AbsGen

The Shapiro/Wright paper is a high point in the Absolute Generality collection. For a start,

1. First, they focus on Dummettian considerations. I’ve already urged here that considerations against the possibility of absolutely general quantification based on Skolemite worries, or on worries about “metaphysical realism”, or indeed on worries about “interpretations”, don’t seem compelling. It seems to me that the key interesting issues hereabouts do indeed arise from considerations about indefinite extensibility (pressed by Dummett, but having their roots, as Shapiro and Wright remind us, in remarks of Cantor’s and Russell’s).
2. It is also the case that, unlike some of the others in the collection, this paper is written with fairly relentless clarity and explicitness (and though it isn’t free of technicalia, the details are kept on a tight rein).

Shapiro and Wright take up a hint in Russell, and (in Sec. 2) consider the following — at least as a first characterization of the scope of the indefinitely extensible:

If the concept P is indefinitely extensible, then there is a one-to-one function from all the ordinals into the Ps.

The argument is this. Suppose, for reductio, that there is a one-to-one function f from the ordinals smaller than some α into the Ps. Then the collection of Ps of the form f(β), where β < α will be (on one generous but reasonable understanding) a "definite" totality of Ps. But recall that by Dummett's informal characterization, an

indefinitely extensible concept is one such that, if we can form a definite conception of a totality all of whose members fall under the concept, we can, by reference to that totality, characterize a larger totality all of whose members fall under it.

So, since by hypothesis P is indefinitely extensible, then there must be, after all, a P which isn’t one of the f(β), where β < α. Choose one, and extend the function f by setting f(α) to have that value. This shows that for any ordinal α, if all the ordinals less than α can be injected into the Ps, then the ordinals less than or equal to α can be injected into the Ps. So, by a transfinite induction along the ordinals, all the ordinals can be injected into the Ps.

Very neat. And though the argument does rest on quite powerful set-theoretic assumptions, it indeed seems rather telling. And by a similar argument,

If there is a one-to-one function from all the ordinals into the Ps, the concept P is indefinitely extensible.

So we get, plausibly, a biconditional connection between the concept P’s being indefinitely extensible and there being an injection of ordinals into the Ps — which makes the case of the ordinals the paradigm case of an indefinitely extensible totality.

Now, as Shapiro and Wright emphasize, this connection doesn’t yet give us an elucidatory account of the notion of indefinitely extensibility (for why is the concept ordinal itself indefinitely extensible?): but — if we are right so far — at least we’ve got a sharp constraint on an acceptable account. But are we right?

The trouble is that the argued connection makes all genuinely indefinitely extensible totalities big, while some Dummettian examples of indefinitely extensible totalities are small. Take for example Dummett’s discussion in his paper on the philosophical significance of Gödel’s theorem. He says that arithmetical truth (of first-order arithmetic) is shown by the theorem to be an indefinitely extensible concept. But why? After all, there’s a perfectly good and determinate Tarskian definition of the set.

But suppose we think of a ‘definite’ totality — more narrowly than before — as one that can be given as recursively enumerable (which is perhaps a thought that chimes with other Dummettian ideas). Then start with some such ‘definite’ set of arithmetical truths A₀, e.g. the theorems of PA. Gödelize to extend the theory to A₁, and keep on going. Any particular theory that is still r.e. can be Gödelized. But note that this time there is evidently a limit on how far along the (full, classical) ordinals we can continue the process — for there are only countably many r.e. sets available to be Gödelized and uncountably many ordinals (even ‘small’ countable ordinals).

So what are we to make of this? Well, one line would be to cleave to the Russellian alignment of the indefinitely extensible with injectability-into-the ordinals, AND similtaneously agree with Dummett that truth-of-arithmetic is indefinitely extensible, by not accepting the classical ordinals in all their glory. The more you restrict the ordinals you accept, the more indefinitely extensible concepts there will be for you. But what of those who are happy with oodles of ordinals? Then the moral seems to be this. There is a difference between saying that the concept P is such that, given any ‘definite’ totality of Ps, we can always find a P that isn’t in that totality (we can always diagonalize out of any given set of Ps), and saying that the totality is (so to speak) indefinitely indefinitely extensible. And that seems right and important.

But how can we develop these ideas of ‘definite’ totalities/indefinite extensibility? The story continues …

I’ve commented at length on the central, load-bearing, section of Parson’s paper. The concluding five and a bit pages I found less engaging. There are some comments on a paper by Rayo and Williamson which I might take up when I get to thinking about Rayo’s related contribution here to Absolute Generality. Then there is an un-worked-out suggestion that we take ‘Everything is identical to itself’ as systematically ambigous. And finally there are some remarks about how those who might worry about the possibility about absolutely general quantification can handle seemingly all-encompassing common-or-garden claims such as that there are no (absolutely no!) talking donkeys. The latter remarks chime with some suggestions of Hellman’s that I’ve already commented on sympathetically, so I won’t expand on them here, but just say that I agree with Parsons that common-or-garden claims about talking donkeys aren’t a serious obstacle to anti-absolutism.

