On Wednesday morning, I emailed you (twice) suggesting that we might do the proof without identifying any LC models and SC models, but just by proving that when

*there is*an LC model having such-and-such properties,*there is*an SC model having so-and-so properties, and so on. When I sat down to write this proof up, though, I wasn't after all able to see how to make this work. So I went back to the idea I had suggested in class.This was: instead of talking about satisfying the limit condition for all formulas, we instead talk about satisfying it with respect to a set of formulas, that grows as our induction proceeds. That's the key idea of the proof reported below. The write-up is a bit verbose, because I wanted that, if you took the time to read this, you'd be able to follow how the proof goes.

Afterwards, I reviewed the proof to think about how it might be simplified. What were the key assumptions we relied on in the final steps? Maybe some of the machinery we had used would be extraneous. This led me to the insight that there is after all a very drastic simplification possible. If you just want to hear what it is, you can skip to the end. But that's no fun.

We want to show that if a formula is valid on the Lewis semantics for counterfactuals (LC-valid) then it is valid on the Stalnaker semantics for counterfactuals (SC-valid).

We'll be focusing on structures that satisfy all the conditions to be a LC model, *plus* also have anti-symmetric closeness relations. These structures will coincide with the structures that satisfy all the conditions to be an SC model, *minus* the limit condition. (The definition of the "base" condition for LC models and SC models differs, but the definitions are equivalent when the closeness relation is anti-symmetric.)

So these structures are a proper subclass of the LC models, that satisfy some but not all of the conditions to be SC models. We'll want to further restrict this class by adding some limit condition(s). But the limit condition for SC models is defined partly in terms of the Stalnaker valuation function \(V\). We could also formulate a limit condition that's defined using the Lewis valuation function \(LV\). For clarity, I'll speak of these as Stalnaker-limit condition(s) and Lewis-limit condition(s).

For reference, here is the clause stating Stalnaker's truth-conditions for counterfactual formulas:

(STC) \(V_{\mathscr{M}}(\phi \mathbin{\square \mkern-2.7mu\raise0.3ex\hbox{${\rightarrow}$}}\psi,w)=\mathsf{True}\) iff for any world \(x\) that is a closest-to-\(w\) world where \(V_{\mathscr{M}}(\phi,x)=\mathsf{True}\), \(V_{\mathscr{M}}(\psi,x)=\mathsf{True}\).

Let's understand this valuation function to be defined even when \(\mathscr{M}\)'s closeness relation doesn't satisfy the Stalnaker-limit condtion. Hence it's defined also on structures that don't officially count as SC models.

Here is the clause stating Lewis's truth-conditions:

(LTC) \(LV_{\mathscr{M}}(\phi \mathbin{\square \mkern-2.7mu\raise0.3ex\hbox{${\rightarrow}$}}\psi,w)=\mathsf{True}\) iff (a) \(LV_{\mathscr{M}}\) doesn't make \(\phi\) \(\mathsf{True}\) at any world; or (b) there is a world \(x\) where \(LV_{\mathscr{M}}(\phi,x)=\mathsf{True}\) and for all \(y\) at least as close to \(w\) as \(x\), \(LV_{\mathscr{M}}(\phi \supset \psi,y)=\mathsf{True}\).

We're going to do an induction on the complexity of formulas, but before we do that, it will help to prepare thoroughly.

To begin, I want to define a notion of a structure that may not satisfy *all* the conditions to be an official SC model, but which approximates them. As our induction progresses we'll get closer and closer to the class of official SC models. I'll call our models "SC-like with respect to a set of formulas \(\Gamma\)." The definition for these models \(\mathscr{M}\) is just like the official definition of an SC model (and so too the defintion of an LC model, plus anti-symmetry), except in place of the Stalnaker-limit condition we have a condition restricted to formulas in \(\Gamma\):

(SL) For all \(\chi \underline{\in \Gamma}\) and \(w\) in \(\mathscr{M}\), either (a) there is no \(u\) where \(V_{\mathscr{M}}(\chi,u)=\mathsf{True}\); or (b) there is a \(u\) where \(V_{\mathscr{M}}(\chi,u)=\mathsf{True}\) and \(u\) is at least as close to \(w\) as any world where that's true.

