The case of the missing ‘If’: Accessibility relations in Stalnaker’s theory of conditionals *

A part of Stalnaker’s (1968) influential theory of conditionals has been neglected, namely the role for an accessibility relation between worlds. I argue that the accessibility relation does not play the role intended for it in the theory as stated, and propose a minimal revision which solves the problem, and brings the theory in line with the formulation in Stalnaker & Thomason 1970.


Accessibility in Stalnaker 1968
A part of Stalnaker (1968)'s influential theory of conditionals has been neglected, namely the role for an accessibility relation between worlds.I believe this is because the accessibility relation does not play the role intended for it in the theory as stated.I propose a minimal revision which solves the problem, and brings the theory in line with the formulation in Stalnaker & Thomason 1970.In Stalnaker's theory, a conditional A > B is evaluated relative to a model structure and a selection function. 1A model structure M is a triple K, R, λ , where K is a set of possible worlds; R is a reflexive binary accessibility relation on K, with αRβ read 'β is possible with respect to α'; and λ is the absurd world where every sentence is true, accessible from no other world under R.A selection function f is a function which takes a sentence and a world as its arguments, and returns a world as its value.Relative to a background model M and accessibility relation f , Stalnaker imposes four conditions on f (the names are later additions): (1) Success: For all antecedents A and base worlds α, A must be true in f (A, α).
(2) Absurdity: For all antecedents A and base worlds α, f (A, α) = λ only if there is no world possible with respect to α in which A is true.
(3) Strong Centering: For all antecedents A and base worlds α, if A is true in α, then f (A, α) = α.
(4) CSO: For all base worlds α and all antecedents B and B , if The accessibility relation R figures only in the second condition, Absurdity.This condition does not seem to be strong enough, however.Later in the paper, Stalnaker proposes that we can define the ordinary modal operators and ♦ out of his conditional semantics.In particular, he proposes this definition for : Now, if is meant to receive its ordinary semantic interpretation, then we should also have the following truth conditions, relative to a model structure K, R, λ : Stalnaker is not explicit that he intends to receive the Standard Semantics, but he uses in an altogether standard way (for instance, in defining the strict conditional).So we should expect Standard Semantics to match Defined Semantics.
But it doesn't: if A is true according to the Defined Semantics, then it is true according to the Standard Semantics, but not vice versa.Suppose that A is true at α on Defined Semantics.Then ¬A > A is true at α, which can only hold if f (¬A, α) = λ , which, according to Absurdity, only holds if all the worlds accessible from α make ¬A false, which guarantees that all the worlds accessible from α make A true, so A is true according to Standard Semantics.But now suppose that A is true on Standard Semantics.Then every world accessible from α makes A true.It is consistent with all the conditions above, however, that f (¬A, α) is some world β = λ which makes ¬A true.Given the assumption that all the worlds accessible from α make A true, β is not accessible from α under R; but nothing rules out a situation in which β is nonetheless selected as the closest ¬A-world.In other words, given the conditions as stated, inaccessible, non-absurd worlds can be selected.In that case, ¬A > A will not be true at α, and so according to the Defined Semantics, A will be false.Thus Defined Semantics comes apart from Standard Semantics.
For a simple model of this divergence, let K = {α, β , λ }, with A true at α and false at This model is clearly consistent with all the constraints above.According to Standard Semantics, A is true at α, since α can only access A-worlds.But according to Defined Semantics, A is false at α, since ¬A > A is false at α, since A is false at f (¬A, α) = β .
