Modal Requirements
3.5. Parametric modal requirements 135
opt for the formulations that keepRreflexive and non-parametric.
3.5.2 Armstrong’s nomological infallibilism
On Armstrong’s (1968, 189,1973, 168) account, non-inferential knowl-edge consists in a “law-like connection” between belief and truth. If we construe “law-like connection” as nomic necessity, his account is a parametric infallibilist requirement.49 In its simpler version:
Armstrong’s simple nomological account ECαiffCαandEα, GCαiffBαandCα,
αRβiffpβ =pαand the laws ofαhold inβ,
whereCis a relevantly general specification of a subject’s conditions and circumstances.
Ris reflexive andGCα∧¬ECα→ Tα, so the requirement entails true belief.
It further requires avoidance of error in any nomically possible case of belief in the same proposition in which Cholds. This is a variant of the infallibly based belief condition, whereCplays the role of the basis.
On Armstrong’s (1973, 170, 182, 197) more refined statement, non-inferential knowledge does not merely require a law-like connection be-tween believingFaandFafor some particular propositionFa(whereFis a predicate and aa singular term), but a more general law-like connec-tion between believingFxandFx for any x. This revised requirement fits our schema as follows. First, assume that propositions are structured.
Second, writep[a/b] for the proposition that results from substituting the singular term/concept for a in p by the term for b in all its occurrences.
GCαandECαare defined as before, andRis modified to:
Armstrong’s nomological account αRβ iff ∃x∃y(pβ = pα[x/y]) and the laws ofαhold inβ.
The requirement is only meant to apply to knowledge of singular propo-sitions (1973, 192–4).
49. Thanks to Pascal Engel here.
3.5.3 Nozick’s sensitivity and tracking with methods
Nozick amends his tracking account by considering the methods with which a belief is formed (Nozick,1981, 179). If in actualitySbelieves that pvia methodm, the new requirements are:
Nozick’s sensitivity with methods If p was not true andS were to use mto arrive at a belief whether (or not)p, thenSwould not believe, viam, thatp.
Nozick’s adherence with methods Ifpwas true andS were to usemto arrive at a belief whether (or not)p, thenSwould not believe, via m, thatp.
We give the Nozickian formalisation in AppendixA. Here we give the quasi-Nozickian one. We abbreviate:
Bmα inα,Sαbelieves thatpαon the basis ofm.
We define:
Quasi-Nozickian M-sensitivity (QNT) EmαiffBmα∧ ¬Tα.
GmαiffBmα.
αRmβ iff pβ = pα, Sβ = Sα, and wβ is at least as close to wα as any world wherepα is falseand Sα usesmto arrive at a belief whether (or not)pα.
Quasi-Nozickian M-adherence (QNT) ImαiffBmα∧ ¬Tα.
GmαiffBmα.
αR′mβiffpβ =pα,Sβ =Sα, andwβis at least as close towαas any world wherepα is falseor Sα fails to usemto arrive at a belief whether (or not)pα.50
50. Here is how the definitions are obtained. WriteBm?pfor “Susemto arrive at a belief whether (or not)p”. For a given caseα, the antecedent of the first conditional is
¬pα∧Bm?pα, the antecedent of the second ispα∧Bm?pα. The conditionals are applied in cases α such that Bmα. (In other cases, the requirements are not satisfied since Gmαfails). Consider first those such thatTα, that is,pα holds. The antecedent of the first conditional is false, the antecedent of the second is true. On the Quasi-Nozickian semantics, a subjunctive conditional is true iffthe corresponding material conditional is trueup to and including the closest world where the antecedent has a different truth value. This leads us up to worlds such that¬pα∧Bm?pαfor the first conditional, but to worlds such
3.5. Parametric modal requirements 137 The two requirements cannot be unified as in the simple version (sec.3.4.4).
Method-relative Adherence and Sensitivity do not operate on the same sphere: on the method-relative version of Adherence, it is not required that one believesp up to the closest worlds in whichpis false, but only up to the closest worlds in which one “ceases” to use the method actu-ally used. This is how method-relative Adherence avoids the sceptical consequences of simple Adherence (see sec.3.4.5on the latter).
