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Inbal.6? %f     /pn  ` ` ̙33` 333MMM` ff3333f` f` f` 3>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>> &(    6" "P " T Click to edit Master title style! !$  0X" " " RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S  0" "`` " >*  0 " "`  " @*@  0x" "` ` "H@___PPT9"@ j*$" Z  Bd޽h ? ̙33 Default Design 0 P(  P P 0` 0   V*   P 0Ld C X0  X*  d P c $ ?<   P 0xh H .*  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S P 6l    V*  F P 6m C X` H@___PPT9"@ x*.&   H P 0re"f ? ̙3380___PPT10.OPO >6d(  d d 0Ȥ 0   D*  d 0 C X0  F*  d 6    D* B d 6\ C X` H@___PPT9"@ t*.&   H d 0re"f ? ̙3380___PPT10.O K0  $8(  $ $  `챀f> L!?IQ  DDeriving Knowledge from Belief Joe Halpern, Dov Samet, Ella SegevEAC EE EE E,%R $  `fԔL!?"+ 0S#,$D   0 www.tau.ac.il/~samet6(2g,cH $ 0޽h ? ̙33___PPT10i.v J +D=' = @B +&(   (  &  < ?"6@ NNN?N<g,$ 0 HTHEAETETUS: 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.0 2 ea  c $|<T<$ 0   T"The Tripartite Theory of Knowledge##(  <KKz?"6@`NNN?N_,$D  0m  3 r 3?3ԔgggAD[X8c?"<?@@`NNN?N B ,$D 0 SqS knows that p iff (i) p is true; (ii) S believes that p; (iii) S is justified in believing that p.r 2aaeaaaeee aeeeee aee eeea`H  0޽h ? ̙33___PPT10.~(+WD' = @B Dx' = @BA?%,( < +O%,( < +D@' =%(%(D' =%(Dp' =A@BB/BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =-g6B fade*<3<*pD' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =-g6B fade*<3<*pD' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =-g6B fade*<3<*,D' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*,H%(D' =-g6B fade*<3<*,HD' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*Hr%(D' =-g6B fade*<3<*Hr++0+ ++0+ ++0+ +PK0 0 |(  |< | B82GH_?"6@ NNN?NZ!w ,$D  0 XSmith saw Jones driving a Ford(2a | S 76   JGettier s refutation of justification&&( _ | H9 ?"6@ NNN?N% ,$ 0 up = Jones owns a FordD 2eae | B C ?"6@ NNN?Nȵz  Z(i) p is true; (ii) S believes that p; (iii) S is justified in believing that p.r[ 2c(g(g(g( c(g(g(g(g(g( c(g(g( g(g(g(c(' | <xP ?"6@ NNN?N% ,$ 0 IQ (ii) Smith believes that p; (iii) Smith is justified in believing that p.R 2 aeeee a e eeea  | H4\ ?"6@ NNN?N) aI ,$  0 8q = Jones owns a Ford or he has a birth mark on his backD9 2ea5e  | <|a ?"6@ NNN?Np ,$   0 0\ Smith does not know p. 2f aaaea  | <m ?"6@ NNN?Nu  G,$  0 b(i) q is true; (ii) Smith believes that q; (iii) Smith is justified in believing that q.(c 2aeee aeeee a e eeea4 | ByGaHU?"6@ NNN?NQ,$D  0 P It was a rented car.(2aB | BDQ)?"0@NNN?N F3 ,$D   0 | <} ?"6@ NNN?Ni,$  0 "\ Smith knows q ?h 2f aeat | <( ?"6@ NNN?N&9,$ 0 (i) p is false;h 2aeeeaT | BĊG̤H]?"6@ NNN?N N ,$D  0 p6Smith never saw Jones back(2aH | 0޽h ??0||| ̙3399___PPT109.μR?+JLDp7' = @B D+7' = @BA?%,( < +O%,( < +D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*|%(D' =-g6B fade*<3<*|D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*|%(D' =-g6B fade*<3<*|D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*|$%(D' =-g6B fade*<3<*|$D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*|$R%(D' =-g6B fade*<3<*|$RD_' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*|%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*|D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*|D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*|%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*|D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*|D_' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*|%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*|D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*|D_' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* |%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* |D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* |D\' =%(D' =%(D' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*|%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*|D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*|D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* |%(D' =-g6B fade*<3<* |D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* |%(D' =-g6B fade*<3<* |D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* |5%(D' =-g6B fade*<3<* |5D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* |5c%(D' =-g6B fade*<3<* |5cD_' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*|%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*|D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*|D_' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*|%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*|D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*|+0+0+| ++0+| ++0+| ++0+ | ++0+ | ++0+ | ++0+| ++0+| ++0+| ++0+| +hK0   ~ (  p  Bd fԔ?"6@ NNN?N0 |$D 0H@___PPT9"@ <4Formal epistemology & . of knowledge and belief in the possible world framework, began with Jaakko Hintikka s book knowledge and belief published in 1962.  2[eeeeeefee8[ .   NԶ ?"6@ NNN?NmT bStalnaker:  On the logic of knowledge and belief j2 2 c g c $c c  )  B fԔ?"6@ NNN?N1 |$D 0H@___PPT9"@ $Edmund Gettier s (1963) classic refutation of the Justified True Belief analysis of knowledge & immediately spawned an epistemological industry...p 2eeefyee&    B fԔ?"6@ NNN?N 7e= |$D 0H@___PPT9"@ TThere was little contact between these two very different epistemological projects. .U(2TaaT H  0޽h ? ̙33  ___PPT10 .5,+D ' = @B D ' = @BA?%,( < +O%,( < +D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D ' =%(D' =%(Dp' =A@BB*BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*++0+ ++0+ ++0+ + K0  o(  @  << ?"6@ NNN?NZ= ,$ 0 b6(K) B(p ! q) ! (Bp ! Bq)  2aeee,  6 Z* -Hintikka: Knowledge and Belief as Modalities&.