Archive for March, 2008

Philosophy of Religion 5: Incarnation

Monday, March 31st, 2008

The concluding part of Chapter 3 of Michael Murray and Michael Rea’s An Introduction to the Philosophy of Religion concerns the incarnation. Just two very quick comments on these pages, one on how the authors approach the issue, and one on their final shot at a supposedly helpful analogy.

Murray and Rea kick off by quoting from at length from the Chalcedonian Creed of AD451, which propounds a doctrine of the incarnation, in effect by contrasting the “correct” view with various possible heretical interpretations. But the creed does seem — unsurprisingly — to be shot through with relics of philosophical views of the time. A student reader might very reasonably ask: why should we take a document that seems to be coloured by the metaphysics of the day as authoritative in shaping our understanding of what we might now mean by talking, say, of Jesus as our Lord? Different believers at different times (or at least, those with a taste for philosophizing about it) will no doubt interpret their religion in the light of the philosophical fashions of the day. Why give any special weight to the intellectual fashions of the fifth century?

The student’s worry here has, it seems to me, some real force. And Murray and Rea don’t really address such worries. They do talk of Christianity as a ‘doctrinal religion’, and argue that “a proper assessment of Christianity will require attention to a proper understanding of the core doctrines”. But it will probably be very unclear even to a believing student why understanding the religious doctrines of the gospels is best done via a later credal gloss which is then to be interpreted rather on the model of trying to make sense of an ancient metaphysics text. I suspect that Murray and Rea are in danger of losing their audience here.

But be that as it may. Let’s briefly consider their own best shot at understanding how it might make sense to speak of Jesus as being one person but as having both a human mind (that can, as scripture tells us, grow in wisdom and suffer temptation) and a divine mind. “Suppose,” they say, “we think that the human mind and the divine mind are related in a way similar to the way in which a person’s conscious mind is said to relate to her ’subconscious’ mind.”

Well, the contents of Joan’s subconscious mental processing are not routinely present to her consciousness (though some of them may be available). But equally, of course, they aren’t present to any other consciousness. There’s just one centre of consciousness here! But presumably, in the case of Jesus’s divine mind, that is a centre of consciousness (if we understand anything at all about the divine mind!). So what is going on here, according to the model, is that we have two centres of consciousness but with only the kind of partial access within Jesus from the human to the divine mind that we have within Joan from the conscious to the subconscious part of her mind. But that now sounds just like two (albeit imperfectly communicating) persons associated with Jesus — in the way there are two persons imperfectly communicating in a commissurotomy patient, according to Murray and Rea. Why isn’t that exactly the Nestorian “two person” heresy that they were struggling to avoid?

Philosophy of Religion 4: Lord, Liar, Lunatic

Sunday, March 30th, 2008

Pp. 75–80 of the Murray/Rea Introduction contain a rather extraordinary episode which I can’t forbear from commenting on.

They consider the following argument — they call it the “Lord-Liar-Lunatic” argument — for believing the Jesus of Nazereth was divine. Jesus claimed to be divine. The claim is either true or false. If the latter, either Jesus knew it was false, and was a liar. “On the other hand, if he unwittingly falsely claimed to be divine, then he was crazy.” But

the influence of Jesus’s teaching … has been enormous. Literally millions of people have found peace, sanity and virtue in orienting their lives round his teachings. … All of these facts together make it seem very likely that Jesus was neither so wicked and egomaniacal as to try deliberately to deceive others into thinking that he was divine, nor so mentally unbalanced as to be fundamentally confused about his own origin, powers and identity. If Jesus was not a liar or a lunatic … then there is only one alternative left: his claim to divinity was true.

Which really is a quite jaw-droppingly awful argument. Suppose we grant that Jesus claimed himself to be divine (I thought that was contended by many biblical scholars, but let it pass). And suppose he did so sincerely even though he wasn’t divine. Then he was badly deluded. But what on earth is the problem with that? History is full of people suffering from “crazy” delusions but functioning very successful in many domains of life.

Murray and Rea argue, in effect, that you can’t be “sane” and so deluded as to believe yourself divine when you aren’t (it isn’t, they argue, the sort of thing you can make a straight mistake about, at least if “divine” is used in the “perfect being” sense). OK: for the sake of argument, let’s agree with Murray and Rea: if Jesus was not divine, he was not fully “sane”. But — to repeat — that of course is entirely compatible with e.g. being an inspirational moral teacher. Bad cognitive mulfunction in one area is compatible with managing spectacularly well in other areas.

