THE SMEDLIUM CASE by Stephen Law

This is adapted from my 2005 Ratio paper “Systems of Measurement”.

Wittgenstein says that we cannot describe the standard metre bar as being one metre long (because of its peculiar role in metric system of measurement). Kripke says we can, and that it is, in fact, only contingently one metre long.

Consider these two systems of measurement

1. The W-system of measurement

Let the reference of ‘one W’ with respect to any arbitrary time t and possible world w be the length that stick W has at t at w (and be empty otherwise). Thus stick W can never be and could never have been anything other than one W long. It is a necessary truth that, if it exists, W is one W long.

Having thus defined ‘one W’, we can now set about expressing the length of a given object as a multiple/fraction of one W.

Notice that in this system of measurement stick W’s length in Ws at any arbitrary time and/or world is stipulatively held constant. Stick W is necessarily one W long. Shrink or stretch it: stick W remains one W long. Indeed, by shortening stick W one alters the W dimensions of other objects.

I shall call this the W system of measurement. It seems plausible that on Wittgenstein’s view the ‘peculiar role’ assigned to the Standard Metre in the metric system is precisely that which is assigned to stick W in the W system.

2. The K-system of measurement

Suppose we introduce the expression ‘one K’ to refer to that length which stick K happens actually to possess at t0. We might then go on to measure length in Ks using stick K, and do so quite accurately, just so long as stick K remains the same length. But then, even though the length of K is used to measure length in Ks – indeed, even though it may be the only thing we use to measure length in Ks — it is nevertheless contingent that stick K is one K long. For stick K might not have been the length it actually is.

Let’s call any system of measurement in which all measures are used in this way K-type.

On Kripke's view, the metric system is like the K-system. ‘One metre’ names a certain length: that length which the Standard Metre happens currently to possess. Thus the Standard Metre is only correctly used to measure length in metres on the condition that it remains that same length.

Intuitively, it seems Kripke is right about this.

But now let’s turn to my Smedlium Case.

The Smedlium Case

Imagine a world quite similar to our own that contains large quantities of a metal-like material – let's call it smedlium – which gradually and unpredictably alters in size. All smedlium objects expand and contract in unison. At one o'clock on one particular day all the smedlium objects are 5% larger than they were at mid-day; at two o'clock they are all 10% smaller, and so on. Despite this peculiarity, smedlium remains a useful material. In fact, it is the strongest and most durable material available. One of the inhabitants of this world builds machinery made wholly out of smedlium. The machines are used in situations where their size relative to non-smedlium objects doesn't matter. The smedlium engineer constructs and calibrates a measuring rule made out of smedlium to use when manufacturing such machines. She measures dimensions in ‘S’s, one S being measured against the length of her smedlium measure. Of course, so far as manufacturing smedlium machines is concerned, a smedlium measure is far more useful than is a rule made out of some more stable material, for it allows the smedlium engineer to ignore the changes in size of the object upon which she is working. For example, she knows that, say, if the hole for the grunge lever measured 0.5 S in diameter at one o'clock, then a grunge lever which measures 0.5 S in diameter at two o'clock will just fit into that hole, despite the fact that the hole is now noticeably smaller than it was at one o'clock.

Now one might think that here at least is one case in which a measuring rod functions as does stick W in the W system, not as does stick K in the K system. Surely, one might argue, what ‘one S’ designates with respect to any arbitrary time and world is the length of the smedlium engineer’s measuring rod whatever it might be at that time and world, not the length that it actually possesses at some particular moment in time. The smedlium system is a W-type system.

And yet, oddly enough, we have the same modal intuitions about the smedlium system as we do about the metric system. It seems that the smedlium measuring rod might cease to possess the measurement one S. It might actually come to possess e.g. the measurement 0.9 S.
Suppose, for example, that mid-way through a month when the smedlium engineer is working on a particularly important project, a saboteur breaks into the smedlium engineer's workshop and indulges in some industrial espionage. The saboteur shaves 10% off the end off the smedlium measuring rod knowing this will cause the smedlium engineer all sorts of problems. Isn't it the case that the smedlium measuring rod no longer possess the measurement one S? To me, this certainly seems the right way to describe the situation. Indeed, it seems right to say that the smedlium measuring rod now has the measurement 0.9 S, given that it is now 10% shorter than it would otherwise have been.

It also seems right to say that the smedlium measure might never have had the measurement one S: it might always have been only 0.9 S long (one might tell a story on which the mould in which stick S was originally cast leaks at one end, producing a sightly shorter stick). So, intuitively, it is contingent that the smedlium measuring rod possesses the measurement one S.

A puzzle for Kripke

So we have the same sort of modal intuitions about the smedlium system as we do about the metric system.

We saw that the Kripkean explanation of why it is contingent that the Standard Metre possesses the dimension one metre is that ‘one metre’ is a rigid designator: it rigidly designates a certain length – a length the Standard Metre happens only contingently to possess.