(&(&H  0޽h ? ̙33z___PPT10Z.L+JD' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*+8+0+ + K0 @ &(  ^  c 6Aglass05an6   < ?"6@ NNN?N\i 7 Z 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. h 2eeee e#e4eee eg ffg ffg ffg *eeeH  0޽h ? ̙33___PPT10i.Uv2+D=' dX|= @B +S\  K0   (    H GwH3?"6@ NNN?N7 X,$D  0 lIn a probabilistic modal logic, if Bp says that the agent assigns probability 1 to p, K, D, 4, and 5 hold. m(2#ae)aeaeaeaeaeaea  <p ?"6@ NNN?NZ= ,$ 0 (K) B(p ! q) ! (Bp ! Bq) (D) Bp ! ~ B ~ p (4) Bp ! BBp (5) ~ Bp ! B ~ BpU 2aeaeeeaeaeeaeaeaeaeeaeaeee>#  6 Z* -Hintikka: Knowledge and Belief as Modalities&.(&(&\   <` ?"6@ NNN?N D ,$   0 ~(K) K(p ! q) ! (Kp ! Kq) (D) Kp ! ~ K ~ p (4) Kp ! KKp (5) ~ Kp ! K ~ Kp U 2aeaeeeeaeaeeeeeaeaeeeeee    @`  NP@ ?"6@ NNN?N0 {P ,$  0  (T) K(p) ! p | 2aeae4  BIG'HÝf3?"6@ NNN?NiH ,$D  0 PPositive introspection(2a4  BMGHf3?"6@ NNN?N% U,$D  0 PNegative introspection(2a4  HQGHq?"6@ NNN?N=(,$D  0 J S5-knowledge" (2 e2  B 3fo?"6@ NNN?N Vr ,$D   0  NU ?"6@ NNN?N  ,$ 0 :(L1) Kp ! Bp (L2) Bp ! KBp. 2a,H  0޽h ?O@ ̙33EE___PPT10D.L+ tDB' = @B DA' = @BA?%,( < +O%,( < +D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-g6B fade*<3<*0D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0@%(D' =-g6B fade*<3<*0@D_' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =%(D' =%(DP' =A@BB5BB0B%(D' =+4 8?PCB ppt_wBCBB*Y3>B ppt_w<*D' =+4 8?PCB ppt_hBCBB*Y3>B ppt_h<*D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*@U%(D' =-g6B fade*<3<*@UD_' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =%(D' =%(DP' =A@BB5BB0B%(D' =+4 8?PCB ppt_wBCBB*Y3>B ppt_w<*D' =+4 8?PCB ppt_hBCBB*Y3>B ppt_h<*D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(D' =%(DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*D' =%(D' =%(DP' =A@BB5BB0B%(D' =+4 8?PCB ppt_wBCBB*Y3>B ppt_w<*D' =+4 8?PCB ppt_hBCBB*Y3>B ppt_h<*D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(DS' =%(D' =A@BBBB0B%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* D ' =%(D' =%(Dp' =A@BB*BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*D&' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =%(Dt' =A@BBBB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =%(D' =%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D)' =4@BB BB%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D ' =%(D' =%(Dp' =A@BB*BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*++0+ ++0+ ++0+ ++0+  ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ +=   (|(  ( ( S <T   T"The Tripartite Theory of Knowledge##(m ( 3 r 3?3ԔgggAD[X8c?"<?@@`NNN?NB ,$@ 0 SqS knows that p iff (i) p is true; (ii) S believes that p; (iii) S is justified in believing that p.r 2aaeaaaeee aeeeee aee eeea`N  ( 6 dp4,$ 0 FWhy isn t correct-belief knowledge?R$(c( ( c ((  ( <l ?"6@ NNN?N; Mu,$  0 =Because a belief's turning out to be true is epistemic luck. ^>(2-g g0c0c - H ( 0޽h ? ̙3355___PPT105.~(+"~D}4' = @B D84' = @BA?%,( < +O%,( < +D ' =%(D ' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-g6B fade*<3<*(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-g6B fade*<3<*(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*(,%(D' =-g6B fade*<3<*(,Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*(,H%(D' =-g6B fade*<3<*(,HDh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*(Hr%(D' =-g6B fade*<3<*(HrDq' =%(D' =%(D' =A@BBBB0B%(D ' =+4 8?XCB ppt_xBCB ppt_xB*Y3>B ppt_x<*(D' =+4 8?`CB ppt_yBCB0-ppt_h/2B*Y3>B ppt_y<*(D' =1:Bhidden*o3>+B#style.visibility<*(%(D ' =%(D' =%(Dp' =A@BB/BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* (%(D' =-g6B fade*<3<* (D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* (D' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<* (D' =%(D' =%(DN' =4@BBBB%()))D' =+4 8?PCB ppt_wBCBB*Y3>B ppt_w<*(HrD' =+4 8?PCB ppt_hBCBB*Y3>B ppt_h<*(HrD ' =+ 4 8? CB ppt_xBC2B]ppt_x+-0.0500*(ppt_x*0.9511+(1-ppt_y)*0.3090)BCdB]ppt_x+-0.1000*(ppt_x*0.8090+(1-ppt_y)*0.5878)BCB]ppt_x+-0.1500*(ppt_x*0.5878+(1-ppt_y)*0.8090)BCB]ppt_x+-0.2000*(ppt_x*0.3090+(1-ppt_y)*0.9511)BCB_ppt_x+-0.2500*(ppt_x*-0.0000+(1-ppt_y)*1.0000)BC,B_ppt_x+-0.3000*(ppt_x*-0.3090+(1-ppt_y)*0.9511)BC^B_ppt_x+-0.3500*(ppt_x*-0.5878+(1-ppt_y)*0.8090)BCB_ppt_x+-0.4000*(ppt_x*-0.8090+(1-ppt_y)*0.5878)BCB_ppt_x+-0.4500*(ppt_x*-0.9511+(1-ppt_y)*0.3090)BCBappt_x+-0.5000*(ppt_x*-1.0000+(1-ppt_y)*-0.0000)BC&Bappt_x+-0.5500*(ppt_x*-0.9511+(1-ppt_y)*-0.3090)BCXBappt_x+-0.6000*(ppt_x*-0.8090+(1-ppt_y)*-0.5878)BCBappt_x+-0.6500*(ppt_x*-0.5878+(1-ppt_y)*-0.8090)BCBappt_x+-0.7000*(ppt_x*-0.3090+(1-ppt_y)*-0.9511)BCB_ppt_x+-0.7500*(ppt_x*0.0000+(1-ppt_y)*-1.0000)BC B_ppt_x+-0.8000*(ppt_x*0.3090+(1-ppt_y)*-0.9511)BCRB_ppt_x+-0.8500*(ppt_x*0.5878+(1-ppt_y)*-0.8090)BCB_ppt_x+-0.9000*(ppt_x*0.8090+(1-ppt_y)*-0.5878)BCB_ppt_x+-0.9500*(ppt_x*0.9511+(1-ppt_y)*-0.3090)BCB]ppt_x+-1.0000*(ppt_x*1.0000+(1-ppt_y)*0.0000)B*Y3>B ppt_x<*(HrD ' =+ 4 8? CB ppt_yBC2B]ppt_y+-0.0500*(ppt_x*0.3090-(1-ppt_y)*0.9511)BCdB]ppt_y+-0.1000*(ppt_x*0.5878-(1-ppt_y)*0.8090)BCB]ppt_y+-0.1500*(ppt_x*0.8090-(1-ppt_y)*0.5878)BCB]ppt_y+-0.2000*(ppt_x*0.9511-(1-ppt_y)*0.3090)BCB_ppt_y+-0.2500*(ppt_x*1.0000-(1-ppt_y)*-0.0000)BC,B_ppt_y+-0.3000*(ppt_x*0.9511-(1-ppt_y)*-0.3090)BC^B_ppt_y+-0.3500*(ppt_x*0.8090-(1-ppt_y)*-0.5878)BCB_ppt_y+-0.4000*(ppt_x*0.5878-(1-ppt_y)*-0.8090)BCB_ppt_y+-0.4500*(ppt_x*0.3090-(1-ppt_y)*-0.9511)BCBappt_y+-0.5000*(ppt_x*-0.0000-(1-ppt_y)*-1.0000)BC&Bappt_y+-0.5500*(ppt_x*-0.3090-(1-ppt_y)*-0.9511)BCXBappt_y+-0.6000*(ppt_x*-0.5878-(1-ppt_y)*-0.8090)BCBappt_y+-0.6500*(ppt_x*-0.8090-(1-ppt_y)*-0.5878)BCBappt_y+-0.7000*(ppt_x*-0.9511-(1-ppt_y)*-0.3090)BCB_ppt_y+-0.7500*(ppt_x*-1.0000-(1-ppt_y)*0.0000)BC B_ppt_y+-0.8000*(ppt_x*-0.9511-(1-ppt_y)*0.3090)BCRB_ppt_y+-0.8500*(ppt_x*-0.8090-(1-ppt_y)*0.5878)BCB_ppt_y+-0.9000*(ppt_x*-0.5878-(1-ppt_y)*0.8090)BCB_ppt_y+-0.9500*(ppt_x*-0.3090-(1-ppt_y)*0.9511)BCB]ppt_y+-1.0000*(ppt_x*0.0000-(1-ppt_y)*1.0000)B*Y3>B ppt_y<*(HrD' =1:Bhidden*o3>+B#style.visibility<*(Hr%(D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* (%(D' =-g6B fade*<3<* (++0+( ++0+( ++0+ ( ++0+ ( +Q K0  i(    < ?"6@ NNN?NZ=  (K) B(p ! q) ! (Bp ! Bq) (D) Bp ! ~ B ~ p (4) Bp ! BBp (5) ~ Bp ! B ~ BpU 2aeaeeeaeaeeaeaeaeaeeaeaeee>#  6 ,$  0 w%Reduction of knowledge to true belief6& (((:  < ?"6@ NNN?N D  (K) K(p ! q) ! (Kp ! Kq) (D) Kp ! ~ K ~ p (4) Kp ! KKp (5) ~ Kp ! K ~ Kp V 2aeaeeeeaeaeeeeaeaeaeeeeee    @`  N ?"6@ NNN?N0 {P   (T) K(p) ! p | 2aeaeD   <  ?"6@ NNN?N bm ,$ 0 f rue elief 2 a)   N` ?"6@ NNN?N  ,$ 0 9T 2a)   NH ?"6@ NNN?N  ,$ 0 9B 2a/   N$; ?"6@ NNN?N7 iMW ,$  0 ?