Another related point. Suppose a world of many messianic preachers, all deluded as to their own divinity (well, there’s been a fair bit of it around over the centuries — it’s a mental virus that can infect people, it seems). Most preach a variety of messages that fall on stony ground. Some preach messages that “catch on” temporarily, but in a quite horribly destructive way. But one, let’s suppose, picking up on ideas already in the air, charismatically preaches in a way that strikes a chord with his contemporary listeners; the message is taken up and propagated; and this time, let’s suppose “millions of people [find] peace, sanity and virtue in orienting their lives round his teachings”. But the fact that one such preacher happens to initiate a benignly propagating message [if that's what we think Christianity is -- of course, that's the subject of a different argument!] isn’t any evidence at all that his pretensions to be divine are any less deluded that those of his colleagues. Given enough different shots at it, and our apparent human propensities to be caught up by religious ideas, one deluded preacher was more or less bound to strike lucky.

Murray and Rea write that “the Lord-Liar-Lunatic argument seems to us … to be stronger than some contemporary critics have given it credit for being”. I do find that an astonishing thing to say. The argument is quite transparently hopeless.

Absolute Generality 25: Indefinitely extensible concepts, "big" and "small"

Sunday, March 30th, 2008

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 …

Three Cheers for Gambero Rosso Italian Wines 2008

Friday, March 28th, 2008

Looking through the last however-many posts, things have been getting a bit wordy and serious here. So time for a quick bit of light relief — though in the form, I’m afraid, of recommending another weighty tome. But what a book! The Gambero Rosso Italian Wines 2008 will damage your wallet a bit if you can’t resist some of the ‘tre bicchieri’ recommendations, but it will sure improve your quality of life.

Since the daughter went off to live in Italy and marry an Italian, I’ve more or less been sticking to drinking the Italian wine (getting up to speed on the culture and all that). The variety is wonderful even from neighbouring vinyards, and the quality can be amazing — though it can be very unexciting too, but that all adds to the thrill of the chase, finding the good stuff (even sometimes buried in Tesco’s). But if you stick to the Gambero Rosso recommendations, you won’t go far wrong.

Philosophy of Religion 3: The Trinity

Friday, March 28th, 2008

The first part of the Murray/Rea book, ‘The Nature of God’, has a couple of chapters on God’s attributes which I’ve just said just a bit about: and now there’s a chapter on more specifically Christian characterizations of God. I’ll take Ch. 3, ‘God triune and incarnate’, in three bites, discussing the Trinity here first.

Just an aside first. It is always difficult to know how to organize introductory books on any area of philosophy. As someone once put it, we need to travel over a wide field of thought criss-cross in every direction. So I fully accept that no linear order is going to be entirely satisfactory. But I do have to say that I find something a bit odd in the Murray/Rea approach here. To get stuck into the Trinity or the mystery (or should that be Mystery, with a capital “M”) of the incarnation before we’ve been given even the flimsiest reason for supposing that there’s anything that has enough of the supposed attributes of God to count as such does seem to be going about things a bit topsy turvy. But ok, let’s read on.

And in fact, the first few pages are rather a good read, because Murray and Rea acknowledge that the doctrine of the Trinity is a pretty rum one, on the face of it beset with internal contradictions, yet is central to traditional Christian doctrine. And they have no trouble trashing a number of once more-or-less popular analogies or models that are supposed to shed light on the doctrine. Things go less well when they go on to offer three more analogies that are supposed to help us out.

