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James Hawthorne Workshop: Conditionals Counterfactuals and Causes In Uncertain Environments

A Qualitative Logic of Comparative Evidential Support Strength An Extension of the Koopman -Keynes Approach to The qualitative Logic of Comparative Evidential Support Underlying the Probabilistic Logic of the Popper Functions. James Hawthorne

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James Hawthorne Workshop: Conditionals Counterfactuals and Causes In Uncertain Environments

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  1. A Qualitative Logic of Comparative Evidential Support StrengthAn Extension of the Koopman-Keynes Approach to The qualitative Logic ofComparative Evidential SupportUnderlying the Probabilistic Logic of the Popper Functions James Hawthorne Workshop: Conditionals Counterfactuals and Causes In Uncertain Environments Düsseldorf 19/5/2011 – 22/5/2011

  2. Main Question Why MeasureEvidential Support Probabilistically? That is: Why measure the support of H by Eon a scale of real numbers between 0 and 1 that satisfies typical probabilistic axioms? More specifically, I advocate a version of Bayesian Confirmation Theory where the confirmation functions are so-called Popper Functions, and I think that evidential support is often best represented by sets of Popper Functions, because individual probabilistic support functions seem to be overly precise measures ofevidential support. But isn’t the fact that probabilistic measures of support are overly precise just a symptom of the fact that probability functions are really ill-suited as measures of confirmation? More generally, why think that a probabilistic measure is at all the right sort of thing to use in measuringevidential support ? The qualitative logic of evidential support I’ll present offers an answer to these questions. =============================================================================== More generally, if the Popper Functions provide a useful version of the notion of conditional probability used in a given project X, then this qualitative logic may provide a useful theoretical account of what the probability functions in project X really represent.

  3. Why measure the support of H by E with Probability Functions? Several Kinds of Answers: 1. Show what results follow: that the resulting probabilistic system supplies an account of confirmation that has intuitively desirable properties. 2. Argue from fundamental principles: (i) argue that the probabilistic axioms themselves are reasonable constraints on a measure of evidential support • (e.g. analyses of axioms ; Dutch Book arguments ; etc.) OR (ii) argue that the axiomatic system captures some deeper, more fundamental logic of evidential support that is itself a compelling account (e.g. Representation Theorem arguments).

  4. Probabilistic Confirmation Functions Conditional Probability Functions (equivalent to those proposed by Karl Popper, Logic of Scientific Discovery, 1959) 0. for some E, F, P[F|E] 1 1. 0 <P[A|B]< 1 2. if B|=A, then P[A|B] = 1 3. if C|=B and B |= C, then P[A|B]=P[A|C] 4. P[(AB)|C] = P[A|(BC)]  P[B | C] 5. if C|=~(AB), thenP[(AB)|C]=P[A|C]+P[B | C] orP[D | C]= 1for all D

  5. Probabilistic Confirmation Functions Popper Probability Functions (only for Sentential Logic Languages) (Karl Popper, Logic of Scientific Discovery, 1959) 0. each P[A|B] a real number and for some E, F, G, H, P[F|E] P[G|H] 1. P[A|A] P[B|B] 2. P[A|(BC)]P[A|(CB)] 3. P[A|C]P[(AB)|C] 4. P[(AB)|C] = P[A | (BC)]  P[B|C] 5. P[A|B] + P[~A|B] = P[B|B] unless P[D|B] = P[B|B] for all D

  6. Probabilistic Confirmation Functions Popper-Field Probability Functions (Extends Popper Functions to Predicate Logic Languages) (Hartry Field, “Logic, Meaning, and Conceptual Role”, JP, 7/1977) Define a PF-ClassM to be a set of functions Psuch that 0. Pis a Popper Function (i.e. satisfies the previous rules) 1. P[((Fc1Fc2)...Fcm) |B] P[xFx|B] 2. if r > P[xFx|B](for r a positive real), then for some P in M defined on a name extension of P’s language L, there are names e1, ..., enin P’s language such that rP[((Fe1Fe2)...Fen)|B] (i.e. P[xFx|B] is a greatest lower bound on values of • P[((Fe1Fe2)...Fen)|B] inPF-ClassM) The Popper-Field Functions on a language and its name extensions is just the union of allPF-Classes M on that language and its name extensions.

