# Strong and weak indication relations

[warning: it’s proving hard to avoid typos in the formulas here. I’ve caught as many as I can, but please exercise charity in reading the various subscripts].

In the Lewisian setting I’ve been examining in the last series of posts, I’ve been using the following definition of indicates-to-x (I use the same notation as in previous posts, but add a w-subscript to distinguish it from an alternative I will shortly introduce):

• $p\Rightarrow^w_x q =_{def} B_x p\rightarrow B_x q$

The arrow on the right is the counterfactual conditional, and the intended interpretation of the B-operator is “has a reason to believe”. This fitted Lewis’s informal gloss “if x had reason to believe p, then x would thereby have reason to believe q”, except for one thing: the word thereby. Let’s call the reading above weak indication. Weak indication, I submit, gives an interesting version of the Lewisian derivation of iterated reason-to-believe from premises that are at least plausibly true in many paradigmatic situations of common belief.

But there is a cost. Lewis’s original gloss, combined with the results he derives, entail that each group member’s reasons for believing A obtains (say: the perceptual experience they undergo) are at the same time reasons for them to believe all the higher order iterations of reason-to-believe. That is a pretty explanatory and informative epistemology–we can point to the very things that (given the premises) justify us in all these apparently recherche comments. If we derive the same formal results on a weak reading of indication, we leave this open. We might suspect that the reasons for believing A are the reasons for believing this other stuff. But we haven’t yet pinned down anything that tells us this is the case.

I want to revisit this issue of the proper understanding of indication. I use $R_x(r, p)$ to formalize the claim that r is a sufficient reason for x to believe that p (relative to x’s epistemic standards and background beliefs).  With this understood, $B_x(p)$ can be defined as $\exists r B(r,p)$.  Here is an alternative notion of indication—my best attempt to capture Lewis’s original gloss:

• $p\Rightarrow^s_x q =_{def} \exists rR_x(r, p)\rightarrow \forall r(R_x(r,p)\supset R_x(r,q))$

In words: p strongly indicates q to x iff were x to have a sufficient reason for believing p, then all the sufficient reasons x has for believing p are sufficient reasons for x to believe q. (My thinking: in Lewis’s original the “thereby” introduces a kind of anaphoric dependence in the consequent of the conditional on the reason that is introduced by existential quantification in the antecedent. Since this sort of scoping isn’t possible given standard formation rules, what I’ve given is a fudged version of this).

Notice that the antecedent of the counterfactual here is identical to that used in the weak reading of indication. So we’re talking about the same “closest worlds where we have reason to believe p”. The differences only arise in what the consequent tells us. And it’s easy to see that, at the relevant closest worlds, the consequent of weak indication is entailed by the consequent of strong indication. So overall, strong indication entails weak indication.

If all the premises of my Lewis-style derivation were true under the strong reading, then the strong reading of the conclusion would follow. But some of the tweaks that I introduced in fixing up the argument seem to me implausible on the strong reading—more carefully, it is implausible that they are true on this reading in all the paradigms of common knowledge. Consider, for example, the premise:

• $A\Rightarrow_x \forall y (x\sim y)$

In some cases the reason one has for believing A would be reason for believing that x and y are relevantly similar (as the conclusion states). I gave an example, I think, of a situation where the relevant manifest event A reveals to us both that we are members of the same conspiratorial sect. But this is not the general case. In the general case, we have independent reasons for thinking we are similar, and all that we need to secure is that learning A, or coming to have reason to believe A, wouldn’t undercut these reasons. (It was the possibility of undercutting in this way that was the source of my worry about the Cubitt-Sugden official reconstruction of Lewis, which doesn’t have the above premise, but rather than premise that x has reason to believe that x is similar to all the others).

So now we are in a delicate situation, if we want to derive the conclusions of Lewis’s argument on a strong reading of indication. We will need to run the argument with a mix of weak and strong indication, and hope that the mixed principles that are required will turn out to be true.

Here’s how I think it goes. First, the first three premises are true on the strong reading, and the final premise on the weak reading.

1. $A \supset B_u(A))$
2. $A\Rightarrow^s_u \forall yB_y(A))$
3. $A \Rightarrow^s_u q$
4. $A\Rightarrow^w_u \forall y [u\sim y]$

Of the additional principles, we appeal to strong forms of symmetry and closure:

• SYMMETRY-S $\forall c \forall x ([A \Rightarrow^s_x c]\wedge \forall y [x\sim y]\supset \forall y[A\Rightarrow^s_y c])$
• CLOSURE-S $\forall a,c (\forall x B_x (a)\wedge \forall x[a \Rightarrow^s_x c]\supset \forall x B_x(c)))$

In the case of closure, strong indication features only in the antecedent of the material conditional, so this is in fact weaker than closure on the original version I presented. These are no less plausible than the originals. As with those, the assumption is really not just that these are true—it is that they are valid (and so correspond to valid inference patterns). That is used in motivating the truth of principles that piggyback upon them are that are also used.

