My favourite session at the recent Arizona Ontology Conference was on Daniel‘s paper Finite Quantities.

Daniel argues that there is suggestive evidence from science to the effect that certain fundamental quantities are quantized rather than continuous.  That, for Daniel, is to say that not all properties of the form <em>having n units of X</em> are instantiated, for certain fundamental X such as mass, charge or perhaps distance.  Rather, for these X, there is some minimal n such that the property <em>having n units of X</em> is instantiated, and for all other instantiated properties of this form, ‘n’ is replaced with some multiple of this minimum.

It is commendable to get clear about what the quantized hypothesis looks like, and Daniel gets quite a lot clearer about it than most other discussions I know of.

However, having clarified that it is not a claim about the necessity (nomic, metaphysical or otherwise) of this restriction on the instantiation of certain properties, or about the non-existence, unreality or other substandardness of such properties (assuming that properties can exist uninstantiated), the view does not seem so very surprising or controversial. 

It strikes me as a much more modest and palatable claim than the claims that quantizers – including Daniel – often <em>sound</em> like they are making.  It sounds considerably less shocking, for instance, than the claim that ‘there is no such thing as’ (say) 1/2 n units of mass, or that although I may express things like “1/2 n units of mass” in <em>language</em> there is ‘no quantity corresponding to these representations’ and that ‘these quantities are not physically real’ (p. 2).

Moreover, clarity as to the exact nature of the quantizer’s thesis seems to make some of Daniel’s argumentative moves puzzling.

One of Daniel’s main opponents in the paper is someone who says that every time (say) a mass of six units is instantiated, the thing which instantiates the property <em>having six units of mass</em> also instantiates <em>having three units of mass</em> (twice over) and <em>having two units of mass</em> (three times over).
  
But let’s be clear about two readings of ‘having three units of mass’.  On the first, it means ‘having at least three units of mass’.  On the second, it means ‘having exactly three units of mass (and no more)’.

Now no-one would deny that everything which instantiates <em>having six units of mass</em> also instantiates <em>having at least three units of mass</em>.  That would be silly.  The quantizer, in this (made up) case, must instead be looking at denying that the property of <em>having exactly three units of mass (and no more)</em> is instantiated by anything.

But once we are clear that this is what is meant, the <em>opponent</em>’s position looks silly.  Obviously something which instantiates <em>having six units of mass</em> does not instantiate <em>having exactly three units of mass (and no more)</em>.

On neither reading, then, does it seem as if a Daniel-style quantizer and the opponent he describes in his paper have a sensible dispute such that they might need to look at the science to resolve it.

This is a comments thread for Jennifer Nagel’s paper “Epistemic Intuitions”:http://www.blackwell-compass.com/subject/philosophy/article_view?article_id=phco_articles_bpl104 in “Philosophy Compass”:http://www.blackwell-compass.com/subject/philosophy/. This article has been made free by Blackwell for the purpose of this thread. Here is the abstract.

bq. We naturally evaluate the beliefs of others, sometimes by deliberate calculation, and sometimes in a more immediate fashion. Epistemic intuitions are immediate assessments arising when someone’s condition appears to fall on one side or the other of some significant divide in epistemology. After giving a rough sketch of several major features of epistemic intuitions, this article reviews the history of the current philosophical debate about them and describes the major positions in that debate. Linguists and psychologists also study epistemic assessments; the last section of the paper discusses some of their research and its potential relevance to epistemology.

In the extended entry are notes on how to comment on threads here at TAR.
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This is a comments thread for Stephen Finlay’s paper “Four Faces of Moral Realism”:http://www.blackwell-compass.com/subject/philosophy/article_view?article_id=phco_articles_bpl100 in “Philosophy Compass”:http://www.blackwell-compass.com/subject/philosophy/. This article has been made free by Blackwell for the purpose of this thread. Here is the abstract.

bq. This article explains for a general philosophical audience the central issues and strategies in the contemporary moral realism debate. It critically surveys the contribution of some recent scholarship, representing expressivist and pragmatist nondescriptivism (Mark Timmons, Hilary Putnam), subjectivist and nonsubjectivist naturalism (Michael Smith, Paul Bloomfield, Philippa Foot), nonnaturalism (Russ Shafer-Landau, T. M. Scanlon) and error theory (Richard Joyce). Four different faces of ‘moral realism’ are distinguished: semantic, ontological, metaphysical and normative. The debate is presented as taking shape under dialectical pressure from the demands of (i) capturing the moral appearances; and (ii) reconciling morality with our understanding of the mind and world.

