theorem, which I’ll write as follows:

(DEDUCTION) Q_{1}, Q_{2}, \ldots, Q_{n} \Rightarrow P iff \Rightarrow (Q_{1} \wedge Q_{2} \wedge \ldots \wedge Q_{n}) \supset P

Finally, we have

(MONOTONICITY) If \Delta \Rightarrow P and \Delta \subseteq \Gamma, then \Gamma \Rightarrow P.

That is: adding premises to an argument can’t take validity away. For all of these results, we can consider model-theoretic and substitutional versions by replacing ‘\Rightarrow‘ with ‘|\!\!\models‘ or ‘\models‘, respectively.

DEDUCTION and MONOTONICITY are relatively trivial, and can be seen to hold by inspecting the definitions of each kind of consequence, plus (for DEDUCTION) the truth-table for the material conditional, \supset. COMPACTNESS is not at all trivial, although in the model-theoretic setting it’s very well established. The substitutional variant is probably going to show itself as the cause of all the bother when all is said and done, but let’s not get ahead of ourselves. Let’s just go through the argument.

I’m actually going to just go through one half of the argument: I’ll argue that, if \Delta |\!\!\models P, then \Delta \models P. But it is trivial to reverse the argument, because all of the premises (other than EQ-R, which is a biconditional) apply equally well to both kinds of consequence. One bit of notation will help: if \Gamma is any finite set, then I take the privilege of rewriting it as \{Q_{1}, Q_{2}, \ldots, Q_{n}\}. OK, here goes.

  1. Suppose \Delta |\!\!\models P
  2. By COMPACTNESS, for some finite \Gamma \subseteq \Delta, \Gamma |\!\!\models P
  3. Rewriting, \{Q_{1}, Q_{2}, \ldots, Q_{n}\} |\!\!\models P
  4. By DEDUCTION, |\!\!\models (Q_{1} \wedge Q_{2} \wedge \ldots \wedge Q_{n}) \supset P
  5. By EQ-R, \models (Q_{1} \wedge Q_{2} \wedge \ldots \wedge Q_{n}) \supset P
  6. By DEDUCTION, \{(Q_{1} \wedge Q_{2} \wedge \ldots \wedge Q_{n}\} \models P
  7. Since \{Q_{1}, \ldots, Q_{n}\} \subseteq \Delta, by MONOTONICITY, \Delta \models P

The argument works just as well in the other direction, and EQ follows. But if both Quine and Boolos are right, EQ doesn’t follow. Something’s gone wrong. COMPACTNESS for substitutional consequence looks like the best bet. But how? What sentences follow substitutionally from an infinite set of sentences but no finite subset?

This is the central puzzle. We’ll have to wait a few more posts before hitting the bottom of this mess.


* Well, maybe just almost any — there might be some wiggle room if some first-order languages have quantifiers semantically restricted to range over just one object.

Oxford Studies in Metaphysics Younger Scholar Prize

Oxford Studies in Metaphysics Younger Scholar Prize

The Younger Scholars Prize program, funded by The Ammonius Foundation (http://www.ammonius.org/)  and administered by the Editorial Board of Oxford Studies in Metaphysicsis an annual essay competition open to scholars who are within 15 years of receiving a Ph.D. or students who are currently enrolled in a graduate program. Independent scholars may also be eligible, and should direct inquiries to the Editor of OSM (see below).  The award is $8,000, and winning essays will be published in Oxford Studies in Metaphysics.

Submitted essays must report original research in metaphysics.  Essays should generally be between 7,500 and 15,000 words; longer essays may be considered, but authors must seek prior approval by providing the Editor with an abstract and a word count prior to submission.  Since winning essays will appear in Oxford Studies in Metaphysics submissions must not be under review elsewhere. To be eligible for next year’s prize, submissions must be received, electronically, by 31 January 2013.  Refereeing will be blind; authors should omit remarks and references that might disclose their identities. Receipt of submissions will be acknowledged by e-mail. The winner will be determined by a committee of members of the Editorial Board of Oxford Studies in Metaphysics and will be announced in late February or early March 2013. (The Editorial Board reserves the right to extend the deadline further, if no essay is chosen.) At the author’s request, the Board will simultaneously consider entries in the prize competition as submissions for publication in Oxford Studies in Metaphysics, independently of the prize.

