Transitivity and intransitivity in evidential support: Some further results William Roche Department of Philosophy, Texas Christian University, Fort Worth, TX, USA, e-mail: w.roche@tcu.edu ABSTRACT: Igor Douven establishes several new intransitivity results concerning evidential support. I add to Douven's very instructive discussion by establishing two further intransitivity results and a transitivity result. Igor Douven (2011) establishes several new intransitivity results concerning evidential support. I aim to add to Douven's very instructive discussion by establishing two further intransitivity results (sections 1 and 2) and a transitivity result (section 3). 1. Douven distinguishes between (i) evidence, (ii) t-evidence, (iii) t'-evidence, and (iv) tt'-evidence. E is evidence for H iff: Pr(H | E) > Pr(H). E is t-evidence for H iff: (a) E is evidence for H and (b) Pr(H | E) > t. Here t is some specified value less than 1 and greater than or equal to .5. E is t'-evidence for H iff: (a) E is evidence for H and (b) either Pr(H | E) – Pr(H) > t' or Pr(H | E) = 1. Here t' is some specified value less than 1 and greater than 0. E is tt'-evidence for H iff: (a) E is t-evidence for H (hence is evidence for H) and (b) E is t'-evidence for H. It is well known that evidence (as defined above) is intransitive-that there are cases in which Pr(Q | R) > Pr(Q), Pr(P | Q) > Pr(P), and yet it is not the case that Pr(P | R) > Pr(P).1 Douven shows that regardless of the values specified for t and t' (but given the constraint that 1 > t ≥ .5 and 1 > t' > 0) t-evidence, t'-evidence, and tt'-evidence too are intransitive. So, where, for example, the value specified for t is .5 there are probability distributions on which Pr(Q | R) > Pr(Q), Pr(Q | R) > .5, Pr(P | Q) > Pr(P), Pr(P | Q) > .5, and yet Pr(P | R) ≯ Pr(P) or Pr(P | R) ≯ .5.2 Tomoji Shogenji (2003) shows that, though evidence (or "probabilistic support") is intransitive, there is a condition for transitivity in evidence. The condition is: 1 See, e.g., Eells and Sober (1983, pp. 43-44), Hanen (1971), Hesse (1970, pp. 50-51), and Shogenji (2003, p. 613). 2 Further, Douven shows that certain alternatives to t'-evidence and tt'-evidence- alternatives involving not the difference measure of confirmation but, for example, the log-likelihood measure or the Kemeny-Oppenheim measure-are intransitive. 2 Transitivity and intransitivity in evidential support: Some further results (C) Pr(P | R  Q) = Pr(P | Q) and Pr(P | R  ¬Q) = Pr(P | ¬Q). Shogenji thus establishes the principle: (TE) If Pr(Q | R) > Pr(Q), Pr(P | Q) > Pr(P), and (C) holds, then Pr(P | R) > P(P).3 Douven shows that regardless of the values specified for t and t' (C) is not a condition for transitivity in t-evidence, t'-evidence, or tt'-evidence. Thus where, say, the value specified for t is .5 there are probability distributions on which Pr(Q | R) > Pr(Q), Pr(Q | R) > .5, Pr(P | Q) > Pr(P), Pr(P | Q) > .5, (C) holds, and yet Pr(P | R) ≯ Pr(P) or Pr(P | R) ≯ .5.4 Consider the condition: (C*) Pr(P | R  Q) ≥ Pr(P | Q) and Pr(P | R  ¬Q) ≥ Pr(P | ¬Q). (C*) is weaker than (C), in that if (C) holds, then (C*) holds, but not vice versa. Elsewhere (Roche 2012) I establish the principle: (TE*) If Pr(Q | R) > Pr(Q), Pr(P | Q) > Pr(P), and (C*) holds, then Pr(P | R) > P(P). (TE*) is stronger than (TE); if (TE*) is correct, then (TE) is correct, but not vice versa. (C*) holds if (C) holds. So, given that regardless of the value specified for t there are probability distributions on which R is t-evidence for Q, Q is t-evidence for P, (C) holds, and yet R is not t-evidence for P, it follows that regardless of the value specified for t there are probability distributions on which R is t-evidence for Q, Q is t-evidence for P, (C*) holds, and yet R is not t-evidence for P. Thus, regardless of the value specified for t (C*) is not a condition for transitivity in t-evidence. Likewise with respect to t'-evidence and tt'-evidence. 3 The labels "(C)" and "(TE)" are mine. 4 Douven also shows that (C) is not a condition for transitivity in t(l)-evidence, tt(l)evidence, t(k)-evidence, or tt(k)-evidence, where "l" refers to the log-likelihood measure of confirmation and "k" refers to the Kemeny-Oppenheim measure. 3 Transitivity and intransitivity in evidential support: Some further results 2. Consider the condition: (C**) Pr(P | R  Q) > Pr(P | Q) and Pr(P | R  ¬Q) > Pr(P | ¬Q). (C**) is stronger than (C*), and neither weaker nor stronger than (C). It can be shown that: (TE**) If Pr(Q | R) > Pr(Q), Pr(P | Q) > Pr(P), and (C**) holds, then Pr(P | R) > P(P).5 (C**) fails to hold if (C) holds. So, any probability distribution on which R is tevidence for Q, Q is t-evidence for P, (C) holds, and yet R is not t-evidence for P is a probability distribution on which (C**) fails to hold. Likewise with respect to t'-evidence and tt'-evidence. Hence, nothing in what Douven says about (C) shows that (C**) is not a condition for transitivity in t-evidence, t'-evidence, or tt'-evidence. It is relatively straightforward to show, however, that with some specifications for t and t' (C**) is not a condition for transitivity in t-evidence, t'-evidence, or tt'-evidence. For example, with t' = .5 it follows from the following probability distribution that (C**) is not a condition for transitivity in t'-evidence: P Q R Pr P Q R Pr T T T 2 41 F T T 1 214 T T F 1 901 F T F 1 45 T F T 1 791 F F T 1 27 T F F 1 844 F F F 314,821,182,887,711 356,242,003,012,980 And with t = t' = .5 it follows from the following probability distribution that (C**) is not a condition for transitivity in tt'-evidence: 5 Start with the argument in Roche (2012) for (TPS*). Replace "z" with "P," "y" with "Q," and "x" with "R." Delete (e), (f), and (g). Then modify the remainder of the argument accordingly. 4 Transitivity and intransitivity in evidential support: Some further results P Q R Pr P Q R Pr T T T 1 43 F T T 1 92 T T F 1 165 F T F 1 133 T F T 1 608 F F T 31,847 1,202,624 T F F 1 250 F F F 1,389,286,211 1,509,816,000 I thank a very generous anonymous reviewer for providing the above two probability distributions (or, strictly speaking, slight variants thereof), and for providing a probability distribution showing that with t = .5 (C**) is not a condition for transitivity in t-evidence. Each of the three distributions was found using the decision procedure PrSAT developed by Branden Fitelson (in collaboration with Jason Alexander and Ben Blum).6 I leave it for further investigation whether with certain other