Higher order ignorance inside the margins


According to the KK-principle, knowledge iterates freely. It has been argued, notably in Greco (J Philos 111:169–197, 2014a), that accounts of knowledge which involve essential appeal to normality are particularly conducive to defence of the KK-principle. The present article evaluates the prospects for employing normality in this role. First, it is argued that the defence of the KK-principle depends upon an implausible assumption about the logical principles governing iterated normality claims. Once this assumption is dropped, counter-instances to the principle can be expected to arise. Second, it is argued that even if the assumption is maintained, there are other logical properties of normality which can be expected to lead to failures of KK. Such failures are noteworthy, since they do not depend on either a margins-for-error principle or safety condition of the kinds Williamson (Mind 101:217–242, 1992; Knowledge and its limits, OUP, Oxford, 2000) appeals to in motivating rejection KK. “Introduction: KK and Being in a Position to Know” Section formulates two versions of the KK-Principle; “Inexact Knowledge and Margins for Error” Section presents a version of Williamson’s margins-for-error argument against it; “Knowledge and Normality” and “Iterated Normality” Sections discuss the defence of the KK-Principle due to Greco (J Philos 111:169–197, 2014a) and show that it is dependent upon the implausible assumption that the logic of normality ascriptions is at least as strong as K4; finally, “Knowledge in Abnormal Conditions” and “Higher-Order Ignorance Inside the Margins” Sections argue that a weakened version of Greco’s constraint on knowledge is plausible and demonstrate that this weakened constraint will, given uncontentious assumptions, systematically generate counter-instances to the KK-principle of a novel kind.

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Fig. 1


  1. 1.

    For the purposes of the present paper, the KK-Principle can be treated as the disjunction of (KK) and \( ({\text{KK}}^{-})\).

  2. 2.

    Even if she is not in a position to know precisely how inexact her knowledge is, it is plausible that there is some i  > 0 such that she is in a position to know that she is not in a position to know the temperature to within i °F.

  3. 3.

    In fact, Greco includes a third conjunct in the RH-clause of the biconditional:

    (NTK.iii) Being in C causes or constitutes S believing that p.

    However, he proposes that this conjunct can be dropped under the idealising assumption that if normally, being in C entails p then normally being in C causes S to believe that p (2014a, 184) (similar qualifications can be found in Dretske (1981) and Stalnaker (2015)). This idealisation can be treated as analogous to Williamson’s idealising assumption in the extension of the margins-for-error argument against \( ({\text{KK}}^{-})\), that S knows that p if S is in a position to know that p. Satisfying (NTK.i-ii) can reasonably be treated as necessary and sufficient for being in a position to know under Greco’s theory. Thus, dropping the idealising assumptions, Greco’s argument in fact only establishes the following version of the KK-principle (which is neither stronger nor weaker than (KK)):

    \( ({\text{KK}}^{\thicksim})\) (S is in a position to know that p) ⊃ (S is in a position to know that (S is in a position to know that p)).

  4. 4.

    Note that, on certain formulations, the information carried by a system in a given state is said to be the strongest such proposition. On this variant, the relevant bridge principle will be that S knows, when in C, that p only if the information carried by C entails that p.

  5. 5.

    We assume that S is in exactly one cognitive state at each world; that is, λC.CS is a partition of the subset of modal space in which S exists.

  6. 6.

    (SR) corresponds to the assumption that the accessibility relation for ■ is shift reflexive: if w′∈N(w), then w′∈N(w′).

  7. 7.


    (P 1 ) KS(KSp) ≡ (■(CS ⊃ KSp) ∧ ∀p (p ⊃ ¬■¬p)) (from (P2), p/KSp)
    (P 2 ) □(Cs ⊃ (KSp ≡ (■(CS ⊃ p) ∧ ∀p (p ⊃ ¬■¬p)))) (NTK)
    (P 3 ) ■(Cs ⊃ (KSp ≡ (■(CS ⊃ p) ∧ ∀p (p ⊃ ¬■¬p)))) (from □φ⊨■φ)
    (P 4 ) KS(KSp) ≡ ■(CS ⊃ (■(CS ⊃ p) ∧ ∀p (p ⊃ ¬■¬p))) ∧ ∀p (p ⊃ ¬■¬p) (from (P 1 ), (P 3 ))
  8. 8.


    (P 1 ) ■(⊤ ⊃ φ) ⊃ ■(⊤ ⊃ ■(⊤ ⊃ φ)) (from (CI))
    (P 2 ) ■(φ) ⊃ ■(⊤ ⊃ ■(φ)) (from (P 1 ), φ⟚⊤ ⊃ φ)
    (P 3 ) ■(φ) ⊃ (■⊤ ⊃ ■■(φ)) (from (P 2 ), K)
    (P 4 ) (■(φ)∧■⊤) ⊃ ■■(φ)) (from (P 3 ), ((φ∧ψ) ⊃ χ)⟚(φ ⊃ (ψ ⊃ χ)))
    (P 5 ) ■⊤ (from ■Nec)
    (P 6 ) ■(φ) ⊃ ■■(φ) (from (P 4 ),(P 5 ), {(φ∧ψ) ⊃ χ), φ}⊨ ψ ⊃ χ)
  9. 9.

