### 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.

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.

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

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

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

Labels: descriptions, Kripke, length, meaning, reference, relative, Rigid designators, Wittgenstein