Abstract
It is well known that the probabilistic relation of confirmation is not transitive in that even if E confirms H1 and H1 confirms H2, E may not confirm H2. In this paper we distinguish four senses of confirmation and examine additional conditions under which confirmation in different senses becomes transitive. We conduct this examination both in the general case where H1 confirms H2 and in the special case where H1 also logically entails H2. Based on these analyses, we argue that the Screening-Off Condition is the most important condition for transitivity in confirmation because of its generality and ease of application. We illustrate our point with the example of Moore’s “proof” of the existence of a material world, where H1 logically entails H2, the Screening-Off Condition holds, and confirmation in all four senses turns out to be transitive.
Similar content being viewed by others
Notes
Cf. Carnap (1962, Preface to the Second Edition) on “concepts of increase in firmness” and “concepts of firmness”.
We leave it open that t may be context-dependent (perhaps higher in higher-stakes contexts and lower in lower-stakes contexts). We are following the standard view here that whether H is rationally acceptable given E is determined solely by Pr(H | E) (and perhaps the context), though the view is not unproblematic. Cf. Shogenji (2012).
The example is taken from Shogenji (2003).
This case is adapted from Dretske (1970, pp. 1015–1016).
We have in mind, of course, non-trivial such conditions and not, say, the condition that E confirms-IF H2 as a condition for transitivity in confirmation-IF in the general case.
See Roche (2012a).
That confirmation-IF is transitive under (C1*) is shown in Shogenji (2003).
We are following Kotzen (2012) in calling (C2) “the Dragging Condition”.
Cf. Kotzen (2012, p. 66).
Cf. Kotzen (2012, p. 72, n. 22).
Moretti (2012, Sect. 5) establishes a principle similar to Theorem 2 where (X) is (C2). It can be put thus: if (a) Pr(H2) > t, (b) Pr(H1 | E) > Pr(H1), (c) Pr(H1 | E) > t, (d) H1 entails H2, and (e) (C2) holds, then Pr(H2 | E) > P(H2).
Theorem 4 is essentially the same as “Closure*” in Chandler (2010, p. 337, n. 5).
Here and throughout the paper when we speak of positive instances of transitivity, we have in mind non-vacuous positive instances.
Each of (C1)–(C3) fails to hold in the zoo case. That (C1) fails to hold follows from the fact that Pr(H2 | E ∧ ¬H1) < Pr(H2 | ¬H1); E increases the probability of ¬H2 given ¬H1, and, so, decreases the probability of H2 given ¬H1. That (C2) fails to hold follows from the fact that Pr(E | H1) = Pr(E | ¬H2), thus Pr(E | H1) ≤ Pr(E | ¬H2); Kotzen (2012, pp. 81–82) shows that (where E confirms-IF H1) if Pr(E | H1) ≤ Pr(E | ¬H2), then (C2) does not hold. Pr(¬H2 ∧ ¬H1 | E) > Pr(¬H2 ∧ ¬H1) and Pr(H2 ∧ H1 | E) = Pr(H1 | E) > Pr(H1) = Pr(H2 ∧ H1), so, since Pr(¬H2 ∧ H1 | E) = 0 = Pr(¬H2 ∧ H1), it follows that Pr(H2 ∧ ¬H1 | E) < Pr(H2 ∧ ¬H1), thus (C3) fails to hold.
The extant literature on transmission failure is extensive. See, e.g., Beebee (2001), Brown (2003, 2004), Cling (2002), Coliva (2011), Davies (1998, 2000, 2003, 2004), Dretske (2005a, b), Ebert (2005), Hale (2000), Hawthorne (2005), Kotzen (2012, Sect. 6), McKinsey (2003), McLaughlin (2003), Neta (2007), Peacocke (2004, Chap. 4, pp. 112–115), Pryor (2004), Sainsbury (2000), Schiffer (2004), Silins (2005, 2007), Smith (2009), Suarez (2000), Tucker (2010a, b), White (2006, Sect. 5), and Wright (1985, 2000a, b, 2002, 2003, 2004, 2007, 2011). For discussion of how to formalize the issue of transmission failure, see Chandler (2010), Moretti (2012), Moretti and Piazza (2011), and Okasha (2004). Cf. Pynn (2011).
