The Interrogative Approach to Inquiry and Belief Revision Theory, Erratum

Thursday, August 06, 2009

In my recent paper about the Interrogative model and Belief Revision (http://www.springerlink.com/content/0u516q07m8233606/?p=2a860bbc6812425d927b02726ff5a3c4&pi=1) I made several mistakes, and one of them is quite serious. Here is the first published erratum.

Nota: In what follows, I'll use LaTeX code whenever special characters are needed. I'll soon post an erratum on this page.

In order of decreasing importance, the first mistake is in Lemma 8, (1a). It states incorrectly  that, in the case where \phi is in Cn(T):

if \phi is in Cn(\phi), then \phi is in \Cn([T]^\phi).

This is incorrect: it holds only when [T]^\phi includes (at least) one  minimal (for set-theoretic inclusion) \phi-implying subset of T. This need not happen, since to re-open a tree, It suffices that [T]^\phi includes one element of every such \phi-implying subset. (A result stating precisely that is on my "to do" list). The other half of Lemma 8 is correct.

(1a) is only used in the proof of Extensionality. Fortunately, the sentence where it is used is redundant. The correct form of (1a) should be:

(1a') If [T]^\phi is non-empty, then \phi is not in Cn(T \ [T]^\phi)

(1a') is straightforward to prove:

Proof: if [T]^\phi is non-empty, then at least one branch of the tree has been re-opened, and by Completeness of the tree method, if a branch is open, then the falsity of \phi is consistent with the set of premises to which the deductive rules can be applied, i.e. (given Bracketing rule R3) with T \[T]^\phi. Hence  \phi is not in Cn(T \ [T]^\phi) QED


[(1a') is used, without justification, in the proof of Extensionality, too.]

The proof that CBP satisfies the Success postulate, though not strictly speaking incorrect, assumes that the (deductive) rules are applied to the conclusion first. This is not necessary. If they are applied in any order, branches close without any pattern of bracket being sufficient to re-open them (left to the reader). So CBP cannot output any sufficient [T]^\phi, and subsequently is empty.


Proposition 7 is a doubly  mistaken:

1- The result itself is *incorrect*: what it should say is that, when T=Cn(T), for any AGM contraction, there is an application of CBP which *includes* the said contraction, while being itself a withdrawal (unless the special condition for Recovery holds).

2-Most importantly, the proof only shows that for any sentence \psi in a contraction T-\phi, there is an application of CBP(T,\phi) such that \psi is in it. It *does not* show that every \psi in T-\phi are in CBP(T,\phi), only that, for every such sentence, there is an application (possibly a different one) having this sentence as an element. Pretty useless...

Now, the real result is not that uneasy to prove. It requires a scheme (a strategy) to "mimic" T-\phi (which is given). The strategy consists in the following advice:

CBP*: "whenever at least one branch is open, do not open any other branch; if no branch is open, choose whatever CBP-complying pattern of brackets you want"

Then, the proof is by recursion:Order sentences in T-\phi, and
build a tree with the first sentence in T-\phi as premise, \phi as conclusion, and apply CBP* to it. The tree being open, CBP* does not issue any bracket, and then agree with T-\phi in including the consequences of the first premise.

Then one assumes that the nth stage has been build, and that T-\phi and CBP*(T,\phi) do agree so far. Then one extends each open branch with the n+1-th sentence in T-\phi, apply rules, etc. Since the tree is still open (why?), etc.

Hence, T-\phi and CBP*(T,\phi) do agree on every premise. The scheme can be carried in a finite number of steps using B, a finite subset of T-\phi such that Cn(B)=T-\phi. The existence of B follows from Compactness of Cn.

CBP* defines not a contraction operator, but a withdrawal. Interesting consequences follow from its being able to include any AGM *contraction*. The reason is that CBP satisfies the *Relevance* axiom (proposed by Hansson, proof obtained in my thesis) which, in presence of Closure, is equivalent to Recovery.

Hnece, the procedure  agrees with the given AGM contraction at every step. In the limit, it is identical with it. Proving that this is the case is a trifle more complicated.

A TeXified erratum should soon be published on this page, as soon as the proof in the thesis is checked, and though to be correct beyond reasonable doubt.