But note that this explanation is unavailable when it comes to explaining why it is contingent that the smedlium measuring rod possesses the dimension one S. Clearly, “one S” doesn’t rigidly designate a length. An object can retain the dimension one S even while altering in length.

This raises a difficulty for Kripke: it seems that, in the smedlium case, the intuition of contingency is going to have to be accounted for in some other way. But if the contingency is to be explained other than by supposing that ‘one S’ is a rigid designator (of a certain length), then presumably that same alternative explanation might be provided in the metric case too.

For more, see the original paper:
http://lawpapers.blogspot.com/search/label/Systems%20of%20Measurement

philosophy of philosophy

The Seminar on the philosophy of philosophy was amazing, its seems to be about everything I’m most interested in. Better still Barry Smith was there, I hope he’ll come every week, listening to Smith and Papineau arguing is like watching a speeded up film of whole debates in contemporary philosophy.
The thing about the philosophy of philosophy is it is productive. If you can make methodological distinctions clearly, then what may seem to be an intractable problem in philosophy maybe unravelled and revealed as a simply clash of methodological choices. If the goal of philosophy is to unravel the riddles of the universe, (rather than, say, create jobs philosophers), then this might be a generative source of successful philosophy. And what can be better than successful philosophy?
One aspect of the seminar that excited my interest without gaining my understanding was this point about Carnap and Ramsay sentences for the meaning of terms.
The background: a verificationist theory meaning (not the best one according to me) attributed to Quine has it that the meanings of a terms (are) (supervene on) (are constituted by) (are what cause) our dispositions to use the terms. Problem, if person A has theory X about f but person B has theory Y about f, they may have different dispositions. Does this mean that A and B are talking about different things when talking about f? I feel there are strong reasons for saying “no, they’ve just got different theories.” Argument: If A persuaded B to change theories, B may change her beliefs about f, but surely not the meaning of f, otherwise it is hard to say what the belief change consists in. But this is just me. Philosophers sometimes when confronted with a certain species of counterexample to their favourite theory will use the argument “oh, you just have a different concept of f”. So for example, if Ruth Millikan claims that 2 out of three people she has interviewed don’t actually share the Gettier intuition, then we can do one of three things. 1. Say that the interviewees have a different concept of knowledge. 2. Change our theory to accommodate the interviewees. 3. Use our theory to claim that the interviewees are wrong.
Another example:
Stich: “Beliefs don’t exist”.
Chorus: “Of course they exist (otherwise how am I supposed to believe that they don’t)”.
Stich: “But our concept of belief has it that beliefs are entities in the brain with causal roles, yet in our brains is just neural nets, nothing like beliefs.”
Chorus: That’s not my concept.

Now the bit I don’t really understand. A term can be fully (cashed out)? (defined)? (extended)? Using a conjunction of all its (applications)? (sentences in which it appears)? (true sentences in which it appears)? Using a Ramsey sentence.
T(f)
E(Q) T(Q)
The idea is, you list all the sentences in which the term f features, this gives you T(f), then you replace (f) with Q. Then your theory claims that E (Q) in which the Ramsay sentence is true. (E = existential quantifier)
(My view of this is it will only work, and it will work very well, if the Ramsey sentence is a list of sentences which are believed to probability 1, or “known” in which the term appears. This will include all known predictions and known hypothetical and counterfactual cases. Therefore it will include intuitions when we intuitively know, but not when we intuitively reckon, think or guess)
Professor Papineau then compares a Carnap version:
T(f) = If E (Q) then T (Q).

The difference is simply that the Ramsey version is false if nothing fits the description, whereas the Carnap version is true even if nothing fits the description.. Stich has a Carnap version of the concept of belief, because his T(belief) contains sentences like “beliefs are entities in the brain with causal roles within action”. The fact that there are no such entities therefore does not force him to change his concept “belief”, instead it leads him to assert that beliefs don’t exist. However when using a Ramsey sentence, then Stich’s “discoveries” that there are no beliefs simply proves that the theory is false, not that beliefs don’t exist.

What is puzzling me is how to proceed. How does this help untangle the riddles of the universe? My opinion is that our concepts are like the Ramsey sentences. Suppose I am such a materialist that I count it as known that beliefs are bits of brain that fulfil a certain causal role. Then my Ramsey sentence will be like this (S1, S2, S3……..SN). I am horrified to discover that nothing fits this description. I am wrong! Something I thought I knew was false! Impossible? Of course not, it happens all the time, except to the very narrow minded and dogmatic. But since S1 …SN count as known, there is a real conflict, which may well result in a strong desire to reject the evidence. But if the evidence is overwhelming, rationality will require that you give up what previously counted as certain. So let us say you drop SX to SN, the sentences related to the belief – brain identity thesis. Is your concept now different? Not that different, you still have a huge store of known sentences involving the term belief. You still know S1 to SX You still know a million facts involving the term “belief”. Eg. I know that John Wright believes that there is a region of London called Mayfair and he wants us all to go there on the 1st of March. I know that if he believes that there are pubs in Mayfair, then he also believes that there are pubs in London. No bit of fancy neuroscience is going to convince me to drop these facts. These parts of the Ramsey sentence remain intact. The term “brain” may have taken a bit more of a knocking, but hey, how much do I know about brains anyway? I am happy to leave that up to the scientists. Currently I know that the hippocampus is enlarged in taxi drivers and certain male birds, and is involved in navigational tasks. However, if some scientists demonstrate that this is false, then this may radically alter my concept of “hippocampus”. Ouch! But hey! Its only the hippocampus, its not my girlfriend.