The “perspectival” closure principle can be used in a strong form:

• CLOSURE+-S $\forall a,b,c\forall x ([a \Rightarrow^s_x \forall y B_y(b)]\wedge [a \Rightarrow^s_x(\forall y[b \Rightarrow^s_y c])]\supset [a\Rightarrow^s_x \forall yB_y(c)])$

The action in my vierw comes with the remaining principles, and in particular, the “perspectival” symmetry principle. Here it is in mixed form:

• SYMMETRY+-M $\forall a \forall c \forall x\forall z[a\Rightarrow^s_z[A \Rightarrow^s_x c]]\wedge [a \Rightarrow^w_z\forall y[x\sim y]]\supset [a\Rightarrow^s_z \forall y[A\Rightarrow^s_y c]$

The underlying thought behind this perspectival principles (as with closure) is that when you have a valid argument, then if you have reason to believe the premises (in a given counterfactual situation), then you have reason to believe the conclusion. That’s sufficient for the weak reading we used in the previous posts. In a version where all the outer indication relations are strong, as with the strong CLOSURE+ above, it relies more specifically on the assumption that where r is a sufficient reason to believe each of the premises of a valid argument, it is sufficient reason to believe the conclusion.

We need a mixed version of symmetry because we only have a weak version of premise (4) to work with, and yet we want to get out a strong version of the conclusion. Justifying a mixed version of symmetry is more delicate than justifying either a purely strong or purely weak version. Abstractly, the mixed version says that if r is sufficient reason to believe one of the premises of a certain valid argument, and there is some reason or other to believe the second premise of that valid argument, then r is sufficient reason to believe the conclusion. This can’t be a correct general principle about all valid arguments. Suppose the reason to believe the second premise is s. Then why think that r alone is sufficient reason to believe the conclusion? Isn’t the most we get that r and s together are sufficient for the conclusion?

So we shouldn’t defend the mixed principle here on general grounds. Instead, the idea will have to be that with the specific valid argument in question (an instance of symmetry), assumptions about who I’m epistemically similar to (in epistemic standards and background beliefs) itself counts as a “background belief”. If that is the case, then we can argue that the reason for believing the first premise of the valid argument (in a counterfactual situation) is indeed sufficient relative to the background beliefs to entail the conclusion. One of the prerequisites of this understanding will be that either we assume that other agents will believe propositions about who they’re epistemically sensitive to in counterfactual situations where they have reason to believe those propositions; or else that talk of “background beliefs” is loose talk for background propositions that we have reason to believe. I think we could go either way.

In order to complete this, we will need iteration, and in the following, strong version:

• ITERATION-S $\forall c \forall x ([A \Rightarrow^s_x c]\supset [A \Rightarrow^s_x [A\Rightarrow^s_x c]]$

I’ll come back to this.

Let me exhibit how the utmost core of a Lewisian argument looks in this version. I’ll compress some steps for readability:

1. $A\Rightarrow_u^s \forall y B_y A$. Premise 2.
2. $A\Rightarrow^s_u(A\Rightarrow_u^s \forall yB_y A)$. From 1 via ITERATION-S.
3. $A\Rightarrow^w_u \forall y [u\sim y]$. Premise 4.
4. $A\Rightarrow^s_u \forall z(A\Rightarrow_z^s \forall yB_y A)$. From 2,3 by SYMMETRY+-M.
5. $A\Rightarrow^s_u\forall z B_z \forall yB_y A$. From 1,4 by CLOSURE+-S.

This style of argument—which can then be looped—is the basic core of a Lewis-style derivation. You can add in premise 3 and use CLOSURE+, and get something similar with q as the object of iterated B-operators, to get the original. And of course you can appeal to premise 1 and CLOSURE to “discharge” the antecedents of interim conclusions like 5 (this works with strong indication relations because it works for weak indication, and strong indication entails weak).

There’s an alternative way of mixing strong and weak indication relations. On this version we use a mixed form of ITERATION, the original weak SYMMETRY+, and then a mixed form of CLOSURE+

• ITERATION-M $\forall c \forall x ([A \Rightarrow^s_x c]\supset [A \Rightarrow^w_x [A\Rightarrow^s_x c]]$
• SYMMETRY+-W $\forall a \forall c \forall x\forall z[a\Rightarrow^w_z[A \Rightarrow^s_x c]]\wedge [a \Rightarrow^w_z\forall y[x\sim y]]\supset [a\Rightarrow^w_z \forall y[A\Rightarrow^s_y c]$
• CLOSURE+-M $\forall a,b,c\forall x ([a \Rightarrow^s_x \forall y B_y(b)]\wedge [a \Rightarrow^w_x(\forall y[b \Rightarrow^s_y c])]\supset [a\Rightarrow^s_x \forall yB_y(c)])$
1. $A\Rightarrow_u^s \forall y B_y A$. Premise 2.
2. $A\Rightarrow^w_u(A\Rightarrow_u^s \forall yB_y A)$. From 1 via ITERATION-M.
3. $A\Rightarrow^w_u \forall y [u\sim y]$. Premise 4.
4. $A\Rightarrow^w_u \forall z(A\Rightarrow_z^s \forall yB_y A)$. From 2,3 by SYMMETRY+-W.
5. $A\Rightarrow^s_u\forall z B_z \forall yB_y A$. From 1,4 by CLOSURE+-M.

