Abstract
Bayesian confirmation theory is rife with confirmation measures. Zalabardo (2009) focuses on the probability difference measure, the probability ratio measure, the likelihood difference measure, and the likelihood ratio measure. He argues that the likelihood ratio measure is adequate, but each of the other three measures is not. He argues for this by setting out three adequacy conditions on confirmation measures and arguing in effect that all of them are met by the likelihood ratio measure but not by any of the other three measures. Glass and McCartney (2015), hereafter “G&M,” accept the conclusion of Zalabardo’s argument along with each of the premises in it. They nonetheless try to improve on Zalabardo’s argument by replacing his third adequacy condition with a weaker condition. They do this because of a worry to the effect that Zalabardo’s third adequacy condition runs counter to the idea behind his first adequacy condition. G&M have in mind confirmation in the sense of increase in probability: the degree to which E confirms H is a matter of the degree to which E increases H’s probability. I call this sense of confirmation “IP.” I set out four ways of precisifying IP. I call them “IP1,” “IP2,” “IP3,” and “IP4.” Each of them is based on the assumption that the degree to which E increases H’s probability is a matter of the distance between p(H | E) and a certain other probability involving H. I then evaluate G&M’s argument (with a minor fix) in light of them.
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Notes
The neutral value for c 1 and c 3 is 0. The neutral value for c 2 and c 4 is 1.
Let c and c* be confirmation measures. Then, c and c* are ordinally equivalent to each other if and only if the following holds for all ordered pairs of propositions <H 1, E 1> and <H 2, E 2>: c(H 1, E 1) > / = / < c(H 2, E 2) if and only if c*(H 1, E 1) > / = / < c*(H 2, E 2).
I say “in effect” because Zalabardo does not argue explicitly that c 4 meets each of his three adequacy conditions. He argues explicitly that (i) c 2 but not c 1 meets his first adequacy condition, (ii) c 4 but not c 3 meets his second adequacy condition, and (iii) c 4 but not c 2 meets his third adequacy condition. I take it to be a tacit premise in his argument, though, that c 4 meets his first adequacy condition, since without that premise (or a premise or premises entailing it), it would not follow that c 4 is adequate but c 1, c 2, and c 3 are not.
G&M focus on logarithmic versions of c 2 and c 4. This difference matters not at all in this discussion.
Zalabardo, though, does not refer to it as his first adequacy condition. When he speaks of his first adequacy condition, he has in mind AC3.2 below. This is a mere terminological matter however. There is no doubt that he holds that any adequate confirmation measure should meet AC1.0. Similar comments are in order with respect to AC2.0 and AC3.0 below.
The same is true, of course, with respect to c 3. Things are different with c 1 and c 2. It follows both by c 1 and by c 2 that the degree to which E 1 confirms H 1 is less than the degree to which E 2 confirms H 2; c 1(H 1, E 1) ≈ 0.320 < 0.66 = c 1(H 2, E 2) and c 2(H 1, E 1) ≈ 1.481 < 7.6 = c 2(H 2, E 2).
There is a bit of an exegetical puzzle here. Each of (1), (2), and (3) in the following has some prima facie plausibility (to say the least):
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1.
G&M claim in effect in the last sentence in the first displayed passage in this subsection that in the modified version of Schlesinger’s case, H 2’s increase in probability due to E 2 from 1/10 to 19/25 is greater than H 1’s increase in probability due to E 1 from 2/3 to 76/77.
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2.
G&M claim in the same paragraph from which that passage is taken that by c 4, it follows that in the modified version of Schlesinger’s case, the degree to which E 1 confirms H 1 is greater than the degree to which E 2 confirms H 2.
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3.
G&M claim in effect in the second sentence in the third displayed passage in this subsection that the sense of confirmation at issue is confirmation in the sense of increase in probability (H’s increase in probability due to E).
It seems to follow from (1), (2), and (3), though, that
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4.
G&M should reject c 4 on the grounds that it issues the wrong verdict in the modified version of Schlesinger’s case.
But, of course, G&M do not reject c 4 on the grounds that it issues the wrong verdict in the modified version of Schlesinger’s case. What gives? I read G&M as follows. The claim referred to in (3) should be understood in terms of IP unprecisified. The claim referred to in 1, in contrast, should be understood in terms of IP precisified along the lines of IP1 below. If G&M are thus read, then 4 does not follow from (1), (2), and (3) and the puzzle is thus resolved.
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1.
c 5 meets AC3.1 in the special case where p(E | H 1) > p(E | ¬H 1) and p(E | H 2) > p(E | ¬H 2) but not in the special case where p(E | H 1) < p(E | ¬H 1) and p(E | H 2) < p(E | ¬H 2).
There is no mention of c 6 in ZA. But ZA could be modified so as to include mention of c 6. Zalabardo (2009, p. 632, fn. 5) notes in a footnote that c 6, as with c 3, fails to meet AC2.0.