Compare Theorems 3 and 7. The latter shows that the antecedent condition that H1 entails H2 is essential to the former, but does not show that the same is true of the antecedent condition that H1 confirms-TSF H2. We noted earlier that it follows from H1 ⊢ H2 that H1 confirms-IF H2 (except for the uninteresting cases), but it does not follow that H1 confirms-TSF H2. So, (b) is not redundant. Moreover, the argument given in Appendix 5 for Theorem 7 does not involve cases where H1 entails H2, and therefore does not itself imply that there can be cases where (a) E confirms-TSF H1, (c) H1 entails H2, and (d) (C1)–(C3) all hold, and yet E does not confirm-TSF H2. It can be shown, however, that such cases are possible—regardless of the value specified for t. Due to space considerations we omit the proof.
The Converse Consequence Condition is introduced and rejected in Hempel (1965).
When H1 and H2 are mutually entailing, Pr(H1) = Pr(H2) and Pr(H1 | E) = Pr(H2 | E), in which case if Pr(H1 | E) > Pr(H1), it follows that Pr(H2 | E) > Pr(H2). Counterexamples to (CCC) are thus cases where H1 and H2 are not mutually entailing. For relevant discussion, see Milne (2000).
(C1*), like (C1), is a condition for transitivity in confirmation-IF in the case where H2 entails H1. But there are cases of transitivity in confirmation-IF in the case where H2 entails H1 where (C1) holds but (C1*) does not.
Cf. Kotzen (2012, Sects. 3 and 4).
In fact, it must be the case that Pr(H2 | ¬H1 ∧ E) > Pr(H2 | ¬H1), which means that (C1*) fails to hold. So, if E confirms-IF H1, confirms-IF&SF H1, or confirms-TSF H1, (C3) entails not-(C1*).
This does not require that H2 also logically entails H1.
It can be shown, further, that even under the condition that E confirms-IF H1, there is no logical entailment between (C2) and (C3).
For ease of expression, we sometimes refer to E as an experience. Strictly speaking, of course, E is a proposition about an experience, not an experience itself.
Some may point out that (C2) is satisfied for the purpose of transitivity in confirmation-TSF because Pr(H2) ≤ t from the second antecedent of transitivity in confirmation-TSF, while Pr(H1 | E) > t from the first antecedent of transitivity in confirmation-TSF. However, we are not assuming here that the second antecedent of transitivity in confirmation-TSF holds, precisely for the reason that Pr(H2) is questionable and questioned. The second antecedent of transitivity in the other three senses of confirmation is unproblematic: H1 confirms-IF H2, confirms-SF H2, and confirms-IF&SF H2, from Pr(H2 | H1) = 1 > P(H2) and Pr(H2 | H1) = 1 > t.
Since confirmation-TSF is transitive in MOORE, it follows that H1 does not confirm-TSF H2. As we noted in Footnote 29 we are not making the assumption that H1 confirms-TSF H2. Our claim of transitivity in confirmation-TSF in MOORE is of the form: if E confirms-TSF H1 and H1 in turn confirms-TSF H2, then E confirms-TSF H2.
We can state the point more simply: if Pr(H1 | E) > t, then Pr(H2) > t. Note that this does not imply the failure of (C2) Pr(H2) < Pr(H1 | E). (C2) can be true while Pr(H1 | E) > t and Pr(H2) > t, and thus the conditional is also true.
Bear in mind here and throughout the remainder of the argument that Pr(H1 | E) and Pr(H2 | H1) are continuous monotonically increasing functions of β, and that Pr(H1) and Pr(H2) are continuous monotonically decreasing functions of β.
See Roche (2012b) for a similar argument for the claim that regardless of the value specified for t the following condition is not a condition for transitivity in confirmation-IF&SF: Pr(H2 | E ∧ H1) > Pr(H2 | H1) and Pr(H2 | E ∧ ¬H1) > Pr(H2 | ¬H1). Note that this condition is stronger than (C1) and neither stronger nor weaker than (C1*), and that, like (C1) and (C1*), it is a condition for transitivity in confirmation-IF.
