Deriving Knowledge from Belief Joe Halpern Dov Samet

Deriving Knowledge from Belief Joe Halpern, Dov Samet, Ella Segev www. tau. ac. il/~samet

The Tripartite Theory of Knowledge THEAETETUS: That is a distinction, Socrates, which I have heard made by some one else, but I had forgotten it. He said that true opinion, combined with reason, was knowledge, but that the opinion which had no reason was out of the sphere of knowledge. S knows that p iff (i) p is true; (ii) S believes that p; (iii) S is justified in believing that p.

Gettier’s refutation of justification p = Jones owns a Ford (i) p is false; It was a rented car. (ii) Smith believes that p; (iii) Smith is justified in believing that p. Smith saw Jones driving a Ford Smith does not know p. q = Jones owns a Ford or he has a birth mark on his back (i) q is true; (ii) Smith believes that q; (iii) Smith is justified in believing that q. Smith knows q ? Smith never saw Jones’ back

Stalnaker: “On the logic of knowledge and belief” Formal epistemology …. of knowledge and belief in the possible world framework, began with Jaakko Hintikka’s book knowledge and belief published in 1962. Edmund Gettier’s (1963) classic refutation of the Justified True Belief analysis of knowledge … immediately spawned an epistemological industry. . . There was little contact between these two very different epistemological projects.

Hintikka: Knowledge and Belief as Modalities (K) B(p → q) → (Bp → Bq)

‘I can't believe that!’ said Alice. ‘Can't you? ’ the Queen said in a pitying tone. ‘Try again: draw a long breath, and shut your eyes. ’ Alice laughed. ‘There's no use trying, ’ she said: ‘one can't believe impossible things. ’ ‘I daresay you haven't had much practice, ’ said the Queen. ‘When I was your age, I always did it for half-anhour a day. Why, sometimes I've believed as many as six impossible things before breakfast. ’

Hintikka: Knowledge and Belief as Modalities (K) B(p → q) → (Bp → Bq) (K) K(p → q) → (Kp → Kq) (D) Bp → B p (D) Kp → K p (4) Bp → BBp (4) Kp → KKp (5) Bp → B Bp (5) Kp → K Kp Positive introspection S 5 -knowledge Negative introspection (T) K(p) → p In a probabilistic modal logic, if Bp says that the (L 1) Kp → Bp agent assigns probability (L 2) Bp → KBp 1 to p, K, D, 4, and 5 hold.

Why isn’t correct-belief knowledge? The Tripartite Theory of Knowledge S knows that p iff (i) p is true; (ii) S believes that p; (iii) S is justified in believing that p. Because a belief's turning out to be true is epistemic luck.

Reduction of knowledge Hintikka: to true belief Knowledge and Belief as Modalities S 4 -knowledge (K) B(p → q) → (Bp → Bq) (K) K(p → q) → (Kp → Kq) (D) Bp → B p (D) Kp → K p (4) Bp → BBp (4) Kp → KKp (5) Bp → B Bp (5) Kp → K Kp ( ) Kp ↔ Bp p (T) K(p) → p Which axioms do the True elief B axioms on the left imply? (L 1) Kp → Bp (L 2) Bp → KBp K, D, 4, 5, and TB imply K, D, 4, T, L 1, L 2

Bad news (for S 4 proponents) Also, S 4 + (. 2) S 4. 2 Also, S 4 + (SB) (Stalnaker) (. 2) K Kp → K K p (SB) Bp → BKp S 4 -knowledge + linked KD 45 -belief can be reduced to KD 45 -belief by defining knowledge as true belief

True belief and negative introspection True belief: Bp p False belief : Bp p A theorem of KD 45 -belief + True belief ↨ negative introspection false belief

Epistemic luck (Bp p) is an axiom Why is true belief considered epistemic luck? i. e. Bp → p is an axiom logically Because epistemic misfortune (false belief) is possible. KD 45 belief for which “Where there is good epistemic misfortune is logically impossible luck there is misfortune” is S 5 -knowledge. (Ancient Chinese proverb)

Semantics of KD 45 -belief Aumann and Heifetz, (2002), “Incomplete Information” in Handbook of Game theory, Ed. Aumann and Hart, “In the formalism of Sections 2 and 6, (describing the semantics of modal probabilistic belief) the concept of knowledge plays no explicit role. However, this concept can be derived from that formalism. ”

Semantics of KD 45 -belief Accessibility relation (D) - serial (4) - transitive (5) - Euclidian . . . A frame for KD 45 . . . … adding S 5 -knowledge… Any frame for KD 45 -belief can be extended in a unique way to a frame for the linked S 5 -knowledge. . .

