Monday, March 31, 2008

Philosophy of Religion 5: Incarnation

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?

Sunday, March 30, 2008

Philosophy of Religion 4: Lord, Liar, Lunatic

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"

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 ...

Friday, March 28, 2008

Three Cheers for Gambero Rosso Italian Wines 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

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!]

Thursday, March 27, 2008

Philosophy of Religion 2: Eternity

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

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

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 ...

Wednesday, March 26, 2008

Absolute Generality 23: Parsons on the Williamson argument again

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

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 ...!

Tuesday, March 25, 2008

Philosophy of Religion 1: The ability to sin

The first chapter of Murray and Rea's An Introduction to the Philosophy of Religion is called 'Attributes of God: independence, goodness and power'. You can get an idea of its style and content from its final paragraph:

As the foregoing makes clear, the topic of the concept of God is a philosophically rich and fascinating one. In engaging the concept philosophically we must first decide which concept of God is salient, and then consider the various puzzles that arise for the divine attributes that follow from that concept. Our attention has been focused on the concept of God that arises from perfect-being theology. On that concept God is, among other things, self-existent (or necessarily existing), a creator and providential superintender, and perfectly good. While these attributes initially seem straightforward, more careful scrutiny shows us that interesting puzzles lie just beneath the surface. Do these puzzles show that there is something incoherent about the concept of God found in perfect-being theology? It is not at all clear that they do. However, it is also not clear that they don't. As we have seen, resolving this question requires taking stands on controversial claims which are currently at the forefront of discussion in the field. For this reason, discussion on these topics will be vigorous and ongoing.

That is, it has to be said, rather flat-footed prose, isn't it? (How did "engaging the concept" or "on that concept God is self-existent" get past the copy editor, I wonder grumpily). And the conclusion is bathetic. I fear there might be some students readers nodding off over this.

But if the style leaves something to be desired, what about the content? Well note that, from the off, this isn't so much philosophy of religion but the philosophy of theological speculation. Some may very well ask: How many of those who have just attended Holy Week services, for example, are committed to "perfect-being theology" in anything like the sense discussed here? Why? Just how is that commitment manifested in the quotidian prayers and practices of the unreflectively pious? I don't know -- and Murray and Rea don't pause to tell us. Which seems pretty unsatisfactory, both philosophically and pedagogically. Philosophically, because the relationships between religion and the religious life, on the one hand, and theological speculation, on the other, are surely rather obscure. We should certainly be asking, for example, what does talk of God's perfect goodness mean e.g. in the thought of someone trying to live a Christian life (there is surely some hermeneutic exploration needed here). And the danger in just diving in to wrestle with theological speculation is that those students of religious inclination will just feel that somehow that this is an intellectual game that doesn't really engage with their beliefs.

OK, but given that Murray and Rea are playing the theological speculation game, how well do they do it?

So far, I'm inclined to say, it all goes a bit quickly, and in places superficially. Here's one example. They are discussing the proposition that a being is morally praiseworthy only if it has the ability to sin. And they offer as a consideration against this proposition that we'd think someone praiseworthy who, faced with a possible occasion for killing finds homicide unthinkable (rather than finds himself genuinely torn over the issue, weighing up the pros and cons and coming down against killing in the end). But to find homicide "unthinkable" is to take the thought that A would be an act of homicide as a conclusive reason against doing A that simply silences any putative reasons for doing A. And someone who finds homicide unthinkable in that sense (most of us, I hope!) isn't thereby rendered unable to pull the trigger -- it is not a kind of paralysis, or like an inability to lift a ton weight. But if I can still pull the trigger, exactly why I don't I count as still having the "ability to sin" here (albeit an ability I at the moment wouldn't dream of exercising)? After all, I have the ability -- don't I? -- to change my mind about what's unthinkable and so do the dastardly deed. Well, there's more to be said: but we certainly need to know a lot more about the relevant notion of ability in play here in the proposition under discussion. But Murray and Rea rush on.

Religion and The Philosophy of Religion

I've commented here a couple of times about some daft newspapers columns about matters of religion and science (see here and here). And -- quite unexpectedly -- I've found myself contributing a number of responses on religion as a newbie panelist for Ask Philosophers (for example, here, here, and here).

