Appendix a List of Axioms, Propositions and Theorems

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

- 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

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

VnEw: [TE.n Ai\.n Bi/\l2k(T)] ~ z

Vn Ew : [T E.n Ai\ .n Bi] /\ Vn Ew : [l 2 k(T)] ~ z

VnEw: [TE.n Ai\.n Bi] /\l2k(T)~ z

TEn [.n Ai\.n Bi]/\l2k(T)~ nEw z

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). Choose v E (c: I n) such that f-L I k = v I k and v ~ A. Then

Vl E w: (v,l) E hA.

3 Show that if A ~ T, then hA U hB is absolute time invariant with respect to (c:, n). This is obviously so because 314 APPENDIX B

(j.t, m) E [c I n] n (hA U hB) {:} [(j.t, m) E [c I n] n hAl V [(j.t, m) E [c I n] n hB] . • Finally we can prove that the proposition:

PROPOSITION BA £T is a (J-algebra.

Proof.

1 By lemma B.2, 0 E £T and W E £T. 2 It now remains to be shown in general that if a E £T then a E £T. Proceed by cases:

(a) If a = hA E £T then a = hA E £T. This is obvious from the time invariance of hA and hA in the lemma above. (b) If a = hA E £T then a = hA E £T. Follows again from the time invariant demonstration in the lemma above. (c) If a = 0 then a = W. (d) If a = W then a = 0. (e) If a = hA U hB E £T then a = hA U hB E £T. Follows also by lemma 2.

3 Finally h Ai U h Bi = hn Ai \ nBi· Follows immediately by lemma B. 2, case 4 . • Proof of proposition 12.4

PROPOSITION B.5 ARA-Knowledge and 84

If knowledge is defined as A nti-realist Reliable A dequate belief (ARA) then knowledge validates 84.

Proof. Assume knowledge is defined as ARA. Then show in turn that the S 4-axioms are satisfied:

1 Show that (c,n) F K~RAA '* A, i.e., if (c,n) E [K~RAA] then (c, n) E [A]. This is immediate by lemma 11.2 since [K~RA A] ~ [A]. APPENDIX B 315 2 Show that (c,n) F K:RA(A => C) => (K:RAA => K:RAC), i.e., if (10, n) E [K:RA(A => C)] and (10, n) E [K:RA A], then (10, n) E [K:RAC]. Assume the first conjunct ofthe antecedent claim (10, n) E [K:RA(A => C)]. Then

(a) 3h <;;;:

[v I k] n [[A] U [C]] -I 0, } { (v,k) 3k' ~ kVI ~ k',V(T,I) E [v I k] : : (c,n) E h, a([A] U [C], Til) = 1

(b) (10, n) E [A] u [C]. Also second conjunct of the antecedent claim (10, n) E [KtRA A] if and only if

(d) (c,n) E [A]. Then prove from (a)-(d) that

(e) 3h" <;;;:

[v I k] n [C] -I 0, } { (v, k) 3k' ~ kVI ~ k', V(T, I) E [v I k] : : (10, n) E h", a([C],T II) = 1

(f) (10, n) E [C]. Condition (f) is immediately implied by (b) and (d). Define h" = h n h' which is non-empty. Realize that by (a) and (c) and the epistemic soundness of the assessment method a

V(T, I) E [v I k] : a (([A] U [C]) n [A],T II) = 1,

and by distribution of the intersection one obtains

V(T,I) E [v I k] : a (([A] n [A]) U ([A] n [C]),T II) = 1, 316 APPENDIX B

which collapses to

V(7,l) E [v I kJ: a ([AJ n [C],7 Il) = 1.

Then clearly [GJ ;2 [AJ n [G],

and again by epistemic soundness

V(7,l) E [v I kJ : a([C],7 Il) = 1.

Since h" = h n h' and (*) hold, (e) is implied.

3 Show that, (c,n) 1= K~RAA {:} K~RAK~RAA, i.e., (c,n) E [K~RAAJ if and only if (c,n) E [K~RAK~RAAJ. (=» Show that, (c,n) 1= K~RAA => K~RAK~RAA, i.e., if (c,n) E [K~RAAJ then (c,n) E [K~RAK~RAAJ. Suppose (c,n) E [K~RAAJ. Then

(a) ~h ~

[v I kJ n [AJ f= 0, } { (v, k) ~k' ~ kVl ~ k', V(7, l) E [v I kJ : : (c, n) E h, a([A], 7 Il) = 1

(b) (c, n) E [AJ. Then prove that

[v I kJ n [K~RA AJ f= 0, { (v, k) ~k' ~ kVl ~ k', V(7, l) E [v I kJ : } , (.,n) E h', a([K~RA A], 7 Il) = 1

(d) (c,n) E [K~RAAJ. Condition (d) is immediately implied by assumption. For condition (c) apply the following reductio argument based on a's reliable performance. Suppose otherwise. Then

Vk' ~ k~l ~ k',~(7,l) E [v I kJ : a([K;tRAA],7 It) = o. APPENDIX B 317 But note that the assessment method a is reliable. Hence the fact that [v I k] n [K~RA A] of. 0 holds implies by definition that

:::Jk' 2 kVl2 k',V(T,I) E [v I k]: a([K:RAAJ,T II) = 1.

which contradicts the reductio clause. (¢::) This direction is again obvious as an iteration of the axiom of truth (T) . • Proof of Proposition 15.1

PROPOSITION B.6 The MMS+ extensions

1 (MMS+l) KfIT A=? KfRT A. 2 (MMS+2) KfIT A=? KtSAA. 3 (MMS+3) KfIT A=? KtRAA.

4 (MMS+4) KfRT A=? KfIT A.

5 (MMS+5) KfRT A=? KtSAA.

6 (MMS+6) KfRT A=? KtRAA.

7 (MMS+7) KtSAA =? KtRAA.

8 (MMS+B) KtRAA =? KtSAA.

Proof. MMS+1+8

1 Show that, (c, n) F KfIT A=? KfRT A. Then

(1) [c I n] n [A] of. 0, (2) "In' 2 n, V(T, n') E [c I n] : (2.a) b(T I n') ~ [A], (2.b) (T,n) E b(T In').

Let discovery method b induce its assessment correlate a. Since, by (1)-(2)

(c,n) E [A] and VI E w: (c,n+l) E [AJ, 318 APPENDIX B construct the assessment correlate in such a way that

Vn' :2: n, VeT, n') E [e I nJ : a([Al, Tin') = 1 ¢:} 8(T I n') ~ [AJ. But by (2)

Vn' :2: n, VeT, n') E [e I nJ : 8(T I n') ~ [AJ so :3k :2: n, Vn' :2: k, VeT, n') E [e I nJ : a([AJ, Tin') = 1 which is the definition of RRT-knowledge.

2 Show that, (e, n) F KfIT A =} KtSA A. Follows from the defini• tions of RIT- and ABA-knowledge. 3 Show that, (e, n) F KfIT A=} KtRA A. Apply the strategy from 1 supplying the obvious modifications. 4 Show that, (e,n) F KfRT A=} K[llT A. Suppose (e,n) E [KfRT AJ. Then:

(1) (e,n) E [AJ and Vl Ew: (e,n+l) E [AJ, (2) :3kVn' :2: k, V( T, n') E [e I nJ : a([AJ, Tin') = l.

Then show that (1) and (2) imply (e, n) E [K[llT AJ hence

(i) [e I nJ n [AJ :f= 0, (ii) Vn' :2: n, VeT, n') E [e I nJ : 8(T I n') ~ [AJ, (iii) (T, n) E 8(T In').

(i) and (iii) are obvious. Next induce the discovery correlate in the following way given (1):

Vn' :2: n, VeT, n') E [e I nJ : 8(T I n') ~ [AJ ¢:} a([Al, (7 In')) = l. By (2) :3kVn':2: k,V(T,n') E [e I nJ : a([Al,T In') = 1 hence

Vn' :2: n, VeT, n') E [e I nJ : 8(T I n') ~ [AJ which is the definition of RIT-knowledge.