  7. Why measure the support of H by E with Probability Functions? Several Kinds of Answers: 1. Show what results follow: that the resulting probabilistic system supplies an account of confirmation that has intuitively desirable properties. 2. Argue from fundamental principles: (i) argue that the probabilistic axioms themselves are reasonable constraints on a measure of evidential support • (e.g. analyses of axioms ; Dutch Book arguments ; etc.) OR (ii) argue that the axiomatic system captures some deeper, more fundamental logic of evidential support that is itself a compelling account (e.g. Representation Theorem arguments).

  8. A More Fundamental Qualitative Logic of Evidential Support • Want to characterize comparative support relations≽of form • H1| E1≽H2| E2 : • conclusionH1is supported bypremise(s)E1 • at least as strongly as • conclusionH2is supported bypremise(s)E2 • (under interpretation ) Think of this notion of comparative support strength under an interpretation of a language (for predicate logic) as a basic semantic notion – i.e. as basic in the same way that the notion of truth under an interpretation of a language (for predicate logic) is a basic semantic notion. What semantic rules should all comparative support relations,≽ , obey? That is, what semantic rules should constrain how logical terms behave within a comparative support relation≽? • (in the way that semantic rules constrain how logical terms behave w.r.t the notion oftruth under interpretation)

  9. The Main Idea The semantic rules that govern ≽ should be intuitively plausible constraints on comparisons of evidential support strength. Each such relation ≽should be (at least) a partial order -- perhaps not every pair of cases of support strength may be directly comparable. The semantic rules that govern ≽ should (if possible) not presuppose the notion of logical entailment – rather, logical entailment should fall out of the relations ≽ as a special case. It should turn out thatquantitative conditional probability functions (i.e. Popper Functions) merely provide a convenient way to represent the comparative support relations.

  10. A Few Definitions Comparative Support Relationunder an interpretation: • H1|E1 ≽H2|E2 : H1is supported byE1at least as strongly asH2issupported byE2 Given acomparative support-strength relation ≽, define four associated relations: • (1) H1|E1 ≻H2|E2 abbreviates H1|E1 ≽H2|E2 and notH2|E2 ≽H1|E1 : • H1is supported byE1more strongly thanH2issupported byE2 • (2) H1|E1H2|E2 abbreviates H1|E1 ≽H2|E2 and H2|E2 ≽H1|E1 : • H1is supported byE1to the same extent thatH2issupported byE2 • (3) H1|E1 ≺≻H2|E2 abbreviates notH1|E1 ≽H2|E2 and notH2|E2 ≽H1|E1 : • the support-strength forH1byE1is not determinately comparable tothat ofH2byE2. • (4) EH abbreviates H|E≽E|E : • Esupportively entailsH It will turn out that: 1. each supportive entailment relation(i.e. for each ≽) is a so-called Rational Consequence Relation 2. indeed, each Rational Consequence Relation is represented by • a supportive entailment relation(for some ≽) 3. Logical Entailment is justsupportive entailment for every ≽

  11. The Main Idea It should turn out thatquantitative conditional probability functions (i.e. Popper Functions) are merely a convenient way to represent thecomparative support relations. (That’s the main point of this project.) That is, ... The rules that constrain the comparative support relations should be probabilistically soundin that each Popper Function should satisfy them. For each Popper Function P, define the corresponding comparative relation≽ such that, for all sentences H1, E1,H2, E2, H1 | E1≽H2| E2iffP[H1 | E1] P[H2| E2]. Then, for each Popper Function P, we want its corresponding comparative relation≽ to be a comparative support relations – i.e. to satisfy the rules for the comparative support relations. And we want the rules that constrain the comparative support relations to beprobabilistically complete in the sense that each comparative support relation≽ that satisfies the specified rules should be representable by a Popper Function P.