In the extended entry are notes on how to comment on threads here at TAR.
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This is a comments thread for John Worrall’s paper “Evidence in Medicine and Evidence-Based Medicine”:http://www.blackwell-compass.com/subject/philosophy/article_view?article_id=phco_articles_bpl106 in “Philosophy Compass”:http://www.blackwell-compass.com/subject/philosophy/. This article has been made free by Blackwell for the purpose of this thread. Here is the abstract.

bq. It is surely obvious that medicine, like any other rational activity, must be based on evidence. The interest is in the details: how exactly are the general principles of the logic of evidence to be applied in medicine? Focussing on the development, and current claims of the ‘Evidence-Based Medicine’ movement, this article raises a number of difficulties with the rationales that have been supplied in particular for the ‘evidence hierarchy’ and for the very special role within that hierarchy of randomized controlled trials (and meta-analyses of the results of randomized controlled trials). The point is not at all to question the application of a scientific approach to evidence in medicine, but, on the contrary, to indicate a number of areas where philosophers of science can contribute to a proper implementation of exactly that scientific-evidential approach.

In the extended entry are notes on how to comment on threads here at TAR.
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Not all of these are from current employers, but that’s the theme.

• As Carrie notes “below”:http://tar.weatherson.org/2008/01/15/more-great-jobs-at-arch/, there are two new postdocs being advertised at Arché, connected to the methodology project. I assume that philosophers from a lot of different areas would be viable candidates for such positions. Actually, I’d rather hope that it doesn’t turn into an entirely M&E focussed discussion. My “most prominent contribution”:http://brian.weatherson.org/counterexamples.pdf (PDF) to methodology debates started from the idea that epistemologists should copy some trends from work in ethics. But such contributions may be better made by an actual ethicist. Anyway, these positions are highly recommended; arguably they are a better way to start your career than most tenure-track jobs.
• I’m not employed by the “Ammounius Foundation”:http://www.ammonius.org/ but they are teaming with one of my colleagues, Dean Zimmerman, to again run an “Essay Competition for Younger Metaphysicians”:http://www.ammonius.org/younger_scholars/2008.html. Actually you don’t have to be a metaphysician to enter the competition, you just have to submit a metaphysics paper. Details “here”:http://www.ammonius.org/younger_scholars/2008.html for whoever is interested.
• Many of you will have seen that Cornell is “in the news”:http://www.nytimes.com/2008/01/12/books/12feud.html?em&ex=1200373200&en=24b96d107ca2b3a2&ei=5087%0A over the most famous book review of the last few years. I had several comments drafted about this, but they were mostly ripoffs from comments on related topics by “Bill Simmons”:http://sports.espn.go.com/espn/page2/story?page=simmons/071026 and “Clive James”:http://torch.cs.dal.ca/~johnston/poetry/bookofmyenemy.html, so maybe I should leave it go.
• We’re going to be trialing a new initiative with “Philosophy Compass”:http://www.blackwell-compass.com/subject/philosophy/, namely using TAR for comments threads on articles. Blackwell have made three articles available free of charge to get this off the ground, and I’ll be posting links and comment threads later this afternoon.

Two more Arché postdocs are being advertised, this time in connection with the new Philosophical Methodology project.  If interested, check out the details on the St Andrews website.  These are very attractive positions, running for four years and based in a world-leading research centre.  I can highly recommend application to anyone at the right career stage and with the right philosophical interests.

Following up on <A href=http://tar.weatherson.org/2008/01/01/signalling-and-job-markets/>Brian’s recent post</a> about candidates having to signal to departments that they’re actually interested, I’ll mention some ideas that my friend and colleague Mike Titelbaum and I were discussing one evening at the APA.

One thing that would remove the need for people to signal like this would be by putting the decisions of who is hired where in the hands of some sort of benevolent third-party (perhaps like the APA or something). Candidates could submit a ranked list of their preferences for which department they’d like to be at, and departments could submit a ranked list of their preferences for candidates (after having seen the files and conducted interviews and such), and hopefully some sort of matching between candidates and departments could be arranged from this information. (We might also want to allow some sort of cut-off where a candidate or department could specify that they’d rather just remain unmatched this year and repeat the search next year, rather than take anything further down on their list.) If this could be centralized, it would eliminate the inefficiencies each year where a position goes unfilled, because a department’s first few choices take other jobs, by which point their later choices have already settled for something else. These situations hurt both candidates and departments, because there are fewer actual jobs to go around, and some departments end up having to repeat the whole search process.