So let’s move on. I’ll set aside Rayo’s technical excursus for now: so that brings us to another monster paper, this time by Shapiro and Wright …

Let’s start by presenting a Williamson-style argument in a slightly different way.

On an interpretative truth-theory for a language L, as we said, we’ll have a clause for a monadic L-predicate P along the lines of ‘for all o, P is true-of o iff Fo‘. But we are now in the business of imagining running through various different possible interpretations for P, which will result in clauses in definitions of different true-of relations, i.e. different relations ….. is true-of ….. on interpretation I. Now, it might well on the one hand seem that we needn’t think of the different interpretations that are in play here as ‘objects’ (whatever exactly that means). But, on the other hand, we might reasonably suppose that the different true-of relations could at least be indexed by some suitably big collection of objects (some class of numbers perhaps, or more generously some sets, for example).

So the clause in a definition for an indexed true-of relation true-of_α will be given in the form ‘for all o, P is true-of_α o iff Fo‘. But now, since the indexing objects are by hypothesis kosher objects, we can unproblematically define a property R which is had by an object o just in case o is an indexing object and not-(P is true-of_o o).

We can then ask: is there an index κ such that for all o, P is true-of_κ o iff Ro? The familiar argument shows that there can be no such index κ (assuming, that is, that κ falls into the range of the universal quantifier ‘for all o‘). But what should we conclude from that?

Well, we could conclude that, after all, the universal quantifier somehow manages to miss including the object κ in its range. But that is hardly the most natural lesson to draw! Rather, the natural moral is a Tarskian hierarchical one, that given some truth-predicates true-of_κ, we can ‘diagonalize out’ and define another truth-predicate which is not one of them.

Now, Parsons almost makes the point. But, what he actually says is that, if you resist the idea that the universal quantifier must fail to cover absolutely everything, then this “forces us to take the Tarskian view now about the predicate ‘P is true of x according to I‘. That amounts again to saying that we have determinate quantification over absolutely all interpretations but do not have an equally general notion of truth under an interpretation.” (Which Parsons suggests is a troublesome line for the believer in absolutely general quantification to manage.) But in fact that doesn’t seem quite right. For there is, we are supposing, a determinate quantification hereabouts, but we are not required to think of it as a being over ‘all interpretations’, so much as being over all the objects that index some initial bunch of interpretations. The claim, however, is that we can always diagonalize out and define a further interpretation.

And now the question arises why, in this setting when we are generalizing about Davidsonian interpretations, we can’t echo the line that Parsons took about one-off interpretations. He said, you’ll recall, that (in the case of unrelativized truth-theories) ‘true of’ had better not be in the language being interpreted on pain of paradox. So, as he put it, “the interpretation does require ‘ideology’ not present in the language interpreted, but it does not require an expansion of ontology”. Now we are going up a level and talking about different definitions of ‘true of’ on different possible interpretations. And again on pain of paradox there will be a ‘true of’ that isn’t already among those different definitions. But why can’t we say again, “this new interpretation does require ‘ideology’ not already present, but it does not require an expansion of ontology”?

So in the end, I’m not sure that Parsons has firmly put his finger on a problem for the defender of absolute quantification, or at any rate a problem that comes from ideas about ‘interpretation’.

Let me add just a quick footnote harking back to Linnebo’s paper which we skipped over. One thing I did note was that he takes the strongest response to the Williamson line of argument to be a type-theoretic one — but Linnebo goes on to discuss a simple theory of types, and in a way this seems now to be going off in the wrong direction. For what we have just seen, in the case of a hierarchy of true-of relations is a ramification into levels and it is that which is doing the paradox-avoiding. But I’ll try to return to this observation.

Parsons, however, doesn’t think that the principal problems about quantifying over everything arise from a supposed commitment to metaphysical realism but are “logical difficulties … [which] arise from considering how sentences or discourses containing quantifiers are interpreted. This apparently innocent talk of interpretation turns out to have considerable weight.” Why?