Similarly, we'll have a notion of LC models \(\mathscr{M}\) that may not be Stalnaker-acceptable *tout court*, but which approximate that state. They'll be "L-good with respect to \(\Gamma\)." These will satisfy a similarly-restricted limit condition, now defined in terms of \(LV\) instead of \(V\):

(LL) For all \(\chi \underline{\in \Gamma}\) and \(w\) in \(\mathscr{M}\), either (a) there is no \(u\) where \(\underline{LV}_{\mathscr{M}}(\chi,u)=\mathsf{True}\); or (b) there is a \(u\) where \(\underline{LV}_{\mathscr{M}}(\chi,u)=\mathsf{True}\) and \(u\) is at least as close to \(w\) as any world where that's true.

More preparation before we begin our induction.

It will be useful to massage the definition (LTC) of the Lewis valuation function for the special cases where \(\mathscr{M}\) is an LC model that's L-good (wrt a set of formulas \(\Gamma\)). Such models will satisfy the limit condition (LL) that we just stated. They'll also satisfy the anti-symmetry condition, which entails that in case (LLb), no distinct world is exactly as close to \(w\) as \(u\) is. Hence for such models, we'll have:

(LL*) For all \(\chi \in \Gamma\) and \(w\), either (a) there is no \(u\) where \(LV_{\mathscr{M}}(\chi,u)=\mathsf{True}\); or (b) there is a \(u\) where \(LV_{\mathscr{M}}(\chi,u)=\mathsf{True}\) and \(u\) is closer to \(w\) than any other world where that's true.

Let's think about how things stand when clause (LL*b) is true: that is, when some \(\phi \in \Gamma\) and there's a closest-to-\(w\) \(\phi\)-world, \(u\). (During this phase of the proof, when I speak of \(\phi\) being \(\mathsf{True}\) at a world, this is always according to \(LV_{\mathscr{M}}\).) Recall clause (LTCb):

there is a world \(x\) where \(LV_{\mathscr{M}}(\phi,x)=\mathsf{True}\) and for all \(y\) at least as close to \(w\) as \(x\), \(LV_{\mathscr{M}}(\phi \supset \psi,y)=\mathsf{True}\)

Since we said that \(u\) is a closest-to-\(w\) \(\phi\)-world, it has to be at least as close to \(w\) as the \(x\) described in (LTCb), so if (LTCb) is correct, \(LV_{\mathscr{M}}(\phi \supset \psi,u)\) has to be \(\mathsf{True}\), and since \(\phi\) is \(\mathsf{True}\) at \(u\), \(\psi\) must therefore also be \(\mathsf{True}\) there.

Conversely, suppose \(LV_{\mathscr{M}}(\psi,u)=\mathsf{True}\). So then \(\phi \supset \psi\) would also be \(\mathsf{True}\) at \(u\). We know that no world distinct from \(u\) is equally close to \(w\). Since \(u\) is a closest-to-\(w\) \(\phi\)-world, for all \(y\) *closer* to \(w\) than \(u\), we have that \(LV_{\mathscr{M}}(\phi,y)=\mathsf{False}\). So \(\phi \supset \psi\) would be \(\mathsf{True}\) at all such \(y\). So for all \(y\) *at least as close* to \(w\) as \(u\) is, \(LV_{\mathscr{M}}(\phi \supset \psi,y)=\mathsf{True}\).

The preceding two paragraphs demonstrate that when clause (LL*b) is true, then: *clause (LTCb) is true iff \(LV_{\mathscr{M}}(\psi,u)=\mathsf{True}\) at the world \(u\) described in (LL*b).*

The truth-conditions for the Lewis valuation function had the form:

(LTC) \(LV_{\mathscr{M}}(\phi \mathbin{\square \mkern-2.7mu\raise0.3ex\hbox{${\rightarrow}$}}\psi,w)=\mathsf{True}\) iff (a) \(LV_{\mathscr{M}}\) doesn't make \(\phi\) \(\mathsf{True}\) at any world; or [clause (LTCb), here omitted].