The important question to answer (ignoring temporarily the question of whether such a process would have negative consequences as well as positive ones) is whether such a process is even possible. Of course, one could just randomly assign candidates to jobs, but that would be no good – we’d want the assignment process to satisfy certain criteria.

1. The process should be able to take any set of rankings of departments and candidates and produce an assignment, with exceptions only if it’s impossible to construct a matching meeting the minimum acceptability cutoffs.

2. The matching should be “stable” in the sense that if C1 is matched with D1 and C2 is matched with D2, then it should not be the case that C1 prefers D2 to D1 and D2 prefers C1 to C2. (This condition guarantees that no department and candidate have an incentive to defect from the centralized assignment. Perhaps this condition can be dropped if the overall system is important enough to people’s long-term careers that there are already strong incentives not to defect.)

3. If one particular list of preferences produces a match between candidate C and department D, then keeping the same lists of preferences while raising C’s position on D’s list, or D’s position on C’s list should also result in C and D being matched matched. (This is the condition that guarantees there is no incentive to falsely list one’s preferences. We might want to further require that these changes in preferences make <i>no</i> change in the overall matching, because these changes should be irrelevant to anyone else’s matching.)

4. Maybe there should be other conditions too – the only potential one that comes to mind is that changing your preferences among candidates or departments that are lower on your list than the one you were matched to shouldn’t change anything, though perhaps this criterion is more arguable.

Once we’ve got a list of criteria like this, it should be possible either to construct an algorithm that meets these criteria, or to prove that no such algorithm exists. By <A href=http://en.wikipedia.org/wiki/Marriage_theorem>Hall’s Marriage Theorem</a>, criterion 1 is always possible as long as there is no set of m departments that find only the same n candidates acceptable, or m candidates that only find the same n departments acceptable, with m>n. By the <A href=http://en.wikipedia.org/wiki/Stable_marriage_problem>Stable Marriage Theorem</a>, there is in fact an algorithm that satisfies criterion 2. The question is whether these two can be combined with criteria 3 and 4.

Now, since criteria 3 and 4 were inspired by <A href=http://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem>Arrow’s Impossibility Theorem</a>, it might seem that such an algorithm is impossible. However, I have hope in this case, because the construction is not as involved in this case. In Arrow’s theorem, the problem is that given a bunch of rankings, no group ranking can be found that is positively influenced by all of them. In this case, we start with a bunch of rankings, but don’t need to produce a ranking – instead we just need to produce a pairing. And in this case, differences in rankings seem like they should only make things easier (because if the two of us have different first choices, you can make both of us happy in this case, while you can’t in Arrow’s situation).

In fact, if we don’t have minimum acceptability cutoffs, and all the candidates agree on the ranking of the departments (or vice versa), I can construct an algorithm that satisfies all these criteria. Just have the departments draft candidates one at a time, with the departments picking from most preferred to least preferred. (Or vice versa, if the departments all agree on a ranking for the candidates.) Since disagreements in ranking look like they should intuitively make things easier, hopefully this means that there’s an algorithm that will work in general, though actually coming up with this algorithm looks much harder.

Anyway, someone working on social choice theory should find the right set of criteria here and publish the proof of the possibility (or impossibility) theorem, and then the APA (and other professional societies) can have the discussion about whether or not to adopt the procedure. One thing that can be said for the current system is that it gives candidates and departments many chances to adjust their rankings of one another – this might be a way to help get around impossibility theorems, which will assume that the ranking is fixed.

 This article on the rise philosophy in Scottish schools is worth a click, not only for the heartening story itself (well done, Lisa) but also for the salutary reminder, from the comments, of how much of a problem philosophy still faces with its public image. 

HT: Nothing of Consequence, via Philosophy, et cetera.

I think that the following three forms are truth-conditionally equivalent.

(1) p or q
(2) p or else q
(3) p or, if not p, q

I think that (2) and (3) are equivalent in a stronger sense; they express the very same proposition. That is, “else” just means “if not”. But it isn’t obvious that (1) and (3) are equivalent. In particular, saying that they are equivalent requires some constraints on our theory of conditionals.

_Digression_. If we have intuitionist doubts about conditional excluded middle then we shouldn’t think that (1) and (3) are equivalent. If we let q = ~p, then (3) is obviously an intuitionist theorem, while (1) is not. It’s an interesting question what the intuitionist should say about (2) in that case. I think they should say it is a theorem, and perhaps say that one reason classical theorists are so tempted by the purported theoremhood of _p or ~p_ is that they confuse it with the genuine theorem _p or else ~p_. But real intuitionists might have alternative views. I’m setting intuitionism aside for this post though. _End of Digression_.