  12. The Main Idea The rules that constrain the comparative support relationsshould beprobabilistically sound and complete in that each Popper Function should corresponds to a comparative support relation, and each comparative support relation ≽ (that satisfies the rules) should be representable by a Popper Function P. ================================================================= The strongest version ofprobabilistic representation for acomparativesupport relation would be this: Strong Probabilistic Representation: • For each comparative support relation ≽(that satisfies the specified rules) there is a (unique) Popper function P such that, for all H1, E1,H2, E2, • P[H1 | E1] P[H2| E2] if and only if H1| E1≽H2| E2 . Such a representation result will be forthcoming, but only for those comparative support relations≽ that provide a complete order on comparative support-strength – i.e., only for those ≽ such that for all H1, E1,H2, E2, • either H1| E1≽H2 | E2 or H2| E2≽H1| E1 (complete comparability) However, that is a very strong constraint on comparative support strength. I’ll characterize comparative support relations that need not satisfy this condition. Strong Probabilistic Representation will also require an Archimedean Condition – more on that next.

  13. The Main Idea The rules that constrain the comparative support relationsshould beprobabilistically sound and complete in that each Popper Function should corresponds to a comparative support relation, and each comparative support relation ≽ (that satisfies the rules) should be representable by a Popper Function P. ========================================================================== A weaker notion of probabilistic representation forcomparativesupport relationswould be this: Moderate Probabilistic Representation (but not permitting infinitesimally greater support): • For each comparative support relation ≽ there is a Popper Function P such that for all H1, E1,H2, E2, • (1) if H1|E1≻H2|E2, then P[H1|E1] > P[H2|E2]; • (2) if H1|E1H2|E2, then P[H1|E1]=P[H2|E2]. (1) and (2) are jointly equivalent to the following conditions: • if P[H1|E1]>P[H2|E2], then H1|E1 ≻H2|E2 or H1|E1≺≻H2|E2. if P[H1|E1]=P[H2|E2], then H1|E1H2|E2 or H1|E1≺≻H2|E2. The fact thatcomparative support strength is a partial order implies that probabilistic representationsof evidential support tend to be overly precise. Thus, evidential support is often represented by a set of conditional probability functions rather than by a single conditional probability function. In order to satisfy condition (1) the comparative support relations have to satisfy an Archimedean Condition – e.g. If H1|E1≻H2|E2, then for an integer n  2 there are n sentences S1, ..., Sn such that: notE2~E2, E2(S1...Sn), (for distinct i, j) E2~(Si·Sj), Si|E2Sj|E2, and H1|E1≻(SiH2)|E2

  14. The Main Idea The rules that constrain the comparative support relationsshould beprobabilistically sound and complete in that each Popper Function should corresponds to a comparative support relation, and each comparative support relation ≽ (that satisfies the rules) should be representable by a Popper Function P. ========================================================================== An even weaker notion of probabilistic representation forcomparativesupport relationswill apply more generally to all such relations – including those that provide only Non-Archimedean partial orders on support strength. Weak Probabilistic Representation (permittinginfinitesimally greater support): • For each comparative support relation ≽ there is a Popper Function P such that for all H1, E1,H2, E2, • (1) if H1|E1≻H2|E2, then P[H1|E1]  P[H2|E2]; • (2) if H1|E1H2|E2, then P[H1|E1]=P[H2|E2]. (1) and (2) are jointly equivalent to the following condition: • if P[H1|E1]>P[H2|E2], then H1|E1 ≻H2|E2 or H1|E1≺≻H2|E2. Weak Probabilistic Representation still requires an axiom about partitions, but one that’s much weaker than the previous Archimedean Condition – e.g. an Arbitrarily Large Partitions Condition: For each integer m  2 there is an integer n  m such that for n sentences S1, ..., Sn and some sentence E: notE~S1, (for distinct i, j) E~(Si·Sj), Si|ESj|E.