( ) 2a   N= ?"6@ NNN?NB b ,$ 0 Kp ! Bp p 2eeeae   3 rlA 3?3ԔgggAD[X8c?"<?@@`NNN?N ,$D 0 -Which axioms do the axioms on the left imply?B.(2"aea  BG fo?"6@ NNN?N(,$D 0 /K, D, 4, 5, and TB imply K, D, 4, T, L1, L2T0 2e a ea  6@M V -Hintikka: Knowledge and Belief as Modalities&.(&(&2  B 3fo?"6@ NNN?Nq 6a,$D  04  HPGQHr?"6@ NNN?NQ ,$D  0 J S4-knowledge" (2 et  N\T ?"6@ NNN?N   :(L1) Kp ! Bp (L2) Bp ! KBp. 2a,H  0޽h ? ̙3344___PPT104.L+oJ6D 1' = @B D0' = @BA?%,( < +O%,( < +D' =%(D' =%(DP' =A@BB5BB0B%(D' =+4 8?PCB ppt_wBCBB*Y3>B ppt_w<*D' =+4 8?PCB ppt_hBCBB*Y3>B ppt_h<*D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D8 ' =%(D' =%(Dp' =A@BB*BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<* D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* DB ' =%(D ' =%(D>' =A@BB@BB0B%()?)?D' =.M7 BBBBB_M -1.94444E-6 -2.59259E-6 L -0.05555 -0.09815 *3>*B ppt_xB ppt_y=@0BBAApBB9㼐BI<* D<' =A@BB@BB0B%()?)?D' =.K7 BBBBB]M 1.94444E-6 -2.59259E-6 L -0.11615 -0.09838 *3>*B ppt_xB ppt_y=@0BBAApBB8nBI<* D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<* D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<* Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D6' =A@BB BB0B%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D]' =%(D' =%(D' =A@BBBB0B@L>%(D' =1:Bvisible*o3>+B#style.visibility<*%(D]' =%(D' =%(D' =A@BBBB0B@L>%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D"' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =%(Dp' =A@BBBB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*+H+0+ ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+ ++0+ ++0+ ++0+ +5K0 ?7   (    3 r~?3Ԕ___ s|9)?"<?@@`NNN?N k,$D 0 eS4-knowledge + linked KD45-belief can be reduced to KD45-belief by defining knowledge as true belieff(2 g c g c g g c g g c g c g   6} ) PBad news, ((   s <APullHair"`Y)y,$D 0a   N ?"6@ NNN?NJH,$ 0 q(for S4 proponents)B 2(g( (   HGNH?"6@ NNN?N:,$D  0 Also, S4 + (.2) S4.2z(2aeaeae  HpGHYH?"6@ NNN?Nv L) ,$D 0 4(.2) ~ K ~ Kp ! K ~ K ~ p $(2aeaeeeeea   HtGH ?"6@ NNN?Nj,$D  0 Also, S4 + (SB) (Stalnaker)z(2aeaea e   HخGH)?"6@ NNN?N4! ,$D  0 (SB) Bp ! BKp (2aeaeea H  0޽h ?O@  ̙33Y&Q&___PPT101&.M0+LDE$' = @B D$' = @BA?%,( < +O%,( < +D' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwipe(up)*<3<* D6' =%(D' =%(D' =A@BB5BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*D' =%(D' =%(DP' =A@BB5BB0B%(D' =+4 8?PCB ppt_wBCBB*Y3>B ppt_w<*D' =+4 8?PCB ppt_hBCBB*Y3>B ppt_h<*D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D6' =%(D' =%(D' =A@BB5BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*D' =%(D' =%(DP' =A@BB5BB0B%(D' =+4 8?PCB ppt_wBCBB*Y3>B ppt_w<*D' =+4 8?PCB ppt_hBCBB*Y3>B ppt_h<*D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(++0+ ++0+  ++0+  ++0+ ++0+ ++0+ ++0+ ++0+ +$K0 b Z 0   (  M  < ?"6@ NNN?N# )True belief and negative introspection ^*(2 g$c$g$c$  < ?"6@ NNN?Nv0% ,$ 0 <True belief: Bp p z 2aeaea$  < ?"6@ NNN?N8(( X,$ 0 FFalse belief : 2a  N ?"6@ NNN?N. N,$ 0 Bp ~p  2eaeab   N$ ?"6@ NNN?N 9B8 ,$ 0 r&A theorem of KD45-belief + True belief0' 2 aegz (   /  ,$D 0-  C xH 3?3ԔgggAD[X8c?"<?@@`NNN?N(  9  2aN     *   B ?"6@ NNN?N  r0~ negative introspection&(2   ZZ\ ?"6@ NNN?N%   >!  2   B| ?"6@ NNN?N  J false belief" 2 g H  0޽h ? ̙33___PPT10.@<+CD' = @B Dg' = @BA?%,( < +O%,( < +D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D ' =%(D' =%(Dp' =A@BB*BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* D' =%(D' =%(Dc' =4@BB*BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*++0+ ++0+ ++0+ ++0+  +HK0 P  (     3 r 3?3ԔgggAD[X8c?"<?@@`NNN?N,$@ 0 EKD45 belief for which epistemic misfortune is logically impossible ~F(2 g c !c c c c    <d ?"6@ NNN?N`5,$ 0 A logically 2 a  H8GaH-?"6@ NNN?NT ,$D  0  Where there is good luck there is misfortune (Ancient Chinese proverb)`/(2 2a,eacg  <H ?"6@ NNN?N^d ,$ 0 -Why is true belief considered epistemic luck?@. 2a ea  <! ?"6@ NNN?Nj^_,$ 0 8Because epistemic misfortune (false belief) is possible.`9 2aaa ea  <T) ?"6@ NNN?NXR HEpistemic luck 2c(   ~0e0e    B}CDE4F @  o 8c8c     ?A)BCD|E||  '92T&DbT1f$}& @    # "0e@       @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN\,$D  0M   N- ?"6@ NNN?N a,$   0 ] S5-knowledge.4 2 g c ,   N2 ?"6@ NNN?N Dc,$  0 <is 2c H   H6G$H?"6@ NNN?N 3,$D   0 ^\~(Bp ~p) is an axiom i.e. Bp ! p is an axiom/(2eaeaee aH  0޽h ?/   ̙3311___PPT101.+YDh/' = @B D#/' = @BA?%,( < +O%,( < +D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D_' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =%(D' =%(DP' =A@BB5BB0B%(D' =+4 8?PCB ppt_wBCBB*Y3>B ppt_w<*D' =+4 8?PCB ppt_hBCBB*Y3>B ppt_h<*D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(Dg' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D ' =%(D' =%(Dp' =A@BB/BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<* D(' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D_' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* D_' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* D_' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* /%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* /D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* /D_' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* ++0+  ++0+  ++0+ ++0+ ++0+ ++0+ ++0+  ++0+  ++0+  +v  D< (  F  <[ ?"6@ NNN?NL8I,$ 0 hSemantics of KD45-belief4 2 c( g(  <a ?"6@ NNN?N]N ,$  0 * 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. x 2(c6cPcccc>  <k ?"6@ NNN?Nrex,$ 0 `Aumann and Heifetz, (2002),  Incomplete Information in Handbook of Game theory, Ed. Aumann and Hart, ti 2e aeaaaH  0޽h ? ̙33  ___PPT10 .Ipի+|D ' = @B D ' = @BA?