That (3) entails (1) just requires that modus ponens is valid. (Assuming classical logic for the non-conditional connectives). Proof:

# Assume that (3) is true.

# Assume that p is true.

# Then (1) follows from line 2 by or-introduction.

# Drop the assumption of p, and assume not-p is true.

# Then if not-p, q is true. (Disjunctive syllogism from 1 and 4)

# So q is true (Modus ponens from 4 and 5)

# So (1) is true (from 6 by or-introduction)

# So (1) follows from (3), since it follows from either (3) combined with either p or ~p

I think this already raises a problem for those like McGee and Lycan who deny modus ponens. But most of us believe in modus ponens, so we can accept this entailment. The bigger issue is whether (1) entails (3).

Assume, as many philosophers do, that A and B can be true while If A, B is not true. Now consider what happens when we let p = ~A, and q = B. In that case (1) will be true, because the second disjunct is true. But neither disjunct of (3) is true, so (presumably) (3) is not true.

On the other hand, if we accept the principle of strong centring for indicative conditionals (i.e. that A & B entails If A, B), we can prove that (1) entails (3). Proof:

# Assume that (1) is true.

# Assume that p is true

# Now (3) follows from line 2 by or-introduction

# Drop the assumption of p, and assume ~p

# So q is true (by disjunctive syllogism from 1 and 4)

# So if ~p, q is true (by strong centring from 5 and 6)

# So (3) is true (by or-introduction from line 7)

# So (3) follows from (1), since it follows from (1) combined with either p or ~p

Since intuitively (1), (2) and (3) are truth conditionally equivalent, this seems to me to be a powerful argument for strong centring.

Mostly inspired by “some”:http://semanticsarchive.net/Archive/TI1OGVlY/iffiness.pdf “papers”:http://www-personal.umich.edu/~thony/note_on_if.pdf by Thony Gillies, I’ve been thinking a lot lately about the scope of modals inside conditionals. I started thinking about these again because of some issues that come up in a paper that John MacFarlane and Niko Kolodny are presenting at the “AOC”:http://aoc.web.arizona.edu/. But that paper isn’t online, so I won’t discuss (yet) the issues they raise, which are largely about deontic modals. Instead I want to go over a puzzle about disjunction that challenges a position I’m attracted to.

The basic setup of the puzzle comes from Thony. Imagine that I’ve lose a marble. The three places in my apartment that marbles go under are the sofa, the table and the desk. I’ve looked under the table, and the marble isn’t there. It seems I can truly say (1).

(1) if the marble isn’t under the desk, it must be under the sofa.

But it also seems I can truly say.

(2) It’s not the case that the marble must be under the sofa (since it might be under the desk).

And from these, by modus tollens, it would seem to follow that the marble is under the desk. But I’m clearly in no position to conclude that. What has gone wrong? There are three possible options that one could take here.

• Conclude that modus tollens is invalid.
• Conclude that the logical form of (1) is not as it appears, and really the modal ‘must’ takes wide scope over the conditional.
• Conclude that there is an equivocation between (1) and (2).

The first option seems desperate, particularly since we can use a similar argument to show that modus ponens is invalid. Obviously some people (e.g. McGee, Lycan) have rejected modus ponens, and for reasons not a million miles from what we’re considering here. But it seems like a move of last resort to me. By the end of this post we’ll come to something that might look like a new cost of the position.

The second option is, however, a very popular position. I used to believe it, but I’ve been convinced by Thony that it isn’t true. The problem concerns what happens with weak modals. So (3) seems to be false.

(3) If the marble isn’t under the desk, it might be under the table.

That’s false because I looked under the table, and I know it isn’t there. But let’s say that the modal ‘might’ takes wide scope, and the conditional is a material implication. So (3) is equivalent to (4).

(4) It might be the case that (the marble is under the desk or the table).

But (4) is true, and we wanted (3) to be false. Perhaps the problem isn’t the assumption that the modal in (3) takes wide scope, but that the conditional is a material implication. So let’s assume that the embedded conditional is some kind of epistemic conditional. (I.e. if p, q is roughly equivalent to must (not p or q).) Then we have a different problem. Imagine that I said (3) before I had looked under the table. Then it would be intuitively true. But (5) on this reading is arguably false.

(5) It might be the case that (if the marble isn’t under the desk it is under the table).