5 Show that, (e, n) F KfRT A =} KtSA A. Apply the same strategy as in case 4 providing the obvious modifications. APPENDIX B 319 6 Show that, (c:,n) F KfRT A=> KtRAA. Immediate by definition. 7 Show that, (c:, n) F KtSA A => KtRA A. Immediate given case 1 provided the obvious modifications. 8 Show that, (c:, n) F KtRA A => KfsA A. Immediate given case 4 provided the obvious modifications . • Appendix C Glossary

he encyclopedic items are sorted in order of appearence rather T than alphabetically.

• KaLC-Knowledge as Limiting Convergence. A thesis originally attributed to the American pragmatism of C. S. Peirce and W. James. Peirce held the view that science may, for all we know, converge asymptotically to the truth in the limit while James said that knowledge and science may reach the truth eventually but we may not know infallibily when. Since then both realistically and anti-realistically minded philosophers of science and epistemologists have elaborated on this approximation thesis. KaLC is the funda• mental thesis of this book. The primary aim is not to say whether converge will or will not occur. It is to investigate the proposal that such convergence, if it occurs, is descriptive of scientific knowledge. Hence, to provide an epistemology and logic of limiting convergence for both realists and anti-realists.

• The Components of Scientific Inquiry-The elements of sci• entific inquiry include:

(a) A world or collection of worlds with which inquiry is concerned. (b) A set of scientific inquiry methods. (c) A set of empirical hypotheses for the scientific inquiry methods to entertain. (d) A relation of correctness. (e) An epistemic commitment for an inquiry method relative to a hypothesis.

• Epistemology-The reply to knowledge skepticism and hence the study of the nature of knowledge and justification in particular with respect to the defining features of knowledge, criteria of acquisition and limits of knowledge and justification.

321 322 APPENDIX C • Skepticism-The foe of epistemology. A family of proposals which hold knowledge to be impossible or at least not demonstratively possible because it is possible that we err. Skeptical arguments often rest on either global or local underdetermination.

• Scientific Knowledge-The knowledge of the laws of nature con• sidered here. The scientific knowledge should not be confused with everyday knowledge. The laws of nature have a complex induc• tive structure. Hence, in order to obtain knowledge of the laws some rather strong forcing and conjection conditions have to be met which are not required for everyday knowledge.

• Laws of Nature-The objects of scientific knowledge. Laws of nature are spatio-temporally symmetrical or invariant. Hence, laws of nature are assumed to hold at all times and in all possible worlds specified by the accessibility relation.

• Everyday Knowledge-The kind of knowledge which is not of the laws of nature and/or does not require heavy forcing and conjection conditions to be met in order to obtain. For instance I know that I am writing these encyclopedic entries now, but I neither need a limiting success criterion to converge to this fact, nor success in all possible worlds admitted by the background knowledge.

• The Tripartite Definition of Knowledge-The standard tri• partite analysis of knowledge suggests that some scientist or sci• entist applying a method 2: knows a hypothesis h if the following conditions obtain: Method 2: knows h iff

(a) 2: believes h, (b) h is true, (c) 2: is justified in believing h.

• Belief, Truth and J ustification-The three necessary and sup• posedly jointly sufficient components of the standard definition have received unequal attention. In the standard tripartite analysis, a belief is usually taken to be a psychological primitive or disposi• tional psychological state existing when both manifested and when APPENDIX C 323 unmanifested. The standard analysis further suggests, as a neces• sary condition, that knowledge of h implies that h is true. The jus• tification condition has probably received the greatest attention of the three components. Knowledge requires that the satisfaction of the belief condition 1 is "adequately" connected to the satisfaction of the truth condition 2. Conditions 1 and 2 are jointly insufficient to secure knowledge since some true beliefs may be the result of lucky conjectures, accidental inferences, evidence collected under obscured perceptual circumstances etc. Such beliefs should in the standard analysis not count as knowledge since 1 and 2 are inade• quately connected to each other due to the questionable means or methods and thus methodology by which the true beliefs have been derived. According to condition 3, if some argument or other jus• tificational structure can be provided that describes why the first two conditions are adequately connected, then the scientist may be said to have secure indication that a known proposition is true or correct.

• Methodology-The study of the methods and methodological recommendations by which science arrives at its posited truth. Methodology may be viewed as corresponding to epistemology's justification. Methodological recommendations may fall in either one of the following two categoies: Hypothetical methodology: The methodological recommenda• tion is advocated in the aim of finding the correct answer. Categorical methodology: The methodological recommendation is advocated to its own end independently of finding the correct answer. • Modal Logics-The term denotes a whole range of logics designed to study various modes. For this investigation the interesting logical modes include:

alethic or metaphysical modes including necessity operators 8, 0 and E3 prefixed formulas of a propositional logic. temporal modes including tense operators F, G, P and H pre• fixed formulas of a propositional logic. epistemic modes including a knowledge operator Kg prefixed formulas of a propositional logic where B denotes an arbitrary inquiry method of either assessment or discovery. 324 APPENDIX C • Formal Learning Theory-Also called the logic of reliable in• quiry, computational epistemology.The formal study of inductive problems and their intrinsic solvability for both ideal and com• putable agents.

• Modal Operator Theory-A term coined by Vincent F. Hen• dricks and Stig Andur Pedersen to denote the cocktail obtained by mixing epistemic, tense and alethic logic in order to study scientific knowledge acquisition and validation in the limit:

agent epistemic logic

tense tense logic = Modal Operator Theory

metaphysics alethic logic

• Global Underdetermination-A hypothesis is globally under• determined if there are two possible worlds such that one of the worlds assign the truth-value true to the hypothesis while the other assigns false in such a way that the evidence received by the inquiry method remains the same forever regardless of which world is the actual world. All the well-known skeptical objections like Cartesian demons, brains in vats, s~ampmen, the Duhem-Quine thesis and webs of beliefs are variants of global underdetermination.

Branching time Scientific Inquiry JJIrro ~ Justification Tense logic ~ ;p Axiom of Wisdom IJfl ~ JJ(r-H-YP-ot:-~-'I' Ontology (.2) Epistemic logic Probability .} (AFK) Modal logic Science @ Formal Learning Theory

• Local U nderdetermination-A hypothesis is locally underde• termined by the evidence in a possible world if there is an infinite sequence of evidence possible for all the method knows, such that each initial segment of this evidence sequence could arise regard• less of whether the hypothesis is true or false (Kelly 96). All the APPENDIX C 325 well-known skeptical objections like Sextus' argument against the skeptics, Hume's problem of induction and Goldman's New Riddle of Induction are variants of local underdetermination ..

• Epistemic Modesty (EM)-The dictum of Academic skepticism which says that all I know is that I know nothing. If epistemic modesty is propositionally formalized as an axiom schema then it logically implies axiom (5) also called the axiom of wisdom. It can then be shown that the Academics cannot converge to the axiom which they take to witness the impossibility of knowledge even if they are allowed to use a limiting convergence criterion and an infallible method.

• Gettier Cases-A set of skeptical counterexamples formulated by Edmund Gettier intended to show that knowledge is not true justified belief. A Gettier case is often a situation in which one can derive a truth from a falsehood: One may be justified in believing what is in fact a falsehood. One may also in fact believe a truth at random but not have reasons for or even reasons against the truth. Then by applying the introduction rule for disjunction gluing the truth and the falsehood together one may derive a truth from a falsehood but one would simultaneously know the truth for the wrong reasons.

• Forcing-The term denotes the family of epistemologies which all pay homage to the idea that the way to combat skepticism citing prima facie possibilities of error as the most substantial arguments against knowledge claims or sound justified opinions is to agree that real possibilities of error undercut knowledge but at the same time argue that the possibilities cited by skepticism fail to be genuinely relevant. Nozick's counterfactual account of knowledge, epistemic logic, Bayesianism and the KaLC-paradigm are all instances of forcing.

• Accessibility Relation-A relation specifying how to access other viable state descriptions of the actual world, relevant circumstances or possible worlds. Depending on the particular epistemological approach, the accessibility relation may be cashed out in terms of proximity, probability, similarity, classical relational properties like 326 APPENDIX C reflexivity, symmetry and transitivity and extensions of the eviden• tial handle. Accessibility relations are often integral parts of forcing proposals.

• Demon World-A possible world discussed often by philosophers inhabited by a vicious skeptical demon committed to show that everything is globally underdetermined all the time-it is a very remote world and can be killed by forcing. Scientists often kill it this way.