%,( < +O%,( < +D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D'  =-o6Bwipe(up)*<3<*++0+ ++0+ ++0+ +F ?7 DIϢ(  2 F # ,0e0e    BCDEdF, @  o 8c8c     ?A)BCD|E||S0. D$xQ~/hB E)vHo79yR@        # "0e@       @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abu g ,$DR 02 G # ׅ80e0e    BCDEdF, @  o 8c8c     ?A)BCD|E||S0. D$xQ~/hB E)vHo79yR@        # "0e@       @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abD O ,$@Q 02  # hm0e0e    BCDEdF, @  o 8c8c     ?A)BCD|E||S0. D$xQ~/hB E)vHo79yR@        # "0e@       @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab,$@ 02  # hm0e0e    BCDEdF, @  o 8c8c     ?A)BCD|E||S0. D$xQ~/hB E)vHo79yR@        # "0e@       @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab} ,$@  0  <@~ ?"6@ NNN?NL8I hSemantics of KD45-belief4 2 c( g(F  <ā ?"6@ NNN?NMe,$ 0 h (D) - serial@ 2ae aJ  <T ?"6@ NNN?N3MeS,$ 0 l(4) - transitive@ 2aeaI  < ?"6@ NNN?NxM,$ 0 k(5) - Euclidian@ 2aea2  BP fo?"6@ NNN?N~!,$D 0 NAccessibility relation 2a@   B fo?"6@ NNN?N~F ,$D 0 \A frame for KD450 2 ae   <$ ?"6@ NNN?N  ,$ 0 ;. 2cH   < ?"6@ NNN?NNh i ; ,$  0 ;. 2cH   < ?"6@ NNN?N ,$ 0 ;. 2cH   < ?"6@ NNN?N2,$ 0 ?." 2gH  <ԧ ?"6@ NNN?N?` a ,,$ 0 ?." 2gH  < ?"6@ NNN?N,$ 0 ;. 2cH  <8 ?"6@ NNN?NMN,$ 0 ;. 2cH  <` ?"6@ NNN?N*PQ,$ 0 ?." 2gH,   0e0e    BCDEdF, A@  o 8c8c     ?A)BCD|E||S0. D$xQ~/hB E)vHo79yR@        # "0e@       @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abJ U r ,$@ 02  # \0e0e    BCDEdF, @  o 8c8c     ?A)BCD|E||S0. D$xQ~/hB E)vHo79yR@        # "0e@       @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab~ ,$D 02  # 50e0e    BCDEdF, @  o 8c8c     ?A)BCD|E||S0. D$xQ~/hB E)vHo79yR@        # "0e@       @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abeW,$D 0B  HDo?"0@NNN?N ,$@  0B  HDo?"0@NNN?N H,$@  0B  HDo?"0@NNN?N$ A,$@  0B  HDo?"0@NNN?N,$@ 0B  BDo?"0@NNN?N ,$@  0B  BDo?"0@NNN?N ,$D  0B  BDo?"0@NNN?N 7,$@  0B  BDo?"0@NNN?NT[,$@  0B  BDo?"0@NNN?N,$@  02 " # hm0e0e    BCDEdF, @  o 8c8c     ?A)BCD|E||S0. D$xQ~/hB E)vHo79yR@        # "0e@       @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abk &  v,$@  0 $ < ?"6@ NNN?N 0 1 ,$! 0 ;. 2cH % < ?"6@ NNN?N< ),$7 0 ;. 2cH & < ?"6@ NNN?N ,$" 0 ;. 2cH ' <l ?"6@ NNN?N  ,$# 0 ?." 2gH ( <x ?"6@ NNN?N-  ,$$ 0 ?." 2gH ) <H ?"6@ NNN?N ,$% 0 ;. 2cH * < ?"6@ NNN?N op,$& 0 ;. 2cH + <H ?"6@ NNN?N rs ,$' 0 ?." 2gH2 , # 0e0e    BCDEdF, @  o 8c8c     ?A)BCD|E||S0. D$xQ~/hB E)vHo79yR@        # "0e@       @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abnl w `,$@6 02 - # \0e0e    BCDEdF, @  o 8c8c     ?A)BCD|E||S0. D$xQ~/hB E)vHo79yR@        # "0e@       @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab  ,$@( 02 . # hm0e0e    BCDEdF, @  o 8c8c     ?A)BCD|E||S0. D$xQ~/hB E)vHo79yR@        # "0e@       @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab,$@) 02 / # 50e0e    BCDEdF, @  o 8c8c     ?A)BCD|E||S0. D$xQ~/hB E)vHo79yR@        # "0e@       @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abxy,$@* 0B 0 HDo?"0@NNN?N ,$@+ 0B 1 HDo?"0@NNN?N 6,$@, 0B 2 HDo?"0@NNN?N7  /,$@- 0B 3 HDo?"0@NNN?N,$@. 0B 4 BDo?"0@NNN?N ,$@/ 0B 5 BDo?"0@NNN?N . p ,$@0 0B 6 BDo?"0@NNN?N  %,$@1 0B 7 BDo?"0@NNN?N v}},$@2 0B 8 BDo?"0@NNN?N ,$@3 0B 9 BDo?"0@NNN?N|,$@  0B : BDo?"0@NNN?N~,$D  0B ; BDo?"0@NNN?N| y,$@4 0B < BDo?"0@NNN?N ,$@5 0B = HDo?"0@NNN?N~  ,$DH 0B > HDo?"0@NNN?N 0  p ,$DJ 0B ? HDo?"0@NNN?N  %,$@I 02 @ # 0e0e    BCDEdF, @  o 8c8c     ?A)BCD|E||S0. D$xQ~/hB E)vHo79yR@        # "0e@       @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab^ P ,$DK 0B A HDo?"0@NNN?N v}},$@L 0B B HDo?"0@NNN?N ,$@N 0B C HDo?"0@NNN?N| y,$@M 0B D HDo?"0@NNN?N ,$DO 0B E HDo?"0@NNN?N< < ,$@P 0$ H 3 r 3?3ԔgggAD[X8c?"<?@@`NNN?N ) ,$DS 0 dAny frame for KD45-belief can be extended in a unique way to a frame for the linked S5-knowledge.e(2a e aeffa eaf I <p ?"6@ NNN?N  ,$  0 ,& adding S5-knowledge& @ 2 a eaH  0޽h ? ̙33  ___PPT10 .XG;+lkDS' = @B D' = @BA?%,( < +O%,( < +DP' =%(D' =%(D' =A@BB*BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D ' =%(D' =%(Dp' =A@BB*BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<* Dh' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D ' =%(D ' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D(' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*DA' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*DY' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D(' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D ' =%(D< ' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*9%(D' =-o6Bwipe(up)*<3<*9D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*:%(D' =-o6Bwipe(up)*<3<*:D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*I%(D' =-s6Bwipe(left)*<3<*IDk' =%(Dj' =%(Dc' =4@BB/BB%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =-g6B fade*<3<*"D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*"D' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<*"Dp' =A@BB/BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-g6B fade*<3<*$D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<*$Dp' =A@BB/BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*&%(D' =-g6B fade*<3<*&D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*&D' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<*&Dp' =A@BB/BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =-g6B fade*<3<*'D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*'D' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<*'Dp' =A@BB/BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-g6B fade*<3<*(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*(D' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<*(Dp' =A@BB/BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =-g6B fade*<3<*)D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*)D' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<*)Dp' =A@BB/BB0B%(D' =1:Bvisible*o3>+B#style.visibility<**%(D' =-g6B