specifications for t and t' (C**) is a condition for transitivity in t'-evidence or tt'-evidence.7 I want to focus on whether with certain other specifications for t (C**) is a condition for transitivity in tevidence. I aim to show that the answer is negative-so that regardless of the value specified for t (C**) is not a condition for transitivity in t-evidence. (For completeness I begin with the case where t = .5. The schema used in addressing that case is also used in addressing the case where t > .5.) Consider the following schema, where α, β ∈ R+ and τ = α + (1/10)β + (1/10)β + (2/10)β + (1/10)β + (9/10)β + 10β: P Q R Pr P Q R Pr T T T α/τ F T T (2/10)β/τ T T F (1/10)β/τ F T F (1/10)β/τ T F T (1/10)β/τ F F T (9/10)β/τ T F F 0 F F F 10β/τ 6 See Fitelson (2008) for a description of PrSAT and some applications. 7 Clearly, the fact that with some specifications for t and t' (C**) is not a condition for transitivity in t'-evidence or tt'-evidence leaves it open that with certain other specifications for t and t' (C**) is a condition for transitivity in t'-evidence and tt'evidence. 5 Transitivity and intransitivity in evidential support: Some further results Suppose t = .5 and α = β = 1. Then it follows that: (1) Pr(Q | R) = ! ! !!" ! ! ! !!" ! ! !!" ! ! !!" ! = ! ! !!" ! ! !!" ! ! !" ! ! !" = ! !! ; (2) Pr(Q) = ! ! !!" ! ! !!" ! ! !!" ! ! ! !!" ! ! !!" ! ! !!" ! ! !!" ! ! !!" ! ! !"! = ! ! !!" ! ! !" ! ! !" ! ! !!" ! ! !" ! ! !" ! ! !" ! ! !" ! !" = ! !" ; (3) Pr(P | Q) = ! ! !!" ! ! ! !!" ! ! !!" ! ! !!" ! = ! ! !!" ! ! !!" ! ! !" ! ! !" = !! !" ; (4) Pr(P) = ! ! !!" ! ! !!" ! ! ! !!" ! ! !!" ! ! !!" ! ! !!" ! ! !!" ! ! !"! = ! ! !!" ! ! !" ! ! !!" ! ! !" ! ! !" ! ! !" ! ! !" ! !" = ! !" ; (5) Pr(P | R  Q) = ! ! ! !!" ! = ! ! ! !!" = ! ! ; (6) Pr(P | R  ¬Q) = ! !" ! ! !" ! ! !!" ! = ! !" ! !" ! ! !" = ! !" ; (7) Pr(P | ¬Q) = ! !" ! ! !" ! ! !!" ! ! !"! = ! !" ! !" ! ! !" ! !" = ! !!" ; (8) Pr(P | R) = ! ! !!" ! ! ! !!" ! ! !!" ! ! !!" ! = ! ! ! !" ! ! !!" ! ! !" ! ! !" = ! ! . By (1) and (2) it follows that R is t-evidence for Q. By (3) and (4) it follows that Q is tevidence for P. By (3), (5), (6), and (7) it follows that (C**) holds. By (8) it follows that R is not t-evidence for P because Pr(P | R) ≯ t. So, with t = .5 there are probability distributions on which R is t-evidence for Q, Q is t-evidence for P, (C**) holds, and yet, though R is evidence for P, R is not t-evidence for P since Pr(P | R) ≯ t. 6 Transitivity and intransitivity in evidential support: Some further results I turn now to the case where t > .5. Suppose α = 1 and β ≥ 1. The aim is to show that with t > .5 there are probability distributions which are instances of the schema above such that R is t-evidence for Q, Q is t-evidence for P, (C**) holds, and yet, though R is evidence for P, R is not t-evidence for P since Pr(P | R) ≯ t. First, observe that each of Pr(Q | R) and Pr(P | Q) approaches 1 as β tends to ∞: (9) lim!→! ! ! !!" ! ! ! !!" ! ! !!" ! ! !!" ! = 1; (10) lim!→! ! ! !!" ! ! ! !!" ! ! !!" ! ! !!" ! = 1. So, regardless of the value specified for t there is a value for β such that Pr(Q | R) > t and Pr(P | Q) > t. The same is true of Pr(P | R), since Pr(P | R), like each of Pr(Q | R) and Pr(P | Q), approaches 1 as β tends to ∞: (11) lim!