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Saturday, February 7, 2009

What Hempel didn't say about Ayer and Church

I’ve started prepping Philosophy of Language class for the next academic year (yeah, I don’t have to teach till Oc!). As for the text, The Philosophy of Language (third edition), edited by Martinich is the classic.

Right now, I’m looking at Hempel’s Empiricist Criteria of Cognitive Significance: Problems and Changes. At some point, he mentions Ayer’s revised verifiability requirement and Church’s argument against it, without explaining either of them. Below, a minor addendum which covers these things.

On p. 29 (Martinich pagination) Hempel discusses Ayer’s formulation of the testability criterion. The first formulation from Language, Truth and Logic is pretty much this:
A sentence S has empirical import if from S in conjunction with suitable subsidiary hypotheses it is possible to derive observation sentences which are not derivable from the subsidiary hypotheses alone.
Hempel then explains why this criterion is too wide: it allows empirical import to any sentence whatsoever. Take S to be any sentence. Let O be an observation statement that doesn’t follow from S. Then clearly, O is derivable from S and a subsidiary hypothesis: “If S, then O”.

This is all well known. What’s slightly less known is the formulation that Ayer gave in the appendix to the second edition of Language, Truth and Logic. Here’s how Hempel phrases it:
In effect, the modification restricts the subsidiary hypothesis mentioned in the previous version to sentences which either are analytic or can independently be shown to be testable in the sense of the modified criterion.
He then adds a footnote which says:
This restriction is expressed in recursive form and involves no vicious circle.
and refers the reader to the second edition of LTL. Next, he elaborates on one difficuly with this new version. He also says:
Another difficulty has been pointed out by Church, who has shown that if there are any three observation sentences none of which alone entails any of the others, then it follows for any sentence S whatsoever that either it or its denial has empirical import according to Ayer’s revised criterion.
Hempel then moves on to other issues, without explaining why Ayer’s revised version involved no vicious circle and how Church argued for his claim.

Now, if you’ve ever been curious as to how exactly Ayer formulated the criterion and why doesn’t it involve a vicious circle, and what exactly Church’s argument was, here are the answers.

Ayer’s formulation:
I propose to say that a statement is directly verifiable if it is either itself an observation-statement, or is such that in conjunction with one or more observation-statements it entails at least one observation-statement which is not deducible from these other premises alone; and I propose to say that a statement is indirectly verifiable if it satisfies the following conditions: first, that in conjunction with certain other premises it entails one or more directly verifiable statements which are not deducible from these other premises alone; and secondly, that these other premises do not include nay statement that is not either analytic, or directly verifiable, or capable of being independtly established as indirectly verifiable. And I can now reformulate the principle of verification as requiring of a literally meaningful statement, which is not analytic, that it should be either directly or indirectly verifiable, in the foregoing sense (p. 181 of the penguin books edition of LTL, 1971).
As for Church’s argument (from his 1949 review):

Step 1. Suppose O1, O2, O3 are three observation statements that are logically independent.

Step 2. ((~O1)^O2) v (O3^~S) is verifiable, for together with O1 it entails O3. Indeed, O1 entails ~((~O1)^O2), and this, by disjunctive syllogism applied to the original sentence gives us O3^~S. Conjunction elimination results in O3.

Step 3. S together with ((~O1)^O2) v (O3^~S)entails O2. This follows by a very similar argument. S forces us to reject the right argument of the disjuction, leaving us with a conjunction whose argument is O2.

Step 4. So S is verifiable, if ((~O1)^O2) v (O3^~S) doesn’t entail O2 without S.

Step 5. If ((~O1)^O2) v (O3^~S) does entail O2 without S, then each of the disjuncts entails O2 without S. So ~S ^ 03 entails O2. In such case, however, ~S comes out verifiable.