Quasi-Nozickian M-tracking (NMT) For anyα, QNMTαiffthere is an msuch thatBmαand:
¬Imβfor any βsuch thatpβ = pα, Sβ = Sα, andwβ is at least as close towαas any world wherepαis falseor Sαusesmto arrive at a belief whether (or not)pα.
¬Emβfor anyβsuch thatpβ =pα,Sβ =Sα, andwβis at least as close to wα as any world where pα is false and Sα uses m to arrive at a belief whether (or not)pα.
Nozick’s M-tracking is the only requirement for which we cannot avoid introducing a parameter on the respect of similarity relationR. Not only the error to be avoided has to be an error made by using the same method, but the range of worlds in which error has to be avoided also depends on the method used in the target case.
Luper-Foy (1984, 28–9) and Williamson (2000, 154) point out prob-lems arising from the fact that methods are mentioned in the antecedent of the sensitivity conditional and thereby affect the range of relevant worlds.51 They suggest that Nozick should drop the reference to meth-ods in the antecedent. (Neither of them endorse the resulting view.) The
that¬pα∨Bm?pαin the second. Then we check that the resulting semantics ensures that sensitivity is violated in casesαsuch thatBmα∧ ¬pα, which they do:Ris reflexive, since wαis as close to itself as any other world, so ifBmα∧ ¬Tαsensitivity fails.
51. Suppose methods are finely individuated so as to require some form of “perceptual equivalence” (as inGoldman,1976). Suppose I am currently seeing a tree. What are the closest worlds in which there is no tree, but I usea similar perceptual experienceto decide whether there is a tree? Presumably worlds in which I hallucinate a tree, and it is likely that I wrongly believe that there is a tree in some such world. So even in ordinary cases, the sensitivity conditionals would take one to “far out” worlds and would hardly ever be satisfied. But on the other hand, Nozick would need finely individuated methods to deal with cases like Goldman’s Dachshund (cf.131above).
amended sensitivity requirement is just like Nozick’s original one but for the relativisation of the relevant type of error and ignorance to methods.
AssumingSbelieves thatpviam:
Revised Nozick’s sensitivity with methods If p was not true, then S would not believe, viam, thatp.
Revised-Nozickian M-sensitivity (RNMS) For anyα, RNMSαiff there is anmsuch thatBmαand¬Emβfor anyβsuch thatαRβ, where:
EmαiffBmα∧ ¬Tα, and
αRβ iffpβ = pα,Sα = Sβ, andwβ is at least as close towα as any¬pα
world.
The respect of similarity relation need not be parametric here.
An analogous revision for Adherence is inadvisable. If no reference to methods is made in the antecedent of Adherence, then Adherence requires one to believe up to the closest¬p world, which may be a “far out” world depending onp. We would be saddled again with the sceptical consequences of adherence (sec.3.4.5).
3.5.4 Basis-relative safety
Several proponents of safety want to relativise safety to bases (Williamson, 2000, 128; Sosa1999a, 378;1999b, 149; 2005, 156;Pritchard,2005, 152–5).
As before, we abbreviate:
BmpiffSbelieves thatpon the basis ofm.
The fixed version of basis-relative safety is given by:
Basis-relative safety (BS) EmαiffBmα∧ ¬Tα GmαiffBmα,
αRβiffpβ =pα,Sβ =Sα,wβis close towα.
That is, one believes on a basis such that one avoids false belief on the same basis at close worlds. We assume a fixed threshold for closeness, but of course a variant with a variable threshold could be formulated. This ver-sion of Safety, fixed threshold included, has been endorsed byPritchard (2005, 162). (Pritchard (2010) has added a proposition-uncentred virtue
3.6. Reliability requirements 139