  15. The Comparative Support Rulesw/ Classical Logical Entailment 0. for some H1, E2, H2, E2, H1|E1≻H2|E2 (non-triviality) 1. B|B ≽ H|E (maximality) 2. If H1|E1≽ H2|E2 , H2|E2≽ H3|E3, then H1|E1≽ H3|E3 (transitivity) 3. If E1|= E2, E2|= E1, then H|E1≽ H|E2 (Antecedent L-Equivalence)! 4. If H2|= H1, then H1|E ≽ H2|E (Consequent L-Consequence) ! {rule 4 yields Reflexivity: H|E ≽ H|E} 5. If H1|E1≽ H2|E2, then ~H2|E2≽ ~H1|E1 or E1D (all D) (negation-symmetry)! 6. If H1|(E1F) ≽ H2|E2 and H1|(E1~F) ≽ H2|E2, then H1|E1≽ H2|E2 (alternate presumption)* 7. If EH, then E~F or (EF)H (Rational Monotonicity)

  16. The Comparative Support Rulesw/o Classical Logical Entailment 0. for some H1, E2, H2, E2, H1|E1≻H2|E2 (non-triviality) 1. B|B ≽ H|E (maximality) 2. If H1|E1≽ H2|E2 , H2|E2≽ H3|E3, then H1|E1≽ H3|E3 (transitivity) 3. H|(E1E2) ≽ H|(E2E1) (Antecedent Commutivity) ! 4.1 (HH)|E ≽ H|E (Consequent Repetition: CR) ! 4.2 H1|E1≽ (H1H2)|E1 (Simplification) ! {4.1-4.2 jointly yield Reflexivity: H|E ≽ H|E} 5.1 If H1|E1≽ ~H2|E2, then H2|E2≽ ~H1|E1 or E1 D (all D) 5.2 If ~H1|E1≽ H2|E2, then ~H2|E2≽ H1|E1 or E1 D (all D) (negation-symmetry) ! 6. If H1|(E1F) ≽ H2|E2 and H1|(E1~F) ≽ H2|E2, then H1|E1≽ H2|E2 (alternate presumption)* 7. If EH, then E~F or (EF)H (Rational Monotonicity)

  17. The Comparative Support Rules 8. If H1|(A1·E1) ≽ H2|(A2·E2), A1|E1≽ A2|E2, then (H1·A1)|E1≽ (H2·A2)|E2 (composition) 8.1 If H1|(A1·E1) ≻ H2|(A2·E2), A1|E1≽ A2|E2, E2 ~(H2·A2), then (H1·A1)|E1≻ (H2·A2)|E2 8.2 If H1|(A1·E1) ≽ H2|(A2·E2), A1|E1≻A2|E2, E2 ~(H2·A2), then (H1·A1)|E1≻ (H2·A2)|E2 8.3 If H1|(A1·E1) ≺≻ H2|(A2·E2), E2 ~(H2·A2), then (H1·A1)|E1≺≻ (H2·A2)|E2 8.4 If A1|E1≺≻ A2|E2, E2 ~(H2·A2), then (H1·A1)|E1≺≻ (H2·A2)|E2 ======================================================================== If (H2·A2)|E2≽ (H1·A1)|E1, A1|E1≽ A2|E2, E2 ~(H2·A2), then H2|(A2·E2) ≽ H1|(A1·E1) If (H2·A2)|E2≽ (H1·A1)|E1, H1|(A1·E1) ≽ H2|(A2·E2), E2 ~(H2·A2), then A2|E2≽ A1|E1 (decomposition)