What we've established is that *when \(\mathscr{M}\) is an LC model that's L-good wrt a set \(\Gamma\) that contains \(\phi\)*, then if clause (LTCa)/(LL*a) is false, and so (LL*b) is true --- that is, there is a world \(u\) where \(LV_{\mathscr{M}}(\phi,u)=\mathsf{True}\) and \(u\) is closer to \(w\) than any other world where that's true --- then in that case clause (LTCb) is true iff \(LV_{\mathscr{M}}(\psi,u)=\mathsf{True}\). So for such models, (LTC) is equivalent to:

(LTC*) \(LV_{\mathscr{M}}(\phi \mathbin{\square \mkern-2.7mu\raise0.3ex\hbox{${\rightarrow}$}}\psi,w)=\mathsf{True}\) iff (a) \(LV_{\mathscr{M}}\) doesn't make \(\phi\) \(\mathsf{True}\) at any world; or (b) there is a world \(u\) where \(LV_{\mathscr{M}}(\phi,u)=\mathsf{True}\) and \(u\) is closer to \(w\) than any other world where that's true, and \(LV_{\mathscr{M}}(\psi,u)=\mathsf{True}\).

I'll repeat again the Stalnaker truth-condition for comparison:

(STC) \(V_{\mathscr{M}}(\phi \mathbin{\square \mkern-2.7mu\raise0.3ex\hbox{${\rightarrow}$}}\psi,w)=\mathsf{True}\) iff for any world \(x\) that is a closest-to-\(w\) world where \(V_{\mathscr{M}}(\phi,x)=\mathsf{True}\), \(V_{\mathscr{M}}(\psi,x)=\mathsf{True}\).

Now for our induction.

We're going to be talking about cases where some structure \(\mathscr{M}\) counts as both an LC model that's L-good wrt a set \(\Gamma\), and also a model that's SC-like wrt \(\Gamma\), and we're asking whether \(LV_{\mathscr{M}}\) and \(V_{\mathscr{M}}\) will agree on their evaluations of some formula \(\chi\) at every world in \(\mathscr{M}\).

We begin with \(\Gamma\) empty, and consider first atomic \(\chi\). In such cases, \(LV_{\mathscr{M}}(\chi,w)\) and \(V_{\mathscr{M}}(\chi,w)\) will both be settled by \(\mathscr{M}\)'s atomic interpretation function, and so will agree for every \(w\).

Now, for all such \(\chi\), **there are three scenarios**. Either: (a) there is no world \(u\) where either \(LV_{\mathscr{M}}(\chi,u)=\mathsf{True}\) or \(V_{\mathscr{M}}(\chi,u)=\mathsf{True}\); or (b) there is such a world, and for every \(w\) there's such a world that's closest to \(w\); or (c) there is such a world, but for some \(w\) there is no such world that's closest to \(w\). In cases (a) and (b), the model will not only satisfy the conditions for being L-good and SC-like with respect to our original \(\Gamma\); since \(LV_{\mathscr{M}}\) and \(V_{\mathscr{M}}\) agree on their evaluation of \(\chi\) at every world, it will also satisfy both of those conditions *with respect to the set \(\Gamma\cup \{\, \chi \,\}\)*. In case (c), it won't satisfy either condition wrt that set. Hence, for the set of all atomic formulas, we know that \(\mathscr{M}\) will be L-good wrt that set iff it's SC-like wrt it.

Similarly if \(\chi\) is complex but contains no \(\mathbin{\square \mkern-2.7mu\raise0.3ex\hbox{${\rightarrow}$}}\)s. For all such \(\chi\), \(LV_{\mathscr{M}}\) and \(V_{\mathscr{M}}\) will agree at every world; and so \(\mathscr{M}\) will be L-good wrt \(\Gamma\cup \{\, \chi \,\}\) iff it's SC-like wrt the same set.

Now, for our induction, let's assume that \(\phi\) and \(\psi\) are formulas that each contain at most \(k\)-deeply embedded \(\mathbin{\square \mkern-2.7mu\raise0.3ex\hbox{${\rightarrow}$}}\)s. Let's assume further that \(LV_{\mathscr{M}}\) and \(V_{\mathscr{M}}\) agree on their valuation of all such formulas at every world in \(\mathscr{M}\); and that \(\mathscr{M}\) is L-good and SC-like with respect to the set \(\Gamma\) of all such formulas. We're going to argue that **in such cases, \(LV_{\mathscr{M}}\) and \(V_{\mathscr{M}}\) also agree on their valuation of \(\phi \mathbin{\square \mkern-2.7mu\raise0.3ex\hbox{${\rightarrow}$}}\psi\) at every world**. If that's true for arbitrary \(\phi\) and \(\psi\), then \(LV_{\mathscr{M}}\) and \(V_{\mathscr{M}}\) agree in their valuation at every world of the formulas we already described, *or* consist of a simple counterfactual, whose \(\mathbin{\square \mkern-2.7mu\raise0.3ex\hbox{${\rightarrow}$}}\)s go at most \(k+1\) deep.