→! ! ! !!" ! ! ! !!" ! ! !!" ! ! !!" ! = 1. But, crucially, the following inequalities hold: (12) Pr(Q | R) – Pr(P | R) > 0; (13) Pr(P | Q) – Pr(P | R) > 0. That (12) holds can be seen by verifying that: Pr(Q | R) – Pr(P | R) = ! ! !!" ! ! ! !!" ! ! !!" ! ! !!" ! − ! ! !!" ! ! ! !!" ! ! !!" ! ! !!" ! = ! !" ! ! !!" ! ! ! !!" ! ! !!" ! ! !!" ! > 0. 7 Transitivity and intransitivity in evidential support: Some further results That (13) holds is clear: Pr(P | Q) – Pr(P | R) = ! ! !!" ! ! ! !!" ! ! !!" ! ! !!" ! − ! ! !!" ! ! ! !!" ! ! !!" ! ! !!" ! > 0. Next, consider the inequalities: (14) Pr(Q | R) – Pr(Q) > 0; (15) Pr(P | Q) – Pr(P) > 0. I noted above that each of Pr(Q | R) and Pr(P | Q) approaches 1 as β tends to ∞. This is not true of Pr(Q) and Pr(P)-quite the opposite in fact. Each of Pr(Q) and Pr(P) approaches 0 as β tends to ∞: (16) lim!→! ! ! !!" ! ! !!" ! ! !!" ! ! ! !!" ! ! !!" ! ! !!" ! ! !!" ! ! !!" ! ! !"! = 0; (17) lim!→! ! ! !!" ! ! !!" ! ! ! !!" ! ! !!" ! ! !!" ! ! !!" ! ! !!" ! ! !"! = 0. I showed above that with α = β = 1, Pr(Q | R) > Pr(Q) and Pr(P | Q) > Pr(P). So, given (9), (10), (16), and (17), and with α = 1 and β ≥ 1, it follows that (14) and (15) hold. It remains to be shown that (C**) holds. First, verify that: (18) Pr(P | R  Q) – Pr(P | Q) = ! ! ! !!" ! − ! ! !!" ! ! ! !!" ! ! !!" ! ! !!" ! = !! ! ! !!" ! ! ! !!" ! ! ! !!" ! ! !! ! ! !!" ! ! ! !!" ! ! !!" ! ! !" ! ! ! !!" ! ! ! !!" ! ! !!" ! ! !!" ! = ! !!" ! ! !!" ! ! !" ! ! ! !!" ! ! ! !!" ! ! !!" ! ! !!" ! > 0. 8 Transitivity and intransitivity in evidential support: Some further results Next, observe that: (19) Pr(P | R  ¬Q) – Pr(P | ¬Q) = ! !" ! ! !" ! ! !!" ! − ! !" ! ! !" ! ! !!" ! ! !"! > 0. So, (C**) holds. The aim, recall, is to show that with t > .5 there are probability distributions which are instances of the schema above such that R is t-evidence for Q, Q is t-evidence for P, (C**) holds, and yet, though R is evidence for P, R is not t-evidence for P since Pr(P | R) ≯ t. Take α = β = 1. Then, as shown above (for the case where t = .5), it follows that Pr(Q | R) = 6/11 > Pr(Q) = 7/62, Pr(P | Q) = 11/14 > Pr(P) = 3/31, (C**) holds, and Pr(P | R) = 1/2. If 6/11 > t > .5, it follows that the probability distribution in question is a probability distribution which is an instance of the schema above such that R is t-evidence for Q, Q is t-evidence for P, (C**) holds, and yet, though R is evidence for P, R is not t-evidence for P since Pr(P | R) ≯ t. If, instead, t ≥ 6/11, then let the value of β increase until Pr(Q | R) > t and Pr(P | Q) > t but Pr(P | R) ≯ t; that there is such a value for β is guaranteed by (9), (10), (12), and (13). It will still be the case that Pr(Q | R) > Pr(Q), Pr(P | Q) > Pr(P), and (C**) holds; this follows from (14), (15), (18), and (19). So, the probability distribution in question will be a probability distribution which is an instance of the schema above such that R is t-evidence for Q, Q is t-evidence for P, (C**) holds, and yet, though R is evidence for P, R is not t-evidence for P since Pr(P | R) ≯ t. The result is that regardless of the value specified for t, that is, regardless of whether t = .5 or t > .5, (C**) is not a condition for transitivity in t-evidence. 