    It may be that the iterated reading of (c) is dispreferred, in favor of the non-iterated reading (on which both normality claims are evaluated with respect to the world of utterance). This is, it should be clear, not the reading on which the case is probative regarding the plausibility of 4 for ■.

  10. 10.

    Goodman and Salow (2018) propose an alternative defense of (KK) via appeal to a normality-based account of knowledge. Their models include a single ordering of worlds for comparative normality, which results in KD45 for the logic of ■. If their models are expanded to allow for contingency in facts about comparative normality (i.e., by introducing distinct orderings for each world), then if the resulting logic of maximal normality (i.e. ■) is weaker than K4, counter-instances to (KK) are predicted.

  11. 11.

    i.e., ■φ⊨φ.

  12. 12.

    The argument against (4) for ■ can be given, mutatis mutandis, for ■Cs.

  13. 13.

    Greco (p.c.) is sympathetic to a version of this response.

  14. 14.

    The obvious, if ad hoc, solution would be to adopt the conjunction of (NTK.i) and p as a sufficient condition instead. Yet, while accounting for factivity, this response still cannot accommodate the apparent incompatibility of knowledge with gettierization. In many cases in which S has a gettierized belief that p, S will be in a cognitive state C such that, normally, if S is in C, then p. The proponent of the unrevised version of (NTK) can at least suggest that in such cases, conditions are abnormal. Hence, S’s failure to know that p is to be explained as a result of the failure of (NTK.ii). Yet, if (NTK.ii) is replaced with p, then we appear committed to ascribing knowledge to S in such cases, contrary to the gettier intuition (since ex hypothesi, (NTK.i) is satisfied and if S has a gettierized belief that p then p).

  15. 15.

    Suppose that the probability of having a hallucinatory appearance as of a red screen is .00000001 (i.e., one-in-ten-million). Assume it is independent of the colour of the screen. Then the probability of having an appearance as of a red screen while the screen is in fact blue is .00000001 × .99999999 = .000000099999999 – or almost 10 times as likely as the screen being red.

  16. 16.

    Smith defends a normality condition on justification, rather than knowledge. Assuming (propositional) justification for p is a necessary condition on knowledge that p, Smith’s condition will entail (WNC).

  17. 17.

    Stalnaker (2015, 38): “Does it make a belief any safer, in a sense of safety that has epistemic merit, if all the very similar cases are Gettier cases(cases of justified true belief without knowledge) rather than cases of false belief? […] However the relevant nearness relation is spelled out, it does not seem reasonable to think that a belief being true by coincidence in nearby situations should contribute to the robustness and stability of the belief in the actual situation.” Thanks to an anonymous Philosophical Studies reviewer for raising this point.

  18. 18.

    Proof: Let Min(A) = {⟨e,f⟩: ¬∃⟨e′,f′⟩ ∈ A: e′ < e}. Min(A) is the set of worlds in A in which the parameter takes an A-minimal value. Let Max(A) = {⟨e, f⟩: ¬∃⟨e′,f′⟩ ∈ A: e < e′}. Max(A) is the set of worlds in A in which the parameter takes an A-maximal value. By (II) KSp only if Min(R(Min(N(⟨e,f⟩))) ∈ p and Max(R(Max(N(⟨e,f⟩))) ∈ p. Yet Max(R(Min(R(Min(N(⟨e, f⟩)))) = Min(N⟨ef⟩) and Min(R(Max(R(Max(N(⟨e, f⟩)))) = Max(N⟨e, f⟩). Thus, since |N(⟨e,f⟩)| > 1, there is no k such that ⟨kf⟩ is R-accessible from every world in both Min(R(Min(N(⟨e, f⟩))) and Max(R(Max(N(⟨e,f⟩))). So there is no k such that Easy(k) is true at both.

  19. 19.

    An anonymous reviewer for Philosophical Studies suggests the following, related principle:

    S knows, in C, that p only if normally, (CS ⊃ S knows that p).

    Assuming that knowledge that p is incompatible with it easily being the case that ¬p, then this principle will have the same consequence regarding knowledge of what could easily have been the case.


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This paper has benefited from helpful discussion with and feedback from Brian Ball, Andy Egan, Juan Sebastian Piñeros Glasscock, Daniel Greco, Alex Roberts, Ginger Schultheis, Timothy Williamson and audiences at Yale, Oxford, and The Joint Session of Mind and the Aristotelian Society 2017.

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Correspondence to Sam Carter.

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Carter, S. Higher order ignorance inside the margins. Philos Stud 176, 1789–1806 (2019). https://doi.org/10.1007/s11098-018-1096-5

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  • Higher-order knowledge
  • KK
  • Positive introspection
  • Iterated knowledge
  • Margins for error
  • Normality