Deriving Knowledge from Belief Joe Halpern , Dov Samet , Ella Segev

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. Sknows that p iff (i) p is true; (ii) Sbelieves that p; (iii) S is justified in believing that p.

(i) p is true; (ii) Sbelieves that p; (iii) S is justified in believing that p. Gettier’s refutation of justification It was a rented car. p = Jones owns a Ford (i) p is false; (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 never saw Jones’ back Smith knows q ?

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-an-hour a day. Why, sometimes I've believed as many as six impossible things before breakfast.’

Hintikka: Knowledge and Belief as Modalities (K)K(p → q) → (Kp → Kq) (D) Kp →  K  p (4)Kp → KKp (5)  Kp → K  Kp (K) B(p → q) → (Bp → Bq) (D) Bp →  B p (4) Bp → BBp (5)  Bp → B  Bp In a probabilistic modal logic, if Bp says that the agent assigns probability 1 to p, K, D, 4, and 5 hold. (T) K(p) → p (L1) Kp → Bp (L2) Bp → KBp Positive introspection S5-knowledge Negative introspection

The Tripartite Theory of Knowledge Why isn’t correct-belief knowledge? Sknows that p iff (i) p is true; (ii) Sbelieves 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 to true belief Hintikka: Knowledge and Belief as Modalities S4-knowledge (K)K(p → q) → (Kp → Kq) (D) Kp →  K  p (4)Kp → KKp (5)  Kp → K  Kp (K) B(p → q) → (Bp → Bq) (D) Bp →  B p (4) Bp → BBp (5)  Bp → B  Bp (T) K(p) → p ( ) Kp ↔ Bp p (L1) Kp → Bp (L2) Bp → KBp Which axioms do the axioms on the left imply? T rue elief B K, D, 4, 5, and TB imply K, D, 4, T, L1, L2

Bad news (for S4 proponents) Also, S4 + (.2)S4.2 Also, S4 + (SB)(Stalnaker) (.2)  K  Kp → K  K  p (SB) Bp → BKp S4-knowledge +linked KD45-beliefcan be reduced toKD45-beliefby defining knowledge as true belief

 negative introspection false belief ↨ True beliefand negative introspection True belief: Bp p Bp p False belief : A theorem of KD45-belief + True belief

Epistemic luck (Bp p) is an axiom i.e.Bp → p is an axiom Why is true belief considered epistemic luck? logically Because epistemic misfortune (false belief) is possible. KD45 belief for which epistemic misfortune is logicallyimpossible “Where there is good luck there is misfortune” (Ancient Chinese proverb) is S5-knowledge.

Semantics of KD45-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 KD45-belief . . . Accessibility relation A frame for KD45 (D) - serial . . (4) - transitive . . . (5) - Euclidian … adding S5-knowledge… . . . Any frame for KD45-beliefcan be extended in a unique way to a frame for the linked S5-knowledge. . . . . .

…uniquely extend… K1, K2 S5 B KD45 B is linked to K1, K2 then K1p K2p ┬ ↨

…uniquely extend… Can S5knowledgebe reduced to KD45 belief? No! Is there a formula X(p,…)in terms of B only such that KD45 +Kp X(p,…)  S5 +Link ↨

Algebras of subsets with operators (Ω,A, O1, O1 …) 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.

(Ω,A,B) (Ω,A,K) KD45 algebra S5algebra (Ω,A,B,K) KD45 + S5+ link Algebras of subsets with operators (0) B(Ω) = Ω • (0)K(Ω) =Ω • (K)K(E → F) → (K(E) → K(F)) = Ω •  K(E) → K  K(E) = Ω • (T) K(E) → E = Ω (K) B(E → F) → (B(E) → B(F)) = Ω (D) B(E) →  B(E)= Ω (4) B(E) → B(B(E)) = Ω (5)  B(E) → B  B(E) = Ω A → Bmeans A  B (L1) K(E) → B(E) = Ω (L2) B(E) → K(B(E)) = Ω

(Ω,A,B) (Ω,A,K) KD45 algebra S5algebra (Ω,A,B,K) KD45 + S5+ link Algebras of subsets with operators Can every KD45 algebra, (Ω,A,B)be extended to a KD45 + S5+ link algebra(Ω,A,B,K)? (0) B(Ω) = Ω • (0)K(Ω) = Ω • (K)K(E → F) → (K(E) → K(F)) = Ω •  K(E) → K  K(E) = Ω • (T) K(E) → E = Ω (K) B(E → F) → (B(E) → B(F)) = Ω (D) B(E) →  B(E)= Ω (4) B(E) → B(B(E)) = Ω (5)  B(E) → B  B(E) = Ω No! (L1) K(E) → B(E) = Ω (L2) 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 believesall events that contain . { E  {o}; E B B(E) = E \ {o}; E B If B is a nonprincipal ultrafilter in A, then (Ω,A,B) is a KD45 algebra.

{ E  {o}; E B B(E) = E \ {o}; E B L2B(E) → K(B(E)) = Ωi.e.B(E)  K(B(E)) L1 T (5) Suppose (Ω,A,B) is extended to (Ω,A,B,K). For each o , {}  Band therefore, {} = B({}) K(Ω\{o}) =K({}) K(B({})) Thus, Ω\{o} K(Ω\{o}) Ω\{o} {o} =K(Ω\{o})  K(K(Ω\{o})) B(K(Ω\{o})) = B({o}) A contradiction

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

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

And what about justification? As long as the axioms of J are consistent and do not involve B. KD45 Axioms of JKp p Bp Jp  S5 +Link ↨

An open problem L is a logic for the modalities M1, M2 ,…, Mn . L+ is a logic for the modalities M1, M2 ,…, Mn , 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 ofM1, M2 ,…, Mn such that L + (Mn+1pX) imply L+ ? ↨

Deriving Knowledge from Belief Joe Halpern , Dov Samet , Ella Segev