That’s arguably false because at that stage I know that _must (marble is under desk or table)_ is false. It’s starting to look like the problem is that ‘might’ takes wide scope, not the interpretation of the conditional. As I mentioned in “a previous post”:http://tar.weatherson.org/2007/11/08/gillies-on-wide-scopism/ these problems are even worse when we use deontic modals. So I think the wide scope solution is flawed.

That leaves us with the equivocation solution. Here’s the version of the equivocation story I prefer. (It’s basically a translation into static semantics of the dynamic semantics story Thony likes.) Epistemic modals have an attached plural variable. In standard settings the values of that variable are those propositions known, or perhaps available to be known, in the conversation. (There are some relativist bells and whistles I like here, but I’m ignoring those for this post.) Roughly, _Must (X) p_ is true iff some of the X collectively entail p, and _Might (X) p_ is true iff p is consistent with the X.

The complication comes in conditionals, particularly in the consequent of conditionals. In that case, I think the X is ‘partially bound’. All the propositions that are known are among the X, but so is the antecedent. So the LF of (1) is something like

(1a) If (marble not under desk), must (Y) (marble under sofa)

where the Y are all the propositions known plus the proposition that the marble isn’t under the desk. And the LF of (3) is something like

(3) If (marble not under desk), might (Y) (marble under table)

where Y is as above. This gives all the right results in these cases, and does so in something like a systematic way.

Now for the problem. Epistemic modals also behave oddly in the second disjunct of disjunctions. So both (6) and (7) sound OK to me.

(6) Either the marble is under the desk or it must be under the sofa.
(7) Either the marble is under the desk or else it must be under the sofa.

I think (7) is a little preferable to (6), a fact that will become a little important in what follows. Both (6) and (7) can be combined, it seems, with (2) to conclude (improperly) that the marble isn’t under the desk. Again we have three options.

• Conclude that disjunctive syllogism is invalid.
• Conclude that the modal in (6)/(7) takes wide scope.
• Conclude that there is an equivocation between (6)/(7) and (2).

Again, both of the first two options seem unhappy. Indeed, denying disjunctive syllogism for natural language ‘or’ seems to be an even more radical step than denying it for natural language ‘if’. (Various heterodox logicians might disagree here, but debating that would take us too far afield.) And the wide-scope move looks just disastrous, since it would make (8) acceptable.

(8) Either the marble is under the desk or it might be under the table.

So we need some kind of equivocation story. Following some (long) conversations with Ishani, I’m inclined to believe the following story.

Start with (7). Arguably the ‘else’ there means ‘if not’. In general, I think, it seems fine to analyse (9) as (10).

(9) p or else q.
(10) p or, if not p, q.

If ‘if’ means material implication, then (10) is just equivalent to _p or q_. But we might not read ‘if’ that way. In particular, we might say that modals in the consequents of conditionals have partially bound variables. So we might read (7) as (11), which we in turn analyse as (12).

(11) Either the marble is under the desk or, if not, it must be under the sofa.
(12) (Marble under desk) or (if (not under desk), must (Y) (under sofa)).

Again, Y consists of those propositions we know plus the antecedent, i.e. that marble isn’t under the desk. So we know the second disjunct is true, so the disjunction is known, so it is assertable. (And note that we don’t have the normal Gricean problems with redundant disjuncts because the first disjunct is a constituent of the second disjunct.)

So far so good. But what should we say about (6)? Well, I think we should say much the same thing. More precisely, I think we should say that there is an unpronounced ‘else’ in (6), and that ‘else’ means ‘if not’, and then the analysis is as for (7). The reason that (6) is a little less happy than (7) is that it only makes sense if we read in this unpronounced element, and the speaker should probably have pronounced this for us. The reason that we can read this element in is that in general _p or q_ and _p or else q_ are so close in meaning that we can freely substitute one for the other.

Note that this story is available to people who think that the problem with the argument from (1) and (2) is that modus tollens is invalid. Such people should think that the argument from (7) and (2) to the conclusion that the marble is not under the desk is not, strictly speaking, an instance of disjunctive syllogism. Rather, it requires the use both of modus tollens and disjunctive syllogism. And such people say that modus tollens is invalid. No wonder the argument fails!

But there is something a little odd about this position. If modus ponens is invalid, then it is possible for _p or else q_ to be true while _p or q_ is false. That’s a surprising, and I think a little unhappy, result. I much prefer the equivocation story.

Coming later: The same arguments run through with deontic modals rather than epistemic modals.