So let's prove our bold claim. For reminder, we have that, when \(\mathscr{M}\) is an LC model that's L-good wrt a set \(\Gamma\) that contains \(\phi\), as we're assuming it is, then:

(LTC*) \(LV_{\mathscr{M}}(\phi \mathbin{\square \mkern-2.7mu\raise0.3ex\hbox{${\rightarrow}$}}\psi,w)=\mathsf{True}\) iff (a) \(LV_{\mathscr{M}}\) doesn't make \(\phi\) \(\mathsf{True}\) at any world; or (b) there is a world \(u\) where \(LV_{\mathscr{M}}(\phi,u)=\mathsf{True}\) and \(u\) is closer to \(w\) than any other world where that's true, and \(LV_{\mathscr{M}}(\psi,u)=\mathsf{True}\).

and we assume the Stalnaker truth-condition not only for official SC models, but also for models that are merely SC-like wrt set \(\Gamma\):

(STC) \(V_{\mathscr{M}}(\phi \mathbin{\square \mkern-2.7mu\raise0.3ex\hbox{${\rightarrow}$}}\psi,w)=\mathsf{True}\) iff for any world \(x\) that is a closest-to-\(w\) world where \(V_{\mathscr{M}}(\phi)=\mathsf{True}\), \(V_{\mathscr{M}}(\psi,x)=\mathsf{True}\).

As I said, we also know that \(LV_{\mathscr{M}}\) and \(V_{\mathscr{M}}\) agree on their evaluations of \(\phi\) and \(\psi\) at every world in \(\mathscr{M}\).

Ok, now to business. For each world \(w\), there are three cases:

Case 1. \(LV_{\mathscr{M}}\) doesn't make \(\phi\) \(\mathsf{True}\) at any world. We said \(V_{\mathscr{M}}\) will agree with it about \(\phi\) at every world, so by (LTC*) and (STC), it will follow that \(LV_{\mathscr{M}}\) and \(V_{\mathscr{M}}\) both make \(\phi \mathbin{\square \mkern-2.7mu\raise0.3ex\hbox{${\rightarrow}$}}\psi\) \(\mathsf{True}\) at \(w\) (and indeed at every world).

In the remaining two cases, we assume that \(LV_{\mathscr{M}}\) (and so too \(V_{\mathscr{M}}\)) makes \(\phi\) \(\mathsf{True}\) at some world.

Case 2. \(LV_{\mathscr{M}}(\phi \mathbin{\square \mkern-2.7mu\raise0.3ex\hbox{${\rightarrow}$}}\psi,w)=\mathsf{True}\). Since clause (LTC*a) is false, clause (LTC*b) must be true, and we have that there's a world \(u\) where \(LV_{\mathscr{M}}(\phi,u)=\mathsf{True}\) and \(u\) is closer to \(w\) than any other world where that's true, and \(LV_{\mathscr{M}}(\psi,u)=\mathsf{True}\). By (STC) we get that \(V_{\mathscr{M}}(\phi \mathbin{\square \mkern-2.7mu\raise0.3ex\hbox{${\rightarrow}$}}\psi,w)=\mathsf{True}\).

Case 3. \(LV_{\mathscr{M}}(\phi \mathbin{\square \mkern-2.7mu\raise0.3ex\hbox{${\rightarrow}$}}\psi,w)=\mathsf{False}\). So clause (LTC*b) must be false, and we have that *either* there's no world \(u\) where \(LV_{\mathscr{M}}(\phi,u)=\mathsf{True}\) --- but we're assuming there is such a world --- *or* there is such a world but there's no closest such world --- but we're assuming \(\mathscr{M}\) is L-good wrt a set \(\Gamma\) that contains \(\phi\), so there has to be a closest such one --- *or* there is a closest such world but \(LV_{\mathscr{M}}(\psi,u)=\mathsf{False}\). But then the right-hand side of (STC) will be false, so \(V_{\mathscr{M}}(\phi \mathbin{\square \mkern-2.7mu\raise0.3ex\hbox{${\rightarrow}$}}\psi,w)=\mathsf{False}\).

