Timothy Williamson, Knowledge and Its Limits, Ch 5, "Margins and Iterations"

(outline for presentation, May 1)

5.1 Against the KK principle

(1_i) M knows that if T is i+1 then ~(2_i).

(2_i) M knows that T is not i.

(3_i) M knows that (2_i).

(L) Luminosity

(KK) For any pertinent p, if M knows p then M knows that M knows p.

[S] Simple Creature Objection to (KK)

[I] Iterated Pertinence Objection to (KK)

(C) If p and all members of X are pertinent, p is a logical consequence of X, M knows each member of X, then M knows p.

[GC] General Closure: We know consequences of what we know.

[IC] Intuitive Closure: Knowing p₁, …, p_n, competently deducing q, and thereby coming to believe q is in general a way of coming to know q. [Deduction is a way of extending knowledge.]

[N] Nozickean Counterfactual Objection to (IC)

[P] Probability Objection to (IC)

[V] Vagueness Objection to (IC)

[R] Reflective Equilibrium

Defending Intuitive Closure

[IC] Intuitive Closure: Knowing p₁, …, p_n, competently deducing q, and thereby coming to believe q is in general a way of coming to know q. [Deduction is a way of extending knowledge.]

[N] Nozickean Counterfactual Objection to (IC) [cf. Ch.7]

[P] Probability Objection to (IC)

Ø  No probability short of 1 turns true belief into knowledge. [cf. Ch.10]

Ø  Reworking (C) to apply to single-premise inferences.

[V] Vagueness Objection to (IC)

Ø  Not vagueness in “know” but limits on M’s eyesight and M’s knowledge of it.

Seductiveness of the KK principle

One cannot knowingly identify a particular counterexample to KK in first-person present tense. [K(Kp ∧ ~KKp) → KKp → ~K~KKp → ~K(Kp ∧ ~KKp)] [Ch.12]

Application: The “Glimpse” Paradox [Ch. 6]

(p) “The exam will be on the penultimate day.”

(l) “The exam will on the last day.”

() “If the exam is on the penultimate day, then the pupils do not know now that it will not be on the last day.”

5.2 KK^k & KK^ω

One knows⁰ p iff p. One knows^k+1 p iff one knows^k that one knows p. (k ∈ N)

(KK^k) is false.

One knows^ω p iff one knows^k p for every k. (k ∈ N)

(KK^ω) is a logical truth [challenge to the generalized argument against luminosity]

Ø  Infinite many premises & attenuated luminosity [not a satisfying response]

Ø  Knowing^ω may fail the gradualness requirement.

5.4 [Belief about the accuracy of one’s own estimates needs margin for error.]

5.5 [Belief about the accuracy of others’ estimates needs margin for error.]

5.3 Reliability/Safety/Margin-of-Error

[O] Obtain; [SO] Safely Obtain; [B] Believe; [K] Know; [FB] Falsely Believe;

[IPK] In a Position to Know; [SfE] Safety from Error; [MoE] Margin of Error;

[c] a small positive real number; [u] any non-negative real number

(L) For all α, if O^α(C), then IPK^α(O(C)). [C is a luminous condition.]

(7) For all α and β, if v(α) = v(β), then O^α(C) iff O^β(C). [C’s obtainment depends only on the value of the relevant parameter.]

(8) For all α and u, if |u – v(α)| < c, and B^α(O(C)), then for some β close to α, v(β) = u and B^β(O(C)). [One’s belief is not perfectly discriminating with respect to the underlying parameter.]

(9) For all α and β, if β is close to α, and K^α(O(C)), then ~FB^β(O(C)). [Connection between knowledge and SfE]

(10) For all α, if IPK^α(O(C)), then for some β, v(α) = v(β) and K^β(O(C)). [Def. IPK: One is in a position to know something determined by the value of a parameter only if one can know without changing the value of the parameter.] [(9) + (10) → “SfE Assumption”: IPK only if SfE]

(11) For all α, if K^α(O(C)), then B^α(O(C)). [Knowledge implies belief.]

(12) For all α and β, if |v(α) – v(β)| < c, then O^α(C) iff O^β(C).

(13) For all u, for some α, v(α) = u. [The parameter varies continuously.]

(14) For all α and β, O^α(C) iff O^β(C). [Triviality: C obtains in all cases or none.]

(15) For all α and β, if |v(α) – v(β)| < c, and IPK^α(O(C)), then O^β(C). [MoE Principle]