  18. The Comparative Support Rules 9. If H1|(A1·E1)≽ A2|E2, A1|E1≽ H2|(A2·E2), then (H1·A1)|E1≽ (H2·A2)|E2 (composition) 9.1 If H1|(A1·E1) ≻ A2|E2, A1|E1≽ H2|(A2·E2), E2 ~(H2·A2), then (H1·A1)|E1≻ (H2·A2)|E2 9.2 If H1|(A1·E1) ≽ A2|E2, A1|E1≻ H2|(A2·E2), E2 ~(H2·A2), then (H1·A1)|E1≻ (H2·A2)|E2 9.3 If H1|(A1·E1) ≺≻ A2|E2, E2 ~(H2·A2), then (H1·A1)|E1≺≻ (H2·A2)|E2 ========================================================================== If (H2·A2)|E2≽ (H1·A1)|E1, A1|E1≽ H2|(A2·E2), E2 ~(H2·A2), then A2|E2≽ H1|(A1·E1) If (H2·A2)|E2≽ (H1·A1)|E1, H1|(A1·E1) ≽ A2|E2, E2 ~(H2·A2), then H2|(A2·E2) ≽ A1|E1 (decomposition)

  19. Comparative SupportBayes’ Theorems Bayes’ Theorem 1: Suppose B~H1. If E|(H1B) ≻ E|(H2B), H1|B ≽ H2|B, then H1|(EB) ≻ H2|(EB). Bayes’ Theorem 2: Suppose B~H1, B~H2, B~(H1H2). If E|(H1B) ≻ E|(H2B), then H1|(EB(H1H2)) ≻ H1|(B(H1H2)) and H2|(B(H1H2)) ≻ H2|(EB(H1H2)). ============================================================== Compare: Suppose P[H1|B] > 0, P[H2| B] > 0, P[H1H2|B] = 0. If P[E | H1B]>P[E | H2B], then P[H1 | EB(H1H2)]P[H1 | EB]P[E | H1B]P[H1 | B] ------------------------- = --------------- = -------------- ------------ P[H2 | EB(H1H2)]P[H2 | EB]P[E | H2B]P[H2 | B] P[H1 | B]P[H1 | B(H1H2)] > ------------ = ----------------------- P[H2 | B]P[H2 | B(H1H2)]

  20. The Comparative Support Rules Extendibility Rule(s): ≽is extendable to a comparative support relation ≽ (i.e. whenever A|B≻C|D, A|B≻C|D ; whenever A|BC|D, A|BC|D ) such that ≽ satisfies all the other rules (1-9 thus far), and also satisfies the following rules: 10.H1|E1≽ H2|E2 or H2|E2≽ H1|E1 (complete comparability) ========================================================= Example of an relation ≽ that satisfies rules 1-9 but is not extendable to a relation ≽ that satisfies Complete Comparability can have H|(BE)≻H|BandH|(B~E)≻H|B {but only if E|B≺≻E|(HB)} ========================================================== Theorem (from:1-5, 7-9): H|B ≺≻ H|(BE) or H|B ≺≻ H|(B~E) or E|B≺≻E|(HB) or H|(BE) ≽ H|B ≽ H|(B~E) or H|(B~E) ≽ H|B ≽ H|(BE).

  21. The Comparative Support Rules Extendibility: ≽is extendable to a comparative support relation ≽ (i.e. whenever A|B≻C|D, A|B≻C|D ; whenever A|BC|D, A|BC|D ) that satisfies all the other rules, and also satisfies the following rules: 10.H1|E1≽ H2|E2 or H2|E2≽ H1|E1 (complete comparability) {follows that} For any integer n > 1, if A1, ..., An, and B1, ..., Bn are such that • C(A1...An), C~(AiAj),An|C≽...≽ A1|C, not C~C, • D(B1...Bn), D~(BiBj),Bn|D≽...≽ B1|D,not D~D, thenAn|C≽ B1|D. (subdivision)* 11. For each integer m  2 there is an integer n  m such that for n sentences S1, ..., Sn and some sentence E: notE~S1, (for distinct i, j) E~(Si·Sj), Si|ESj|E. (arbitrarily large partitions) or, alternatively, 11.+If H1|E1≻H2|E2, then for some integer n  2 there are n sentences S1, ..., Sn such that: notE2~S1, (for distinct i, j) E2~(Si·Sj), Si|E2Sj|E2, E2(S1...Sn), and H1|E1≻(SiH2)|E2. The Representation Theorems hold for all comparative support relations that satisfy these rules. Strong Representation holds for each ≽satisfying rules 0-10 and 11+. Moderate Representation holds for each ≽satisfying rules 0-9 and extendable to rules 10-11+. Weak Representation holds for each ≽satisfying rules 0-9 and extendable to 10-11.

