Iteration vs. Entrenchment

I’m going to have one more run at a form of the Lewisian derivation that justifies the strong conclusions (e.g. that the reason for believing A would be a reason for believing each of the iterated B-claims.

I’ll be using strong-indication again, though since this is the only indication relation I’ll use in this discussion, I’ll drop the superscript disambiguation:

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

Remember that R is the relation of something being sufficient reason to believe, *relative to background beliefs and epistemic standards*. Let’s introduce a new operator $E_x$, which will say that the embedded proposition is a background belief or epistemic standard for x—or as I’ll say for short, is entrenched for x.

We have the first three premises on a strong reading of indication again. But I’ll now change the fourth premise from an indication principle to one about E:

1. $A \supset B_u(A))$
2. $A\Rightarrow_u \forall yB_y(A))$
3. $A \Rightarrow_u q$
4. $E_u \forall y [u\sim y]$

A linked change is that we abandon IITERATION for a principle that says that propositions about what indicates what to a person is part of their epistemic standards/background beliefs:

• ENTRENCHMENT $\forall c \forall x ([A \Rightarrow_x c]\supset E_x[A\Rightarrow_x c]$

The core derivation I have in mind goes like this:

1. $A\Rightarrow_u \forall y B_y A$. Premise 2.
2. $E_u(A\Rightarrow_u \forall yB_y A)$. From 1 via ENTRENCHMENT.
3. $E_u \forall y [u\sim y]$. Premise 4.
4. $E_u \forall z(A\Rightarrow_z \forall yB_y A)$. From 2,3 by NEWSYMMETRY+.
5. $A\Rightarrow_u\forall z B_z \forall yB_y A$. From 1,4 by NEWCLOSURE+.

What then are these new principles of NEWSYMMETRY+ and NEWCLOSURE+ and how should we think about them? NEWSYMMETRY+ is another perspectival form based on the validity of strong symmetry:

• SYMMETRY-S $\forall c \forall x ([A \Rightarrow_x c]\wedge \forall y [x\sim y]\supset \forall y[A\Rightarrow_y c])$

NEWSYMMETRY+ is then an instance of a principle that propositions that are entrenched for an individual are closed under valid arguments, with SYMMETRY-S providing the relevant valid argument:

• NEWSYMMETRY+ $\forall c \forall x\forall z[E_z[A \Rightarrow_x c]]\wedge [E_z\forall y[x\sim y]]\supset [E_z \forall y[A\Rightarrow_y c]]]$

NEWCLOSURE+ is based again validity of closure for the B-operator under strong indication, which is again something that really just reduces to modus ponens for the counterfactual condition hidden inside the indication relation:

• CLOSURE-S $\forall a,c (\forall x B_x (a)\wedge \forall x[a \Rightarrow_x c]\supset \forall x B_x(c)))$

But the principle we use isn’t just the idea that some operator or other is closed under closure. The thought is instead a principle about reason-transmission that goes as follows. Suppose two propositions entail a third, and r is sufficient reason (given one’s background beliefs and standards) to believe the first proposition. Then, if the second proposition is entrenched (part of those background beliefs and standards), r is a also sufficient reason (given one’s background beliefs and standards) to believe the third proposition. The underlying valid argument relevant to this is CLOSURE-S, which makes this, in symbols:

• NEWCLOSURE+ $\forall a,b,c\forall x ([a \Rightarrow_x \forall y B_y(b)]\wedge [E_x(\forall y[b \Rightarrow_y c])]\supset [a\Rightarrow_x \forall yB_y(c)])$

NEWCLOSURE+ seems to me pretty well motivated. NEWSYMMETRY+ just as good as anything we’ve worked to so far. STANDARDS now replaces ITERATION. Unlike ITERATION, there’s no chance of deriving it from principles about counterfactuals and the transparency of whatever B stands for. Instead, it simply represents it’s own transparency assumption: that true propositions about the epistemic standards and background beliefs of an agent are themselves part of an agent’s epistemic background. It is weaker than a transparency assumption about beliefs or reasons to believe used in motivating ITERATION since it has a more restricted domain of application. It is stronger than earlier transparency assumptions insofar as it requires that the propositions to which it applies are not merely believed (or things we have reason to believe) but have the stronger status of being entrenched.

