Modal Requirements
3.4. Simple modal requirements 121 where the belief is held by a different subject.) Some de se propositions
like “I exist” or “I have a belief” satisfy it.
3.4.3 Goldman’s relevant-alternatives condition and Lewis’s infallibilist contextualism
Goldman’s (1976, 778–84) relevant-alternatives condition for percep-tual knowledge is a restricted form of the infallible belief condition. Gold-man introduces two weakenings of infallible belief. First, one should avoid error only atrelevantalternative cases, not all possible cases (1976, 774–8). Second, one should only avoid error in cases that are perceptu-ally equivalent to the target case, not all relevant cases. Roughly, a case β is perceptually equivalent to a target caseα iffthe subject in αcannot distinguish being inαfrom being inβ(1976, 778–84). We define:
Goldman’s relevant alternatives (GRA) αRβ iff pβ = pα, β is a relevant alternative toαandβis perceptually equivalent toα.23
Goldman’s characterisation of perceptual equivalence makes clear that a case perceptually equivalent to αmust involve a (living) subject posi-tioned in a certain way relative to certain objects and undergoing some perceptual experience — at least as long as α is itself a case involving some perceptual experience. It is not clear whether the subject has to be the same as in the target case; so we leave this aspect open here. Note also that the relevant-alternative relation is a relation between casesand not betweenworlds.24 That would allow the requirement to exclude some actual cases of errors about the same proposition as being nevertheless irrelevant. But Goldman does not discuss such issues.
Lewis (1996, 551–3) proposes a contextualist variant of Goldman’s relevant-alternative condition. (The idea was earlier put forward by
23. Wherever the subject of α is not existent, not alive or without experience, we assume that any case is equivalent to it. This ensures that R is reflexive. This also implies that sleeping subjects do not have perceptual knowledge, but the consequence is acceptable — they have only mnemonic knowledge, which is not covered by the account.
24. Like the closeness relation in Williamson’s safety, see3.4.6below.
Stine(1976) and Lewis(1979) and partly anticipated by Goldman(1976, 776–7).) In Lewis’s version, each context fixes a set of relevant alternative possibilities. So we need to relativise the requirement to conversational contexts. Perceptual equivalence is formulated more narrowly as having the same experiences and memory as in actuality. (It is also unclear whether Lewis requires the subject to be different across perceptually equivalent cases: I’ll leave it open here as well.25)
Lewis’s contextualist infallibilism (LCI) For any contextcand caseα,α satisfiesLCIrelatively tociff:
Bαand¬Eβfor anyβsuch thatαRβ, where
αRβiffβis a member of the set of alternatives for contextc, andSβ has the same experiences and memories inβasSαinα.
Note though that on this formulation, the relations R are typically not reflexive. On pain of trivialisation, some contexts will count some cases as irrelevant. Letγbe a case irrelevant inc. By the definition ofR,¬γRγ. So Ris not reflexive. This has the disastrous consequence that factivity fails in Lewis’s account, as we argue in ch.6(sec.6.6.2).
3.4.4 Nozick’s sensitivity and tracking
The simple version of Nozick’s truth-tracking account (1981, 172–
7) requires true belief, sensitivity and adherence, where the latter are characterised as follows, for a subjectSand a propositionp:
Sensitivity Ifpwere not true,Swould not believe thatp.26 Adherence Ifpwere true,Swould believe thatp.27
25. See his formulations inLewis(1996, 552–3), for instance: “I say that the unelim-inated possibilities are those in which the subject’s entire experience and memory are just as they actually are.” (553) The natural reading is that the subject of the possibility in question is the same as the actual one (since “they” binds “the subject’s entire expe-rience”). On the other hand, Lewis evokes the possibility that one does not know who he is (p. 555). He may have realised that on his account that is possible only if relevant alternative cases includes ones in whichsomeone elsehas the same evidence.
26. The sensitivity requirement was earlier suggested byDretske(1971, 1) and Arm-strong(1973, 169).
27. Nozick’s actual requirement is “ifpwere true,Swould believepand would not believe¬p.” I ignore the complication.
3.4. Simple modal requirements 123 (Dretske(1971, 1–8) had suggested a similar view.28)
Let us write “pq” for the subjunctive conditional “ifpwere true,q would be true.” Nozick adopts a variant of the standard Stalnaker-Lewis semantics for subjunctive conditionals (Stalnaker,1968;Lewis,1973). The semantics assume a three place relation xy zover worlds such that for any w, w is a complete preorder over a set of worlds including w.29 Whenw1 w w2 holds we say thatw1 isat least as closetowasw2. When w′ is out of the order ofw,w′ ww′ fails and we say thatw1isinaccessible from w. wis assumed to be as close to itself as any world in the order:
for any w′ such thatw′ w w′,w w w′. Each world w′ in the order ofw determines asphereof worlds that are at least all close towatw′.