  22. The Comparative Support Rules The version of the rules that does not presuppose the deductive logical entailment relation also needs explicit rules for quantifiers. The following rules are analogous to those proposed by Hartry Field for the Popper Functions. (Here ‘CS’ stands for “Comparative Support”.) Define a CS-ClassM to be a set of relations ≽such that 0. ≽satisfies the comparative support rules 1-9. 1. ((Fc1Fc2)...Fcm) |B≽ xFx| B • if A|B≻xFx|C, then for some ≽ in M defined on a name extension of ’s language L, there are names e1, ..., en in ≽’s language such that • A|B ≽((Fe1Fe2)...Fen)|C (unless for all n  2, for any n sentences S1, ..., Sn such that notC~S1, (for distinct i, j) C~(Si·Sj), Si|CSj|C, C(S1...Sn), we have (SixFx)|C≻A|B). 2. if ((Fe1Fe2)...Fen)|C ≻ A|B for all ≽ a name extension of ≽in M, for all names e1, ..., en in ≽’s language, then xFx|C ≽A|B (unless A|B≻xFx|C, butfor all n  2, for any n sentences S1, ..., Sn such that notC~S1, (for distinct i, j) C~(Si·Sj), Si|CSj|C, C(S1...Sn), we have (SixFx)|C≻A|B). {need both 1 and 2 only because the relation ≽ need not be a complete order} The Comparative Support Relations on a language and its name extensions is just the union of allCS-Classes M (for that language and its name extensions)that are also extendableso as to satisfy rules 10 and 11 (rule 11+ for the ArchimedeanComparative Support Relations).

  23. Representation Theorems The rules that constrain the comparative support relationsareprobabilistically sound and complete in that each Popper Function corresponds to a comparative support relation, and each comparative support relation ≽ (that satisfies the rules) is representable by a Popper Function P as follow: Strong Probabilistic Representation (for Completely Comparable Archimedean CSRs): For each comparative support relation ≽that satisfies 0-10 and 11+, there is a unique Popper function P such that, for all H1, E1,H2, E2, P[H1 | E1] P[H2| E2] if and only if H1| E1≽H2| E2 . Moderate Probabilistic Representation (for Archimedean CSRs): For each comparative support relation ≽ that satisfies 0-9 and is extendable to 10-11+, there is a Popper Function P such that for all H1, E1,H2, E2, • (1) if H1|E1≻H2|E2, then P[H1|E1] > P[H2|E2]; • (2) if H1|E1H2|E2, then P[H1|E1]=P[H2|E2]. (1) and (2) are jointly equivalent to the following conditions: • if P[H1|E1]>P[H2|E2], then H1|E1 ≻H2|E2 or H1|E1≺≻H2|E2. if P[H1|E1]=P[H2|E2], then H1|E1H2|E2 or H1|E1≺≻H2|E2. Weak Probabilistic Representation (for CSRs that may permitinfinitesimally greater support): For each comparative support relation ≽ that satisfies 0-9 and is extendable to 10-11,there is a Popper Function P such that for all H1, E1,H2, E2, • (1) if H1|E1≻H2|E2, then P[H1|E1]  P[H2|E2]; • (2) if H1|E1H2|E2, then P[H1|E1]=P[H2|E2]. (1) and (2) are jointly equivalent to the following condition: • if P[H1|E1]>P[H2|E2], then H1|E1 ≻H2|E2 or H1|E1≺≻H2|E2.

