Measuring rods

In his popular book on relativity theory, Einstein says that distance is just what measuring rods measure. I am having a hard time making sense of this in Einstein’s operationalist setting.

Either Einstein is talking of real measuring rods or idealized ones. If real ones, then it’s false. If I move a measuring rod from one location to another, its length changes, not for relativistic reasons, but simply because the acceleration causes some shock to it, resulting in a distortion in its shape and dimensions, or because of chemical changes as the rod ages. But if he’s talking about idealized rods, then I think we cannot specify the relevant kind of idealization without making circular use of dimensions—relevantly idealized rods are ones that don’t change their dimensions in the relevant circumstances.

If one drops Einstein’s operationalism, one can make perfect sense of what he says. We can say that distance is the most natural of the quantities that are reliably and to a high degree of approximation measured by measuring rods. But this depends on a metaphysics of naturalness: it’s not a purely operational definition.

Verifiability

I’ve been reading Ayer’s Language, Truth and Logic, and been struck by how hard it is define verifiability, which is crucial since Ayer thinks that statements are meaningful if and only if they are verifiable or analytic (or, I suppose, denials of analytic statements, though oddly he doesn’t mention that possibility). Ayer rightly notes that we don’t want verifiability to mean conclusive verifiability. In the body of the text, he offers three criteria for the (weak) verifiability of p, apparently without realizing they are quite different:

1. it is possible for an experience to “make [p] probable”

2. a possible observation would be “relevant to the determination of [p’s] truth or falsehood”

3. together with some additional premises it entails an observation statement that cannot be derived from the premises alone.

In the 1946 introduction, he realizes that (3) makes every proposition verifiable (just make the additional premise be “if p, then o” for any observation statement o) and offers this:

4. a statement is directly verifiable if it is an observation statement or if conjoined with one or more observation statements it entails an observation statement that is not entailed by the latter observation statements,

then gives a complicated account of indirectly verifiable statements, and finally defines verifiable statements as ones that are directly or indirectly verifiable.

Here’s a curious thing: (1), (3), (4) as well as the complex account of indirect verifiability can apply to a statement p without applying to its negation. For instance, consider any crazy “metaphysical” non-verifiable statement q such as his example “the Absolute is lazy” and let o be any observation statement. Then q ∧ o satisfies (3) and (4), since it entails o by itself (and hence as conjoined with some other independent observation statement). On the other hand, the negation of q ∧ o is equivalent to ∼q ∨ ∼o and does not satisfy (3), or (4), or the indirect verifiability criterion. Similarly, q ∨ o satisfies (1), since the experience reported in o would make q ∨ o conclusively probable, but the negation does not seem to satisfy (1).

But we would expect that if a statement is meaningful so is its negation. Certainly, Ayer thinks so: non-meaningful statement are neither true nor false on his view.

Account (2) is nicely symmetric between truth and falsity, and hence escapes this worry. But it has the consequence that the conjunction of two meaningful statements can be meaningless. For again let q be any crazy non-verifiable statement, and let o be the statement “It’s bright here”. Then q ∨ o is verifiable by (1), since observing brightness is clearly relevant to the truth of the disjunction, and so is q ∨ ∼o, since observing darkness is clearly relevant. Thus, both q ∨ o and q ∨ ∼o are meaningful. But (q ∨ o)∧(q ∨ ∼o) is logically equivalent to q. Thus, either the conjunction of two meaningful statements can be meaningless, or every statement is meaningful.

Maybe the best move for the positivist is to allow that meaningfulness is not closed under negation?