References
Beebee, H. (2001). Transfer of warrant, begging the question, and semantic externalism. Philosophical Quarterly, 51, 356–374.
Brown, J. (2003). The reductio argument and transmission of warrant. In S. Nuccetelli (Ed.), New essays on semantic externalism and self-knowledge (pp. 117–130). Cambridge, MA: MIT Press.
Brown, J. (2004). Wright on transmission failure. Analysis, 64, 57–67.
Carnap, R. (1962). Logical foundations of probability (2nd ed.). Chicago: University of Chicago Press.
Chandler, J. (2010). The transmission of support: A Bayesian re-analysis. Synthese, 176, 333–343.
Cling, A. (2002). Justification-affording circular arguments. Philosophical Studies, 111, 251–275.
Cohen, S. (2005). Why basic knowledge is easy knowledge. Philosophy and Phenomenological Research, 70, 417–430.
Coliva, A. (2011). Varieties of failure (of warrant transmission: What else?!). Synthese. doi:10.1007/s11229-011-0006-6.
Davies, M. (1998). Externalism, architecturalism, and epistemic warrant. In C. Wright, B. Smith, & C. Macdonald (Eds.), Knowing our own minds (pp. 321–361). Oxford: Oxford University Press.
Davies, M. (2000). Externalism and armchair knowledge. In P. Boghossian & C. Peacocke (Eds.), New essays on the a priori (pp. 384–414). Oxford: Oxford University Press.
Davies, M. (2003). The problem of armchair knowledge. In S. Nuccetelli (Ed.), New essays on semantic externalism and self-knowledge (pp. 23–55). Cambridge, MA: MIT Press.
Davies, M. (2004). Epistemic entitlement, warrant transmission and easy knowledge. Proceedings of the Aristotelian Society, Supplementary Volumes, 78, 213–245.
Douven, I. (2011). Further results on the intransitivity of evidential support. Review of Symbolic Logic, 4, 487–497.
Dretske, F. (1970). Epistemic operators. Journal of Philosophy, 67, 1007–1023.
Dretske, F. (2005a). The case against closure. In M. Steup & E. Sosa (Eds.), Contemporary debates in epistemology (pp. 13–26). Malden, MA: Blackwell.
Dretske, F. (2005b). Reply to Hawthorne. In M. Steup & E. Sosa (Eds.), Contemporary debates in epistemology (pp. 43–46). Malden, MA: Blackwell.
Ebert, P. (2005). Transmission of warrant-failure and the notion of epistemic analyticity. Australasian Journal of Philosophy, 83, 505–521.
Hale, B. (2000). Transmission and closure. Philosophical Issues, 10, 172–190.
Hawthorne, J. (2004). Knowledge and lotteries. Oxford: Clarendon.
Hawthorne, J. (2005). The case for closure. In M. Steup & E. Sosa (Eds.), Contemporary debates in epistemology (pp. 26–43). Malden, MA: Blackwell.
Hempel, C. (1965). Studies in the logic of confirmation. In C. Hempel, Aspects of scientific explanation and other essays in the philosophy of science (pp. 3–46). New York: Free Press.
Kotzen, M. (2012). Dragging and confirming. Philosophical Review, 121, 55–93.
Kukla, A. (1998). Studies in scientific realism. Oxford: Oxford University Press.
McKinsey, M. (2003). Transmission of warrant and closure of apriority. In S. Nuccetelli (Ed.), New essays on semantic externalism and self-knowledge (pp. 97–115). Cambridge, MA: MIT Press.
McLaughlin, B. (2003). McKinsey’s challenge, warrant transmission, and skepticism. In S. Nuccetelli (Ed.), New essays on semantic externalism and self-knowledge (pp. 79–96). Cambridge, MA: MIT Press.
Milne, P. (2000). Is there a logic of confirmation transfer? Erkenntnis, 53, 309–335.
Moretti, L. (2002). For a Bayesian account of indirect confirmation. Dialectica, 56, 153–173.
