Notes for Week 11 of Confirmation 11/21/07 Branden Fitelson 1 Two Reflections on Our Discussions from the Past Few Weeks 1.1 Reflections on Maher on (NC) As you’ll recall, Maher’s “counterexample” to (NC) was based on intuitions about what raises the probability of what, relative to “what we (actually) know” (K α ). As I pointed out, this is unfortunate, since he needs an ex- ample that holds, relative to no background knowledge (or empty /a priori background knowledge K ⊤ ). Sim- ilarly, Maher’s argument for the existence of inductive probabilities only establishes the existence of (some) inductive probabilities, relative to substantive, empirical background conditions [Pr(p | K α )]. Again, this is unfortunate, since his main applications of inductive probability to confirmation theory involve principles like (NC) which are explicitly to be interpreted as involving inductive probabilities, relative to no/empty/a priori background evidence [Pr(p | K ⊤ )]. Apparently, he seems to think that some sort of analogical argu- ment will allow us to go from facts about Pr(p | K α ) to facts about Pr(p | K ⊤ ). But, the analogical argument will also have to (somehow) establish the existence of Pr(p | K ⊤ ) from the existence of Pr(p | K α ). Moreover, the analogical argument will also have to (somehow) tell us something about the values (or ranges of values) of Pr(p | K ⊤ ), for some p’s. For instance, Maher needs certain p’s to have low Pr(p | K ⊤ ) values, in order for his “counterexample” to (NC) to have any force. [This adds up to some very heavy lifting for an analogical argument!] This is all review from what I said in my notes a few weeks back. Now, I want to point out another problematic fact about Maher’s discussions on inductive probability and confirmation. Let’s think about Maher’s explicatum for Pr(p | K ⊤ ). Maher’s λ/γ-continuum has adjustable parameters γ F and γ G , which correspond to Pr(Fa | K ⊤ ) and Pr(Ga | K ⊤ ), repsectively, for an L 2,2 with two predicates F and G and a constant a, which (let’s assume) appears in some evidence statement E of interest. Maher tells us that: The choice of γ F and γ G will depend on what the predicates ‘F ’ and ‘G’ mean and may require careful deliberation. For example, if ‘F ’ means ‘raven’ then, since this is a very specific property and there are vast numbers of alternative properties that seem equally likely to be exemplified a priori, γ F should be very small, surely less than 1/1000. A reasoned choice of a precise value would require careful consideration of what exactly is meant by ‘raven’ and what the alternatives are. Later, in his “counterexample” to (NC), Maher sets the values of γ F and γ G very low, and shows that this suffices to generate a counterexample to (NC) for his explicatum for “confirmation relative to no background evidence.” Maher claims that the values of γ F and γ G will depend on “exactly is meant by ‘raven’ and what the alternatives are”. This sounds (to my ear) like the values depend on contextual factors, which are pragmatic (this is what Carnap said about his later systems). But, recall that Maher also claims that (1) The probability that the ball is white, given that it is white or black is 1 2 . [Pr(W a | Wa ∨ Ba) = 1 2 .] is just plain true. For Maher, the truth-value of (1) doesn’t depend on any contextual or pragmatic factors (e.g., on what other alternative colors might be “in play”, etc.). But, he’s now implying that the truth-value of (2) The probability that the ball is white, given that it is white or non-white is 1 2 . [Pr(W a |⊤) = 1 2 .] does depend on “what the alternatives are”. Presumably, this means that the value of Pr(W a |⊤) depends on how many alternative (non-white) colors for a are “in play” in the context at hand. Let’s think about that. By Maher’s own axioms for Pr(·|·), we can show (provided only that Pr(W a∨Ba|⊤) ≠ 0 and Pr(W a&Ba|⊤) = 0, which both seem to be presupposed by Maher in this context) that the following must be true: (3) Pr(W a | Wa ∨ Ba) = Pr(W a & (W a ∨ Ba) |⊤) Pr(W a ∨ Ba |⊤) = Pr(W a |⊤) Pr(W a |⊤) + Pr(Ba |⊤) . This is somewhat odd. Moreover, (1) and (3) can both obtain only if we assume a naïve equiprobability model for the a priori probabilities of the Ca’s, for all alternative colors C (e.g., if W and B are the only alternatives, then Pr(W a |⊤) = Pr(Ba |⊤) = 1 2 is true, etc.). But, what’s the motivation for that ? Looks like we’re back to the “Principle of Indifference” for the determination of the γ-values. But, why think indifference makes any more sense here (for the Ca’s) than it did for the state descriptions of L 2,2 , which led to m † ? I think Carnap’s concession that the γ’s are only fixable pragmatically (a posteriori ) is telling. But, this seems to be abandoning “a priori” inductive probabilities altogether, which Maher doesn’t seem to want to do. 1