…uniquely extend… K 1, K 2 B B is linked to K 1, K 2 KD 45 K 1 p K 2 p ↨ ┬ then S 5

…uniquely extend… Can S 5 knowledge be reduced to KD 45 belief? Is there a formula X(p, …) in terms of B only such that No! ↨ KD 45 + Kp X(p, …) S 5 +Link

Algebras of subsets with operators (Ω, A , O 1 …) Ω – set of possible worlds A – algebra of subsets of Ω (called events) Oi : A → A By Jónsson-Tarksi’s theorem, “of subsets” is w. l. o. g.

Algebras of subsets with operators (Ω, A , B) (Ω, A , K) KD 45 algebra S 5 algebra (0) B(Ω) = Ω (K) B(E → F) → (B(E) → B(F)) = Ω (0) K(Ω) = Ω (D) B(E) → B( E) = Ω (5) K(E) → K K(E) = Ω (4) B(E) → B(B(E)) = Ω (T) K(E) → E = Ω (K) K(E → F) → (K(E) → K(F)) = Ω (5) B(E) → B B(E) = Ω (Ω, A , B, K) A → B means A B KD 45 + S 5 + link (L 1) K(E) → B(E) = Ω (L 2) B(E) → K(B(E)) = Ω

Algebras of subsets with operators (Ω, A , B) (Ω, A , K) Can every KD 45 algebra, KD 45 algebra S 5 algebra (Ω, A , B) (0) K(Ω) = Ω be extended to a (K) K(E → F) → (K(E) → K(F)) = Ω (D) B(E) → B( E) = Ω (5) K(E) → K K(E) = Ω KD 45 + S 5 + link algebra (4) B(E) → B(B(E)) = Ω (T) K(E) → E = Ω (Ω, A , B, K)? (0) B(Ω) = Ω (K) B(E → F) → (B(E) → B(F)) = Ω (5) B(E) → B B(E) = Ω (Ω, A , B, K) No! KD 45 + S 5 + link (L 1) K(E) → B(E) = Ω (L 2) B(E) → K(B(E)) = Ω

Ω – an infinite set of possible worlds A - an algebra of events. B - a subset of A of “big” events. At o the agent believes all big events. At each o the agent believes all events that contain . B(E) = { E { o}; E B E { o}; E B If B is a nonprincipal ultrafilter in A, then (Ω, A , B) is a KD 45 algebra.

L 1 (5) L 2 B(E) → K(B(E)) = Ω T i. e. B(E) K(B(E)) Suppose (Ω, A , B) is extended to (Ω, A , B, K). For each o , { } B and therefore, { } = B({ }) K(B({ })) = K({ }) K(Ω{ o}) B E { }; E o Thus, B(E) = { E {o E Bo} Ω{ o} K(Ω{ })o}; Ω{ { o} = K(Ω{ o}) K( K(Ω{ o})) A contradiction B( K(Ω{ o})) = B({ o})

Ω = {0, 1, 2, …} Ω A = 2 Ex. 1 B – a non-principal ultrafilter on Ω Ω = {0, 1, 2, …} Ex. 2 A – the algebra generated by finite sets B – all co-finite sets Ω = [0, 1] Ex. 3 A – the σ-algebra generated by countable sets B – all co-countable sets

Define for each in [0, 1] μ (E) = { 1; B(E) 0; B(E) Then μ is a σ-additive measure on A , Ω = [0, 1] (E) = 1} and B(E) = { | μ A – the σ-algebra generated by countable sets Probabilistic certainty B – all co-countable sets can be incompatible with knowledge.

And what about justification? As long as the axioms of J are consistent and do not involve B. ↨ KD 45 Axioms of J Kp p Bp Jp S 5 +Link

An open problem L is a logic for the modalities M 1, M 2 , …, Mn. L+ is a logic for the modalities M 1, M 2 , …, Mn+1. Every L-algebra can be extended in a unique way to an L+ algebra. ↨ Does this imply that there is a formula X in terms of M 1, M 2 , …, Mn such that L + (Mn+1 p X) imply L+ ?