I don't pretend, though, to be any kind of expert on the philosophy of religion: far from it (not that I needed to be to make the entirely obvious points in those postings I've just linked to). But I guess if I'm going to keep sounding off like this I ought to know just a bit more about the hot topics and fancy moves in the philosophy of religion these days. So this afternoon I picked up a copy of the brand new An Introduction to the Philosophy of Religion by Michael J. Murray and Michael Rea. It's published in the same CUP series as my Gödel book -- which in part is what drew my attention to the book.

The publisher's website calls it "a balanced and broad introduction", though a quick browse through makes it look as if "fence-sitting" might be a bit more accurate. And apparently it will be "a valuable accompaniment to undergraduate and introductory graduate-level courses", but it looks more like a first year intro. undergrad text to me. But ooops, I mustn't rush to judgement! Having shelled out for the book (albeit with a hefty press author's discount!) I better read it now: so I'll dive in and report back here.

At least it will make light relief from thinking about Absolute Generality, and other logic stuff.

Sunday, March 23, 2008

Absolute Generality 21: Parsons on metaphysical realism

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!

Saturday, March 22, 2008

Absolute Generality 20: Linnebo on sets, properties, etc.

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.

Friday, March 21, 2008

Absolutely Generality again: new readers start here!

Back in December, I was blogging about the Rayo/Uzquiano volume Absolute Generality. Then other things intervened. But it is time, at last, to get back to reading the rest of the volume.

So far, I've said something about the papers by Fine, Glanzberg, Hellman, Lavine and McGee. I've just put together what I wrote, only lightly edited, into a single document here, so you don't have to trawl back through the blog archive. As you'll see, however, I did admit defeat with Lavine's badly-written paper, trying to understand what he says in the end about schematic generality. I might return to that. But next up, over the next few days, we'll take a look at Oystein Linnebo's paper (all comments as we go along will, of course, be gratefully received).

Tuesday, March 18, 2008

"The Atheist Delusion"?

In the Guardian Review this last weekend, John Gray has a long piece entitled "The Atheist Delusion" in which he lambasts Dennett, Dawkins and the other usual suspects for their simple-mindedness about religion, for their intellectual sloppiness (e.g. the "memes" stuff), for promulgating quasi-religious myths of their own, for failing to face up to the evils perpetrated in the twentieth century in the name of atheist ideologies, and more besides.

Yet Gray himself ends up writing

Not everything in religion is precious or deserving of reverence. There is an inheritance of anthropocentrism, the ugly fantasy that the Earth exists to serve humans, which most secular humanists share. There is the claim of religious authorities, also made by atheist regimes, to decide how people can express their sexuality, control their fertility and end their lives, which should be rejected categorically. Nobody should be allowed to curtail freedom in these ways, and no religion has the right to break the peace.

But hold on. To reject the claims of religious authorities "categorically" is, precisely, to presuppose that the cardinals (or the ayatollahs, or whoever) do not have access to ultimate truths about what God categorically requires of us. And that presumption needs argument. Which is what our atheist writers aim to give us.

Perhaps there are deep human needs that, for many, are assuaged by participating in "communities of faith", and which if not given expression in the more gentle pieties of some traditional faiths can find very much uglier outlets (whether in fundamentalist religions or in fundamentalist poltical movements). But to think of a religion as just an often benign repository of moral precepts and myths and ritual practices for communities to live by (with no more, though not less, authority than comes from that very human role), is to deprive religions of the particular divine authority that they can clamorously assert for themselves. If that denial of special authority -- which Gray seems to endorse -- is to be justified, then the theological arguments of those who claim very much more for religion need to be countered. And when those who claim more are particularly vocal, they need to be countered not just inside the ivory tower, philosopher talking to philosopher, but countered, vigorously, accessibly, relentlessly, in the wider public conservation. Which is what Dennett, Dawkins and the other usual suspects are, really rather admirably, trying to do.

Monday, March 17, 2008

Logical Options, in retrospect

A quick end-of-term report on using Bell, DeVidi and Solomon's Logical Options as reading for a seminar, in case my experiences with this book are useful to others choosing a text.