  1. (I paraphrase:) The Father, Son and Holy Spirit are like three members of a family. They are each divine, but the Godhead, i.e. the society of three persons, is the one God. Well, likewise Zeus, Hera, Athena, Apollo and company are a family. The individuals are each divine: can we then say the society of gods, the Pantheon, is the Godhead, god-as-one? Well, that doesn’t sound at all right about the Greek Pantheon. And following the analogy would have us talking about The Father, Son and Holy Ghost as three gods. Murray and Rea note the worry, saying that the family analogy “pushes in the direction of polytheism”. (They limply suggest that the defender of the analogy might say that the criticism requires “a serious analysis of what exactly it means to be a polytheist”. But such a defender would just be missing the point. We plainly don’t need any heavy duty analysis of polytheism to see that the family analogy assimilates Father, Son and Holy Ghost too closely to the Greek Pantheon in a way no traditional Trinitarian would want.)
  2. Just as a single human being can have multiple personalities, so too a single God can exist in three persons. The trouble with that formulation, of course, is that a personality (in the usual sense) isn’t a person; so we need to say something stronger, namely that where there is a single human being there may be more than one person. And sure, some have claimed that that is possible, e.g. in the case of some commissurotomy patients (though the claim is highly contentious). But even if we accept it, it doesn’t seem to help very much. For if a society of three people inhabiting three different bodies in a family relation isn’t a good model of the Trinity, why is a society of three people fighting over the same body? (Murray and Rea are in fact equally suspicious. But having so far come up with two dud analogies they say “these two analogies seem to have a great deal of heuristic value”. But that’s cheating. An analogy that you can’t make work is an analogy that doesn’t work: you have to go back to drawing board, not wistfully wonder whether it might have “heuristic value”, whatever that is.)
  3. The Father, Son and Holy Spirit are the same God but different persons in just the way that a statue and its constitutive lump are the same material object but different form-matter compounds. The trouble here is that Murray and Rea explain “are the same material object” as “share all of their matter in common”. So, when the wraps are off, their idea is that the Father, Son and Holy Spirit “are the same God” in just the way that a statue and its constitutive lump share all of their matter in common. But how can that be, unless we construe “are the same God” as “share all of their spirit-stuff in common”, or something like that? And of course we haven’t the foggiest what that begins to mean. Ah, say Murray and Rea quickly, “Of course, God is not material, so this can only be an analogy.” But if it isn’t an analogy we come near to being able to make use of, this is just useless arm-waving.

So, as far as Murray and Rea’s arguments go, the doctrine of the Trinity (as a bit of metaphysics) ends up as utterly obscure as it was at the outset. No surprise there then. As to the question of the religious content of the doctrine, what it means in a religiously led life to walk with God and acknowledge Jesus as his Son, and so forth, all that sadly goes unexplored.

What was a surprise was an argument they report from Richard Swinburne that purports to show that there are a priori reasons — quite independent of scripture — for the doctrine of the Trinity. God is perfectly loving; but need not have created anything. But perfect love requires a beloved, one existing even if he didn’t create anything, so this would have to be another divine person. But truly perfect love requires not only one beloved but also a third object of love — an additional person whom lover and beloved can cooperate together in loving. Hence the Trinity. Wow! Murray and Rea report Swinburne’s argument for the delights of threesomes with a straight face (though they don’t buy it). Mockery might seem a more apt response.

[And now I better take a bit of break from the Introduction to the Philosophy of Religion, and get back to talking about the terrific Shapiro/Wright paper ... But I'll no doubt not be able to resist returning to Murray and Rea soon!]

Philosophy of Religion 2: Eternity

Thursday, March 27th, 2008

The second chapter of the Murray/Rea Introduction is on “Attributes of God: eternity, knowledge and providence”. It has to be said, again, that the discussion seems to be pretty remote from engaging with the religious meaning of talk of seeking eternal life or of not being able to hide from God’s knowledge, and so forth. But we’ll just have to let this pass. Taking the investigations into philosophical theology on their own terms, how do they fare?

Here’s an example — a small one, but not entirely untypical perhaps. The context is talking about omniscience:

There are set-theoretic reasons for thinking that it makes no sense to talk about “every proposition”. For example, one might think that it makes sense to talk about every proposition only if there is a set of all propositions; but there are good reasons for thinking that there can’t be a set of all propositions. Here’s why: let P be the set of all propositions. Now consider the conjunction C of all members of P. C won’t be a member of P, since no conjunction has itself as a conjunct. Thus P can’t be the set of all propositions. … Hence it looks like there is no set of all propositions; and so it looks as if we can’t say things like “God believes every proposition.” If this argument is sound, then, the common-sense definition of omniscience will have to be modified.

Two comments — ignoring the point that the argument really ought to have have been directed explicitly against “God believes every true proposition”. First, no indication at all is given of why one one might think that it makes sense to talk about every proposition only if there is a set of all propositions (a student might very reasonably ask why, if she wants to talk about all donkeys, she has to believe in something else as well, namely a set of donkeys). So we are actually given no reason to suppose that generalizing over propositions is illegitimate, even if there is no set of all propositions. Second, although there is indeed a plausible argument against the claim that there is a set of all propositions (the Cantorian argument exploited by Patrick Grim in his The Incomplete Universe), this isn’t it. For a start, suppose you think of propositions as individuated by the set of possible worlds they are true at. Then the conjunction of (P & Q) with P and with Q is the same proposition as (P & Q) — so a conjunction can “contain itself as a conjunct” in one perfectly good sense. If Murray and Rea don’t like that entirely familiar but abstract Lewisian view of propositions, then they had better explain what other notion of proposition they are working with, and then they need to explain why on their (less abstract?) account, the operation of forming infinitary conjunctions is well defined. But of course they don’t.