Inquiries should be directed to the Editor, Dean Zimmerman, at dwzimmer@rci.rutgers.edu, or by post through regular mail at:

Professor Dean Zimmerman
OSM Younger Scholars Prize
Philosophy Department
Rutgers University
Davison Hall, Douglass Campus
26 Nichol Avenue
New Brunswick, NJ  08901-2882

Impossible Lewisian Modal Realism

Many people think that talking about possible worlds is useful in philosophy.  A good number of those people think that talking about impossible worlds is also useful.  In most cases, talking about impossible worlds as well as possible worlds is innocuous.  On most of our views about what worlds are, impossible worlds are no more ontologically problematic than possible worlds: sets of propositions all of which can’t be true together are no more mysterious than sets of propositions all of which can be true together; if talk of possible worlds is a merely pragmatically useful fiction, talk of impossible worlds can be such without any further mystery.  And so if you hold some such view about worlds, the question as to whether we should talk about impossible worlds depends solely on whether to do so is useful – there is no metaphysical problem in doing so.

 

Not so, seemingly, if you are a Lewisian realist about worlds.  For Lewis, a world at which there are blue swans is a world with blue swans as parts, and so a world with round squares is a world with round squares as parts.  And so, to believe in the latter world is to believe in round squares.  And this is to raise a metaphysical problem, for now one must admit into one’s ontology objects which could not exist.  In brief, impossible worlds for Lewis are problematic because of how he thinks worlds represent: they represent something being the case by being that way, whereas his opponents think worlds represent in some indirect manner, by describing things to be that way, or picturing them to be that way, or etc.  Impossible worlds are not metaphysically mysterious on the latter views because there is no metaphysical puzzle in there being a description of something that couldn’t exist, or a picture of something that couldn’t exist; but they are a metaphysical puzzle for Lewis, because there is a metaphysical puzzle in there being something that couldn’t exist.

 

Nonetheless, some think this is a price worth paying: they like Lewis’s account of possibilia but are impressed by the arguments for the need for impossibilia, so want to extend Lewis’s ontology to include impossible worlds.  I’ve heard this move a few times in conversation, but the one person I know of who has defended it in print is Ira Kiourti.  (Yagisawa defends a view similar to Lewisian realism with impossible worlds, but with some crucial differences.)  Now, there are some big, and familiar, problems with believing in genuine impossible worlds that they each try to deal with, but I am yet to be convinced can be solved.  (See my critical study of Yagisawa.)  But I want to raise a problem for Lewisian realism with impossible worlds that I haven’t seen discussed and which I don’t even know how one would start to answer.

 

I can see how Lewisian realism with impossible worlds is supposed to deal with impossibilities like ‘There is a round square’ or ‘Frank is taller than Jim and Jim is taller than Frank’.  I just need to believe in impossible objects – a round square and a man that is both taller and shorter than some other man – and then I can believe in worlds composed in part of such objects.  Now, personally I can’t conceive of such objects – but so what?  If I’ve got good reason to believe in them, I can postulate them.  But I don’t see how we are meant to account for an impossibility like ‘2+2=5’.  For Lewis, ‘2+2=4’ is necessary not because there’s a number system that is a part of each world and which behaves the same way at each world; rather it’s necessary that 2+2=4 because the numbers are not part of any world – they stand beyond the realm of the concreta, and so varying what happens from one portion of concrete reality to another cannot result in variation as to whether 2+2 is 4.  And so, since contingency just is, for Lewis, variation across certain portions of concrete reality – namely, the worlds, which are just big concrete objects – there is simply no room for contingency with respect to the mathematical truths.  Necessary truths about the realm of concreta are necessary because each relevant portion of concrete reality is a certain way: that is, no matter what variation you get across these concrete portions of reality, things are that way.  But necessary truths about the realm of pure abstracta are necessary because they have nothing to do with how concrete reality is: so irrespective of what variation you get across these concrete portions of reality, things are that way.  (See my paper on Lewis and reduction, esp. footnotes 3-5, and the corresponding discussion in the text.)