3. In section 1, I argued that, though (C*) is a condition for transitivity in evidence, (C*) is not a condition for transitivity in t-evidence, t'-evidence, or tt'-evidence. In section 2, I argued that, though (C**) is a condition for transitivity in evidence, (C**) is not a condition for transitivity in t-evidence. I now aim to show that there is a condition for transitivity in t-evidence (regardless of the value specified for t, but given the constraint that 1 > t ≥ .5). The condition is: (C***) Pr(P | R  Q) ≥ Pr(P | Q), Pr(P | R  ¬Q) ≥ Pr(P | ¬Q), and Q entails P. 9 Transitivity and intransitivity in evidential support: Some further results The aim is to show: (TE***) If R is t-evidence for Q, Q is t-evidence for P, and (C***) holds, then R is t-evidence for P. Three comments. First, (C***) is simply the conjunction of (C*) and the condition that Q entails P. Second, (C***) could instead be put as: Pr(P | R  ¬Q) ≥ Pr(P | ¬Q) and Q entails P. When Q entails P, it follows that Pr(P | R  Q) = Pr(P | Q) = 1, hence Pr(P | R  Q) ≥ Pr(P | Q). Third, (TE***) could instead be put as: If R is t-evidence for Q, Q entails P, Pr(P) < 1, and (C*) holds, then R is t-evidence for P. If Q is t-evidence for P and (C***) holds, then Q entails P, Pr(P) < 1, and (C*) holds, and vice versa. Suppose, now, (a) R is t-evidence for Q, thus (a1) Pr(Q | R) > Pr(Q) and (a2) Pr(Q | R) > t. Suppose (b) Q is t-evidence for P, hence (b1) Pr(P | Q) > Pr(P) and (b2) Pr(P | Q) > t. Suppose (c) (C***) holds, thus (c1) (C*) holds and (c2) Q entails P. By (a1), (b1), (c1), and (TE*), it follows that Pr(P | R) > P(P). By (c2) and the theorem that if Q entails P, then Pr(P | R) ≥ Pr(Q | R), it follows that Pr(P | R) ≥ Pr(Q | R). By (a2), it then follows that Pr(P | R) > t. Thus, since Pr(P | R) > P(P) and Pr(P | R) > t, it follows that R is tevidence for P. So, (TE***). Let's consider an example. Suppose t = .7. Suppose we have a nonstandard deck of cards, in that there are 100 cards in the deck, 50 of the cards are heart (hence red) court cards, 15 of the cards are diamond (thus red) court cards, 20 of the cards are diamond non-court cards, 5 of the cards are black court cards, and 10 of the cards are black noncourt cards. Suppose we randomly draw a card from the deck. Let R be the claim that the card drawn is a court card, Q be the claim that the card drawn is a heart, and P be the claim that the card drawn is a red card. Suppose, then, the following probability distribution: P Q R Pr P Q R Pr T T T .5 F T T 0 T T F 0 F T F 0 T F T .15 F F T .05 T F F .2 F F F .1 Then: 10 Transitivity and intransitivity in evidential support: Some further results Pr(Q | R) ≈ .714 > Pr(Q) = .5; Pr(P | Q) = 1 > Pr(P) = .85; Pr(P | R  Q) = 1 ≥ Pr(P | Q) = 1; Pr(P | R  ¬Q) = .75 ≥ Pr(P | ¬Q) = .7; Pr(P | R) ≈ .929 > Pr(P) = .85. R is t-evidence for Q, Q is t-evidence for P, (C***) holds, and, as (TE***) implies, R is t-evidence for P. Two final points are in order. First, there is a clear sense in which (TE***) is a "transmission" principle as opposed to a "closure" principle. Consider the principle: (CE) If Pr(Q | R) > t, and Q entails P, then Pr(P | R) > t. (CE) is a closure principle.8 It can be understood as saying that (where t is the threshold for rational acceptability) if Q is rationally acceptable