Here’s how I think the dialectic goes in the compressed but elegant Section 3 of Parsons’s paper (with some changes in notation):

1. Quantifiers are standardly interpreted as ranging over some domain, predicates are interpreted by subsets of the domain etc. A domain is understood to be a set. In standard set theory, no set contains absolutely everything. (Going for a set + classes theory just shuffles the problem upstairs.) So quantifications aren’t over absolutely everything.
2. But in fact, Parsons says, it isn’t specific issues about sets or classes that generate the type of difficulty we encounter here. For consider any style of semantic interpretation for one-place predicates that assigns the open wff F the entity E(F), and which tells us that ‘Fa‘ is true just when o I E(F), where o is the denotation of a, and I is some appropriate relation. (So if E(F) is a property, I is the instantiation relation; if E(F) is an extension, I is set membership; and so on.) Now, suppose that the language in question can itself talk about the entity E(F) and the relation I, so that now — within the language itself — we have ‘Fa‘ is true iff ‘a I E(F)‘ is true. Now consider the one-place predicate ‘R‘ defined so that ‘Rx’ iff ‘not-(x I x)’, and suppose a is the term E(R). Then, we’ll have ‘Ra‘ is true iff ‘a I E(R)‘ iff ‘a I a‘ iff ‘not-Ra‘. Contradiction. So either there just is no such object as E(R), in which case we have a problem about giving a familiar sort of semantics for the language: or it is not available in the domain of quantification to be picked out, and the language’s quantifiers don’t range over everything.
   
3. But ahah! Maybe the trouble in (1) comes from the idea that semantic interpretation requires us to assign an entity to be the domain. Recall, e.g., Cartwright’s familiar animadversions against what he calls the All-in-One principle, the idea that a domain is another object, additional to the objects it contains. And maybe the trouble in (2) comes from the idea that semantic interpretation requires us to assign an entity to be the interpretation of a predicate. Recall, e.g. the possibility of a metaphysics-light Davidsonian style of interpretation where predicates are interpreted by translation. [Then the residue of the generalized Russell paradox, with E(F) being simply F, and R the ‘true of’ relation is just a familiar sort of semantic paradox. This indeed will lead us to say that the ‘true of’ had better not be in the language being interpreted on pain of paradox. “So,” says Parsons neatly, “the interpretation does require ‘ideology’ not present in the language interpreted, but it does not require an expansion of ontology. So far so good for the idea that the domain of the variables includes absolutely everything.”]
4. But what, however, if one wants to generalize about Davidson-style interpretations (though, as Parsons notes, it is a moot question when we really need to). Do we get back to the sort of contradiction that we met when considering the ontologically loaded notion of interpretation deployed in (1) and (2)?
5. If we are going self-consciously to relativize interpretative truth-theories (in a way that Davidson doesn’t) preparatory to generalizing about them, then we’ll have clauses for a predicate P like this ‘(for all o), P is true of o according to interpretation I iff Fo‘. Now suppose that an interpretation can itself be an object which P can be true of. And put Ro iff not-(P is true of o according to o). Now consider an interpretation J such that (for all o) P is true of o according to interpretation J iff Ro iff not-(P is true of o according to o). Identify o with the interpretation J and we have a contradiction again. [Thus Williamson’s version of the Russell paradox argument.]
6. One response is to continue to allow that J is an object but conclude that it can’t fall into the range of the quantifiers, so that the quantifiers can’t be running over absolutely everything. So we again get an argument against absolutely general quantification, even though we are no longer thinking that interpretations as themselves ontologically loaded and as assigning objects as domains to quantifiers or entities as interpretations to predicates.

So far, so good! But, as I just said, that’s only one response to the Williamson argument. It isn’t the only possible one. Parsons mentions (at least) one other line of response at the end of his Section 3, though concludes that “the friends of absolute quantification” face difficulties in the other direction(s) too. But why?

Well, here things get a bit murkier. I’ll need to think for a while more …!

It is a pleasure, as always, to turn to a paper by Charles Parsons (a long time ago, his “Frege’s Theory of Numbers” was one of the papers that grabbed me when I very first started philosophy and it helped get me really enthused by Frege’s project). Nice too, in a volume of overlong papers that ‘The Problem of Absolute Universality’ sticks to a reasonable length.

In his first section, Parsons quickly reviews a few reasons for supposing that sometimes, at any rate, we aim to make claims that do involve absolutely general quantification — offering the usual sort of cases. For example, there are logical examples like Everything is self identical (which surely is indeed intended to be about everything.) And there are more humdrum examples like There are no unicorns. (If we suppose ‘there are’ ranges only over some domain D, then the statement could be true even if there are unicorns outside D. Then, asks Parsons, “can we exclude this outcome short of admitting a D that is absolutely everything?”).

So what are the problems about taking such cases at face value? Parsons takes the main problems to be logical in character, and more about them in due course. But first, in his second section, he discusses “a problem of a more metaphysical character”.

