WHO’S AFRAID OF UNDERMINING? Why the Principal Principle need not contradict Humean Supervenience Peter B. M. Vranas The University of Michigan

OVERVIEW Part 1: The two theses Ê The Principal Principle (PP) Ë Humean Supervenience (HS) Part 2: Lewis’s argument (and its failure) Ê Step A: HS entails undermining Ë Step B: Undermining contradicts PP Part 3: Three attempts to rescue the contradiction (and their failure)

THE PRINCIPAL PRINCIPLE Basic idea. (MP) Cr(A| )=x. Generalization of MP. (PP) If E is admissible with respect to, then: Cr(A|E )=x. Consequence of PP. Let T=. (EP) If ET is admissible with respect to, then: Cr(A|ET)=f(A).

HUMEAN SUPERVENIENCE Analogy: Among dot-matrix pictures, no two differ without differing in the point-by-point arrangement of dots. Formulation: (HS) Among worlds like ours, no two differ without differing in the spacetime- point-by-spacetime-point arrangement of local properties. Support: Lewis has argued that HS holds for laws of nature, counterfactuals, causation,... Restriction to worlds like ours: crucial.

PART 2 Part 1: The two theses Ê The Principal Principle (PP) Ë Humean Supervenience (HS) Part 2: Lewis’s argument (and its failure) Ê Step A: HS entails undermining Ë Step B: Undermining contradicts PP Part 3: Three attempts to rescue the contradiction (and their failure)

LEWIS’S ARGUMENT, STEP A: HS ENTAILS UNDERMINING H: the past and present history of the world. F: a possible future with positive chance. Given HS, HF determines (among worlds like ours) a chance function f’. If f’ differs from the actual chance function f, then F is an under- mining future (F undermines T= ). Theorem: HS entails that for every time instant there is an undermining future; i.e., there is F, possible and with positive chance, such that: (UND) HF entails ~T (among worlds like ours).

LEWIS’S ARGUMENT, STEP B: UNDERMINING CONTRADICTS PP (EP)If ET is admissible, then: Cr(A|ET)=f(A). (UND)HF entails ~T (among worlds like ours). Suppose F undermines T (so f(F)>0) and HT is admissible. Then: Ê From EP: Cr(F|HT)=f(F)>0. Ë From UND: HFT impossible, so Cr(F|HT)=0. Contradiction? No! HFT need not be false in every possible world; only in worlds like ours.

PART 3 Part 1: The two theses Ê The Principal Principle (PP) Ë Humean Supervenience (HS) Part 2: Lewis’s argument (and its failure) Ê Step A: HS entails undermining Ë Step B: Undermining contradicts PP Part 3: Three attempts to rescue the contradiction (and their failure)

FIRST ATTEMPT TO RESCUE THE CONTRADICTION Let U be: HFT is false in worlds like ours. (1) HS is a priori true. So:(2) U is a priori true. So:(3) HFT is a priori false. So:Cr(HFT)=0, although HFT not impossible. My reply. (a) Lewis concedes that the truth of HS is an empirical matter; thus (1) is false. (b) (2) does not follow from (1).

SECOND ATTEMPT TO RESCUE THE CONTRADICTION Let U be: HFT is false in worlds like ours. (1) Cr(UHFT)=0 because UHFT a priori false. (2) Cr(U)=1 because U necessary. So: Cr(HFT)=0 although HFT not impossible. My reply. Let G be false and G=‘G false in  ’. (1) Cr(GG)=0 because GG a priori false. (2) Cr(G)=1 because G necessary. So: Cr(G)=0 for every false G. Reductio!

THIRD ATTEMPT TO RESCUE THE CONTRADICTION (EP) If ET is admissible, then: Cr(A|ET)=f(A). Suppose F undermines T (so f(F)>0) and UHT is admissible. Then: Ê From EP: Cr(F|UHT)=f(F)>0. Ë UHFT a priori false, so Cr(F|UHT)=0. Contradiction? No! UHT is inadmissible because it gives directly evidence about the future. This solution differs from saying that HT is inad- missible, as Lewis does, who thus rejects EP, for which the admissibility of HT is essential.

CONCLUSION: AGAINST IMPERFECTIONISM Lewis (1994): l A feature of Reality deserves the name of chance to the extent that it obeys PP. l Because of undermining, nothing perfectly deserves the name of chance. l But an imperfect candidate (namely, the New Principle) may deserve the name well enough. If my argument succeeds, then we need not espouse Lewis’s imperfectionism. Proponents of PP need not be afraid of undermining.