• Background knowledge-The set of relevant or viable circum• stances, alternatives or possible worlds in which the inquiry method is to succeed in order to obtain knowledge. The background as• sumptions may admit different possible worlds depending on the nature of the forcing condition. The forcing condition may cash relevance out in similarity, vicinity, probability, classical relational properties etc. The background knowledge in the KaLC-paradigm is determined to be all the possible worlds which have the same evidential handle as the actual world observed until "now".

• The shrinking property-A special property of the background knowledge such that every later background knowledge (or world fan) is included in every earlier background knowledge (or world fan).

• Conjection-The set of possible worlds over which the inquiry method chooses to project its conjecture relative to the forcing con• ditions.

• Correctness Relation-A relation which specifies the sense in which a hypothesis is correct in a possible world. In the current framework a hypothesis may either be true or empirically adequate in a possible world so the arbitrary correctness relation C((c, n), h) is introduced to denote either one when necessary. The truth and the empirical adequacy enjoy an asymmetrical relationship. A true hypothesis is also empirically adequate but an empirically adequate hypothesis is not necessarily true since truth may depend on other features than the evidence. APPENDIX C 327 • Evidence Stream-An evidence stream is what provides the in• quiry method with evidence. Formally evidence streams are w• sequences of natural numbers denoted by c, T, v... An evidence stream has the form

and encodes other types of discrete evidence from atoms to black ravens depending on the particular interpretative reduction.

A finite initial segment of an evidence stream is called the han• dle and is denoted c I n. The set of all infinite evidence streams which extends the handle is called the fan and is denoted (c In).

• Interpretative Reduction-The process of reducing raw data experience to informative evidence. This reduction can be highly theoretical. A radiologist examining an X-ray of a patient's lungs may observe a black dot on the pleura. Given his knowledge of radiology and background knowledge of medical possibilities he may eventually interpret the dot as a benign or malign tumor. The observable black dot on the X-ray image says very little in itself, if anything. Almost everybody would be able to observe the dot, but far from everybody would be able to identify it as a tumor of a benign or malign nature. The debate between realism and anti-realism only enters after the reduction.

• Possible Worlds-In general, and fairly uncontroversially, a pos• sible world is a viable alternative state description of, or alternative to, the actual world. A possible world in the KaLe-paradigm is an ordered pair consisting of an evidence stream and a state coor• dinate. A possible world has the form (c, n). The set of all possible worlds is denoted W.

• World Fan-The world fan is the set of possible worlds which extends the handle and is denoted [c I n]. The world fan constitutes the background knowledge in the KaLe-paradigm.

• Hypotheses-Hypotheses are sets of possible worlds. Attention is restricted to the set of absolute time-invariant £. An absolute 328 APPENDIX C time invariant empirical hypothesis has the following relation to consistency with the evidence: If the hypothesis is absolute time invariant with respect to a possible world and if the hypothesis is consistent with the current evidence, then it is possible that the hypothesis remains consistent with the evidence in all future. Ab• solute time-invariant empirical hypotheses correspond to the laws of nature.

Modal Learning Theory ~ Possible world cr==:'l ~ Conjection Scientific f' 9 inquiry KK Skepticism @ Diachronic Methodology /CJ) F: Transmissibility Forcing G 0 Modal logics V

• Inquiry Methods-The methods used to conduct scientific in• quiry. A scientist or an agent may apply such a method or the method may conduct inquiry on its own much like a computer. Thus, since a method can also operate as an independent cognitive device, "scientist", "agent" and "method" are used largely inter• changeably. The inquiry methods considered here may be either one of the following two kinds even though the primary interest is in the former: Discovery methods. A discovery method is a function which takes finite segments of evidence as inputs and outputs hypothe• ses. Discovery methods are usually denoted by I, /5. Assessment methods. An assessment method is a function which takes finite segments of evidence and hypotheses as inputs and outputs truth-values 0 or 1; 0 signifies incorrectness while 1 sig• nifies correctness. Assessment methods are usually denoted by a,{3. Arbitrary inquiry methods of either assessment or discovery are denoted by 2,8 . • Inducement-Assessment and discovery methods can construct or simulate each other by watching over their respective outputs on the evidence received. APPENDIX C 329 • Methodological Recommendations-Prescriptions or program bits controlling the inquiry methods' behavior. A methodological recommendation is boosting if it aids the method in validating epis• temic axioms. A methodological recommendation is a debilitation if it bars the method from validating epistemic axioms. Finally, a methodological recommendation is called neutral if it is neither boosting nor debilitative. Hence, if a methodological recommen• dation is debilitative it is because it prevents the method from being epistemically stronger than it in fact is while obeying the rec• ommendation. The methodological recommendations investigated here pertaining to discovery and assessment respectively include:

Discovery: * Consistency. A discovery method is consistent if its conjec• tures are consistent with the evidence seen so far. * Perfect memory. A discovery method has perfect memory if it remembers the evidence seen so far. * Consistent expectation. A discovery method is consistently expectant if its conjectures always are directed into the fu• ture and respects the evidence seen so far (note that con• sistent expectation and perfect memory are inconsistent, so a discovery method cannot comply with both prescriptions at the same time). * Infallible. A discovery method is infallible if its conjectures entail the correctness of the hypothesis in question. Assessment: * Epistemic soundness. An assessment method is reliable by definition so epistemic soundness is a structural criterion rather than a recommendation since it forces the assessment method to respect the entailment based output-conventions of the discovery method .

• Success Criteria-The criterion species the way in which an in• ductive problem is solvable. For the assessment paradigm, the suc• cess criteria include verification, refutation and decision of a hy• pothesis relative to the evidence. For the discovery case, the success criterion amounts to identification of a hypothesis, i. e. the method produces a conjecture entailing a hypothesis. A discovery method is always non-trivial in the sense that it never identifies absurdities. 330 APPENDIX C By definition, an assessment method cannot conjecture absurdities.

• Convergence Criteria-The criterion specifies the time by which the answer to an inductive problem is to be had. The convergence criteria considered here include the following:

(a) Certainty convergence. The method is to converge by some finite time and halt with the answer. (b) Limiting convergence. The method is allowed to vacillate a finite but not specifiable number of times prior to the modulus of convergence. Thus, the method will eventually converge but one may just not know exactly when.

• p,-operator-Also called minimimalization. The p,-operator is a mathematical operator which searches for the least object such that some relation is satisfied. The p,-operator is used for defining the convergence moduli for both assessment and discovery methods.

• Convergence Modulus-The earliest possible time after which all the inquiry methods' conjectures remain the same.

• Logical Reliability-An inquiry method is logically reliable if it is guaranteed to succeed in all the possible worlds admitted by the background knowledge for some specified notion of successful convergence.

• Epistemic Axioms-These axioms are designed to determine the epistemic strength of a knowledge acquiring inquiry method. The axioms considered here most notably include:

(D) also called the axiom of consistency. If a method knows a hypothesis, then it does not also know its negation. (T) also called the axiom of truth. If a method knows some hypothesis, then the hypothesis is true. (K) also called the axiom of deductive cogency. The knowledge of a method is closed under implication. APPENDIX C 331 (4) also called the K K -thesis or the axiom of self-awareness. If a method knows a hypothesis, then the method knows that it knows the hypothesis. (5) also called the axiom of wisdom or negative introspection. If a method does not know a hypothesis, then the method knows that it does not know the hypothesis. (AFK) also called the axiom of futuristic knowledge. If a method knows some hypothesis, then the method knows the hypothesis in all future. An axiom introduced to separate the cognitive powers of convergent scientific realism from conver• gent scientific anti-realism. This axiom is independently enter• tained by (Fagin et al. 95).

• Epistemic Strength-The epistemic strength of a knowledge op• erator is determined by the epistemic system corresponding to the operator. The systems considered here most notably include the following in order of increasing strength:

KD4 is self-explanatory. 84= (T) + (K) + (4). 85= (T) + (K) + (5). • Forcing Quantifiers-The forcing quantifiers specify the conjec• tional scope in time and in possible worlds:

The forcing time quantifier ranges over all possible later times from a particular point in time onwards, i. e. Vn' 2': n ... where n' and n are state instances. The forcing world quantifier ranges over all possible worlds in accordance with the background knowledge, i. e. V( 7, n') E [6' I n] ... where (7, n') is a possible world and [6' I n] is the world fan or the background knowledge.