The main advantage of this version of the argument would be that the version of ITERATION it requires is weaker. Otherwise, we are simply moving the bump in the rug from mixed SYMMETRY+ to mixed CLOSURE+. And that seems to me a damaging shift. We use mixed SYMMETRY+ many times, but the only belief we have ever to assume is “background” to justify the principle is the belief that all are similar to me. In the revised form, to run the same style of defence, we would have to assume that belief about indication relations of more and more complex contents are backgrounded. And that simply seems less plausible. So I think we should stick with the original if we can. (On the other hand, the principle we would need here is close to the sort of “mixed” principle that Cubitt and Sugden use, and they are officially reading “indication” in a strong way. So maybe this should be acceptable).

So what about the ITERATION-S, the principle that the argument now turns on? As a warm up, let me revisit the motivation for the original, ITERATION-W, which fully spelled out would be:

• $[\exists r R_u(r, A)\rightarrow \exists r R_u(r,c))]$
$\supset[\exists s R_u(s,A)\rightarrow$
$\exists s R_u(s,[\exists r R_u(r, A)\rightarrow \exists r R_u(r,c)])]$

Assume the first line is the case. Then we know that at the worlds relevant for evaluating the second and third lines, we have both $\exists r R_u(r,c)$ and $\exists r R_u(r, A)$. By an iteration principle for reason-to-believe, $\exists s_1R_u(s_1,\exists r R_u(r,c))$ and $\exists s_2 R_u(s_2,\exists r R_u(r, A))$. And by a principle of conjoining reasons (which implicitly makes a rather strong consistency assumption about reasons for belief) $\exists s R_u(s,\exists r R_u(r,A)\wedge \exists r R_u(r, c))$. But a conjunction entails the corresponding counterfactual in counterfactual logics for strong centering, and so plausibly the reason to believe the conjunction is a reason to believe the counterfactual: $\exists s R_u(s,\exists r R_u(r,A)\rightarrow \exists r R_u(r, c))$. That is the rationale for the original iteration principle.

Unfortunately, I don’t think there’s a similar rationale for the strong iteration principle. The main obstacle is the following: point: one particular sufficient reason for believing A to be the case (call it s) is unlikely to be one’s reason for believing a counterfactual generalization that covers all reasons to believe that A is the case. In the original version of iteration, this wasn’t at issue at all. But the rationale I offered uses a strategy of finding a reason to believe a counterfactual by exhibiting a reason to believe the corresponding conjunction, which entails the counterfactual. In order to find a reason to believe the conjunction of the relevant counterfactual below (the one appearing in the third line) But an essential part of that strategy was arguing that a certain thing was a reason to believe a counterWhen you write down what strong iteration means in detail, you see (in the third line below) that this is going to have to be argued for. I can’t see a strategy for arguing for this, and I the principle itself seems likely to be false to me, as stated.

• $[\exists r R_u(r, A)\rightarrow \forall r (R_u(r, A)\supset R_u(r,c))]$
$\supset[\exists s R_u(s,A)\rightarrow$
$\forall s( R_u(s,A)\supset R_u(s,[\exists r R_u(r, A)\rightarrow \forall r (R_u(r, A)\supset R_u(r,c))]]$

That’s bad news. Without this principle, the first mixed version of the argument I presented above doesn’t go through. I think there’s a much better chance of mixed iteration being argued for, which is what was needed for the second version of the argument. But that was the version of the argument that required the dodgy mixed closure principle. Perhaps we should revisit that version?

I’m closing this out with one last thought. The universal quantifier in the consequent of the indication counterfactual is the source of the trouble for strong ITERATION. But that was introduced as a kind of fudge for the anaphor in the informal description of the indication relation. One alternative is use a definite description in the conclusion of the conditional—which on Russell’s theory introduces the assumption that there is only one sufficient reason (given background knowledge and standards) for believing the propositions in question. This would give us:

• $p\Rightarrow^d_x q =_{def}$
$\exists rR_x(r, p)\rightarrow \exists r(R_x(r,p)\wedge \forall s (R_x(s, p)\supset r=s) \wedge R_x(r,q))$

Much of the discussion above can be rerun with this in place of strong indication. And I think the analogue of the strong ITERATION has a good chance of being argued for here, provided that we have a suitable iteration priciple for reason-to-believe. For weak iteration, we needed only to assume that when there is reason to believe p, there is reason to believe that there is reason to believe p. In the rationale for a new stronger version of ITERATION that I have in mind we will need that when s is a reason to believe that p, then s is a reason to believe that s is a reason to believe that p. Whether this will fly, however, turns both on being able to justify that strong iteration principle and on whether indication in the d-version, with its uniqueness assumption, finds application in the paradigmatic cases.

For now, my conclusion is that the complexities involved here justifies the decision to run the argument in the first instance with weak indication throughout. We should only dip our toes into these murky waters if we have very good reason to do so.