 

In that case, while I can add to my ontology round squares if I wish, and hence believe in a big concrete object with a round square as a part, and thereby have a world that represents the impossible circumstance of there being a round square, I don’t see even what weird metaphysical move to make to get a Lewisian world that represents 2+2 as being 5.  Worlds don’t have numbers as parts: they are sums of concrete individuals; and if we give that up, we don’t have something worth calling an extension of Lewisian realism (Yagisawa gives this up, which is why I’m ignoring his view here).  A Lewisian world nonetheless represents that there are numbers, because numbers exist from the standpoint of that world.  And while the concrete objects that exist in one Lewisian world are never the same as the concrete objects that exist at another world, the numbers that exist from the standpoint of each world are just the same.  And that’s why mathematical truths are necessary, for Lewis: because a world represents some mathematical claim as being the case just because the numbers represented as existing from the standpoint of that world are as the claim says they are.  And since the same numbers exist from the standpoint of any two worlds, no two worlds differ in what mathematical claims they represent as being the case.  Given Lewis’s account of what a world is and how they represent something to be the case, there is simply no room for variation across worlds in what mathematical claims are represented as true: hence there is no room for contingency in mathematical claims, but nor is there room for a world that represents some impossible mathematical claim as true, no matter what we think about the extent of what concrete worlds there are.  So the Lewisian simply can’t extend her ontology to admit worlds that account for every impossible situation.

Two postdoctoral positions on the nature of representation

Two postdoctoral positions at the University of Leeds are currently being advertised in connection with the ERC-funded project The Nature of Representation. They are fixed term for 4 years, to start on 1 September 2013

One focuses on the philosophy of language. Official details here.

The other focuses on the philosophy of mind.  Official details here.

Deadline for applications is 19th December. Contact Robert Williams at j.r.g.williams@leeds.ac.uk for further details.

New papers on reference; indeterminacy; personal identity.

I’ve got a new paper on Donald Davidson’s theory of reference (to appear in a Companion to Davidson’s philosophy). It has three sections: (1) The role of Reference in T-theories; (2) Inscrutability of Reference; (3) Explanations and Reference. It has quite a bit of stuff on exactly how to think about his attitude to reference (especially given he effectively endorses some kind of inscrutability of reference). I end up puzzling a lot about how to reconcile two things: his official relativization strategy, and the analogy he often draws to instrumentalism. This is something that I hope to think more about in connection with the new project. The paper is here: Davidson on Reference

The two other papers both feature indeterminate personal identity. Indeed, they used to be one (rather disjunctive) paper, and concern two radically different approaches to the same puzzle: how to act under indeterminacy. 

The first—Decision making under indeterminacy—investigates a treatment of action under indeterminacy which builds in an inconstant pattern of action. This connects to recent debate on `imprecise’ or `mushy’ credences. One of the interpretative suggestions I float there (the mind-making proposal).

The second–Nonclassical minds and indeterminate survival—kicks off with a review of some work I’ve been doing for a while, on how our account of rational beliefs, desires, and other states of mind need to adapt when truth and logic go nonclassical. The core of the paper is a case study of the kind of commitments that are tacit in Lewis’s attempt to reconcile `personal identity being what matters’ and `psychological relations being what matters’ in the face of Parfit’s challenges. 

As ever, thoughts and feedback really welcome!

How can you know you’re present?

I’ve posted a new paper online: ‘How Can You Know You’re Present?

Abstract:
Some argue that non-presentist A-theories face an epistemic objection: if they were true, then we could not know whether we are present.  I argue that the presentist is in no better an epistemic position than the non-presentist.  In §1 I introduce the sceptical puzzle: I look at two ways in which the non-presentist could claim that our experiences give us evidence for our presentness, but find each wanting.  In §2, I argue that the puzzle also faces the presentist, and that a number of potential solutions either fail or are equally available to the non-presentist.  I conclude by defending one solution to the puzzle.