• Perspectives on Scientific Inquiry-The points of view from where scientific inquiry is analyzed. There are two perspectives currently discussed:

1st person perspective. A perspective on scientific inquiry is 1st person if it is considered what an agent can solve, can do or defend considering the available means for an end given the epistemic environment he is sunk into. 332 APPENDIX C 3rd person perspective. A perspective on scientific inquiry is 3rd person if it is considered what an agent could solve, could do or defend considering the best means for an end independently of the epistemic environment he is sunk into.

• Synchronic Principle-An epistemic or doxastic or combined principle is synchronic if the consequent obtains by the very same time the antecedent obtains. Axiom (T) is for instance a synchronic principle both from a 1st and a 3rd person perspective.

• Diachronic Principle-An epistemic or doxastic or combined principle is diachronic if the consequent either obtains later or would have obtained later than the antecedent even if things had been otherwise. Axiom (4) or also called the K K -thesis is for in• stance a synchronic principle from a 1st person perspective (if it holds at all) but a diachronic principle from the current 3rd person perspective given consistent expectation.

1st person perspective Synchronic Discovery Reliability 3

Accessi bi I tyi

Epistemology Convergence modulus

Assessment Relevance ~ 3rd person perspective • Knowledge Transmissibility-The property denotes a situation in which one inquiry method acquires knowledge of some hypoth• esis only knowing that another inquiry method has knowledge of the hypothesis in question. In other words, the property signifies the possibility of transmitting knowledge. In the KaLe-paradigm there are two types of inquiry methods and two types of correctness relations. Thus, there are four transmissibility cases:

Uniform transmissibility. Methods and correctness fixed. Semi-uniform transmissibility 1. Methods mixed but correct• ness fixed. APPENDIX C 333 Semi-uniform transmissibility 2. Methods fixed but correctness mixed. Non-uniform transmissibility. Methods and correctness mixed.

• When vs. what-

When: A scientific inquiry method has converged if there is a modulus of convergence after which it does not change its mind pertaining to its conjecture. What: What makes a method converge is a question unsuited for the logic of convergence. What may depend on crucial ex• periments, psychological dispositions, social imprimatur, cul• tural norms and traditions, psychological certainty and a host of other issues often globally underdetermined by the evidence. Appendix D Resources

here is a substantial amount of literature on most of the subjects T covered in this book. It is consequenly close to impossible to list all the supplementary texts from which this book has benefitted and from which both new and advanced readers may profit.

Resources ,/ Non-technical '" Technical Epistemology Belief Revision Bayesianism Methodology Formal Learning Epistemic Logic Philosophy of Tense Logic Science Alethic Logic

Science Mathematics - Mathematical Logic - Recursion Theory -www- - Topology l. PUBLISHED RESOURCES • Epistemology-The epistemological literature is obviously quite vast in size. Thus, a very limited set of suggestions will be made. For an introduction to epistemology one may refer to (Dancy 85). For an encyclopedia of epistemology with many entries consider ( Dancy & Sosa 96 (eds.)). Next, an in-depth introduction to knowl• edge versus skepticism is provided by (Nozick 81) in which one also finds the counterfactual analysis of knowledge. An introductory book to epistemology (but also to methodology and philosophy in general) is (Glymour 92). A sharp presentation in which one also finds an introduction to the philosophy of formal learning theory. Finally, a source book on various contemporary epistemological is• sues is the one compiled by (Goodman & Snyder 93). In this com• pilation many of the original texts on autoepistemology, reliabilism, epistemic justification, skepticism, etc. are published . • Philosophy of Science-Like with the epistemology resources there are many valuable texts in philosophy of science. For an en-

335 336 APPENDIX D cyclopedia refer to (Newton-Smith 00 (ed.)) in which articles on almost every aspect of the philosophy of science figures. Two books on philosophy of science include (Van Fraassen 80) and (Hacking 83) in which one finds original arguments concerning scientific re• alism and anti-realism, scientific practice, the aim of science, etc. • Science and Philosophy-There is often a significant discrep• ancy between what philosophers have to say about science and what scientists themselves have to say about their own endeav• our. Often scientists do not want to engage in discussions which philosophers deem relevant exactly because scientists deem them irrelevant. From time to time, however, one can find scientists deal• ing with philosophical issues. Notable scientists and Nobel Prize takers in physics like (Feynman 53), (Feynman 67), (Feynman 95), (Feynman 99) and (Weinberg 94) discuss laws of nature, the aim of science and the relation to philosophy. Also (Jammer 66), (Rhodes 86) and (Hawking 88) in more specialized books include various philosophical perspectives on science. • Methodology-There are various approaches to methodology and many different questions in methodology. For an encyclopedia of methodology refer to (Newton-Smith 00 (ed.)). For two different influential methodological approaches, pertinent to, but different from the current perspective see: Belief Revision-For a thorough presentation of belief revi• sion refer to (Gardenfors 88). Then for a related, but different perspective on scientific inquiry from the 1st person point of view consider (Levi 91). Levi's approach also has a Bayesian inspiration. Bayesianism-There are at least three fundamental introduc• tions to Bayesian methodology and probability in methodology: (Jeffrey 83), (Howson & Urbach 89) and (Earman 92).

• Formal Learning Theory-The field may admittedly seem like a rather extravagant technical discipline long before the significant philosophical results begin to emerge. Lately however, a variety of texts have surfaced which place the area in a broader philosophi• cal context. For a general introduction to formal learning theory and computational epistemology and how it relates to more stan• dard epistemological and methodological issues refer in particular to (Glymour 91), (Glymour 92, chapter 10), (Kelly 94), (Kelly & APPENDIX D 337 Schulte 96, although less introductory), (Schulte & Cory 96) and also (Kelly et al. 97), the encyclopedic article (Kelly 98) and the paper comparing the computational epistemological approach to Rorty's approach to philosophy of science (Kelly 99). From here there is Kelly's excellent book in which the entire formal learning framework is outlined in detail along with all the philosophically pertinent results (Kelly 96). The book does however assume rudi• mentary familiarity with some mathematical machinery (see be• low). Now, Kelly's book, among other things, extends or genralizes some results found earlier in the formal learning literature in partic• ular (Osherson et al. 86) albeit the latter does not put the paradigm into a broader philosophical perspective but stays within the lan• guage acquisition paradigm (Gold 65), (Gold 67). Recently (Martin & Osherson 98) but also (Kelly et al. 96) apply learning theory to study the theory of belief revision. Cory Juhl, applies learning the• ory to study Reichenbach's straight rule of induction (Juhl 94) but also Putnam's original argument against Carnap (Juhl 95a) and Bayesianism in (Juhl95b), (Juhl95b). Oliver Schulte utilizes com• putational epistemology to investigate the following three epistemic aims: convergence to a correct theory, fast convergence to a cor• rect theory and steady convergence to a correct theory (avoiding retractions or mind-changes) using various versions of Goodman's New Riddle of Induction and inference of conservation principles in particle physics as illustrations (Schulte 96). Papers which de• rive from (Schulte 96) include the technical one (Schulte 99a) and (Schulte 99b) to the same end as the former but at a much lower level of technicality.

• Epistemic Logic-There is a huge amount of literature on epis• temic logic, especially from conference proceedings and transac• tions. Valuable books on the subject include (Hintikka 62) and also (Lenzen 78). In recent years epistemic logic has been studied quite intensively by computer scientists rather than philosophical logicians. For an overview, see (Gabbay et al (eds.) 95) and most notably (Fagin et al. 95) the latter providing a combined textbook and research monograph with both introductory material and ad• vanced computational applications.