On the standard Stalnaker-Lewis semantics,30 the subjunctive con-ditional p q holds at w iff there is no accessible p-world, or there is some accessible p-world w′ such that p → q holds throughout the sphere of worlds closer to w than w′ (Lewis, 1973, 16). We say that a semantics of subjunctive conditionals is variable iffthe sphere of worlds through with p → q has to be true varies with the antecedent proposition.
The Stalnaker-Lewis semantics is variable for conditionals with false an-tecedents. Where p and p′ are both false, p q requires the truth of p → qup tosome p-world, whilep′ q′ requires the truth ofp′ → q′ up tosome p′world.
TheLimit assumptionis the idea that for any worldwand antecedent p, there is a set ofp-worlds that are the closest to w(Lewis, 1973, 19-20).
That is, there are worldsw′ such thatpholds atw′and for any worldw′′
28. For all Dretske says in his (1971), he could have had a fixed-threshold semantics for subjunctive conditionals that would result in a safety-like condition; in particular, we could formalise the requirement in Dretske (1971, 1–2) roughly as Sosa’s reliable indication safety (see below3.5.4). It is only Dretske’s (1970) endorsement of what are in fact consequences of the sensitivity requirement — the failure of deductive closure and the related diagnosis of sceptical arguments — that justify the attribution of a sensitivity-like view to him. Curiously, the alternative-based semantics for subjunctive conditionals that he sketches in his (1970) does not appear in his (1971). I do not attempt to formalise Dretske’s account here.
29. Preorders allow ties but no incomparabilities. See appendixAfor details.
30. Revision note. This paragraph and the next have been revised. Thanks to Tim-othy Williamson for pointing out to me that original formulations relied on the Limit assumption.
in the order ofw, ifpholds atw′′thenw′ ww′′. If the limit assumption holds, the semantics is equivalent to saying that p q holds at w iff q holds at all p-worlds closest tow. Though the assumption is dubious (Lewis, 1973, 20), it will simplify our discussion to make it. As far as I can see, the assumption does not affect our conclusions.
With this semantics and the limit assumption, sensitivity is an infalli-bilist requirement characterised as follows:
Sensitivity (SE) αRβiffpβ =pα, Sα = Sβ, andwβ is at least as close towα as any¬pαworld.31
SE requires belief inpα in the actual world, and that the subject avoids false belief that pα in a “sphere” of cases whose worlds are at least as close as the closest¬pα worlds. Sincewα is maximally close to itself,αis included in the sphere, and sopα must be true in α. Now consider any other caseβsuchαRβ. Ifwβ is a world in whichpαis true, then βcannot be an error case, sincepβ = pα and pα is true. (Thus SE does not require belief in the interveningpα worlds that are closer than the first¬pαones.) The crucial case is when wβ is one of the closest not-pα worlds. Since pβ = pα and ¬pα inwβ, β is a case of error iff it is a case of belief. Thus the requirement is satisfied only if the subject does not believepα in the closest worlds in which it is false.32
If we assume, asLewis(1973, 14) does, that eachwis theuniqueworld closest to itself, a subjunctive conditional with a true antecedent is true if its consequent is true. This would trivialise Nozick’s adherence condi-tion. A stronger truth condition for conditionals with truth antecedents is needed. There are several options, which we review in appendix A.
Nozick’s own proposal is that when pis true, p q is true iff q holds at allp-worlds up to butexcludingthe first ¬pones (Nozick, 1981, 680n).
For simplicity, I will use the slightly stronger quasi-Nozickian semantics
31. On Nozick’s view,αRβprobably also requires thattα=tβ, or some such restriction.
Presumably, whether ifphad been false I would have wrongly believed it to be true in the distant past or future does not matter to whether I knowpnow.
32. SEis not strictly equivalent to Nozick’s third condition, since it additionally im-plies true belief (Nozick’s conditions 1 and 2). But that is all the better, I think, and at any rate harmless.