Appendix A List of Axioms, Propositions and Theorems 1 Truth implies empirical adequacy. 2 Empirical adequacy does not imply truth. 3 If a discovery method 8 discovers h in [.s I n] in the limit, then there exists a limiting assessment method 0: which verifies h in [.s I n] in the limit. 4 If an assessment method 0: decides h in [.s I n] in the limit, then there exists a limiting discovery method 8 which discovers h in [.s I n] in the limit. 5 (.s,n) F KsA =* A, x E {RIT,RRT,ASA,ARA}, SE{8,0:}. 6 (.s, n) F K;(A =* C) =* (K;A =* K;C), x E {RIT,RRT,ASA,ARA}, - SE{8,0:}. 7 (.s, n) F K;A {:? K;K;A, x E {RIT,RRT,ASA,-ARA}, S E {8, o:} given consistent expectation for 8. 8 (.s, n).Ii -,K;A =* K;-,K;A, x E {RIT, RRT, ASA,ARA}, SE{8,0:}. 9 (.s,n).Ii S~A =* A, y E {V3, V3K, W, WK}, S = 8. 305 306 APPENDIX A 10 (c, n).Ii S~(A =? C) =? (S~A =? S~C), y E {V:3, V:3K} , ~ ~ 3=8. 11 (c, n) .Ii S~A =? -,S;-,A, y E {V:3, V:3K} , ~ 3=8. 12 (c, n) p S~(A =? C) =? (S~A =? S~C), y E {W,WK}, ~ ~ 13 (c, n) p S~A =? -,S;-,A, y E {W,WK}, ~ 3 = 8. 14 (c, n) p DA =? DA. 15 (c, n) p DA =? BA. 16 (c, n) p BA =? DA. 17 (c, n) p DA =? G A. 18 (c,n) p DA =? FA. 19 (c, n) p BA =? GA. 20 (c, n) p BA =? FA. 21 (c, n) p DA =? GA. 22 (c,n) p DA =? FA. 23 (c, n) pHD A =? A. 24 (c, n) p P D A =? A. 25 Empirical adequacy and tense: (a) (c, n).Ii A =? GA. (b) (c,n).Ii A =? FA. (c) (c, n).Ii a =? Ga. (d) (c, n).Ii a =? Fa. (e) (c, n) .Ii A =? H A. APPENDIX A 307 (f) (c,n) ~ A =} PA. (g) (c,n) Fa=} Ha. (h) (10, n) Fa=} Pa. 26 Truth and tense: (a) (c,n) ~ A =} GA. (b) (c,n) ~ A =} FA. (c) (c,n) Fa=} Ga. (d) (c,n) Fa=} Ga. (e) (c,n)~A=}HA. (f) (10, n) ~ A =} PA. (g) (c,n) Fa=}Ha. (h) (10, n) Fa=} Pa. 27 (c,n) F KtA =} FKtA (AFK) x E {RIT, RRT}. SE{8,a}. 28 (c,n) F KtA =} GKtA (AFK) x E {RIT,RRT}. SE{8,a}. 29 (c,n) ~ KsA =} FKsA, x E {ASA, ARA}, SE{8,a}. 30 (c:, n) ~ KsA =} GKsA, x E {ASA,ARA}, SE{8,a}. 31 Methodological recommendations: + Permissive, - Restrictive, * Insufficient: 308 APPENDIX A (D) (T) (K) (4) (5) -Consistency (+) + (+) * / -Perfect memory (+) + (+) / - Consistent expectation (+) + (+) + / - Infallibility (+) + (+) * / - Epistemic soundness II (+) (+) + + / 32 (MMS+1) (c,n) F KfIT A=} KfRT A. 33 (MMS+2) (c,n) F KfIT A=} KtSAA. 34 (MMS+3) (c,n) F KPT A=} KtRAA. 35 (MMS+4) (c,n) F KfRT A=} KfIT A. 36 (MMS+5) (c,n) F KfRT A=} KtSAA. 37 (MMS+6) (c,n) F KfRT A=} KtRAA. 38 (MMS+7) (c,n) F KtSAA =} KtRAA. 39 (MMS+8) (c,n) F KtRAA =} KtSAA. 40 (MMS-9 ) (c,n) ~ KtSAA =} KfIT A. 41 (MMS- lO ) (c,n) ~ KtSAA =} KfRT A. 42 (MMS-ll ) (c,n) ~ KtRAA =} KfIT A. 43 (MMS- 12 ) (c,n) ~ KtRAA =} KfRT A. 44 (c, n) F K~KBA =} KeA , 8,3 E {a,,B,1',8}, x, y, Z E {RIT, RRT, ASA, ARA}. In particular (a) Theorem 15.4: Uniform Knowledge Transmissibility: APPENDIX A 309 - Methods fixed K~K3A KeA - Correctness fixed II 1. x,y = RIT z=RIT KfITA 2. x,y = ASA z=ASA KtSAA 3. x,y = RRT z=RRT K;:RT A 4. x,y= ARA z=ARA K~RAA (b) Theorem 15.5: Semi-Uniform Knowledge Transmissibility 1. - Methods fixed I K~K3A KeA - Correctness mixed III 1. x = RIT,y = ASA z=ASA KtSAA 2. x = RIT,y = ASA z=RIT 3. x = ASA,y = RIT z=ASA KtSAA 4. x = ASA,y = RIT z=RIT KfITA 5. x = RRT,y = ARA z=ARA KtRAA 6. x = RRT,y = ARA z=RRT 7. x = ARA,y = RRT z=ARA KtRAA 8. x = ARA,y = RRT z=RRT KfRTA (c) Theorem 15.6: Semi- Uniform Knowledge Transmissibility 2. - Methods fixed I K~K3A KeA - Correctness mixed III 1. x = RIT,y = ASA z=ASA KtSAA 2. x = RIT,y = ASA z=RIT 3. x = ASA,y = RIT z=ASA KtSAA 4. x = ASA,y = RIT z=RIT KfITA 5. x = RRT,y = ARA z=ARA KtRAA 6. x = RRT,y = ARA z=RRT 7. x = ARA,y = RRT z=ARA KtRAA 8. x = ARA,y = RRT z=RRT KfRTA (d) Theorem 15.7: Non-Uniform Knowledge Transmissibility. 