• Tense Logic-Tense logic, just like epistemic logic has numerous sources also due to the many proceedings and transactions. Never• theless, important books with overviews and not too technical in- 338 APPENDIX D clude (Gamut 91), (0hrstrom & Hasle 95). Then for a thoroughly technical treatment of tense logic refer to (Burgess 84) . • Alethic Logic-By far, alethic logic has been the field of modal logic which has received the greatest attention. This, once again, implies an endless list of valuable contributions. But perennial texts include (Hughes & Cresswell 68), (Chellas 80), (Bull & Segerberg 84) and a new technically very elegant presentation by (Chagrov & Zakharyaschev 97) with many interesting novel applications. • Mathematical Machinery-The underlying mathematics of the book include: Mathematical Logic-There are many good texts in mathe• matical logic but a fine and complete presentation of the most important themes in mathematical logic is to be found in (Bell & Machover 77) which include everything from soundness and (in-) completeness theorems for sentential logic and full first or• der logic over Boolean algebras, filters, ideals etc. to recursion theory. Recursion Theory-Recursion theory may be viewed as one of the main extensions of mathematical logic. An elementary introduction to recursion theory, the theory of computability and their relations to logic is to be found in (Epstein & Carnielli 89). Then, less wordy but with largely the same content refer to (Cutland 80). The two highly estimated and advanced texts in recursion theory include (Rogers 87) and (Odifreddie 89). Topology (point-set )-Part of the formal framework of this book rests on a particular topological space called the Baire space. It is not really used for anything here as it is in for in• stance (Kelly 96). All the same and for a reader who appreciates a complete picture, an elementary introduction to topology is found in (Mendelson 87). More advanced texts, some of which are advanced way beyond what is needed here, include (Kelley 55), (Cech 69), and (Moschovakis 94). Besides (Kelly 96), ( Schulte & Cory 96) likewise give the interpretation of point-set topology as the theory of inquiry for logically omniscient agents with no limitations on memory capacity. This view turns topol• ogy into a powerful tool for epistemology. The latter paper is a friendly introduction to the connection between topology and epistemology. The paper features topological interpretations of Popper's falsifiability criterion. APPENDIX D 339 2. WWW-RESOURCES Finally there are a lot of resources in philosophy on the web. Below are listed some of the websites with both philosophical and logical material.

• Philosophy Resources on the Internet

The Companion to Modal Operator Theory-A Program in Philosophy Online: http://www.mot.ruc.dk Episteme Links: http://www.epistemelinks.com Garth Kemerling's Philosophy Pages: http://people.delphi.com/gkemerling/ph/ Philosophy in Cyberspace: http://www - personal.monash.edu.aur dey/phil/ Paul Wong's Philosophy Resources: http://arp.anu.edu.aurwongas/ The Stanford Encyclopedia of Philosophy: http://plato.stanford.edu/entries/ The Internet Encyclopedia of Philosophy: http://www.utm.edu/research/iep/ Vincent F. Hendricks Page: http://www.mail.ruc.dkrvincent

• Logic (Mathematics and Computer Science) Resources on the Internet

- Episteme Links-Logic: http://www.epistemelinks.com/Topi/LogiTopi.htm Philosophy in Cyberspace-Logic: http://www.geocities.com/Athens/Acropolis/4393/logic.htm The Stanford Encyclopedia of Philosophy: http://plato.stanford.edu/entries/ - Bertrand Russell Archives: http://www.mcmaster.ca/russdocs/russelll.htm The Logic Daemon: http://logic.tamu.edu/ 340 APPENDIX D Advances in Modal Logic: http://turing.wins.uva.nl;-mdr/AiML/ The Syllogistic Machine: http://home3.swipnet.se;-w - 33039/Syllog.machine.html A Survey of Venn Diagrams: http://www.combinatorics.org/Survey/ds5/VennEJC.html Association of Symbolic Logic: http://www.aslonline.org The Danish Network for the History and Philosophy of Math• ematics: http://mmf.ruc.dk/mathnet/ History of Mathematics-History of Probability and Statistics: http://alephO.clarku.edu;-djoyce/mathhist/ statistics.html MIT Encyclopedia of Cognitive Science: http://mitpress.mit.edu/ Online Computing Dictionary: http://www.instantweb.com;-foldoc/ The 'lUring Test Page: http://cogsci.ucsd.edu;-asaygin/tt/ttest.html Index

Accessibility relation, 77, 132, 186, 221, Axiom of Self-awareness (4), 58, 132, 262 200, 251, 253 (glossary), 325 and forcing, 251, 253 Achinstein, P., 19 Axiom of Truth (T), 199, 247 Adequacy, empirical, 6, 102, 105 and forcing, 247 and time, 233, 238 Axiom of Verum (V), 195, 215 definition of, 104 Axiom of Wisdom (5), 40, 200, 287 Alethic operators, 224 Axiom schema, 41, 287 B (Empirical necessity), 224 Axiom Schema of Epistemic Modesty [J (Temporal necessity), 224 (EMAS), 41, 212, 287 o (Universal necessity), 224 Axiom of Futuristic Knowledge, 233, Algebra, 187 243, 286 Boolean, 187 cr-, 187 Bacon, F., 298 All things being relative, 91 Baire space, 151, 191 Anderson, C., 94 Bayes' Theorem, 147 Angluin, D., 153 Bayesianism, 33, 45, 145 Anti-matter, 94 "almost sure" convergence, 149 Arago, D. F., 298 on countable additivity, 150, 159 Arbitrary inquiry method, 116 on finite additivity, 159 Arcesilaus, 2, 22, 40 on priors, 145 Aristotle, 39, 217, 222 on reliability, 148 Arlo-Costa, H., 259 problem of induction, 151 Armstrong, D., 18, 135 updating, 147 Belief, 163, 248 Aspect, A., 94 (glossary), 322 Assessment, 59 certainty, 163 success criteria descriptive, 163 decision, 59, 115 normative, 163 refutation, 59, 114 rational, strong, 163 verification, 59, 114 Belief Revision (AGM), 30, 33 Assessment method, 114 Bell, J., 96, 338 convergence modulus, 116 Benacerraf, P., 103 decision, limiting, 115 Berkeley, G., 22, 42, 54 reliable, 181 Bicchieri, C., 211 Assessment vs. discovery, 62, 116, 277, Big Bang, 240 287 Bird, A., 18, 64, 256 Astronomy, Ptolemaic vs. Copernican, Blum, L., 153 298 Blum, M., 153 Asymmetry, alethic, 104, 106, 269, 283 Bohm, D., 96 ATI, 84, 87 Bohr, N., 80, 92, 94, 95 Autoepistemic logic, 164 Bonjour, L., 28, 140 Autoepistemic stability, 167 Born, M., 80, 92 Autoepistemology, 162 Boyd, R., 6, 100 Axiom of (.2), 200 Boyle's law, 17 Axiom of (.3), 200 BraUner, T., 222 Axiom of (.4), 200 Brains in vats, 22, 50, 132 Axiom of Cogency (K), 200, 250 Braithwaite, R., 18 and forcing, 250 Branching time, 77, 221, 223 Axiom of Consistency, 248 Brouwer, L. E. J., 224 and forcing, 248 Bull, R., 337 Axiom of F H MV, 220, see also Axiom Bumping pointer architecture, 159 of Futuristic Knowledge, 236 Burgess, J., 337

