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Confirmation and the ravens paradox 1

Confirmation and the ravens paradox 1. Seminar 3: Philosophy of the Sciences Wednesday, 21 September 2011. Required readings. Peter Godfrey Smith. Theory and Reality. Section 3.1-3.3 (can be downloaded from HKU library) Clark Glymour ‘Why I am not a Bayesian’ (on course website ).

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Confirmation and the ravens paradox 1

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  1. Confirmation and the ravens paradox 1 Seminar 3: Philosophy of the Sciences Wednesday, 21 September 2011

  2. Required readings Peter Godfrey Smith. Theory and Reality. Section 3.1-3.3 (can be downloaded from HKU library) Clark Glymour ‘Why I am not a Bayesian’ (on course website)

  3. Optional readings Paul Horwich ‘WittgensteinianBayesianism’ (on course website) J. A. Cover and Martin Curd ‘Commentary on confirmation and relevance’, section 5.1, pp 627-638 (on course website) Hawthorne and Fitelson. ‘The paradox of confirmation’, Philosophy Compass. 2006. pp 93-113(can be downloaded from HKU library)

  4. Tutorials Tutorials will start on this Friday 23 September Class 1: 1 PM - 2 PM seminar room 305 Class 2: 4 PM – 5 PM seminar room 305 Required reading: ‘The Problem of Induction’, Section I, Chapter 7 of Richard Feldman’s book Epistemologypp 130-141 (on course website) Required reading and seminar handouts must be brought along to tutorials

  5. Questions to be addressed Q1) What is it for evidence E to confirm hypothesis H (that is, what is it for E to be evidence in favour of H)? Q2) Which propositions confirm which propositions? Q3) Which propositions do scientists and ordinary people take to confirm which propositions? Note: Skeptics like Hume might still be interested in Q3

  6. Instances of Q2 • Does a being a black raven confirm all ravens are black? • Does a being a white shoe confirm all ravens are black? • Does there was scratching noises coming from the cupboard last night and the cheese in the cupboard has now disappeared confirm the cheese was eaten by a mouse? • Do our data concerning changes in temperature and climate confirm the theory of man-made global warming?

  7. The instantial theory of confirmation (ITC) For any predicates F and G, and any name a, i) Fa.Ga confirms x(Fx  Gx) (All Fs are Gs); and ii) Fa.~Ga disconfirms x(Fx  Gx). Def: ‘.’ means ‘and’ Problem 1 with ITC: There is no obvious way to extend ITC to deal with plausible cases of confirmation like c) and d) described on slide 6.

  8. Problem 2: The ravens paradox(Carl Hempel) Let ‘R’ symbolise ‘is a raven’, and ‘B’ symbolise ‘is black’. • By ITC, ~Ba.~Ra confirms x(~Bx ~Rx) • x(~Bx  ~Rx) is necessarily equivalent to x(Rx  Bx) • By (1), (2) and (EQ), ~Ba.~Raconfirms x(Rx  Bx)

  9. The ravens paradox(cont) (EQ) If E confirms H1, and H1 is necessarily equivalent to H2, then E confirms H2 (PC) therefore follows from (ITC) and (EQ). But how can a white shoe provide evidence that all ravens are black?? (PC) That a is non-black and non-raven confirms that all ravens are black

  10. A response to the paradox Since ‘(xFx).(x(Fx  Gx))’ is a better symbolisation of ‘All Fs are Gs’ than ‘x(Fx  Gx)’, ITC should be replaced with ITC*. ITC*) For any predicates F and G, and any name a, i) Fa.Ga confirms (xFx).(x(Fx  Gx)); and ii) Fa.~Gadisconfirms (xFx).(x(Fx  Gx)).