Background: all our first-year students do a formal logic course using my (tree based) Introduction to Formal Logic. Then they also have to do a second year logic paper which is largely philosophical logic, but which also contains a component of more formal logic work. The formal syllabus covers [1] more on systems of logic other than trees, [2] more on interpretation of the quantifiers, [3] the idea of a formal theory, [4] modal logic, [5] the idea of intuitionistic logic. I had already lectured a bit on [4] and Michael Potter on [1] and [5]. I put on eight seminars, two each on the first four topics, for two groups of self-selected logic fans.

Logical Options seemed a perfect choice for a book to follow. It's tree based, so it follows on nicely from our first year course: and it covers [1] to [5] and a bit more.

I didn't think, though, that the book worked that well. I found myself producing quite extensive handouts for topics [1] to [3] (they are linked on my website), because Bell, DeVidi and Solomon's discussions are often just too terse to be useful. For just one example, no one would really understand the idea of a natural deduction system from their presentation (they give a set of rules, but oddly describe neither a Gentzen nor a Fitch method of setting out proofs using those rules). Some students found Logical Options quite useful as a summary discussion after they had read my more long-winded handouts: but no-one -- and these are very clever and very eager students -- said that they found the book worked well for them as a stand-alone read. So, as they say, close but no cigar.

I will probably give similar seminars next year, but organized a bit differently -- I'll expand/improve my handouts, and then use them as the main materials, with Logical Options as supplementary material rather than (as this year) officially focus on the book.

Thursday, March 13, 2008

Phew ... the end of another term

Two good seminars yesterday to mark the end of term. First, a small group of survivors staggered to the end of Hodges's Shorter Model Theory, helped enormously by Nathan Bowler taking us through the final chapter on Morley's theorem -- and for a moment, at any rate, I had a sense (thanks to Nathan) of understanding something of the route to that high point in model theory.

I'm glad I made the journey: but I'm not sure that Hodges is the best guide for readers who aren't hardcore mathmos (he tends to go straight up cliff faces by the short route, offering few comforting resting points for those of us whose grip is less secure, so we can get our orientiation and recover our breath).

Then in the evening, to the Serious Metaphysics Group (which, if last night's meeting was anything to go by, might be serious, but is certainly not solemn), to hear Tim Crane talking about "Three Dogmas of Quinean Ontology" -- Tim seemed misguided (placing more weight on the idea of "believing in Xs" than I think it will bear), but the occasion was good fun and thought provoking. I must get along again in the future.

And, apart from marking three more M.Phil. essays and going to a couple of hopefully brief exam meetings, that's it for this term. Terrific! So I hope I will now have time to get back to actually doing philosophy and to posting here ...

Sunday, March 09, 2008

Theories, models and Galois connections

Back in 1969, F. William Lawvere (in his Dialectica paper 'Adjointness in Foundations') remarked on what he called "the familiar Galois connection between sets of axioms and classes of models''.

But even if it has long been familiar to some category theorists and theoretical computer scientists, the idea that Lawvere is referring to here seems not to have been picked up that widely. Certainly, I've not been able to find a neat stand-alone presentation which is accessible e.g. to philosophers doing a first course in model theory. So I'm trying to put together some notes to fill the gap. (If anyone can point me to helpful aids to thinking about these things that I might have missed in googling around, then I'd be very pleased to hear about them.)

Don't expect novel fireworks, though! The name of the game is to get at routine points about theories and their models by a slightly unfamiliar route (and I'm still trying to work out just what points fall out of this approach in a natural way). However, approaching familiar territory from an unfamiliar angle can be illuminating. So I think it is probably going to be worth the effort.

Monday, March 03, 2008

Job done

I've now done the corrections for the second printing of the Gödel book. A slightly disconcerting business as -- even on a quick trawl through a few chapters -- I found more that I felt needed changing (on top of the necessary corrections that others had alerted me to) than I had expected. I'm left, then, with the feeling that there must be a lot more that ought to be corrected/improved. But I had to call a halt to the revisions, and at least the second printing will be better than the first, even if not perfect. It should be available in a few weeks. Watch this space for more info about that too.