How not to get published by Jonny Blamey

Some time ago I promised to send my two envelope solution to a journal and post the referees reports on this blog. Here’s the whole story

Last May I gave a talk at a conference and someone told me afterwards that he thought my thesis would have some bearing on the Two Envelope paradox. I saw the solution straight away. Using Ramsey’s measure for degree of belief, the paradox becomes a simple disagreement in probability assignments. I was looking for a problem with which to demonstrate my new techniques in decision theory and here was one that fell into my lap.

I put my solution on the blog and received a huge amount of negative comments. As one of the labels was “rigid designator” I got one comment saying that I obviously didn’t understand the two envelope paradox since “rigid designator” was irrelevant to the problem. I got some other comments along the lines that I didn’t understand probability, maths, philosophy, the principle of indifference, Bayes theorem, the nature of argument and anything at all. My lack of knowledge is on Socratic proportions.

At another conference I met someone who had published work on the two envelope paradox. Friends with him was Sorin Bangu who gave a talk on the principle of indifference. I asked Dr Bangu to post on Bloggin the question. By November an article came out in Mind on the two envelope paradox using the concept of rigid designation by a pair of scholars from Dr Bangu’s university. Does this make my use of rigid designation retro actively valid?

Meanwhile someone on the blog challenged me to send my solution to a journal, specifically the BJPS and when it was rejected, to publish the referees reports on the blog. Instead of a flat rejection the BJPS asked me to rewrite in response to the referees reports. The referees reports said my solution was an ingenious contribution to the literature, but had a flaw. If I could respond to the comments they would consider publishing.

This I did, but in response to something that David Papineau said I was working on trying to express the solution in more natural terms. My Dad came over to visit and we spent an afternoon drawing graphs of possible pairs of envelopes, pigs, people etc. who fit the description. We found that the more densely packed and widely spread the envelopes, the more closely the graph fits 1/x for the sum of the pair and 1/SQRx for the second envelope given the first. If you normalise this then for any N you can show that the probability that swapping will double your money is 1/3. What no one in the literature had grasped was that the envelopes necessarily come in pairs. Once you accept this fact it is possible to find the probability distribution. It was assumed in the literature that you can have any prior probability distribution you like, which is absurd. Why should you act on an arbitrary probability distribution? And how can the probability that an envelope contains 2x P(2x) be greater than P(4x) + P(x)? And if you accept two end points below and above which there can be no envelope, then an equal distribution is impossible. P(2x, x) = 1/3 and P(1/2x, x) = 2/3 is the only pair of conditionals that works.

I tried to put this into easy accessible language and resubmitted to the BJPS. They were late coming back to me, saying that they had to wait ‘til after Christmas for one referee. Finally on my birthday they sent me a rejection with only an extract from one referee’s report.

Begin Quoted Text----------------------------------------------------------Sorry to be a bit slow on this. I have read the paper, and don't think it's really good enough to recommend. There is an interesting central idea, but (a) the paper has too much irrelevant detail in the first six pages and (b) I don't think the main contention of the paper is strongly enough supported. There are also some minor (but distracting) inaccuracies.----------------------------------------------------------End Quoted Text----------------------------------------------------------

I asked them to sent me the full reports, and they said they would, but it turned out that one referees report was lost and the other referee did not want his comments exposed to judgement.

Meanwhile I sent off a really accessible version for the Jacobsen essay prize and did get to see some of the examiners reports from that one. So I’ll paste those below as a substitute.

EXAMINERS REPORTS

3. Envelope stall.
This short essay simply presents a well-known ‘paradox’. No references are given and there is nothing original said about the issues that it raises.

I expected to warm most to essay 3. But, aside from the fact that it makes no reference at all to the literature, and the fact that the answer surely (at least partly) lies in the fact that no one could really think, in anything like a real situation, that there is a uniform prior over all possible pay-offs (s/he assumes this away), I just couldn't see what basis there was for what seems to be the crucial premise namely the distinction between probabilistic and 'financial' terms (p.3).

3. Sam’s Envelope Stall
Lively, but the suggested solution is not cogent. And this is a topic on which there is a huge and sophisticated literature which the author simply ignores.

END

I hope this story is useful to those researchers hoping to get published.