Is Lewis’s ontology qualitatively or ideologically parsimonious?

David Lewis believes in lots of things.  He believes in human beings, and animals and plants; he believes in tables, and statues and universities; he believes in planets, and solar systems and galaxies.  And he believes in sets of such things, and sets of sets of such things, and sets which have only other sets as members.  And so on.  But so far, so mundane: there’s nothing there that plenty of philosophers don’t believe in.  But Lewis also believes in unicorns, and gods, and ghosts, and golden mountains.  Lewis thinks there’s a talking donkey who spends his days giving a completely accurate account of your life.  Lewis thinks that somewhere there is an infinite sequence of intrinsic duplicates of you doing a conga line.
That’s a pretty wild ontology.  Unless you’re a philosopher who believes in something that, as a matter of fact, just could not exist, then Lewis believes in everything you believe in and – chances are – an awful lot more.  How is this ontological extravagance to be justified?  Lewis offers two different answers to this justificatory challenge.  His more commonly mentioned answer is as follows.
The Cost-Benefit Response:
It is indeed a lavish ontology that is proposed.  It is a cost to accept that there are so many things: it is a pro-tanto reason not to accept the proffered theory that it posits so many things.  But this cost is outweighed by the benefits afforded by the theory.  If it is true then it provides for a reduction of the modal, an ontological identification of propositions and properties with sets of individuals, and so on.  These benefits outweigh the admitted ontological costs.  So on the balance of costs versus benefits, the theory should be accepted, and the lavish ontology embraced.
Here Lewis is admitting that his ontology comes at a price, but that it is a price worth paying.  But elsewhere he refused to admit that there is even a price to be paid.  He offers instead the following answer to the justificatory challenge.
The No-Cost Response:
The extra things postulated are just more things of the same kind that we all already believed in.  To believe in more kinds of thing is a cost, but to believe in more tokens of a kind of thing you already believe in is no additional cost.  Thus the postulation of this additional ontology is not even a cost that needs to be paid.  It is not even a pro-tanto reason not to accept the proffered theory that it posits so many things, given that they are things of a kind with things postulated by the theory’s salient rivals anyway.
The former response sees the ontology as a cost to be outweighed, the latter doesn’t even acknowledge it as a cost.  Lewis distinguishes between a principle of quantitative parsimony which tells you to minimise the number of things postulated, and principle of qualitative parsimony which tells you to minimse the number of kinds of things postulated.  He admits the latter as a good rule, but doesn’t think he is breaking it; he admits to breaking the former, but doesn’t recognize it as a good rule to be obeyed.
I’m not interested here in which response to the justificatory challenge Lewis would do better to rely on.  My question here is: is Lewis correct when he says, in the No-Cost Response, that his theory is a pro-tanto offense only against quantitative parsimony and not against qualitative parsimony?
Joseph Melia argued that Lewis was wrong: that his ontology sinned against qualitative parsimony as well.  Indeed, that Lewis’s ontology maximally sins against qualitative parsimony, since it admits the existence of things for any kind of thing that there could be.  The only way to do worse on qualitative parsimony would be to believe in some kinds of thing that couldn’t exist.  But provided that we’re only concerned with theories that refrain from postulating impossibilia, Lewis’s proposed ontology is maximally qualitatively unparsimonious: for every kind of thing there could be, Lewis believes in things of that kind.
John Divers responds on Lewis’s behalf.  Lewis believes in sets and individuals, the end.  Actuality consists of individuals and sets, and the admission of the reality of logical space requires merely the postulation of moreindividuals and sets.  Thus the number of kinds of thing you need to acknowledge by accepting Lewis’s ontology is the same as what we’d need to acknowledge to give a good account of actuality anyway: two.  Thus Lewis does not sin against qualitative parsimony, as he claimed.
How are we to judge this dispute between Divers, on behalf of Lewis, and Melia?  It comes down, seemingly, to a really thorny issue: at what level do we draw the kinds?  Sure, at one level Lewis is merely asking us to believe in things of a kind with what we already believe: individuals (we all believe in those, right?), and the sets that you get by taking those individuals as ur-elemete (and most of believed in sets anyway – and if you don’t, well just believe in Lewis’s ontology minus the sets!).  But on another level, Lewis isn’t just introducing us to new individuals, he’s introducing us to new kinds of individuals.  He believes in unicorns; so there’s a kind of thing – unicorn – that Lewis is asking us to believe in that we didn’t already believe in.