Moretti, L. (2012). Wright, Okasha and Chandler on transmission failure. Synthese, 184, 217–234.
Moretti, L., & Piazza, T. (2011). When warrant transmits and when it doesn’t: Towards a general framework. Synthese,. doi:10.1007/s11229-011-0018-2.
Neta, R. (2007). Fixing the transmission: The new Mooreans. In S. Nuccetelli & G. Seay (Eds.), Themes from G. E. Moore: New essays in epistemology and ethics (pp. 62–83). Oxford: Oxford University Press.
Okasha, S. (1999). Epistemic justification and deductive closure. Crítica, 31, 37–51.
Okasha, S. (2004). Wright on the transmission of support: A Bayesian analysis. Analysis, 64, 139–146.
Peacocke, C. (2004). The realm of reason. Oxford: Oxford University Press.
Pryor, J. (2004). What’s wrong with Moore’s argument? Philosophical Issues, 14, 349–378.
Pynn, G. (2011). The Bayesian explanation of transmission failure. Synthese,. doi:10.1007/s11229-011-9890-z.
Roche, W. (2012a). A weaker condition for transitivity in probabilistic support. European Journal for Philosophy of Science, 2, 111–118.
Roche, W. (2012b). Transitivity and intransitivity in evidential support: Some further results. Review of Symbolic Logic, 5, 259–268.
Sainsbury, R. M. (2000). Warrant-transmission, defeaters and disquotation. Philosophical Issues, 10, 191–200.
Schiffer, S. (2004). Skepticism and the vagaries of justified belief. Philosophical Studies, 119, 161–184.
Shogenji, T. (2003). A condition for transitivity in probabilistic support. British Journal for the Philosophy of Science, 54, 613–616.
Shogenji, T. (2012). The degree of epistemic justification and the conjunction fallacy. Synthese, 184, 29–48.
Silins, N. (2005). Transmission failure failure. Philosophical Studies, 126, 71–102.
Silins, N. (2007). Basic justification and the Moorean response to the skeptic. In T. Gendler & J. Hawthorne (Eds.), Oxford studies in epistemology (Vol. 2, pp. 108–140). Oxford: Oxford University Press.
Smith, M. (2009). Transmission failure explained. Philosophy and Phenomenological Research, 79, 164–189.
Suarez, A. (2000). On Wright’s diagnosis of McKinsey’s argument. Philosophical Issues, 10, 164–171.
Tucker, C. (2010a). When transmission fails. Philosophical Review, 119, 497–529.
Tucker, C. (2010b). Transmission and transmission failure in epistemology. In J. Fieser & B. Dowden (Eds.), Internet encyclopedia of philosophy (published October 2010). Retrieved February 3, 2012 from http://www.iep.utm.edu/transmis/.
White, R. (2006). Problems for dogmatism. Philosophical Studies, 131, 525–557.
Wright, C. (1985). Facts and certainty. Proceedings of the British Academy, 71, 429–472.
Wright, C. (2000a). Cogency and question-begging: Some reflections on McKinsey’s paradox and Putnam’s proof. Philosophical Issues, 10, 140–163.
Wright, C. (2000b). Replies. Philosophical Issues, 10, 201–219.
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 (Ed.), New essays on semantic externalism and self-knowledge (pp. 57–77). Cambridge, MA: MIT Press.
Wright, C. (2004). Warrant for nothing (and foundations for free)? Proceedings of the Aristotelian Society Supplementary Volumes, 78, 167–212.
Wright, C. (2007). The perils of dogmatism. In S. Nuccetelli & G. Seay (Eds.), Themes from G. E. Moore: New essays in epistemology and ethics (pp. 25–48). Oxford: Oxford University Press.
Wright, C. (2011). McKinsey one more time. In A. Hatzimoysis (Ed.), Self-knowledge (pp. 80–104). Oxford: Oxford University Press.
Acknowledgments
We thank an anonymous reviewer for very helpful comments on a prior version of the paper.