341 342 Index

Calculus, natural deduction, 42, 47, 54, Correctness relation, 6, 71, 99, 283 127 (glossary), 326 Cantor, G., 53 C((e:, n), h), 115 Capra, F., 37 Correctness, semantic, 190 Cardinality, 157 Cosmic coincidence, 100 Carnap, R., 31, 42, 61 Countable additivity, 149, 159 Carneades, 2, 22, 40 Counterexamples, 137 Carnielli, W., 338 Cresswell, M., 337 Cartesian certainty, 179 Criteria of successful convergence, 59, incorrigibility, 182 154 indubitability, 179 Crucial experiment, 298 infallibility, 179 Curd, M., 30, 32 Cartwright, N., 18 Cut-rule, 226 Cataleptic impressions, 300 Cyphre, Louis, 111 Cech, E., 338 Certainty, 300 Dancy, J., 335 Chagrov, A., 337 Data, 74 Davidson, D., 23 Chellas, B., 337 de Broglie, L., 92 Chisholm, R., 134 De Niro, Robert, xiii, 55, 173 Chronicle, 222 Deductivism, hypothetico-, 56 C,224 Defeasibility, Theory of, 137 c,224 and Gettier cases, 138 Clairvoyance, 137 defeaters, genuine, 139 Closure, 190 defeaters, misleading, 139 Cognitive goal, 1, 6, 31, 104, 128, 286, Demarcation, 19 291 Demon, 48, 140 Complexity, 157,244 Demon world, 48 Compton, A. H., 80 (glossary), 326 Computability theory, 152 Deontic logic, 248 Computational epistemology, see Logi- Depeche Mode, 281 cal reliabilism Descartes, R., 2, 22 Conditional, subjunctive, 48 Diachronic principle, 255, 264, 289 Conditionalization, 147 (glossary),332 Confirmation theory, 60, 175 Diagonalization, 53, 162 Conjection, 46, 159, 264 see also Sextus Empiricus, 40 (glossary), 326 Differential certainty, see Moore, G. E. conjection set, 38 Diodorus Cronos, 222 Conjectures and refutations, 159 Dirac, P., 94 Conservation principles, 58 Discovery, 61 Consistency, 33, 84, 96 abductive (A'D), 63 Consistency (discovery), 120 context of discovery, 62 Constructive empiricism, 103 inductive (I'D), 63 and time, 233 inference to the best explanation Context of discovery, 60 (Il3£),64 Context of justification, 60, see also As• Discovery method, 111 sessment convergence modulus, 112 Convergence argument, 100 non-trivial, 112 Convergence criteria, 55, 157 stabilization modulus, 114 (glossary), 330 Dogmatism, 39, 52, 175, 182 in the limit, 56, 57 Doob martingale convergence theorems, with certainty, 55 150 Convergence, modulus of, 39, 66, 112 Doob, J. L., 150 (glossary), 330 Doxa,41 Correctness, 29, 71, 99 Doyle, J., 165 implication, 283 Duhem's thesis, 298 Index 343 Duhem, P., 22, 298 Fagin, R., 7, 132, 236, 337 Duvall, Robert, 131 Falsificationism, 30 Fan, 38 Earlier-later relation, -<, 224 Fan, (e In), 77 Earman, J., 19, 150, 336 Feldman, R., 143 on Bayesianism, 149 Feyerabend, P., 75 on laws of nature, 19 Feynman, R., 15, 336 Eddington, G., 298 on laws of nature, 245 Eigen variable, 42, 54 on science in the limit, 302 Einstein, A., 17, 61, 62, 88, 298 on scientific method, 61 mass-energy equivalence law, 17, on the K K -thesis, 302 90 on truth, 106 on quantum mechanics, 96 Filter, 127 on the uncertainty principle, 94 Finite additvity, 147 Electron spin, 80 Finite initial segment Empty join, 123 see also Handle, 77 Episteme, 41 Fitting, M., 259 Epistemic axioms, 199 Fitzgerald, G., 89 (glossary), 330 Fix-point, 167 Epistemic logic, 219, 262 Forcing, 4, 66, 213, 243 and epistemology, 262 (glossary),325 and forcing, 132 and logical sufficiency, 131 Epistemic Modesty and Axiom of Wis• in possible worlds dom, see Axiom Schema of W-stabilization, 247 Epistemic Modesty (EMA8) WK Epistemic operators, 174 V3-stabilization, 246 KARA 180 V3K K~SA'174 in time, 7, 243 6 ' Kf'IT, 178 in worlds, 4, 243, 246 K;;RT, 181 Forcing epistemology, 4, 46, 131, 243 Epistemic soundness (assessment), 126, Epistemic logic, 132 261 Probability, 131 Epistemic strength, 199, 237, 260, 284, Virtues, 52 288 Forcing function, 192 (glossary), 331 Forcing quantifiers Epistemic systems, 201 (glossary), 213, 243, 331 84,201 Formal learning theory, 43, 62, see also 84·2,201 Kelly, K. T. 84·3,201 (glossary), 324 84·4,201 Formal Prerequisites 85,201 algebra, xxv Ve-rum, 215 logic, xxiii Epistemic utility, 161 set theory, xxiii Epistemology, 2, 4, 6, 27, 31, 37, 46, 47, Foucault, L., 298 52, 104, 131, 233, 243, 284, Frege, G., 108 300 Frequentism, 143, 144 (glossary), 321 Fresnel, A. J., 298 EPR-paradox, 94 Fresnel, G., 298 Epstein, R., 338 Future contingents, 221 Equivalence, empirical, 104 Evidence, 74 Gabbay, D., 337 Evidence stream, e, 74 Gaifman-Snir theorem, 149 (glossary), 327 Galileo, 298 Evidence vs. data, 74 Game theory, 211 Externalism, 139, 155 Gamut, L. T. F., 337 344 Index GlIrdenfors, P., 336 Hypotheses, 83 General Theory of Relativity, 91, 95 (glossary), 327 Gettier cases, 28, 46, 47, 135, 137, 138, absolute time invariant empirical, 175, 178, 182 84 (glossary), 325 absurd, 295 Gettier, E., 22, 28, 46, 137, 138 e,86 Ginet, C., 137 eventually stable, 294 Global state, 219 'H., h, 83 Glymour, C., 153, 335, 336 initially stable, 295 on confirmation theory, 31 time invariant empirical, 83, 84 on empirical discovery, 61 on primitivism, 21 Idea, abstract, 54 on skepticism, 46 Ideal, 127 Gold, E. M., 57, 62, 152, 337 Imperatives, 31 Goldman, A., 135 categorical, 32 Goodman, M., 335 hypothetical, 31 Goodman, N., 22, 58 Implication, alethic, 7, 106, 283 Goudsmit, S. A., 81 Inducement, 9, 117, 287 Gravitation, 91 (glossary), 328 Gravitation, Newtonian vs. Einsteinian, assessment and discovery, 118 92, 298 discovery and assessment, 117 Greek alphabet, xxvii Induction, problem of, 42, 151 Gutting, G., 62 Induction, straight rule of, 43 Inductive discovery, see Discovery, in• ductive (IV) Hacking, 1., 336 Infallibility (discovery), 122 Halpern, J., 7, 133,201,219,236,337 Inquiry method, see also Assessment Handle, 38 method, see also Discovery gln,77 method set of all handles, w

logical vs. empirical (when vs. (glossary), 322 what), 291 Knowledge, rule of generalization, (N), on alethic logic, 7 186, 195 on background knowledge, 45, 78 Knowledge, scientific, 8, 17 on correctness relations, 6, 104 (glossary), 322 on empirical adequacy, 104 Knowledge, standard definition of, 27 on epistemic logic, 7 (glossary), 322 on epistemic-tense logic, 233 Knowledge, transmissibility, 9, 267, 285 on epistemology, 29 (glossary), 332 on global underdetermination, 24 non-uniform, 269, 276 on knowledge, 7, 285, 288 semi-uniform 1, 268, 274 on knowledge operators, 173 semi-uniform 2, 269, 275 on laws of nature, 8, 17 uniform, 268, 273 on local underdetermination, 25 Kramers, W., 80 on methodology and justification, Kripke, S., 20, 186 28, Ill, 116, 119, 260, 288 Kripke-structure, 186, 221 on modal logic, 185 Kronig, R., 80 on scientific knowledge, 8 Kuhn, T., 22, 79, 291, 299 on skepticism, 45, 287 on paradigm, 79 on success criteria, 60 on values, 299 on tense logic, 7, 222, 225 Kutschera, F. von, 200 on transmissibility, 277 Lakatos, 1., 299 on truth, 106 Language acquisition, 62, 152 portrayal of scientific inquiry, 39 Laplace operator, 94 the formal paradigm, 72 Laudan, L., 6, 30 KaLe Laws of nature, 8, 17, see Symmetry (glossary), 321 (glossary), 322 Kant, 1., 27, 56, 101, 108, 158 and metaphysical universality, 18, Kelley, J., 338 246 Kelly, K. T., 33, 153, see also Logical and temporal universality, 18, 245 Reliabilism, 287, 336 examples of, 17 characterization theorems, 157 necessitarian theories, 18 on assessment and discovery, 63 regularity theories, 18 on certainty convergence, 55 vs. regularities, 18 on countable additivity, 150 Lehrer, K., 137, 155 on formal learning theory, 153 Lemmon, E. J., 199 on knowledge, 154, 155 Lenzen, W., 7, 28, 132, 133, 337 on relativism, 293 on epistemic logic, 133, 262 the logic of reliable inquiry, 152 Levesque, H., 248 Keynes, J. M., 3, 57 Levi, 1., 32, 161, 258, 336 Kitcher, P., 33, 57, 75 Lewis, C. Day, 185 K K-thesis, see Axiom of Self-awareness Lewis, C. 1., 27 (4) Lewis, D., 44, 73, 200, 259 Klein, P. D., 137 Light, 92 Knowledge, 173 Lipton,. P., 42, 64 types of Local state, 219 ARA,180 Locality assumption, 94 A8A,174 Logic, alethic, 224 RIT,178 (glossary), 323 RRT,181 Logic, epistemic, 7, 45, 132, see also Knowledge, background, 71, 77, 120 Epistemic logic (glossary), 326 (glossary), 323 the shrinking property, 78, 237 on rationality, 133 and world fan, [10 I n], 78 on the analysis of knowledge, 133 Knowledge, everyday, 20, 244 Logic, inductive, 60 346 Index