At one level, everything is of a kind: entity.  Read thus, the rule of qualitative parsimony only ever tells us to (ceteris paribus) choose a theory that doesn’t postulate anything at all over one that does: it will never select between theories that each say that there is something.  That’s pretty useless.  At the other extreme, there’s a kind for every way for things to be: hence, a kind F for every predicate F (at least, every satisfiable predicate).  Read thus, the rule of qualitative parsimony will collapse into the rule of quantitative parsimony, for every new token thing you admit will also be to admit a new kind of thing.
For there to be an interesting rule of qualitative parsimony, we have to find a middle level: a way of dividing things into kinds such that it isn’t automatic that everything is of a kind nor that no two things are of a kind.  (Or better: that for any two things, there’s a kind that one falls under that the other doesn’t.)  But then the question is: at what level do we draw the kinds?  How can we do this in a principled manner?  Divers and Melia draw the kinds at different levels, but who is right?  What facts about reality even speak to one way of drawing the kinds as the correctway (or at least, the correct way for the purposes of weighing theories with respect to qualitative parsimony)?
If you believe in ontological categories, you’ve got an answer: draw the kinds at the level of the categories.  So the principle of qualitative parsimony amounts to saying: (ceteris paribus) choose the theory that postulates the fewest ontological categories.  So take someone like E.J. Lowe, who thinks the things in reality divide into four ontological categories: the substantial particular, the substantial universal, the non-substantial particular, and the non-substantial universal.  On the current proposal, Lowe should view the principle of qualitative parsimony as telling him: believe in whatever kinds of thing you like provided the things fall into one of these four categories – but (ceteris paribus) don’t accept a theory that postulates a fifth category of thing, and (ceteris paribus) prefer a theory that postulates fewer categories of thing.
But personally, I don’t find this very helpful.  The same problem as before just comes back at a different point.  When I think of Lowe’s four ontological categories (e.g. – I’m picking on Lowe’s view, but I think the same thing about every proposal on ontological categories that I’ve encountered), I simply wonder why that is the right way to divide things up.  By a non-substantial universal, Lowe means an Armstrongian universal like redness; by a non-substantial particular he means a trope, like the redness of this postbox.  Why isn’t that one ontological category: property?  By a substantial particular he means kinds like electron.  Why aren’t the universals, tropes and kinds all part of the same ontological category: abstracta?  This is just exactly the same problem as before: where to make the divisions.  But instead of asking directly where to make the divisions for the purposes of qualitative parsimony, we’re assuming we make the divisions at the level of ontological categories and instead asking where to make thosedivisions instead.  I don’t find the detour illuminating, having as little an intuitive grasp of where the ontological categories are as I have of what matters with respect to qualitative parsimony.
I suggest a rethinking of the principle of qualitative parsimony.  I think we should qualitative parsimony as derivative on a more fundamental norm of theory choice: ideological parsimony.  Qualitative parsimony is a virtue just insofar as it facilitates ideological simplicity.
So consider a debate between a compositional nihilist and a universalist.  The former, let us suppose, claims an advantage with respect to qualitative parsimony, since the universalist believes in a kind of thing – a complex object – that the nihilist does not believe in.  The universalist responds, suppose, that she is at no disadvantage with respect to qualitative parsimony since she is only believing in more things of the same kind the nihilist believes in: concrete individuals.  I think that it’s fruitless to try and settle whether, for the purposes of theory choice by qualitative parsimony, mereologically simple concrete individuals are of a kind with mereologically complex complex individuals.  In some sense, complex objects are a new kind of thing, and in another sense they aren’t: the question we should be asking, I think, is whether their admission requires more ideological resources.  And in this case, it plausibly does, because while the nihilist can eschew the ideology of mereology, the universalist needs to admit amongst their fundamental ideological primitives some mereological notion.  Thus, as Ted Sider (inspired by Cian Dorr) argues, there is a pro tanto reason to be compositional nihilists, for it minimizes the ideological complexity in reality.  I think that a drive to ideological simplicity is really what’s behind the drive to qualitative parsimony, and this lets us get a grip on what the relevant level of kinds is: admitting the Xs constitutes admitting a new kind of thing, in the relevant sense, when describing reality if there are Xs requires greater primitive ideological resources than describing reality does if there are no Xs.