Author information
Authors and Affiliations
Corresponding author
Appendices
Appendix 1: Proof of Theorem 2
Let (X) be any of (C1)–(C3). Suppose (a) E confirms-IF&SF H1, so (a1) Pr(H1 | E) > Pr(H1) and (a2) Pr(H1 | E) > t. Suppose (b) H1 confirms-IF&SF H2, therefore (b1) Pr(H2 | H1) > Pr(H2) and (b2) Pr(H2 | H1) > t. Suppose (c) H1 entails H2 and (d) (X) holds. By (a1), (b1), (c), (d), and Theorem 1, it follows that Pr(H2 | E) > Pr(H2). By (c) and the theorem that if H1 entails H2, then Pr(H2 | E) ≥ Pr(H1 | E), it follows that Pr(H2 | E) ≥ Pr(H1 | E). By (a2), it then follows that Pr(H2 | E) > t. So Pr(H2 | E) > Pr(H2) and Pr(H2 | E) > t. So E confirms-IF&SF H2.
Appendix 2: Proof of Theorem 3
Let (X) be any of (C1)–(C3). Suppose (a) E confirms-TSF H1, so (a1) Pr(H1 | E) > Pr(H1), (a2) Pr(H1 | E) > t, and (a3) Pr(H1) ≤ t. Suppose (b) H1 confirms-TSF H2, therefore (b1) Pr(H2 | H1) > Pr(H2), (b2) Pr(H2 | H1) > t, and (b3) Pr(H2) ≤ t. Suppose (c) H1 entails H2 and (d) (X) holds. By (a1), (a2), (b1), (b2), (c), (d), and Theorem 2, it follows that Pr(H2 | E) > Pr(H2) and Pr(H2 | E) > t. By (b3), Pr(H2) ≤ t. Hence Pr(H2 | E) > Pr(H2), Pr(H2 | E) > t, and Pr(H2) ≤ t. Hence E confirms-TSF H2.
Appendix 3: Proof of Theorem 5B
Suppose a card is randomly drawn from a standard deck of cards. Let E be the claim “The card drawn is a Heart”, H1 be the claim “The card drawn is a Red”, and H2 be the claim “The card drawn is a Diamond”. Then, E confirms-IF H1, since Pr(H1 | E) = 1 > Pr(H1) = 1/2, and H1 confirms-IF H2, given that Pr(H2 | H1) = 1/2 > Pr(H2) = 1/4, and both (C2) and (C3) hold, since Pr(H2) = 1/4 < Pr(H1 | E) = 1 and Pr(H2 ∧ ¬H1 | E) = 0 = Pr(H2 ∧ ¬H1). But E does not confirm-IF H2; Pr(H2 | E) = 0 < Pr(H2) = 1/4.
Appendix 4: Proof of Theorem 6
Consider the following schema, to be referred to (for lack of a better name) as “Schema”, where β ∈ ℝ+, β ≥ 1, and τ = 1 + (2/10)β + (1/10)β + (9/10)β + (1/10)β + (1/10)β + 10β:
Schema
E | H1 | H2 | Pr | E | H1 | H2 | Pr |
---|---|---|---|---|---|---|---|
T | T | T | 1/τ | F | T | T | (1/10)β/τ |
T | T | F | (2/10)β/τ | F | T | F | (1/10)β/τ |
T | F | T | (1/10)β/τ | F | F | T | 0 |
T | F | F | (9/10)β/τ | F | F | F | 10β/τ |
On each instance of Schema, it follows that:
By (1) and (2) it follows that (C1) holds. By (3) it follows that (C2) holds. By (4) it follows that (C3) holds.
The aim is to show that regardless of the value specified for t there are instances of Schema on which E confirms-IF&SF H1, H1 confirms-IF&SF H2, and yet, though E confirms-IF H2, E does not confirm-IF&SF H2 because Pr(H2 | E) ≯ t.
First, observe that each of Pr(H1 | E) and Pr(H2 | H1) approaches 1 as β tends to ∞:
So, regardless of the value specified for t there is a value for β such that Pr(H1 | E) > t and Pr(H2 | H1) > t.