Our problem is that for statements of an absolutely general kind to have a definite truth-value, it appears that there has to be a final answer to the question what objects there are, … That is metaphysical realism.

If we are suspicious about metaphysical realism, so understood, we should therefore be suspicious about quantifiers which purporting to really capture everything, once and for all.

But why be suspicious about realism, so understood? Parsons mentions the possibility of going for a trope ontology rather than object/property ontology and ending up with a different catalogue of the fundamental constituents of the universe. But it isn’t at all clear why that possibility counts against quantifying over everything, as I said before in talking about a similar discussion in Hellman’s paper. For a start, note that Parsons says the problem is that there has to be a final answer to the question of what objects there are. But the tropist and the traditionalist (if we can call her that) needn’t disagree at all about the objects that there are. In particular, the tropist needn’t deny any of the objects that the traditionalist posits; it’s just that he has his own special story about what objects are (how they are constructed from tropes).

Perhaps Parsons mis-spoke and meant to say that absolutely general quantification involves fixing not the objects but the entities in some all-embracing cross-category sense. But I suppose that well brought up Fregeans might start getting unhappy about the coherence of that idea. And in any case, since the traditionalist can treat tropes as logical constructions, the tropist and the traditionalist don’t have different stories about what entities (broad sense) there are either, but rather a different story about the entities are interrelated by metaphysical dependence relations (whatever they exactly are).

Now in fact Parsons himself raises the same concerns about the trope example. But he comments

The mere fact of the possibility of construction is not sufficient, since constructions that may be offered will not necessarily satisfy the metaphysical intuitions that drive the alternative framework. That’s a reason for thinking that even if this possibility gives us a way of talking about everything in the world that does not commit us to metaphysical realism, even making sense of it gets us into heavy-duty metaphysics.

But I’m not getting the force of that. For remember the dialectical situation. Someone purports to quantify over everything. The objector says “Ahah! Do you realize you are committed to metaphysical realism in a bad way?”. The proponent of unrestricted quantification says “Why so?”. The objector responds “You are committed to thinking the world carves up into entities in a unique way: and what about e.g. the choice between a traditionalist and trope ontology”. We’ve imagined a come-back: “You’ve not shown that that’s a substantive choice about what there is, rather than a choice about how we organize the world into basic entities and constructions out of them”. And now Parsons is offering the opponent the retort: “Hmmmm, even making sense of that gets us into heavy-duty metaphysics”. To which the original proponent might reasonably protest that it was the objector who started playing the heavy-duty metaphysics game, so he can hardly complain about that. Rather it is the objector who needs to say more about e.g. the trope example and why (i) on the one hand it is supposed to be a contentful and a genuinely different story about what there is (not just a different story about “dependence”), yet (ii) on the other hand there is some kind of free choice about whether to adopt it rather than the traditional story, i.e. there isn’t an objective fact of the matter about which is the right story, which explains why we shouldn’t be metaphysical realists.

So for the moment, until he hears rather more, the proponent of absolutely general quantification can reasonably suppose himself to live to fight another day!

And having said that I would write about Linnebo’s paper next, I find myself rather regretting that promise, and this will have to be a non-comment!

Linnebo begins by announcing that the “strongest argument against the coherence of unrestricted generalization” is Williamson’s variant Russell paradox about interpretations; and he then takes the most promising line of reply on the market to involve adopting a kind of hierarchical type theory. Then Linnebo locates what he thinks to be a problem with the usual kind of “type-theoretic defences”. So he changes tack, and offers a different, more revisionary response to Williamson’s argument, which depends on rethinking the very idea of an interpretation (so that now predicates get not extensions but rather properties as their semantic values). He then needs a whole framework for talking about properties as well as sets (which has, he says, some similarity with Fine’s deviant project in his 2005 paper “Class and membership”), and this gives us a new sort of hierarchy of semantic theories. As with the old-style type-theoretic defences, we can now use this new hierarchical apparatus to blunt Williamson’s argument.

Now, how exciting/illuminating you find all this — given our interest here is in absolute generality — will depend, in part, on whether you think that the Williamson variant on Russell’s paradox gets to the very heart of a certain kind of argument against the possibility of absolute general quantification, or alternatively think that it somewhat muddies the waters by dragging in tangential issues (e.g. in its talk of interpretations as objects, etc.). My hunch so far has been the latter, though of course I’m open to persuasion; and the very complexities of Linnebo’s excursus don’t do much to dislodge that hunch so far.

I might return to Linnebo later, since Rayo’s paper seems to take us back into similar territory. But at the moment, I really don’t think I have anything useful to say. So let’s for now rapidly move on to consider Parsons’s paper.