Logic, internet resources, 339 neutrality, 5, 260 Logic, modal, 199 perfect memory (discovery), 120 (glossary), 323 Methodology, 2, 31, 60, 119, 128, 287 Logic, of consistency, 258 (glossary), 323 Logic, of truth, 258 and justification, 28, 260, 287 Logic, tense, 221 categorical, 32 (glossary), 323 classical view, 30 Logical positivism, 74, 101 hypothetical, 32 Logical Reliabilism, 152 Minimimalization, 112 (glossary), 330 (glossary), 330 bumping pointer, 159 MMS, 271 characterization of, 153 Modal language £, 185 characterization theorems, 157 alethic operators, 225 forcing, 153 Boolean operators, 185 on complexity, 157 epistemic operators, 185 on convergence criteria, 157 primitive symbols, 185 on reliability, 154 tense operators, 225 on success criteria, 154 wjJ, 186 Lorentz, H., 82, 89 Modal operator theory, xiv Lorentz-Fitzgerald contraction, 89 (glossary), 324 LRC-circuit, 51 and alethic logic, xiv and epistemic logic, xiv and formal learning, xiv Mach, E., 88 and tense logic, xiv Machover, M., 338 Modal Semantics, 187 Malcolm, N., 162,253,290 denotation function, cp, 187 Martin, E., 36, 58, 153, 337 model, M =< W, cp,:::: >, 187 knowing that one knows, 255 model restriction, 187 Mass, 90 truth-conditions for Master Argument, 222 alethic operators, 225 Mathematics, 103 Boolean operators, 189 Matrix mechanics, 92 epistemic operators, 192 Maxwell, G., 75 stabilization operators, 247 Maxwell, J. C., 88, 299 tense operators, 225 electrodynamics, 88 Modesty, epistemic (EM), 40, 287 McDermott, J., 165 (glossary),325 McGee, V., 265 Modus Ponens, 186, 270 Meaning postulates, 102 Monotonicity, 165 Meaning variance, 293 Monotonicity, non-, 165 Meaning, semantical, 190 Montaigne, M. de, 22 Mendelson, B., 338 Moore's paradox, 162 Merleau-Ponty, M., 35 Moore's principle, 163 Methodological recommendations, 2, 72 Moore, G. E., 162 (glossary), 329 dialectic angles, 162 boost, 5, 260 Moore, R. C., 164,259 boosting, 5, 260 Moschovakis, Y., 338 trivially, 261 Moses, Y., 236, 337 consistency (discovery), 119 Multi-agent systems, 219 consistent expectation (discovery)., Multiple method systems (MMS), 271 121 Myopia, 32, 161 debilitate, 5, 260 debilitation, 5, 260 Nagel, E., 19 epistemic soundness (assessment), Natural kind terms, 20 126, 261 Naturalism, epistemological, 143, 168 infallibility (discovery), 122 Necessitation, rule of, 270 neutral, 5, 260 Nesting, 157 Index 347

Newton, I., 17, 298 Philosophy, internet resources, 338 absolute space, 87 Physics absolute time, 87 high-energy, 74 mechanics, 87 Newtonian, 87 second law of motion, 17 quantum, 80 Newton-Smith, W., 336 Plato, 25, 39, 71, 99, 253 Non-monotonic reasoning, 165 Poincare, H., 91 autoepistemic, 165 Points, 219 default, 165 Poole, D., 62 Noumenon, 71, 101 Popper, K., 9, 42, 159, 297 Nozick, R., 2, 22, 47, 59, 136, 259, 291, Possible world, (e, n), 76 335 (glossary),327 counterfactual reliabilism, 48 Pragmatism, 42, 57, 100 on global underdetermination, 50 Primitivism, 21, see Heidegger, M., on local underdetermination, 51 see Merleau-Ponty, M., see on the KK-thesis, 254 Sartre, J. P. truth-tracking, 49, 59 Prior, A., 221 truth-tracking (weak), 59, 60 Priors, 145 Probability measure, 147 Observables, 75 equally dogmatic, 149 Ockhamistic semantics, 7, 222 Probability theory, 143, 145 Odifreddie, P., 338 Proof structure, 186 transformation rules, 186 0hrstr!2!m, P., 337 Propensitism, 143 Ontology, 31 long-run theories, 143 of convergence, 72 single-case theories, 143 Operators, alethic, see Alethic operators Proper names, 20 Operators, epistemic, see Epistemic op- Putnam, H., 2, 6, 20, 22, 75, 100 erators and brains in vats, 50 Operators, stabilization, see Forcing and formal learning theory, 153 Operators, tense, see Tense operators Opinion, merger of, 146, 150 Quantum field theory, 61, 94 Optics, particle vs. wave, 298 Quantum mechanics, 92 Osherson, D., 34, 36, 58, 62, 152, 153, Quine, W. v. 0., 22, 143, 298 337 formal learning theory, 2, 152 Ramsey principle, 182 knowing that one knows, 255 Ramsey, F., 258 Ramsey, F. P., 135, 182 Pacino, AI, 1, 243 Rationality, 133 Paradigm, 79 Realism, anti-, 6, 100, 238 Paradigm, inductive, 153 knowledge and time, 233 Particle Realism, scientific, 6, 99 meson, 137 knowledge and time, 233 photon, 74, 95 Recursive enumerable (r. e.), 153 positron, 94 Recursive function identification, 45 Pauli, W., 80 Reduction, interpretative, 74 Peano, G., 108 (glossary), 327 Pedersen, S. A., xviii Reichenbach, H., 9, 19, 43, 60, 153, 157 Peirce, C. S., 1, 25, 57, 100, 294 Reiter, R., 165 Perfect memory (discovery), 120 Relativism, epistemic, 141 Perfect recall, 220 Relativism, scientific, 291 Permutation, 270 Relevant possibilities of error, 51, 131 Perspective, 1st vs. 3rd person, 168, 264, Reliabilism, 47, 135 289 classification, 160 (glossary), 331 counterfactual, 47, 48 Phenomena, 71 epistemic, 135 348 Index