The same is true of Pr(H2 | E), since Pr(H2 | E), like each of Pr(H1 | E) and Pr(H2 | H1), approaches 1 as β tends to ∞:
But, crucially, the following inequalities hold:
Next, consider the inequalities:
We noted above that each of Pr(H1 | E) and Pr(H2 | H1) approaches 1 as β tends to ∞. This is not true of Pr(H1) and Pr(H2)—quite the opposite in fact. Each of Pr(H1) and Pr(H2) approaches 0 as β tends to ∞:
With β = 1, Pr(H1 | E) = 6/11 > Pr(H1) = 7/62 and Pr(H2 | H1) = 11/14 > Pr(H2) = 3/31. So, given (5), (6), (12), and (13), and with β ≥ 1, it follows that (10) and (11) hold.Footnote 32
The argument now runs as follows. Take β = 1. Then Pr(H1 | E) = 6/11 > Pr(H1) = 7/62, Pr(H2 | H1) = 11/14 > Pr(H2) = 3/31, and Pr(H2 | E) = 1/2. If 6/11 > t > .5, we have an instance of Schema on which E confirms-IF&SF H1, H1 confirms-IF&SF H2, and yet, though E confirms-IF H2, E does not confirm-IF&SF H2 because Pr(H2 | E) ≯ t. If, instead, t ≥ 6/11, then let the value of β increase until Pr(H1 | E) > t and Pr(H2 | H1) > t but Pr(H2 | E) ≯ t; that there is such a value for β is guaranteed by (5), (6), (8), and (9). It will still be the case that Pr(H1 | E) > Pr(H1) and Pr(H2 | H1) > Pr(H2); this follows from (10) and (11). The resulting distribution will be an instance of Schema on which E confirms-IF&SF H1, H1 confirms-IF&SF H2, and E confirms-IF H2 but does not confirm-IF&SF H2 given that Pr(H2 | E) ≯ t.
The result is that none of (C1)–(C3) is a condition for transitivity in confirmation-IF&SF regardless of the value specified for t.Footnote 33
Appendix 5: Proof of Theorem 7
Consider Schema, and take β = 1. Then, as noted above, Pr(H1 | E) = 6/11 > Pr(H1) = 7/62, Pr(H2 | H1) = 11/14 > Pr(H2) = 3/31, and Pr(H2 | E) = 1/2. If 6/11 > t > .5, we have an instance of Schema on which E confirms-TSF H1, H1 confirms-TSF H2, and yet, though E confirms-IF H2, E does not confirm-TSF H2 because Pr(H2 | E) ≯ t. If t ≥ 6/11, then, as explained above, let the value of β increase until Pr(H1 | E) > t and Pr(H2 | H1) > t but Pr(H2 | E) ≯ t. Given (10) and (11), it will still be the case that Pr(H1 | E) > Pr(H1) and Pr(H2 | H1) > Pr(H2). Given (12) and (13), it will still be the case that Pr(H1) ≤ t and Pr(H2) ≤ t. The resulting distribution will thus be an instance of Schema on which E confirms-TSF H1, H1 confirms-TSF H2, but, since Pr(H2 | E) ≯ t, E does not confirm-TSF H2. Therefore, regardless of the value specified for t, none of (C1)–(C3) is a condition for transitivity in confirmation-TSF.
Appendix 6: Proof of Theorem 8
We showed above in the proof of Theorem 6 that regardless of the value specified for t there is an instance of Schema on which E confirms-IF&SF H1, H1 confirms-IF&SF H2, and E confirms-IF H2 but does not confirm-IF&SF H2 given that Pr(H2 | E) ≯ t. It follows immediately that regardless of the value specified for t there is an instance of Schema on which E confirms-SF H1, H1 confirms-SF H2, and E confirms-IF H2 but does not confirm-SF H2 because Pr(H2 | E) ≯ t. None of (C1)–(C3), therefore, is a condition for transitivity in confirmation-SF regardless of the value specified for t.
Rights and permissions
About this article
Cite this article
Roche, W., Shogenji, T. Confirmation, transitivity, and Moore: the Screening-Off Approach. Philos Stud 168, 797–817 (2014). https://doi.org/10.1007/s11098-013-0161-3
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11098-013-0161-3