This is rather sloppy writing and sloppy thinking, of just the kind we are trying to get our students to avoid!

Ok, let’s now take something more central. Issues about eternity and providence involve — at least on Murray and Rea’s construal — issues about the metaphysics of time. So they talk a little about different metaphysical theories, outlining what they call “eternalism” and presentism. I doubt whether students will understand much of the two positions from the over-brisk presentation. And the level of discussion is feeble. Eternalism supposedly holds

the familiar subjective experience of the flow of time, the transition from one moment to the next, is mere illusion. … Likewise, eternalism leaves no room for the idea that the past is gone or that the future is open and unsettled.

Wrong both times, of course. A B-theorist like Mellor in Real Time II can and does give an account of the subjective experience of the flow of time — it is no illusion that we have experiences that can reasonably be so-called, and those experience are not illusory in that they tell us nothing false about the world. And for an eternalist, of course the past is gone — it is past, it is out of our causal reach, there is nothing we can do about it, it is over! And an eternalist doesn’t have to be a determinist: he is as able to hold that the future is “open”, as unsettled-by-the-present, as a presentist.

And what about the eternal? Murray and Rea really struggle with the idea the God is eternal but atemporal (so eternal in some sense other than everlasting). How about this: “The idea underlying the doctrine of divine eternity is that God’s life is sort of like an infinitely thick specious present.”? Sort of like? Since when has “A is sort of like B” passed muster as an acceptable form of philosophical analysis?

And what do they mean by the “specious present”? Well, we are told that “the metaphysical present is a durationless instant, an infinitesimal moment of time” — thereby revealing that Murray and Rea don’t know what “infinitesimal” means. By constrast, our experience of the present “has some temporal thickness”: when we hear a friend speak, there is a good sense in which we are conscious “all at once” of a word or phrase, even though the event we are conscious of has duration. (There is no mystery about this, of course: there is a story to be told about the output of information-processing about such events into a short-term buffer.) And “this sort of temporarily thick experience of the present is what people refer to as the (experience of) the ’specious present’”.

So the idea, is it, that while we have information available to us “all at once” (in a snapshot, so to speak) about a relatively short duration, God has information available “all at once” (in a snapshot, so to speak) about a much longer duration? But of course, just having a snap-shot experience doesn’t in itself constitute any kind of life, however much information is available in that experience, or however wide its scope. For a conscious life in any ordinary sense of ‘life’ is constituted by temporal sequences of such experiences. But God is atemporal, it is being supposed. However, I forget: God’s life is only sort of like the specious present.

Spluttering into my coffee again

Thursday, March 27th, 2008

Earlier than usual coffee this morning, waiting for the new Cambridge Apple Store to open for the first time around the corner. (I foolishly thought I’d be able to wander in to take a look at a real-world MacBook Air. Duh! There was a queue hundreds long waiting in line to get in. I’m geeky but I’m not that geeky yet.)

Anyway, reading the Guardian. More foolish sounding off about religion, this time by Seumas Milne.

This has been the decade of liberal rage against religion … the anti-religious evangelists are increasingly using atheism as a banner for the defence of the global liberal capitalist order and the wars fought since 2001 to assert its dominance.

Ye gods. Here, just for a start, is that well-known evangelist Richard Dawkins in full tilt “defending” the Iraq war … in the Guardian.

Later: I eventually got to the Apple Store, and got my hands on a MacBook Air. A thing of real beauty and very covetable (and somehow feels remarkably solid and sturdy in the hand, the keyboard feels lovely too, and the screen is stunning quality). Interestingly, the display desk with seven of eight of them was surrounded by groups of teenage girls (and judging by the photo booth snaps that had been left on the machines, had been for hours). Hardcore Macheads might raise their eyebrows about some of the limitation of MBA. But I suspect there are going to be a lot of style-conscious kids with indulgent parents who are going to pressing oh-so-hard for one!

Absolute Generality 24: Parsons concluded

Thursday, March 27th, 2008

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 …

Absolute Generality 23: Parsons on the Williamson argument again

Wednesday, March 26th, 2008

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-ofα will be given in the form ‘for all o, P is true-ofα 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-ofo o).

We can then ask: is there an index κ such that for all o, P is true-ofκ 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-ofκ, 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.

Absolute Generality 22: Parsons on varieties of Russell’s paradox

Wednesday, March 26th, 2008

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 …!