Reinterpreting the Lewis-Cubitt-Sugden results

In the last couple of posts, I’ve been discussing Lewis’s derivation of iterated “reason to believe” q from the existence of a special kind of state of affairs A. I summarize my version of this derivation as follows, with the tilde standing for “x and y are similar in epistemic standards and background beliefs”.

We start from four premises:

  1. \forall x (A \supset B_x(A))
  2. \forall x (A\Rightarrow_x \forall yB_y(A))
  3. \forall x (A \Rightarrow_x q)
  4. \forall x ([A\Rightarrow_x \forall y [x\sim y]]

Five additional principles are either used, or are implicit in the motivation for principles that are used:

  • ITERATION \forall c \forall x ([A \Rightarrow_x c]\supset [A \Rightarrow_x [A\Rightarrow_x c]]
  • SYMMETRY \forall c \forall [A \Rightarrow_x c]\wedge \forall y[x\sim y]]\supset [\forall y[A\Rightarrow_y c]]
  • CLOSURE \forall a,c (\forall x B_x (a)\wedge \forall x[a \Rightarrow_x c]\supset \forall x B_x(c)))
  • SYMMETRY+ \forall a \forall c \forall x\forall z[a\Rightarrow_z[A \Rightarrow_x c]]\wedge [a \Rightarrow_z\forall y[x\sim y]]\supset [a\Rightarrow_z [\forall y[A\Rightarrow_y c]]
  • CLOSURE+ \forall a,b,c\forall x ([a \Rightarrow_x \forall y B_y(b)]\wedge [a \Rightarrow_x(\forall y[b \Rightarrow_y c])]\supset [a\Rightarrow_x \forall yB_y(c)])

In the last post, I gave a Lewis-Cubitt-Sugden style derivation of the following infinite series of propositions, using (2-4), SYMMETRY+, CLOSURE+, ITERATION:

  • A \Rightarrow_x q
  • A\Rightarrow_x \forall y B_y(q)
  • A\Rightarrow_x (\forall z B_z(\forall y B_y(q)))
  • \ldots

A straightforward extension of this assumes (1) and CLOSURE, obtaining the following results in situations where A is the case:

  • \forall x B_x(q)
  • \forall x B_x(\forall y B_y(q))
  • \forall _x B_x(\forall z B_z(\forall y B_y(q)))
  • \ldots

The proofs are valid, so each line in these two infinite sequences hold no matter how one reinterprets the primitive symbols, so long as the premises are true under that reinterpretation.

As we’ve seen in the last couple of posts, for Lewis, “indication” was a kind of shorthand. He defined it as follows:

  • p\Rightarrow_x q := B_x(p)\rightarrow B_x(q)

where \rightarrow is the counterfactual conditional.

Now, this definition is powerful. It means that CLOSURE needn’t be assumed as a separate premise—it follows from the logic of counterfactuals. And if “reason to believe” is closed under entailment, then we also get CLOSURE+ for free. As noted in edits to the last post, it means that we can get ITERATION from the logic of counterfactuals and a transparency assumption, viz. B_x(p)\supset B_x(B_x(p)).

The counterfactual gloss was also helpful in interpreting what (4) is saying. The word “indication” might suggest that when A indicates p, A must be something that itself gives the reason to believe p. That would be a problem for (4), but the counterfactual gloss on indication removes that implication.

Where Lewis’s interpretation of the primitives is thoroughly normative, we might try running the argument in a thoroughly descriptive vein (see the Stanford Encyclopedia for discussion of an approach to Lewis’s results like this.).

To read the current argument descriptively, we might start by reinterpreting B_x(p) as saying: x believes that p, and indication to be defined out of this notion counterfactually just as before. The trouble with this is some of the premises look false, read this way. For example, CLOSURE+ asks us to consider scenarios where x’s beliefs are thus-and-such, where the propositions x believes in that scenario entails (by CLOSURE) the proposition that the conclusion tells us x believes. unless the agent actually believes all the consquences of things she believes, it’s not clear why we should assume the condition in the consequent of CLOSURE+ holds. Similar issues arise for SYMMETRY+ and ITERATION.

One reaction at this point is to argue for a “coarse grained” conception of belief that makes it closed under entailment. That’s a standard modelling assumption in the formal literature on this topic, and something that Lewis and Stalnaker both (to a first approximation) accept. It’s extremely controversial, however.

If we don’t like that way of going, then we need to revisit our descriptive reinterpretation of the primitives. We could define them so as to make closure under such principles automatic. So, rather than have B_x(p) say that x believes p, we might read it as saying that x is committed to believe p, where x is committed to believe something when it follows from the things they believe (in a fuller version, I’d refine this characterization to allow for circumstances in which a person’s beliefs are inconsistent, without her commitments being trivial, but for now, let’s idealize away that possibility and work with the simpler version). Indication becomes: were x to be committed to believe that p, then they would be committed to believe that q.

If you read through the premises under this descriptive reinterpretation, then I contend that you’ll find they’ve got as good a claim to be true as the analogous premises on the original normative interpretation.

These interpretations need not compete. Lewis’s normative interpretation of the argument may be sound, and the commitment-theoretic reinterpretation may also be sound. In paradigmatic cases where there is a basis for common knowledge in Lewis’s sense, we may have an infinite stack of commitments-to-believe, and a parallel infinite stack of reasons-to-believe.