logical, 31, 152 finite, xxvii nomic sufficiency, 135 infinite, xxvii probabilistic, 145 uncountable, xxvii relativistic, 293 Sextus Empiricus, 2, 22, 25, see also Research programme, 299 induction, problem of, see Resources, 335 also underdetermination, lo• Alethic Logic, 337 cal, 39, 40, 212 Bayesianism,336 Shrinking property (SP), 78, 229, 231, Belief Revision Theory, 336 234 Epistemic Logic, 337 (glossary), 326 Epistemology, 335 Simplicity, 29 Formal Learning Theory, 336 Single-case problem, 143 Mathematical Machinery, 338 Skepticism, 2, 21, 39, 102, 287, 300 Methodology, 336 (glossary), 322 Recursion Theory, 338 Skepticism, Academic, 2, 22, 40, 212, Science and Philosophy, 336 300 Tense Logic, 337 and the axiom of wisdom (5), 40, Topology, 338 212 Rhodes, R., 336 Skepticism, Pyrrhonian, 22, see also Sex- Rigid designator, 20 tus Empiricus, 41, 212 Rogers, R., 338 Snyder, R., 335 Rorty, R., 22 Social constructivism, 104 Rourke, Mickey, 71 Socrates,r37 Run, 219 Sommerfeld, A. J. W., 80 Sosa, E., 335 Special Theory of Relativity, 88 S4MMS,273 Speed of light, 89 Sankey, R., 28 Spread, 224 Sartre, J. P., 35 Stabilization operators, 246, see also Savage, L., 146 Forcing Schmitt, F., 143 Stabilization, modulus of, 66 Schott Bros. Inc., 20 Stalnaker, R., 167 SchrOdinger equation, 94 State coordinate, 76 SchrOdinger, E., 82, 92, 93 Statistics, classical, 145 Schulte, 0., 33, 58, 153, 336 Stob, M., 62, 152 Science and ATI, 87 Stoicism, 39, 300 Science and background knowledge, 80 Subjectivism, 21, 293, see also Rela• Science and shrinking, 78 tivism Scientific inquiry, 27 Substitution, uniform, 186 (glossary), 321 Success criterion, 59, see also Assess• 1st person perspective, 168, 264, ment and Discovery 289 (glossary), 329 3rd person perspective, 168, 264, Supervenience, 24 289 Suppes, P., 31 gaining truth, avoiding error, 32, Swapman,23 49,60 Swinburne, R., 29, 32 messianic, 32 Swoyer, C., 18 myopic, 32 Symmetry, 19, see Laws of nature, 245 Scientific respect, 71 spatial, 19, 245 Scott, D., 218 temporal, 20, 245 Sea fight tomorrow, 222 Synchronic principle, 255, 264, 289 Segerberg, K., 337 (glossary), 332 Seidenfeld, T., 164 Set Talbott, W. J., 135 co-finite, xxvii Tautology, 190 countable, xxvii Tense operators, 221 Index 349

F (Future-tense), 221 Van Fraassen, B., 29, 102, see also Ade- G (Universal future-tense), 221 quacy, empirical, 336 H (Universal past-tense), 221 and time, 233 P (Past-tense), 221 empirical adequacy, 6 Tense structure, (11',--<), 223 on laws, 19 Tensor-theory, 246 on methodology, 29 Term, theoretical, 63 on observability, 75 Theory formation, 63, see Discovery representation, 105 Theory, Newtonian emission vs. Wave, Vardi, M., 219, 236, 337 298 Venus, 298 Theory-ladenness, 74 Verisimilitude, 100 Thermodynamics, 245 Von Mises, R., 144 Thomas, E. D., 82 von Wright, G. W., 132 Thomson, W., 89 TI,84 Wave Time, moment in, t, 224 equation, 93 Tooley, M., 18 ,p-function, 93 Topology, 191 Wave mechanics, 92 Trajectory, 38 Wave-particle duality, 92 (Trans), 270 Weinberg, S., 18, 100, 245, 301, 336 Transcendental deductions, 158 on clairvoyance, 140 Truth, 6 on laws of nature, 18 (glossary), 322 on philosophy, 35 definition of, 106 on science in the limit, 301 theories of, 106 on symmetry and invariance, 19 tracking, 49, 60 on the K K-thesis, 301 tracking (weak), 60 on truth, 106 Truth and convergence, 99 Weinstein, S., 62, 152 Truth, mathematical, 103 What, 298 Tweety,164 (glossary), 333 When, 293 (glossary), 333 Uhlenbeck, G., 81 Whisk, 188 Uncertainty principle, 93 William of Ockham, 222 Underdetermination, 21, 23 Wittgenstein, L., 22, 35 Underdetermination, global, 23, 44, 50, World an Sich, 101 102, 142, 300 World fan, [c I n], 78 (glossary), 324 (glossary), 327 Underdetermination, local, 25, 151 (glossary), 324 X-rays, 74, 78 Unger, P., 301 Xmas tree, 78 Universal clock, 223 Urbach, P., 18, 145,336 Yukawa, H., 139

Validity, in a model, 190 Zakharyaschev, M., 337 Validity, logical, 190 Zeeman effect (anomalous), 78, 80 Values, 299 Zeeman, P., 86 Nomenclature

Notation II Description Page negation (not) xxxi 1\ conjunction (and) xxxi V disjunction (or) xxxi =} material implication (if. .. then .. .) xxxi {=} biconditional implication (. .. if and only iff .. .) xxxi \:j universal quantifier (for all ... .) xxxi 3 existential quantifier (the exists .. .) xxxi 3! unique quantifier (there exists one and only one .. .) xxxi E set membership xxxi complement xxxi C subset relation xxxii C proper subset relation xxxii n intersection xxxii U union xxxii \ difference xxxii 00 infinity xxxii n intersection of a sequence of sets xxxii U union of a sequence of sets xxxii P(A),2A powerset operation xxxii x Cartesian product xxxii R relation xxxiii B Boolean algebra < B, n, U,-; 0,1 > xxxiii a a-algebra xxxiv f function (~,--~,>-->, ...... ,) xxxiv w,N the set of all natural numbers xxxiv f-+ subjunctive conditional 48

351 352 NOMENCLATURE

Notation II Description Page WW w-sequence of natural numbers 74 c, T, B, ... evidence stream, (co, C1, c2, .. .E:k •.. ) 74 n,n',m, ... state coordinate 76 (c, n) possible world 76 c In initial segment of evidence stream c 77 w B) truth-conditions for biconditional implication 190 (c, n) FM a semantical correctness 190 [A]M semantical meaning of a formula A 190 FMA validity 190 NOMENCLATURE 353 Notation II Description Page

'PM,Ce,n) (KtSA h) truth-conditions for A8A -knowledge 191 f6 forcing function 192 'PM,Ce,n) (KfuTh) truth-conditions for RIT-knowledge 193 'PM,Ce,n) (K~RA h) truth-conditions for ARA-knowledge 193 'PM,Ce,n) (K;;RT h) truth-conditions for RRT-knowledge 194 T axiom of truth 200 K axiom of deductive cogency 200 D axiom of consistency 248 4 axiom of self-awareness (KK-thesis, pos. introspec. ) 200 5 axiom of wisdom (neg. introspection) 200 .2 characteristic axiom of 84·2 200 .3 characteristic axiom of 84·3 200 ·4 characteristic axiom of 84·4 200 84 T,K,4 201 84·2 T,K,4+ .2 201 84·3 T,K,4+ .3 201 84·4 T,K,4 +.4 201 85 T,K,5 201 ('f,-<) tense structure 223 t moment in time 224 c chronicle 224 C the set of all chronicles 224 F singular future operator 225 G universal future operator 225 P singular past operator 225 H universal past operator 225 D universal necessity 225 0 temporal necessity 225 8 empirical necessity 225 AFK Axiom of Futuristic Knowledge 233 St3h V~-stabilization operator 246 St3IC h V~-stabilization operator in K. 247 Srh W-stabilization operator 247 SrICh W-stabilization operator in K. 247 MMS+ 1- 12 multiple method extensions 271 References

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1. G. Schurz: The Is-Ought Problem. An Investigation in Philosophical Logic. 1997 ISBN 0-7923-4410-3 2. E. Ejerhed and S. Lindstrom (eds.): Logic, Action and Cognition. Essays in Philo- sophical Logic. 1997 ISBN 0-7923-4560-6 3. H. Wansing: Displaying Modal Logic. 1998 ISBN 0-7923-5205-X 4. P. Hajek: Metamathematics of Fuzzy Logic. 1998 ISBN 0-7923-5238-6 5. H.J. Ohlbach and U. Reyle (eds.): Logic, Language and Reasoning. Essays in Honour of Do v Gabbay. 1999 ISBN 0-7923-5687-X 6. K. Dosen: Cut Elimination in Categories. 2000 ISBN 0-7923-5720-5 7. R.L.O. Cignoli, I.M.L. D'Ottaviano and D. Mundici (eds.): Algebraic Foundations ofmany- valued Reasoning. 2000 ISBN 0-7923-6009-5 8. E.P. Klement, R. Mesiar and E. Pap: Triangular Norms. 2000 ISBN 0-7923-6416-3 9. V.F. Hendricks: The Convergence of Scientific Knowledge. A View From the Limit. 2001 ISBN 0-7923-6929-7 10. J. Czelakowski: Protoalgebraic Logics. 2001 ISBN 0-7923-6940-8 11. G. Gerla: Fuzzy Logic. Mathematical Tools for Approximate Reasoning. 2001 ISBN 0-7923-6941-6

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