fade*<3<**D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<**D' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<**Dp' =A@BB/BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*+%(D' =-g6B fade*<3<*+D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*+D' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<*+Dc' =4@BB/BB%(D' =1:Bvisible*o3>+B#style.visibility<*-%(D' =-g6B fade*<3<*-D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*-D' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<*-Dc' =4@BB/BB%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D' =-g6B fade*<3<*.D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*.D' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<*.Dc' =4@BB/BB%(D' =1:Bvisible*o3>+B#style.visibility<*/%(D' =-g6B fade*<3<*/D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*/D' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<*/Dc' =4@BB/BB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-g6B fade*<3<*0D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*0D' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<*0Dc' =4@BB/BB%(D' =1:Bvisible*o3>+B#style.visibility<*1%(D' =-g6B fade*<3<*1D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*1D' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<*1Dc' =4@BB/BB%(D' =1:Bvisible*o3>+B#style.visibility<*2%(D' =-g6B fade*<3<*2D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*2D' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<*2Dc' =4@BB/BB%(D' =1:Bvisible*o3>+B#style.visibility<*3%(D' =-g6B fade*<3<*3D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*3D' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<*3Dc' =4@BB/BB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-g6B fade*<3<*4D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*4D' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<*4Dc' =4@BB/BB%(D' =1:Bvisible*o3>+B#style.visibility<*5%(D' =-g6B fade*<3<*5D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*5D' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<*5Dc' =4@BB/BB%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D' =-g6B fade*<3<*6D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*6D' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<*6Dc' =4@BB/BB%(D' =1:Bvisible*o3>+B#style.visibility<*7%(D' =-g6B fade*<3<*7D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*7D' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<*7Dc' =4@BB/BB%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =-g6B fade*<3<*8D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*8D' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<*8Dc' =4@BB/BB%(D' =1:Bvisible*o3>+B#style.visibility<*;%(D' =-g6B fade*<3<*;D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*;D' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<*;Dc' =4@BB/BB%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =-g6B fade*<3<*<D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*<D' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<*<Dc' =4@BB/BB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =-g6B fade*<3<*,D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*,D' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<*,Dp' =A@BB/BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =-g6B fade*<3<*%D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*%D' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<*%D&' =%(D&' =%(D]' =4@BBBB%(D' =,54*3>#Bstroke.color=@BPB<*5D' =1:B true*a3>Bstroke.on<*5D]' =4@BBBB%(D' =,54*3>#Bstroke.color=@BPB<*4D' =1:B true*a3>Bstroke.on<*4D]' =4@BBBB%(D' =,54*3>#Bstroke.color=@BPB<*6D' =1:B true*a3>Bstroke.on<*6D]' =4@BBBB%(D' =,54*3>#Bstroke.color=@BPB<*0D' =1:B true*a3>Bstroke.on<*0D]' =4@BBBB%(D' =,54*3>#Bstroke.color=@BPB<*1D' =1:B true*a3>Bstroke.on<*1D]' =4@BBBB%(D' =,54*3>#Bstroke.color=@BPB<*2D' =1:B true*a3>Bstroke.on<*2D]' =4@BBBB%(D' =,54*3>#Bstroke.color=@BPB<*"D' =1:B true*a3>Bstroke.on<*"D]' =4@BBBB%(D' =,54*3>#Bstroke.color=@BPB<*-D' =1:B true*a3>Bstroke.on<*-D]' =4@BBBB%(D' =,54*3>#Bstroke.color=@BPB<*,D' =1:B true*a3>Bstroke.on<*,D]' =4@BBBB%(D' =,54*3>#Bstroke.color=@BPB<*7D' =1:B true*a3>Bstroke.on<*7D]' =4@BBBB%(D' =,54*3>#Bstroke.color=@BPB<*;D' =1:B true*a3>Bstroke.on<*;D]' =4@BBBB%(D' =,54*3>#Bstroke.color=@BPB<*<D' =1:B true*a3>Bstroke.on<*<D]' =4@BBBB%(D' =,54*3>#Bstroke.color=@BPB<*8D' =1:B true*a3>Bstroke.on<*8D]' =4@BBBB%(D' =,54*3>#Bstroke.color=@BPB<*3D' =1:B true*a3>Bstroke.on<*3D]' =4@BBBB%(D' =,54*3>#Bstroke.color=@BPB<*.D' =1:B true*a3>Bstroke.on<*.D]' =4@BBBB%(D' =,54*3>#Bstroke.color=@BPB<*/D' =1:B true*a3>Bstroke.on<*/D ' =%(D' =%(D[' =4@BB BB%()))D' =1:Bvisible*o3>+B#style.visibility<*=%(D' =-g6B fade*<3<*=D' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*?%(D' =-g6B fade*<3<*?D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*>%(D' =-g6B fade*<3<*>D' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =-g6B fade*<3<*@Dt ' =%(D ' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*A%(D' =-g6B fade*<3<*AD+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*C%(D' =-g6B fade*<3<*CD+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*B%(D' =-g6B fade*<3<*BD+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =-g6B fade*<3<*DDA' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*E%(D' =-g6B fade*<3<*ED+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*G%(D' =-g6B fade*<3<*GD+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*F%(D' =-g6B fade*<3<*FD ' =%(D' =%(Dp' =A@BB/BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-g6B fade*<3<*HD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*HD' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<*H++0+ ++0+ ++0+ ++0+ ++0+  ++0+  ++0+  ++0+  ++0+  ++0+ ++0+ ++0+ ++0+ ++0+$ ++0+% ++0+& ++0+' ++0+( ++0+) ++0+* ++0++ ++0+H ++0+I +.   ` $u (  $ $ Z` >L!?"6@ NNN?N "?,$@ 0 then K1p K2p(2c g o c g o c 4 $ TZi ?"6@ NNN?N  ,$  0 >!  2$> $  `b f>L!?"6@ NNN?N r ,$  0 <,% 2c$F $ Zpq f>L!?"6@ NNN?N ,$ 0 Jextend&  (2c$J $ Zt f>L!?"6@ NNN?Njx,$ 0 N& uniquely (2 c$  $ # lx >L!?"6@ NNN?NJ++I,$D  0 B KD45 ^ (2gcgc  $ # l >L!?"6@ NNN?N87,$D  0  K1, K2 S5 (2gocgocg  $ Z4 >L!?"6@ NNN?N J ,$D  0 B is linked to K1, K2(2gcgocgoH $ 0޽h ? ̙33  ___PPT10 . T+CD' = @B DC' = @BA?%,( < +O%,( < +D ' =%(D ' =%(D' =A@BB/BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-g6B fade*<3<*$D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<*$D' =A@BB/BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-g6B fade*<3<*$D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<*$D"' =%(D' =%(Dr' =A@BB BB0B%(D' =1:B 0.25*3>%Bstyle.opacity= B PPT<*$D' =-6B image0BIE*x3=4Bopacity: 0.25 BIE<*$D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* $%(D' =-o6Bwipe(up)*<3<* $D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* $%(D' =-o6Bwipe(up)*<3<* $D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* $%(D' =-o6Bwipe(up)*<3<* $Dh' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-g6B fade*<3<*$D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-g6B fade*<3<*$D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-g6B fade*<3<*$++0+$ ++0+$ ++0+$ ++0+$ ++0+$ ++0+$ ++0+ $ ++0+ $ ++0+ $ +i* bZp ( (  ( ( Z f>L!?"6@ NNN?N  Jextend&  (2c$ ( Z f>L!?"6@ NNN?Njx N& uniquely (2 c$  ( Z ԔL!?"6@ NNN?N ,$  0 nIs there a formula X(p,& ) in terms of B only such thatp8(2c g c g c "  ( C xL 3?3ԔgggAD[X8c?"<?@@`NNN?N/3,$D 0 -Can S5 knowledge be reduced to KD45 belief?.(2c g g g c g c   (  ` f>L!?"6@ NNN?N3 ' ,$ 0 ,KD45 + Kp X(p,& )(2g c g c gc,4  ( TZ ?"6@ NNN?N E,$ 0 >!  2$T ( Z f>L!?"6@ NNN?N ;,$ 0 XS5 +Link4 2g c : ( ZX f>L!?"6@ NNN?N 3 @,$  0 >  2(o ( # l?f>L!?"<?@@`NNN?Nto? ,$D   0 [No! GXH ( 0޽h ? ̙33___PPT10. T+sTD' = @B DV' = @BA?%,( < +O%,( < +D"' =%(D' =%(Dr' =A@BB BB0B%(D' =1:B 0.25*3>%Bstyle.opacity= B PPT<*(D' =-6B image0BIE*x3=4Bopacity: 0.25 BIE<*(D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* (%(D' =-g6B fade*<3<* (D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* (8%(D'  =-o6Bwipe(up)*<3<* (8D(' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* (%(D' =-g6B fade*<3<* (D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* (%(D' =-g6B fade*<3<* (D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D'  =-s6Bwipe(left)*<3<*(D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-g6B fade*<3<*(D_' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*(D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*(++0+( ++0+ ( ++0+ ( ++0+ ( ++0+ ( ++0+( ++0+( ++0+( +'    (    < ?"6@ NNN?NS1U  ^"Algebras of subsets with operators #(2#c(,  C x 3?3ԔgggAD[X8c?"<?@@`NNN?N`,$D 0 "(, A , O1, O1 & ) 2cc c c  <d ?"6@ NNN?Ni` ,$ 0 4  set of possible worldsP 2    <  ?"6@ NNN?N`),$ 0 VA  algebra of subsets of  (called events)2, 2c +   < ?"6@ NNN?N`a ,$ 0 Oi : A ! A  2ckc  c    H\GB9HYfo?"6@ NNN?N ` ,$D  0 lBy Jnsson-Tarksi s theorem,  of subsets is w.l.o.g. F7 2c gc,2  # l f>L!?"6@ NNN?N~ -,$D 0H  0޽h ? ̙33ZR___PPT102. 6k+HD' = @B Dq' = @BA?%,( < +O%,( < +D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*Da' =%(D ' =%(DY' =4@BB BB%()))D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(DP' =A@BB5BB0B%(D' =+4 8?PCB ppt_wBCBB*Y3>B ppt_w<*D' =+4 8?PCB ppt_hBCBB*Y3>B ppt_h<*D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(+P+0+ ++0+ ++0+  ++0+  ++0+ ++0+ +G F > ,(    <$4 ?"6@ NNN?N#  ,$ 0 6(K) B(E ! F) ! (B(E) ! B(F)) =  (D) B(E) ! ~ B(~ E) =  (4) B(E) ! B(B(E)) =  (5) ~ B(E) ! B ~ B(E) = t 2cgcg ggcgcggcgcgcgc g ggcgcgggg`!   N/ ?"6@ NNN?N,$ 0 0(0) B() =  2cgcgg4   <t` ?"6@ NNN?Ns  $ 0XP___PPT92*  (0) K() =  (K) K(E ! F) ! (K(E) ! K(F)) =  ~ K(E) ! K ~ K(E) =  (T) K(E) ! E = r3 2 2 2cgcggggcgcgg gggggcgcggv " @`   <T ?"6@ NNN?NS1U  ^"Algebras of subsets with operators #(2#c(   Nt ?"6@ NNN?N5]O,$  0 `(L1) K(E) ! B(E) =  (L2) B(E) ! K(B(E)) =  P1 2cg4  NPGd3Hpfo?"6@ NNN?N .#h,$D  0 &A ! B means ~ A B(2a aaal uOh 'uOh,$D 0<   T` o?"6@ NNN?NOh r KD45 algebraJ 2gcg $ N ?"6@ NNN?Nu< (, A , B) 2aaaaea l 3h (3h,$D 0P  TT o?"6@ NNN?N3h  S5 algebra` 2ggcg % N ?"6@ NNN?N^ (, A , K) 2aaaaea l %  )9  ,$D  0l   T o?"6@ NNN?N  KD45 + S5 + linkv 2gggcg & Nx ?"6@ NNN?N/% E  P(, A , B, K) 2aaaaeaeea H  0޽h ? ̙33R'J'___PPT10*'.@<+σYD%' = @B Di%' = @BA?%,( < +O%,( < +D' =%(D' =%(Dc' =4@BB/BB%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =-g6B fade*<3<*'D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*'D' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<*'D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =%(D' =%(DP' =A@BB5BB0B%(D' =+4 8?PCB ppt_wBCBB*Y3>B ppt_w<*D' =+4 8?PCB ppt_hBCBB*Y3>B ppt_h<*D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =%(Dc' =4@BB/BB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-g6B fade*<3<*(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*(D' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<*(D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(Dc' =4@BB/BB%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =-g6B fade*<3<*)D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*)D' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<*)D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* +P+0+ ++0+ ++0+ ++0+  ++0+ ++0+ +G K0 S#K# "(    < ?"6@ NNN?N#   6(K) B(E ! F) ! (B(E) ! B(F)) =  (D) B(E) ! ~ B(~ E) =  (4) B(E) ! B(B(E)) =  (5) ~ B(E) ! B ~ B(E) = t 2cgcg ggcgcggcgcgcgc g ggcgcgggg`!  N ?"6@ NNN?N 0(0) B() =  2cgcgg4   <" ?"6@ NNN?Ns  `XP___PPT92*  (0) K() =  (K) K(E ! F) ! (K(E) ! K(F)) =  ~ K(E) ! K ~ K(E) =  (T) K(E) ! E = Z3 2 2 2cgcggggcgcgg gggggcgcggv " @`  <J ?"6@ NNN?NS1U  ^"Algebras of subsets with operators #(2#c(  NR ?"6@ NNN?N5]O `(L1) K(E) ! B(E) =  (L2) B(E) ! K(B(E)) =  h1 2cg4pF uOh  uOh<   T` o?"6@ NNN?NOh r KD45 algebraJ 2gcg   N,* ?"6@ NNN?Nu< (, A , B) 2aaaaea F 3h   3hP   Tp o?"6@ NNN?N3h  S5 algebra` 2ggcg   Nw ?"6@ NNN?N^ (, A , K) 2aaaaea F %   9  l  TP o?"6@ NNN?N  KD45 + S5 + linkv 2gggcg  N ?"6@ NNN?N/% E  P(, A , B, K) 2aaaaeaeea o  # l̘?f>L!?"