310 APPENDIX A - Methods mixed K~K3A KeA - Correctness mixed II l. x = RIT,y = ARA z=ARA KfRAA 2. x = RIT, y = ARA z= RIT 3. x = ARA,y = RIT z=RIT Kf-1TA 4. x = ARA,y = RIT z=ARA KfRAA 5. x = RRT,y = ASA z=RRT 6. x = RRT,y = ASA z=ASA KfsAA 7. x = ASA,y = RRT z=RRT Kf-RTA 8. x = ASA,y = RRT z=ASA KfsAA 45 Theorem 15.8: Transmissibility of Realistic and Anti-realistic Knowl­ edge: (a) A realist's knowledge is non-transmissible to an anti-realist. (b) An anti-realist's knowledge is transmissible to a realist. Appendix B Additional Proofs Proof of Proposition 11.1 PROPOSITION B.1 For every T : ET is a (J -algebra. It has to be shown that ET indeed is a (J -algebra. The following two lemmata will simply the final proof. LEMMA B. 2 The following properties hold: 1 hA n hB = hAnB. 2 hA U hB = hAUB. 3 hA n hB = hA\B· 4 n h n Ai\ n Bi = h n (Ai\Bi)· n<w i<n i<n n<w 5 0 E [T· 6 WE [T. Proof. 1 Prove that hA n hB = hAnB. Now (JL, l) E hA n hB {:} (JL, l) E hA /\ (JL, l) E hB {:} [JL E A /\ l 2: k(JL)]/\ [JL E B /\ l 2: k(JL)] {:} JL E AnB /\l2: k(JL) {:} (JL, l) E hAnB. 2 Prove that hA U hB = hAUB. By similar construction (JL, l) E hA U hB {:} (JL, l) E hA V (JL, l) E hB {:} [JL E A /\ l 2: k(JL)] V [JL E B /\ l 2: k(JL)] {:} [JL E A V JL E B]/\ l 2: k(JL) {:} 311 312 APPENDIX B [p E A U B]/\ l 2 k(p) ~ (p, l) E hAUB. 3 Prove that hA n hB = hA\B. Again (p, l) E hA n hB ~ (p, l) E hA /\ (p, l) tt hB ~ [p E A /\ l 2 k(p)]/\ [(p tt B)V (p E B /\ l < k(p)] ~ [(p E A /\ (l 2 k(p)) /\ P tt B)]V [(p E A /\ l 2 k(p)) /\ (p E B /\ l < k(p))] ~ pEA /\ (p tt B /\ l 2 k(p)) ~ P E A\B /\ l 2 k(p) ~ (p, l) E hA\B. By definition hA U hB is closed under fT = {hA U hB I A,B ~ T}. 4 Prove next that n h n Ai\ n Bi = h n (Ai\Bi)· Now assume that n<w i<n i<n n<w (T,l) E n h n Ai\ n Bi {::} n<w i<n i<n Vn E w : (T, l) E h n Ai\ n Bi ~ i<n i<n VnEw: [TE.n Ai\.n Bi/\l2k(T)] ~ z<n z<n Vn Ew : [T E.n Ai\ .n Bi] /\ Vn Ew : [l 2 k(T)] ~ z<n z<n VnEw: [TE.n Ai\.n Bi] /\l2k(T)~ z<n z<n TEn [.n Ai\.n Bi]/\l2k(T)~ nEw z<n z<n (T,l) E h n (n Ai\ n Bi)· n<w i<n i<n 5 0 E fr =? 0 = hA n hB E fr· 6 W E fr =? W = hA U hB E fr· APPENDIX B 313 • LEMMA B.3 All hypotheses in £T are absolute time invariant. Proof. 1 Show that if A ~ T, then hA is absolute time invariant with respect to (c:, n). Hence, if (f-L, m) E [c: I n] n hA, then :JT E (c: I n)Vl E W : (T, k + l) E hA where k == max{n,m}. Now (f-L,m) E [c: I n] n hA ~ (f-L I n = c: I n)!\ (:JT E A: (f-L,m) E hT ) ~ (f-L I n = c: I n) !\ (f-L = T !\ m 2:: k(T)). This implies Vl 2:: k,max{n,m} 2:: k(T) : (f-L,l) E hA· 2 Show that if A ~ T, then hA is absolute time invariant with respect to (c:, n). Hence, if (f-L, m) E [c: I n] n hA, then :JT E (c: I n)Vl Ew: (T,k+l) E hA where k = max{n,m}. Now (f-L,m) E [c: I n] n hA ~ (f-L I n = c: I n) !\ [f-L \f A V (f-L E A!\ m < k(T))]. There are hence two possibilities (a) f-L \f A=> Vl E w: (f-L,l) E hA- (b) f-L E A!\ m < k(T).