  24. EXAMPLE: De Finetti Lottery–like cases When we permit a broad enough class of CSRs such that we merely have Weak Probabilistic Representation (where some CSRs may permitinfinitesimally greater support): For each comparative support relation ≽ that satisfies 0-9 and is extendable to 10-11,there is a Popper Function P such that for all H1, E1,H2, E2, • (1) if H1|E1≻H2|E2, then P[H1|E1]  P[H2|E2]; • (2) if H1|E1H2|E2, then P[H1|E1]=P[H2|E2]. (1) and (2) are jointly equivalent to the following condition: • if P[H1|E1]>P[H2|E2], then H1|E1 ≻H2|E2 or H1|E1≺≻H2|E2. Then, there exist CSRs, ≽ , such that: For a countably infinite set of sentences {S1, ..., Sn, ...} and a sentence F: notF~S1 (i.e. S1|F≻~F|F ), (for each distinct i, j) F~(Si·Sj), Si|FSj|F. Follows that for this CSRs, ≽ : For each integer n  2, for the n sentences S1, ..., Sn and for a sentence E [i.e. for the sentence (F·(S1S2...Sn)) ] we have: notE~S1 (i.e. S1|E≻~E|E ) and (for distinct i, j) E~(Si·Sj), E(S1S2...Sn) Si|ESj|E.

  25. A Qualitative Logic of Comparative Evidential Support StrengthAn Extension of the Koopman-Keynes Approach to The qualitative Logic ofComparative Evidential SupportUnderlying the Probabilistic Logic of the Popper Functions James Hawthorne Workshop: Conditionals Counterfactuals and Causes In Uncertain Environments Düsseldorf 19/5/2011 – 22/5/2011

  26. System R = P + (RM)The Preferential and Rational Consequence RelationsWeaker than Usual Axioms 0. for some E, F, E |/~F (Nontriv) 1. A|~A (Reflex) 2. if C|=B, B |= C,B |~ A, then C |~ A (LCE) • 3. if C |~ B, B |= A, then C |~ A (RW) 4. if (CB)|~A and (C~B) |~ A, then C |~ A (WOR) 5. if C |~ (BA), then (CB) |~ A (VCM) ============================================================================================================================================ {O: if (C~B)|~B,C |~ A, then C |~ (BA) (WAND)} P: 6. if C|~ B,C|~ A, then C |~ (BA) (AND) ============================================================================================================================================= {Q: if C|~ A and (C~B) |/~ A, then (CB) |~ A (NR)} R: 7. if C |~ A,C|/~~B, then (CB) |~ A (RM)

  27. Probabilistic Confirmation Functions Conditional Probability Functions (equivalent to those proposed by Karl Popper, Logic of Scientific Discovery, 1959) 0. for some E, F, P[F|E] 1 1. 0 <P[A|B]< 1 2. if B|=A, then P[A|B] = 1 3. if C|=B and B |= C, then P[A|B]=P[A|C] 4. P[(AB)|C] = P[A|(BC)]  P[B | C] 5. if C|=~(AB), thenP[(AB)|C]=P[A|C]+P[B | C] orP[D | C]= 1for all D

  28. Probabilistic Confirmation Functions Conditional Probability Functions (JaninaHosiasson-Lindenbaum, JSL, 12/1940) 1. 0 <P[A|B]< 1 {and for some E, F, P[F|E] 1} 2. if B|=A, then P[A|B] = 1 3. if C|=B and B |= C, then P[A|B]=P[A|C] 4. P[(AB)|C] = P[A|(BC)]  P[B | C] 5. if C|=~(AB), thenP[(AB)|C]=P[A|C]+P[B | C] unless|=~C {thus, unlessP[D | C]= 1for all D}

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