<?@@`NNN?N  ,$D   0 [No! GB  3 r 3?3ԔgggAD[X8c?"<?@@`NNN?NS ,$D   0 |Can every KD45 algebra, (, A , B) be extended to a KD45 + S5 + link algebra (, A , B, K)? bb(2 c g c    c c(c  c g c g g g c    c c(c  c g c g g c 48H  0޽h ? ̙33?#7#___PPT10#.@<+i7Dc!' = @B D!' = @BA?%,( < +O%,( < +Dq' =%(%(D' =%(Dr' =A@BB BB0B%(D' =1:B 0.25*3>%Bstyle.opacity= B PPT<*D' =-6B image0BIE*x3=4Bopacity: 0.25 BIE<*Dr' =A@BB BB0B%(D' =1:B 0.25*3>%Bstyle.opacity= B PPT<*D' =-6B image0BIE*x3=4Bopacity: 0.25 BIE<*Dr' =A@BB BB0B%(D' =1:B 0.25*3>%Bstyle.opacity= B PPT<*D' =-6B image0BIE*x3=4Bopacity: 0.25 BIE<*Dr' =A@BB BB0B%(D' =1:B 0.25*3>%Bstyle.opacity= B PPT<*D' =-6B image0BIE*x3=4Bopacity: 0.25 BIE<*Dr' =A@BB BB0B%(D' =1:B 0.25*3>%Bstyle.opacity= B PPT<*D' =-6B image0BIE*x3=4Bopacity: 0.25 BIE<*De' =4@BB BB%(D' =1:B 0.25*3>%Bstyle.opacity= B PPT<*D' =-6B image0BIE*x3=4Bopacity: 0.25 BIE<*De' =4@BB BB%(D' =1:B 0.25*3>%Bstyle.opacity= B PPT<* D' =-6B image0BIE*x3=4Bopacity: 0.25 BIE<* De' =4@BB BB%(D' =1:B 0.25*3>%Bstyle.opacity= B PPT<*D' =-6B image0BIE*x3=4Bopacity: 0.25 BIE<*Du' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<*D' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<*D_' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ +XF )!  (    < ?"6@ NNN?N_,$  0 PB - a subset of A of  big events. ) 2g 33c c c c g 33 c 0  <  ?"6@ NNN?Nl  ,$  0 RB(E) =0 2ea  <" ?"6@ NNN?N I ,$   0 9{ 2cP  <\  ?"6@ NNN?NT } ,$  0 E {wo}; r 2a  <A ?"6@ NNN?N K ,$ 0  E \ {wo}; r 2a\   NL!?"<?@@`NNN?N ,$D  0 If B is a nonprincipal ultrafilter in A, then (, A , B) is a KD45 algebra.lM(2c g 33c  c c    c c c  c g c g c B    Nh ?"6@ NNN?Nm,$  0 L  an infinite set of possible worldsf' 2   g &n  <o ?"6@ NNN?NU,$  0 A - an algebra of events. X 2c g 33c c   H 3Ԕ?"6@ NNN?N ,$D 0V  0޽h ?"` ̙33//___PPT10/.$^F+ATD-' = @B D,' = @BA?%,( < +O%,( < +D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =-s6Bwipe(left)*<3<*)D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<**%(D' =-g6B fade*<3<**D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<**h%(D' =-g6B fade*<3<**hDs' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D7' =%(D' =A@BBBB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D_' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*+h+0+ ++0+ ++0+ ++0+ ++0+ ++0+  ++0+  ++0+ ++0+ ++0+ ++0+ +ԅK0 ,,@'0,(     f| f>L!?"6@ NNN?N] ,$ 0 Suppose (, A , B) is extended to (, A , B, K). For each w wo , {w} B and therefore, {w} = B({w})g0F2 cc ccgccc ccgcgg cccc g 33o033cccgcc4 C  # l f>L!?"6@ NNN?N- t ,$ 0  K(B({w})) 2cgcgcc  # lD f>L!?"6@ NNN?N+ yr ,$  0 = K({w}) 2cgcc.  # l  f>L!?"6@ NNN?N*}q ,$  0  K(\{wo}) 2cgccc   ` f>L!?"6@ NNN?N  c ,$  0 $\{wo} K(\{wo})6 2cccgccc,   # l! f>L!?"6@ NNN?N- vt ,$  0  \{wo} 2ccc;  Z + f>L!?"6@ NNN?N j ,$  0 ?Thus, 2c  # lh1 f>L!?"6@ NNN?N  ,$ 0 ${wo} = ~ K(\{wo})D 2ccccgccc f  # l|> f>L!?"6@ NNN?N ;',$ 0 X" K(~ K(\{wo})) 2cgccc   # lK f>L!?"6@ NNN?N ,$ 0 x B(~ K(\{wo})) 2cgccgccc  # lV f>L!?"6@ NNN?N,$ 0 = B({wo}) 2cgccE  Zp^ >L!?"6@ NNN?NF=,$D  0 IA contradiction 2c58   '    < c ?"6@ NNN?Nl   RB(E) =0 2ea ! <$g ?"6@ NNN?N I  9{ 2cPX " <j ?"6@ NNN?N T }  E {wo}; r 2a` # <hq ?"6@ NNN?N  K   E \ {wo}; r 2a0 $ Nx ?"6@ NNN?N a   lE BH 2g 330 % Ns ?"6@ NNN?N v  lE BH 2g 33 & H 3Ԕ?"6@ NNN?N    `8GH 3oL!?"6@ NNN?N|L 6,$D  0 TL2 B(E) ! K(B(E)) =  i.e. B(E) K(B(E)) R+(2ccV   `G7bH`3oL!?"6@ NNN?N&B,$D  0 TT 6(2gcj   `G%H3oL!?"6@ NNN?N ,$D  0 h(5) H(2cgcA   `DGHU3oL!?"6@ NNN?N0 .,$D  0 ?L1  (2cV  0޽h ?"`O@ ̙33nXfX___PPT10FX.$^F+ w DS' = @B DuS' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BB@BB%()?)?D' =.O7 BBBBBaM 8.33333E-7 -4.81481E-6 L 8.33333E-7 -0.48217 *3>*B ppt_xB ppt_y=@0BBAApBBBv<*'D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*1%(D' =-g6B fade*<3<*1D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*1Z%(D' =-g6B fade*<3<*1ZD' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Zg%(D' =-g6B fade*<3<*ZgD' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D_' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =%(D' =%(DP' =A@BB5BB0B%(D' =+4 8?PCB ppt_wBCBB*Y3>B ppt_w<*D' =+4 8?PCB ppt_hBCBB*Y3>B ppt_h<*D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D ' =%(D' =%(Dp' =A@BB*BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D' =%(D' =%(DP' =A@BB5BB0B%(D' =+4 8?PCB ppt_wBCBB*Y3>B ppt_w<*D' =+4 8?PCB ppt_hBCBB*Y3>B ppt_h<*D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D_' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =%(D' =%(DP' =A@BB5BB0B%(D' =+4 8?PCB ppt_wBCBB*Y3>B ppt_w<*D' =+4 8?PCB ppt_hBCBB*Y3>B ppt_h<*D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D' =%(D' =%(DP' =A@BB5BB0B%(D' =+4 8?PCB ppt_wBCBB*Y3>B ppt_w<*D' =+4 8?PCB ppt_hBCBB*Y3>B ppt_h<*D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*+`+0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ +O= =5  (  B  B Ԕ?"6@ NNN?NN:>,$D 0 ^| = {0, 1, 2, & } A = 2 B  a non-principal ultrafilter on j?0F2c cc,,g 33"T H  B Ԕ?"6@ NNN?N :U ,$D 0 d = {0, 1, 2, & } A  the algebra generated by finite sets B  all co-finite setsrR0F2c$c$c,g$3333 ,92  B Ԕ?"6@ NNN?NE :,$D  0 N = [0, 1] A  the -algebra generated by countable sets B  all co-countable setsBT0F2c$c c$c,g$33B$;  Z f>L!?"6@ NNN?Nt|8,$ 0 ?Ex. 1 2c;  Z f>L!?"6@ NNN?Nd|8,$ 0 ?Ex. 2 2c;  Zd  f>L!?"6@ NNN?N |8 ,$  0 ?Ex. 3 2cH  0޽h ? ̙33..___PPT10..$^F+L@kD-' = @B D,' = @BA?%,( < +O%,( < +D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*?