given R, and Q entails P, then P is rationally acceptable given R. All positive instances of (CE) are thus cases in which P is rationally acceptable given R. Some such cases, though, are cases in which R is evidence against P in that Pr(P | R) < Pr(P). By contrast, all positive instances of (TE***) are cases in which (a) P is rationally acceptable given R and (b) R is evidence for P in that Pr(P | R) > Pr(P).9 Second, though all positive instances of (TE***) are cases in which P is evidence for R in that Pr(P | R) > Pr(P), it does not follow that all positive instances of (TE***) are cases in which Pr(P | R) – Pr(P) ≥ Pr(Q | R) – Pr(Q). In fact, some positive 8 See Chandler (2010, p. 337, n. 5). 9 I leave it for further investigation how (C***) and (TE***) relate to the issue Crispin Wright (2002, 2003) has in mind in speaking of when it is that warrant transmits across entailment. For discussion of how to formalize the issue Wright has in mind, see Chandler (2010), Moretti (2010), and Okasha (2004). Cf. Pynn (2011). 11 Transitivity and intransitivity in evidential support: Some further results instances of (TE***) are cases in which Pr(P | R) – Pr(P) < Pr(Q | R) – Pr(Q).10 The case given in the prior paragraph is a case in point; Pr(P | R) – Pr(P) ≈ .079 < Pr(Q | R) – Pr(Q) ≈ .214.11 References Chandler, J. (2010). The transmission of support: a Bayesian re-analysis. Synthese, 176, 333-343. Douven, I. (2011). Further results on the intransitivity of evidential support. Review of Symbolic Logic, forthcoming. Eells, E., & Sober, E. (1983). Probabilistic causality and the question of transitivity. Philosophy of Science, 50, 35-57. Fitelson, B. (2008). A decision procedure for probability calculus with applications. Review of Symbolic Logic, 1, 111-125. Hanen, M. (1971). Confirmation and adequacy conditions. Philosophy of Science, 38, 361-368. Hesse, M. (1970). Theories and the transitivity of confirmation. Philosophy of Science, 37, 50-63. Moretti, L. (2010). Wright, Okasha and Chandler on transmission failure. Synthese, forthcoming. Okasha, S. (2004). Wright on the transmission of support: a Bayesian analysis. Analysis, 64, 139-146. Pynn, G. (2011). The Bayesian explanation of transmission failure. Synthese, forthcoming. Roche, W. (2012). A weaker condition for transitivity in probabilistic support. European Journal for Philosophy of Science, 2, 111-118. Shogenji, T. (2003). A condition for transitivity in probabilistic support. British Journal for the Philosophy of Science, 54, 613-616. 10 More generally, there are cases in which Pr(Q | R) > Pr(Q), Q entails P, (C*) holds, and yet, though Pr(P | R) > Pr(P), Pr(P | R) – Pr(P) < Pr(Q | R) – Pr(Q). I owe this point to Tomoji Shogenji. 11 I wish to thank Igor Douven, Tomoji Shogenji, and an anonymous reviewer for helpful comments on a prior version of this paper. 12 Transitivity and intransitivity in evidential support: Some further results Wright, C. (2002). (Anti-)sceptics simple and subtle: G. E. Moore and John McDowell. Philosophy and Phenomenological Research, 65, 330-348. Wright, C. (2003). Some reflections on the acquisition of warrant by inference. In S. Nuccetelli, editor. New Essays on Semantic Externalism and Self-Knowledge. Cambridge, Mass.: MIT Press, pp. 57-77.