But notice! What the first Lewis argument gives us is reason to believe that others have reason to believe such-and-such. It doesn’t tell us that we have reason to believe that others are committed to believe so-and-so. So for some of the commitments that people take on in such situations (commitments about what others are committed to believe) might be unreasonable, for all these two results tell us. This will be my focus in the rest of this post, since I am particularly interested in the derivation of infinite commitment-to-believe. I think that the normative question: are these commitments epistemically reasonable? is a central one for a commitment-theoretic way of understanding what “public information” or “common belief” consists in.

Let me first explore and expose a blind alley. Lewis himself extracts descriptive predictions about belief from his account of iterated reasons for belief in situations of common knowledge, he adds assumptions about all people being rational, i.e. believing what they have reason to believe. He further adds assumptions about us having reason to believe each other to be rational in this sense, and so on. Such principles of iterated rationality are thought by Lewis to only be true for the first few iterations. They generate, for a few iterations, that we believe that q, believe that we believe q, believe that we believe that we believe q, etc. And in parallel, we can show that we have reason to believe each of these propositions about iterated belief—so all the belief we in fact have will be justified.

But while (per Lewis) these predictions are by designed supposed to run out after a few iterations, we need to show how everything we are committed to believing we have reason to believe. One might try to parallel Lewis’s strategy here, adding the premise that people are committed to believing what they have reason to believe. One might hope that such bridge principles will be true “all the way up”, and so allow us to derive the analogue of Lewis’s result for all levels of iteration. But this is where we hit the end of this particular road. If someone (perhaps irrationally) fails to believe that the ball is red despite having reason to believe that the ball is red, the ball being red need not follow from what they believe. So we do not have the principles we’d need to to convert Lewis’s purely normative result into one that speaks to the epistemic puzzle about commitment to believe.

Now for a positive proposal. To address the epistemic puzzle, I propose a final reinterpretation of the primitives of Lewis’s account. This time, we split the interpretation of indication and of the B-operator. The B-operator will express commitment-to-believe, just as above. But the indicates-for-x relation does not simply express counterfactual commitment, but has in addition a normative aspect. p will indicate q, for x, iff (i) were x to be committed to believing p, then x would be committed to believing q; and (ii) if x had reason to believe p, then x would have reason to believe q.

Before we turn to evaluating the soundness of the argument, consider the significance of the consequences of this argument under the new mixed-split reinterpretation. First, we would have infinite iterated commitment-to-believe, just as in the pure descriptive interpretation (that’s fixed by our interpretation of B). But second, for each level of iteration of mutual commitment-to-believe, we can derive that A indicates (for each x) that proposition. But indication on this reading, unlike on the pure descriptive reading,  has normative implications. It says that when the group members have reason to believe that A, they will have reason to believe that all are committed to believe that all are committed… that all are committed to believe q. So on the split reading of the argument, we derive both infinite iterated commitment to believe, and also that group members have reason to believe that propositions that they are are committed to believe.

An instance of indication, on this reading, is a conjunction of what indication signified on the two earlier readings of the argument. This means that, for example, ITERATION on the new split reading is good just in case ITERATION on those two earlier readings was good. It can be argued for from the logic of counterfactual conditionals so long as we have transparency both of reasons for belief and of commitment to believe. It requires no separate discussion, therefore. SYMMETRY has the same status, and CLOSURE on the split reading follows from CLOSURE on the descriptive reading alone. I contend that SYMMETRY+ AND CLOSURE+ on the split reading are also perfectly acceptable. You might think there is a special issue with CLOSURE+, since it features the B-operator within an indication operator–but really, what really drives CLOSURE+ is not the details of the particular B-operator, but the general principle that indication is closed under valid arguments. And that motivates CLOSURE on the new reading as well as both of the old ones. Of all the premises, it’s only (2-4) that make claims that are stronger than those on the earlier readings. For example, (2) now says, in part, that were one to have reason to believe that A, then one would have reason to believe that everyone is committed to believe A. This is genuinely new, compared to the assumptions made in the earlier readings. But these are assumptions that are pretty plausibly met by paradigms of bases for common knowledge.

What I’ve argued is that if the pure-descriptive version of Lewis’s argument is sound, and the pure-normative version of Lewis’s argument is sound, then the mixed-split-interpretation version of Lewis’s argument is sound. The conclusion of the argument under this mixed reading scratches an epistemological itch that neither the pure descriptive reading nor the pure epistemological reading (even supplemented with assumptions of iterated rationality) could help with.

That matters to me, in particular, because I’m interested in iterated commitment-to-believe as an analysis of public information/common belief, and I take the epistemological challenge as a serious one. At first, I thought that I could wheel in the Lewis-Cubitt-Sugden proof to address my concerns. But I had two worries. One was about the soundness of that proof, given its reliance on the dubious premise (A6). That worry was expressed two posts ago, and addressed in the last post. But another was the worry raised in the current post: that on the intended reading, the Lewis-Cubitt-Sugden proof really doesn’t show that we have reason to believe all those propositions we are committed to, if we have common belief in the commitment-theoretic sense. But—I hope–all is now well, since the split reinterpretation of the fixed up proof delivers everything I need: both infinite iterated commitment to believe, and the reasonability of believing each of those propositions we are committed to believing.



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