Hence in each case, \(LV_{\mathscr{M}}(\phi \mathbin{\square \mkern-2.7mu\raise0.3ex\hbox{${\rightarrow}$}}\psi,w) = V_{\mathscr{M}}(\phi \mathbin{\square \mkern-2.7mu\raise0.3ex\hbox{${\rightarrow}$}}\psi,w)\).

This establishes our bold claim.

So now we have that \(LV_{\mathscr{M}}\) and \(V_{\mathscr{M}}\) agree in their valuations of formulas that have at most \(k+1\)-deeply embedded \(\mathbin{\square \mkern-2.7mu\raise0.3ex\hbox{${\rightarrow}$}}\)s, except that in the case where \(\mathbin{\square \mkern-2.7mu\raise0.3ex\hbox{${\rightarrow}$}}\)s are embedded \(k+1\) deep, we only have this result for *simple* counterfactuals, not for truth-functional compounds that contain them. But since the definitions of \(V_{\mathscr{M}}\) and \(LV_{\mathscr{M}}\) coincide for other connectives, we can easily proceed to this being true for *all* formulas \(\chi\) with at most \(k+1\)-deeply embedded \(\mathbin{\square \mkern-2.7mu\raise0.3ex\hbox{${\rightarrow}$}}\)s.

Ok, now again as with the atomic \(\chi\), **there are three scenarios\(\dots\)** We don't know that the limit condition(s) will hold for our new \(\chi\)s, but we can prove as before that a model is L-good wrt each \(\Gamma\cup \{\, \chi \,\}\) iff it's SC-like wrt that same set.

Where have we gotten? Our induction shows that this is true:

(Result) If \(\mathscr{M}\) is a structure that's SC-like wrt the set of all SC wffs, then it's also an LC model that's L-good wrt that same set, and for such models, \(V_{\mathscr{M}}\) and \(LV_{\mathscr{M}}\) agree on their evaluation of every formula at every world.

(The same is true when you swap the positions of "SC-like" and "L-good", but this is the direction we'll make use of.)

Now we can wrap everything up: Suppose a formula \(\chi\) is LC-valid, then it's \(\mathsf{True}\) (according to \(LV\)) at every world in every LC model, hence in every LC model that's L-good wrt the set of all wffs. Suppose for reductio that \(\chi\) is \(\mathsf{False}\) (according to \(V\)) at some world in some SC model. Every SC model is a structure that's SC-like wrt the set of all wffs, and so as we've shown would also have to be an LC model that's L-good wrt the same set; but we said there were no such models where \(\chi\) was \(\mathsf{False}\) (according to \(LV\)) at some world. And we also showed that for such models, \(V\) and \(LV\) agree on their evaluation of every formula at every world. So \(\chi\) cannot after all be \(\mathsf{False}\) (according to \(V\)) at some world in some SC model.

Hence if a formula is LC-valid, it must also be SC-valid.

Yes. :-) Can you see how?

Ok. I thought one way to simplify would be that instead of progressively refining our class of models to be both more L-good (that is, L-good wrt a bigger set of formulas) and more SC-like, we could instead just start with the set of models that were *most* so-and-so, for one of these properties, and progressively prove that that exact class was more and more such-and-such, for the other property, as our induction proceeded.

Given the direction of the claim (Result) we use in the final wrap-up, I thought we should start with the set of models that was most SC-like, that is, SC-like wrt all formulas, and then progressively prove that it's more L-good.

Then I realized that we really never make use of the fact that the models are L-good with respect to any set of formulas. We only make use of the fact that they are LC models where \(V\) and \(LV\) agree. So why don't we just forget about thinking in parallel about the Lewis limit condition (part of our definition of L-good) and the Stalnaker limit condition (part of our definition of SC-like). Instead, let's just begin with the set of LC models that have anti-symmetric closeness relations and are most SC-like, that is, satisfy the *Stalnaker* limit condition for all formulas. That's all we really need. These will be LC models and they will also be SC models. We can then inductively prove that for such models, \(LV\) and \(V\) agree on their evaluation of every formula at every world. This proof will be much simpler.