In that case, it doesn’t look too good for Lewis, for even though he’s only introducing us to new individuals and sets of individuals, as Divers says, it nonetheless looks as though we’re going to need new ideological resources to describe those individuals.  We’re going to need new primitive predicates to describe things that instantiate alien properties since, ex hypothesi, those predicates aren’t definable in terms of a logical construction of actually instantiated predicates.  We’re going to need new spatio-temporal ideology to describe those worlds where things aren’t related spatio-temporally but rather are related in a manner ‘analogous’ to spatio-temporal relatedness.  We’re going to need new ideology to describe the ectoplasm ghosts the absence of which allows actuality to be a physicalistically acceptable world.  So it’s looking like Melia is right: the postulation of these new kinds of thing is a sin against qualitative parsimony.  Divers is right that it’s just more individuals, but that doesn’t matter, since they are individuals that are not describable just with the ideological resources we would have needed to describe actuality.
But whether this is really so depends on another question that I don’t know the answer to.  When judging what ideological resources you need, do you only count what you need to describe what there is, or do you need ideology enough to describe the ways things could have been?  For Lewis of course, there’s no difference: what there is includes all that there could have been.  But what about for those of us who think that how things are as a whole could have been different?  Does the mereological nihilist who thinks there could have been composite objects but there just happen not to be get to claim an ideological advantage over the universalist, or does one need to reject the very possibility of composition to claim such an advantage?
Parity with ontological parsimony suggests that you should only count the ideology you need to describe things as they are.  After all, no one would think that it is a sin against ontological parsimony to think that there could have been immaterial minds; it’s only believing in them that counts against ontological parsimony.  In which case, why should the possibility of having to describe things using some mereological notion matter: it only matters whether describing things as they are requires such notions.
Nonetheless, I can’t shake the feeling that ideological parsimony is different from ontological parsimony in this respect.  That the contingent mereological nihilist is at no advantage over the universalist, only the necessitarian nihilist.  After all, a theory of reality is not complete without a description of how things could have been: so your fundamental theory of reality will have to talk about what could have occurred but doesn’t – and so if there could have been complex objects, you will need to invoke mereological notions to describe that possibility.  So you can’t completely eschew speaking mereologically: your fundamental theory will still need its mereological primitives, even if it only ever uses them within the scope of a modal operator.  I find it intuitive that in that case you still incur the ideological cost: you still have to see reality in mereological terms, even if just to say that actuality is mereologically less complex than it could have been.  To really not have anything to do with the ideology of mereology you must not need to resort to it at any point in your description of reality – whether of how things are or how they could be – you must be a necessitarian nihilist.  (I’m assuming here that how things could be really is a part of the theory of reality.  If you were an expressivist or other kind of anti-realist about the modal I suppose you would deny this.  But since those views are false . . .)

If that is right, then things start to look better for Lewis.  In believing in possibilia, Lewis just thinks that the story of how things are and could be is the story of how things are unrestrictedly: so for him, the ideology needed to describe how things are, simpliciter, is the ideology required to describe how things actually are and how they could have been.  But if we were committed anyway to the ideological resources needed to describe both reality and the possible ways reality could be, this won’t be an ideological expansion, and Lewis won’t be sinning against ideological parsimony – hence against qualitative parsimony – after all.