NEWCLOSURE+ is quite close in form to Cubitt and Sugden’s A6, except their principle used (what I notate as) the B operator throughout, where at a crucial point I have an instance of the E operator. An advantage that this gives me is that the E-operator doesn’t feature in the conclusion of the argument, so we are free to reinterpret it however we like to get the premises to come out true—trying to do reinterpret B would change the meaning of the conclusions we are deriving. So, for example, I complained against theirs that crucial principles seemed bad because some of your beliefs or reasons for beliefs might not be resilient under learning new information. But we are free to simply build into E that it applies only to propositions that are resiliently part of one’s background beliefs/standards (or maybe being resilient in that way is part of what it is for something to be treated as a standard/be background).

Having walked through this, let me illustrate the fuller form of the derivation, using all the premises.

1. $A\Rightarrow_u \forall y B_y A$. Premise 2.
2. $A\Rightarrow_u q$. Premise 3.
3. $E_u(A\Rightarrow_u q)$. From line 2 via ENTRENCHMENT.
4. $E_u \forall y [u\sim y]$. Premise 4.
5. $E_u \forall z(A\Rightarrow_z q)$. From lines 3,4 by NEWSYMMETRY+.
6. $A\Rightarrow_u\forall z B_z q$. From 1,5 by NEWCLOSURE+.
7. $E_u(A\Rightarrow_u\forall z B_z q)$. From line 6 via ENTRENCHMENT.
8. $E_u \forall y(A\Rightarrow_y \forall z B_z q)$. From lines 4,7 by NEWSYMMETRY+.
9. $A\Rightarrow_u\forall y B_y \forall z B_z q$. From 1,8 by NEWCLOSURE+.
10. ….

The pattern of the last few lines loops to get that A indicates each of the iterations of B-operator applied to q. And we can then appeal to Premise 1, A and CLOSURE to “detach” the consequents of lines 6,9, etc.

But for our purposes here and now, the more significant thing is lines 6 and 9 (and 12, 15 etc) prior to detachment. For these tell us that a sufficient reason for believing A is itself a sufficient reason for believing each of these iterated B propositions.

So to sum up: if we are content to work with weak indication relations, we can get away with the premises I used in other posts, including ITERATION and previous versions of SYMMETRY+ and CLOSURE+. If we want to work with strong indication, and get information about what is a reason for what, then we need to make changes, and the above is my best shot (especially in the light of the utter mess we got into in the last post!). Interestingly, while NEWSYMMETRY+ and NEWCLOSURE+ it seems to me are more or less equally plausible with the older analogues, the replacement for ITERATION (the principle I’m here calling ENTRENCHMENT) isn’t directly comparable to the earlier, though it’s still broadly a principle of transparency.

There is a delicate dialetical interplay between ENTRENCHMENT and the analysis of the indication relation. The stronger and more demanding indication is, the more plausible ENTRENCHMENT becomes, since the fewer instances fall under it. If we read indication as weak indication throughout, then ENTRENCHMENT would say that every counterfactual relating reasons for belief to reasons for other beliefs is part of the background beliefs/epistemic standards. That’s wildly strong! It’s pretty strong in strong indication version too. It becomes much more plausible if this were restricted to, for example, epistemic connections between propositions that are obvious to the agent.

In the settings I have considered in the previous posts, the counterfactual analysis earned its keep in part because ITERATION (which is here replaced by ENTRENCHMENT) could be treated as an iterated counterfactual. That’s no longer a consideration. The other advantage of having the counterfactual analysis is that it made CLOSURE an instance of modus ponens. But that’s not a reason for accepting the analysis of indication as a counterfactual—it’s just a reason for accepting that indication entails the counterfactual. The final reason for offering the counterfactual analysis is simply that it allows a reduction in the number of primitive notions around: in the original setting, it allows a reduction to just the B operator. That’s a consideration, but in the current context we’re having to work with E’s as well as B’s, so ideological purity is lost.

Once we need ENTRENCHMENT, it seems to me that it would be easier to defend the package presented here if we abandoned the counterfactual analysis of indication, and used it as a primitive notion, while adding as a premise the validity of the following principle which links a now-primitive indication relation to what we were previously calling strong indication:

• $p\Rightarrow^s_x q \supset \exists rR_x(r, p)\rightarrow \forall r(R_x(r,p)\supset R_x(r,q))$

The soundness of the overall argument now turns on whether there exists a triple: of reason-relation, indication relation and entrenchment relation that makes true all the premises.

As a final note: the link between the counterfactual and primitive indication has two roles. One is simply a matter of reading off the significance of the final results. The other is to make CLOSURE valid. But it only makes CLOSURE valid if the B-operator is defined in the Lewisian way as having-reason-to-believe. As per that earlier post, a different counterfactual–concerning commitments to believe–matters for CLOSURE in that setting. So one would add that entailment as an extra premise about the now-primitive indication relation.