%(D' =-g6B fade*<3<*?D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*;%(D' =-g6B fade*<3<*;D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*;R%(D' =-g6B fade*<3<*;RD' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* :%(D' =-g6B fade*<3<* :D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*:T%(D' =-g6B fade*<3<*:T+P+0+ ++0+ ++0+ ++0+ ++0+ ++0+ +@:K0 QI  ((    <# ?"6@ NNN?N=: ],$ 0 4Define for each w in [0,1] 2aaefaefaf3  <- ?"6@ NNN?N : ,$  0 9{ 2c<   <1 ?"6@ NNN?NL) l ,$  0 :0; 2a ! <L!?"6@ NNN?N u3 ,$  0 ^LThen w is a -additive measure on A ,' 2cf3ccc cB ( B@j Ԕ?"6@ NNN?NE : ^ = [0, 1] A  the -algebra generated by countable sets B  all co-countable setsTS0F2c$c cc#c,g$33B$H  0޽h ? ̙33%w%___PPT10W%.$^F+q =D"' = @B D~"' = @BA?%,( < +O%,( < +D' =%(D' =%(D2' =A@BB@BB0B%()?)?Dy' =.A7 BBBBBSM -2.5E-6 2.9077E-6 L -0.00486 -0.63498 *3>*B ppt_xB ppt_y=@0BBAApBBIB<*(D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D'' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-g6B fade*<3<*$D7' =%(D' =A@BBBB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =-s6Bwipe(left)*<3<*"D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =-g6B fade*<3<*!D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =-s6Bwipe(left)*<3<*#D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =-g6B fade*<3<*'D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =-s6Bwipe(left)*<3<*%D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*&%(D' =-g6B fade*<3<*&+h+0+ ++0+ ++0+  ++0+! ++0+" ++0+# ++0+$ ++0+% ++0+& ++0+' ++0+( +  8 0  0 (  0! 0 Z f>L!?"6@ NNN?N YAnd what about justification? (2c, 0 ZL f>L!?"6@ NNN?N Mx _ ,$ 0 LKD45 Axioms of J Kp p Bp Jp'(2g cgf ccgccgfc,4 0 TZ ?"6@ NNN?N] *a ,$ 0 >!  2$:  0 ZЩ f>L!?"6@ NNN?NF G  ,$ 0 >  2(B  0@ TDԔL!?"0@NNN?Nz 2  ,$D 0T  0 Zh f>L!?"6@ NNN?N 1 ,$ 0 XS5 +Link4 2gc  0 Z f>L!?"6@ NNN?NL  4 (2c  0  fԵGN[HN ԔL!?"6@ NNN?NK %,$D  0 ?As long as the axioms of J are consistent and do not involve B.p@(2cgf#cgcH 0 0޽h ?/  0 0 ̙33___PPT10.I Q+&D' = @B Dk' = @BA?%,( < +O%,( < +D(' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-g6B fade*<3<*0D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-g6B fade*<3<*0D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 0%(D' =-g6B fade*<3<* 0D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* 0%(D' =-g6B fade*<3<* 0D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 0%(D' =-g6B fade*<3<* 0D_' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 0%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* 0D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* 0++0+0 ++0+0 ++0+ 0 ++0+ 0 ++0+ 0 +: p h  4 (  4 4 Z f>L!?"6@ NNN?NIo KAn open problem (2c0 4 Z f>L!?"6@ NNN?Ns7,$ 0 pL+ is a logic for the modalities M1, M2 ,& , Mn , Mn+1 . 9 2ck ckckc$ckc$ckc$, < 4 Z f>L!?"6@ NNN?N1s7,$ 0 @`L is a logic for the modalities M1, M2 ,& , Mn . 1 2!ckckc$ckc$+ 4 Z$ f>L!?"6@ NNN?Nsu ,$ 0 AEvery L-algebra can be extended in a unique way to an L+ algebra.PB 27go g 4  f f>L!?"6@ NNN?N sQ>$ 0TL___PPT9.&@ fDoes this imply that there is a formula X in terms of M1, M2 ,& , Mn such that L + (Mn+1p X) imply L+ ?&r 27ckckc$ckckc c ckck:A   4  4 TZ ?"6@ NNN?N K#,$ 0 >!  2$H 4 0޽h ? ̙33rj___PPT10J.K0r^+9D' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-g6B fade*<3<*4D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-g6B fade*<3<*4D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-g6B fade*<3<*4D(' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-g6B fade*<3<*4D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 4%(D' =-g6B fade*<3<* 4++0+4 ++0+4 ++0+4 ++0+4 ++0+ 4 + 0 >6p (  X  C P<   6  S 4zPH .*   How can we say in our framework that there is not ep. Luck? By saying the ep. Misfortune is logically impossible. (without saying why). ,2=H  0re"f ? ̙3380___PPT10.Ǐ 0  (  X  C P<     S "PH .*   uThe title is on deriving knowledge from belief. S5 can be derived sematically (it is not, of course just true belief)B )H  0re"f ? ̙3380___PPT10.T 0  ,(  X  C P<     S PH .*   .We know that relational models don t tell us all about logic. But this is true also when they satisfy the completeness thm. wH  0re"f ? ̙3380___PPT10.ԑ( 0 08(  ^  S P<     c $PH .*   .We know that relational models don t tell us all about logic. But this is true also when they satisfy the completeness thm. wH  0re"f ? ̙3380___PPT10.ԑr .#G.* .1[ 82<@ӫC3oLuHǵ$|.2b0RVy #Q v? Oh+'0\ hp    Samet Dovamedov296Microsoft PowerPointP@pd.@t~n!@kTGJg   4  --- @ !S---- @ !CS---- @ !/---- @ !'---- @ !#---- @ !---- @ !.---- @ !J---- @ !e---- @ !---- @ !---- @ !---- @ !---- @ !---- @ !---- @ ! ---- @ !(---- @ !@---- @ ![---- @ ! v---- @ !---- @ !#---- @ !/---- @ !3---- @ !J:---- @ !c---'@Times New Roman-. 42 ADeriving Knowledge from Belief,"0"-!2*."System-@Times New Roman-.  2 bJoe .-@Times New Roman-. 2 bHalpern/!".-@Times New Roman-.  2 b,.-@Times New Roman-. 2 b Dov Samet," 2.-@Times New Roman-.  2 b#, .-@Times New Roman-. 2 `Ella r).-@Times New Roman-. 2 Segevr".--- @ !{e)---- $)e)e---- @ !zV9--f-- $9V9V--'@Times New Roman-. 42 PDeriving Knowledge from Belief,"0"-!2*.-@Times New Roman-. 33 2 SJoe .-@Times New Roman-. 332 SHalpern/!".-@Times New Roman-.  2 S,.-@Times New Roman-. 332 S Dov Samet," 2.-@Times New Roman-.  2 S2, .-@Times New Roman-. 332 oElla r).-@Times New Roman-. 332 Segevr".-@Times New Roman-. %2 www.tau.ac.il/~samet,,-" 2.--- @ !XX%---- $%X%X---- @ !WI4--f-- $4I4I--'@Times New Roman-. 33%2 www.tau.ac.il/~samet,,-" 2.-՜.+,0d    On-screen Show Sametee_eA DavidArialTimes New RomanSymbolITC Zapf ChanceryDefault DesignSlide 1#The Tripartite Theory of Knowledge&Gettiers refutation of justificationSlide 4Slide 5Slide 6Slide 7#The Tripartite Theory of KnowledgeSlide 9 Slide 10 Slide 11 Slide 12 Slide 13 Slide 14 Slide 15 Slide 16 Slide 17 Slide 18 Slide 19 Slide 20 Slide 21 Slide 22 Slide 23 Slide 24 Slide 25  Fonts UsedDesign Template Slide Titles_F dovdov  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~Root EntrydO)PicturesCurrent UserSummaryInformation(PowerPoint Document(RjDocumentSummaryInformation8