  11. A problem with this response Given (SPC), (ITC*) entails (PC). (SPC) If E confirms H1, and H1 entails H2, then E confirms H2. Argument: • By ITC*, ~Ba.~Ra confirms (x~Bx).(x(~Bx ~Rx)) • (x~Bx).(x(~Bx  ~Rx)) entails x(Rx  Bx) • By (1), (2) and (SPC), ~Ba.~Ra confirms x(Rx  Bx)

  12. Hempel’s response to the paradox (PC) is true. It seems false because a) We falsely think that ‘x(Rx  Bx)’ is only about ravens, when in fact it is about all objects, as it is reformulation as ‘x(~Rx v Bx)’ reveals. b) We falsely think that PC is false, because we fail to distinguish it from the false PC*. (PC*) That a is non-black and non-raven confirms that all ravens are black, given the background information that a is a non-raven.

  13. Two kinds of confirmation relation 3-place: E confirms H relative to background knowledge K 2-place: E confirms H absolutely Connection between them: E confirms H absolutely iff E confirms H relative to no information (or relative to a logical truth T) ITC is intended as a theory of absolute confirmation

  14. The hypothetico-deductive theory of confirmation (HDT) • E confirms H if E can be divided into two parts, E1 and E2, such that a) E1 does not entail E2, but b) the conjunction of H and E1 does entail E2. • E disconfirms H if E entails ~H • Otherwise E neither confirms or disconfirms H In favour of HDT: HDT fits many episodes in the history of science well.

  15. Problems with HDT Problem 1: HDT entails PC, and hence faces the ravens paradox Problem 2: HDT cannot account of confirmation of statistical theories such as the hypothesis that anyone who smokes has a 25% chance of developing lung cancer.

  16. Problem 3: Irrelevant conjunction • Suppose evidence E, made up of E1 and E2, is such that i) E1 does not entail E2, but ii) H.E1 does entail E2. • Then H.S.E1 entails E2, where S is any hypothesis at all. • Hence, according to HDT, E confirms H.S. • Moreover, by SPC, E confirms S. But S can be anything at all!

  17. The probability raising theory of confirmation PRT for absolute confirmation: • E confirms H iff P(H|E) > P(H) • E disconfirms H iff P(H|E) < P(H) where ‘P(H)’ means ‘the probability of H’, and ‘P(H|E)’ means ‘the probability of H given E’.

  18. The probability raising theory of confirmation (cont) PRT for relative confirmation: • E confirms H relative to background knowledge K iffP(H|E.B) > P(H|K) • E disconfirms H relative to background knowledge K iffP(H|E.B) < P(H|K)

  19. Quantitative probability raising theories of confirmation Def: c(H,E,K) = the degree to which E confirms H relative to background knowledge K A popular account of c among PRT theorists: D) c(H,E,K) = P(H|E.K) - P(H|K)

  20. Good’s response to the raven paradox Good’s claim: Whether E=Ra.Ba confirms H= x(Rx  Bx) relative to K depends on what the background knowledge K is. Example: E won’t confirm H if K is the knowledge that either • There are 100 black ravens, no non-black ravens and 1 million other birds • There are 1000 black ravens, 1 white raven, and 1 million other birds

  21. Good’s example E won’t confirm H if K is the knowledge that either • There are 100 black ravens, no non-black ravens and 1 million other birds • There are 1000 black ravens, 1 white raven, and 1 million other birds In this case P(E|H.K) < P(E|~H.K), from which it can be proved that P(H|E.K) < P(H|K).

  22. Good on absolute confirmation Good also claimed that it might be that Ra.Ba fails to confirm x(Rx  Bx) absolutely. Discuss unicorn case.

  23. The standard Bayesian strategy to solve the ravens paradox Show that given plausible assumptions about our background knowledge, Ra.Ba confirms x(Rx  Bx) relative to K more than ~Ra.~Ba. The result if established can then be used to explain why (PC) seems false.

  24. Hawthorne and Fitelson’s attempt Given the assumptions about K given by (K-ass), H+F show that the following theorem holds. K-ass: i) P(H|Ba.Ra.K), P(H|~Ba.~Ra.K), and P(~Ba.Ra|K) aren’t 0 or 1; and ii) P(~Ba|K) > P(Ra|K). Theorem: If P(H|Ra.K) ≥ P(H|~Ba.K), then P(H|Ba.Ra.K) > P(H|~Ba.~Ra.K).

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