2 The missing 'If' The accessibility relation thus seems not to be doing enough in Stalnaker's semantics.
The proper role for the accessibility relation seems to me to be the following: Accessibility: For all antecedents A and worlds α, if there is some world β such that αRβ and β makes A true, then α must access That is, f should always take us to an accessible world, if it's possible to do so while satisfying Success, and otherwise must take us to λ .Accessibility does not follow from Stalnaker's conditions: the toy model just given is a countermodel, since α does not access any ¬A-world under R, but f (¬A, α) = λ .There are two reasons to think that Accessibility gives the accessibility relation its proper role in Stalnaker's semantics.First, Accessibility is exactly what we need to render Defined Semantics equivalent to Standard Semantics. 2Second, Accessibility is entailed by the semantics given in Stalnaker & Thomason 1970, which is a formal companion to Stalnaker 1968 whose goal is to extend the formal treatment of the theory given in Stalnaker 1968.In the propositional fragment of Stalnaker & Thomason (1970)'s system, the four conditions on selection functions and the semantic clauses are all equivalent to those in Stalnaker 1968.There is one crucial difference, however: in the definition of selection functions, Stalnaker & Thomason (1970) stipulate that for all A and α, if f (A, α) = λ , then αR f (A, α).I'll refer to this stipulation by its heading in Stalnaker & Thomason 1970, namely D3.4.D3.4 entails Accessibility, given Stalnaker's other conditions.Actually, it is strictly more than we need: all that we need is the stipulation that, if f (A, α) = λ , 2 Accessibility obviously renders these equivalent.To see that the failure of Accessibility lets these diverge, suppose Accessibility is false.Then for some antecedent A and world α, either there is some world β such that αRβ and β makes A true, f (A, α) = γ for some world γ, and ¬αRγ.Note first that, by Absurdity, γ = λ , since there is an accessible A world from α. Now let {γ} be the proposition true just at γ and λ and false everywhere else in K (for convenience, we also use {γ} to denote the sentence true just at γ and λ and false everywhere else in K).By Success, f ({γ}, α) is either γ or λ .But it can't be λ .For since A is true in λ and {γ} true in λ , by CSO it would follow that γ = λ , contrary to what we have shown.So f ({γ}, α) = γ.It follows that {γ} > ¬{γ} is false at α, and so ¬{γ} is false at α according to Defined Semantics.But now note that since ¬αRγ and ¬αRλ , we have ¬{γ} true at α according to Standard Semantics.Or there is no world β such that αRβ and β makes A true, and f (A, α) = γ = λ .By Success, A is true in γ, and since γ = λ , ¬A is thus false in γ, so A > ¬A is false in α.Thus ¬A is false in α according to Defined Semantics; but it is true in α according to Standard Semantics.then there is a world possible with respect to α in which A is true (this follows from D3.4 given Success, but not vice versa).Note that this, in turn, is just the converse of Absurdity.And so all we need to get to Accessibility from Stalnaker (1968)'s formulation is an extra 'if' in Absurdity, as follows: Absurdity Biconditional: For all antecedents A and base worlds α, f (A, α) = λ if and only if there is no world possible with respect to α in which A is true.Given Stalnaker's other conditions, Absurdity Biconditional entails Accessibility; 3 and D3.4 follows immediately from Accessibility.
Thus we should replace Absurdity with Absurdity Biconditional.This gives the accessibility relation the role that it seemed destined for, and it brings Stalnaker (1968)'s informal exposition in line with Stalnaker & Thomason (1970)'s formal development.This has two upshots, both of them small but worth noting.First, we should treat Stalnaker & Thomason (1970)'s exposition, rather than Stalnaker (1968)'s, as canonical.Second, in the subsequent literature, Stalnaker's accessibility relation has been largely ignored (perhaps partly because of the limited role that it played in the theory as stated).But in more recent literature, it has been recognized that an accessibility relation of some kind may play an important role in the theory of conditionals.For instance, an accessibility relation may play a role in distinguishing indicative from subjunctive conditionals (Stalnaker 1975, von Fintel 1998); and an accessibility relation may play a role in accounting for reverse Sobel sequences (von Fintel 2001, Gillies 2007, Williams 2008, Moss 2012).The idea that the interpretation of conditionals depends not only on a selection function of some sort but also on an accessibility relation was already present in Stalnaker 1968 -albeit missing an 'if'.