This section is devoted to proving the following result.
Theorem 50. The axiom schema (a)–(f) are true in every atomic change model. Proof. We show that every atomic change model satisfies the semantic property corresponding to each axiom scheme. The result then follows from Theorem65.
Pick any atomic change modelM= (W,{Rϕ}ϕ∈L,X,ι). SinceMis clear, to save ink we writew|= ϕinstead ofM,w|= ϕ. Imaginative inclusion is imme- diate sincei(ϕ∨ψ) =i(ϕ)∪i(ψ)by the semantics of IE. Unconditionalisation is also immediate sincew|=ϕ,→ψiffi(ϕ)⊆i(ψ)by definition. Veracity holds by assumption on counterfactual models (Definition46).
Success. Firstly letLbe the set of literals and≤the fixed linear order over the literals used in the construction ofϕ’s standard disjunctive normal formϕ∗. Given any atomic change modelMand finite set of contentsc⊆X, we define thestandard literal formofc(with respect toM), denotedc`, as follows
c`=^{l∈L:ι(l)∈candl≤l0for every literall0withι(l) =ι(l0)}.
The idea is thatc`is a sentence conjoining a representative literal for each ele- ment ofc. Note thatc`is well-formed asc, by semantic definition (Definition
3), is finite. Now pick anyϕ∈ Land supposewRϕv, i.e.v= (w−c−)∪c+for somec∈ci(ϕ).
We show thatv|=c`by showing thatv|=lfor every literallofc`. Pick any conjunctlofc`. Iflis an atomic sentencepthenι(p)∈cby definition ofc`, and sop ∈ c+. Then asv = (w−c−)∪c+, we havec+ ⊆ v, sop ∈ v, i.e. v |= p. And iflis¬pfor some atomic sentencepthen as¬pis a literal ofl,ι(¬p)∈c, and sop∈c−. Then asc∈ci(ϕ), by Fact48ι(p)∈/csop∈/c+. Thusp∈/v, i.e.
v6|=p, and so by classical logicv|=¬p. Either way, then,v|=c`.
Now by construction ofϕ’s standard disjunctive normal formϕ∗and Lemma
12, we may write ϕ∗ asWc∈i(ϕ)c
`
. So as v |= c`, andci(ϕ) ⊆ i(ϕ), v |= c` for somec ∈ i(ϕ), sov |= ϕ∗. And by the standard normal form theorem,
v|=ϕ∗≡ ϕ, so by veracity,v|= ϕ∗↔ϕ, and sov|=ϕ, as required.
Weak centering.Supposew|=ϕ. Then by the standard disjunct normal form theorem (Lemma9) and veracity,w|=c`for somec∈i(ϕ). And aswis a model of propositional logic,cmust be consistent, i.e. c ∈ ci(ϕ). We now show that
wRϕw, that is,w= (w−c−)∪c+.
For the left-to-right inclusion, pick anyp ∈ w. First suppose toward a con- tradiction thatp ∈ c−. Then ι(¬p) ∈ c, so asw |= c`,w |= l for some lit- erall with ι(l) = ι(¬p). Then by veracity w |= l ↔ ¬p, so w |= ¬p, i.e.
p∈/w, contradicting our original assumption. Then asp ∈(w−c−), a fortiori
p∈(w−c−)∪c+, as required.
Conversely, pick anyp∈(w−c−)∪c+. We showp∈w. Ifp∈w−c−then
p∈wand we are done. So supposep∈c+. Thenι(p)∈c, so asw|=c`,w|=l for some literallwithι(l) =ι(p). By veracity,w|=l↔p, sow|= p, i.e. p∈w.
Cumulative transitivity. We prove the stronger result that for anyw,v ∈ W, ifwRϕvandv|=ψthenwRϕ∧ψv.
By the semantics of IE, i(ϕ∧ψ) = i(ϕ)di(ψ), so we have to show that
v= w−(x∪y)−
v= (w−c−)∪c+
for somec∈i(ϕ). We will takex=c. Now, by the standard disjunctive normal form theorem,v|=ψ≡ψ∗, and sov|=ψ∗↔ψby veracity. Then asv |= ψ, we havev |= ψ∗. Recall from the proof ofsuccessabove that
ψcan be written asWd∈i(ψ)d
`
. Then by classical disjunction,v |= d`for some
d ∈ i(ψ). And asvis a model of propositional logic,dmust be consistent, so
d∈ci(ψ). We will takey=d. That is, we show that for every atomicp,
p∈viff p∈wandι(¬p)∈/c∪dorι(p)∈c∪d.
For the left-to-right direction, pick any atomic sentencep ∈v. Then asv= (w−c−)∪c+, eitherp∈wandι(¬p)∈/c, orι(p)∈c. If the latter,ι(p)∈c∪d
and we are done. So suppose p ∈ wand ι(¬p) ∈/ c. It remains to show that
ι(¬p)∈/d. Suppose for reductio thatι(¬p)∈d. Then asv|=d`, by construction ofd`we havev |= lfor some literallwithι(l)∈ candι(l) =ι(¬p). Then by veracity,v |= l ≡ ¬p, and so v |= ¬p. By classical logic, v 6|= p, i.e. p ∈/ v, contradicting the fact thatp∈v.
For the right-to-left direction, first supposeι(p) ∈c∪d. Ifι(p)∈cthen as
c+ ⊆ v, we have p ∈ v. So supposeι(p) ∈ d. Then asv |= d`, by a similar argument as above,v |= p, i.e. p ∈ v, as required. Now suppose p ∈ wand
ι(¬p) ∈/ c∪d. A fortiori,ι(¬p) ∈/ c, sop ∈ w−c−. Then aswRϕv, we have
w−c−⊆v, sop∈vand we are done.
It is remarkable that the atomic change model—a model constructed ac- cording to a particular intuition of atomic change, with no regard for general validities—would turn out to be a model of the desirable axiom schema (a)–(f). When constructing the atomic change model, I did not set out to create a model of those schema. But surprisingly, that is what we have ended up with.
Even though the atomic change model validates many desirable inference patters of counterfactuals, it is far too soon to move from this fact to the idea that the atomic change model is the one correct model of our counterfactual reasoning. For one thing, our result does not make any claims to uniqueness: many other intuitive approaches to counterfactuals may well satisfy the intu- itive inference patterns above.
And another thing, our intuitive interpretation of counterfactuals seems to involve more parameters than the atomic change semantics provides. The fol- lowing section considers one such parameter: our use ofimplicithypothetical contexts when we interpret counterfactuals.