Hence, (2) in ZA is false. Hence, ZA, as with GMA, is unsound.
There is a worry to the effect that the move from AC3.1 to AC3.3 is ad hoc. G&M would need to answer this worry if they were to go with my suggested fix.
The title of Crupi and Tentori (2013) is “Confirmation as partial entailment: a representation theorem in inductive logic.”
I am assuming, as is standard, that c 4 should be understood so that if H is entailed by E and thus Pr(E | ¬H) = 0, then c 4(H, E) = ∞. See G&M (2015, p. 62, n. 4) and Iranzo and Martinez de Lejarza (2013, sec. 3) for relevant discussion.
It is straightforward to show that c 1 and c 2 meet each of IP1b-IP1d and that c 3, c 5, c 6, and c 7 do not. It is also straightforward to show that c 1 but not c 2 meets the following generalization of IP1d:
$$ \mathrm{IP}1\mathrm{d}\ast :\mathrm{If} p\left({H}_1|{E}_1\right)= p\left({H}_2|{E}_2\right)\ \mathrm{and}\ p\left({H}_2\right)> p\left({H}_1\right),\mathrm{then} c\left({H}_1,{E}_1\right)> c\left({H}_2,{E}_2\right) $$Suppose, for example that
$$ \begin{array}{c}\hfill p\left({H}_1|{E}_1\right)=0= p\left({H}_2|{E}_2\right)\hfill \\ {}\hfill p\left({H}_1\right)=0.01<0.02= p\left({H}_2\right)\hfill \end{array} $$Then, c 1(H 1, E 1) = −0.01 > −0.02 = c 2(H 2, E 2) whereas c 2(H 1, E 1) = 0 = c 2(H 2, E 2). It seems clear that IP1 should be modified so as to include IP1d*, for prior probabilities always matter when the issue is absolute distance. It thus seems clear that c 2 is inadequate in the context of IP1.
It is straightforward to show that c 6 meets each of IP2b-IP2d and that c 1, c 2, c 5, c 6, and c 7 do not.
It is straightforward to show that c 5 meets each of IP3b-IP3e and that c 1, c 2, c 3, c 6, and c 7 do not.
It is straightforward to show that c 7 meets each of IP4b-IP4e and that c 1, c 2, c 3, c 5, and c 6 do not.
References
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Acknowledgements
I wish to thank an anonymous reviewer and Tomoji Shogenji for very helpful comments on prior versions of the paper.
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Appendices
Appendix 1
Suppose that:
IP1 implies that c(H 1, E 1) < c(H 2, E 2) whereas IP2 implies that c(H 1, E 1) > c(H 2, E 2). Hence, IP2 disagrees with IP1 on c(H 1, E 1) versus c(H 2, E 2). IP3 implies that c(H 1, E 1) = c(H 2, E 2). Hence, IP3 disagrees with each of IP1 and IP2 on c(H 1, E 1) versus c(H 2, E 2). IP4 implies that c(H 1, E 1) = c(H 2, E 2). Hence, IP4 disagrees with each of IP1 and IP2 on c(H 1, E 1) versus c(H 2, E 2). IP3 implies that c(H 3, E 3) = c(H 4, E 4) whereas IP4 implies that c(H 3, E 3) > c(H 4, E 4). Hence, IP4 disagrees with IP3 on c(H 3, E 3) versus c(H 4, E 4). Hence, no two of IP1, IP2, IP3, and IP4 agree with each other on all cases. QED
Appendix 2
1.1 2.1
All instances of Schema 1 are such that
All instances of Schema 2 are such that
Given that (17) and (18) hold on all instances of Schema 1 and that (19) and (20) hold on all instances of Schema 2, it follows that (4) holds on all instances of Schema 1 and on all instances of Schema 2. QED
1.2 2.2
It follows on Schema 1 that if β = 1, then:
Next, observe that:
That (5) holds on Schema 1 follows from (21) and (22). That (6) and (7) hold on Schema 1 follows from (23) and (24). QED
1.3 2.3
It follows on Schema 2 that if β = 1, then
Next, observe that
That (8) holds on Schema 2 follows from (25) and (26). That (9) holds on Schema 2 follows from (27). That (10) holds on Schema 2 follows from (28). QED
1.4 2.4
First, observe that on Schema 1
Second, observe that on Schema 2
Third, it can be verified that the following holds for any admissible value for β:
That (11) holds follows from (29). That (12) holds follows from (30). That (13) holds follows from (31). QED
1.5 2.5
First, observe that on Schema 1
Second, observe that on Schema 2
Third, it can be verified that the following holds for any admissible value for β:
That (14) holds follows from (32). That (15) holds follows from (33). That (16) holds follows from (34). QED
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Roche, W. Confirmation, Increase in Probability, and the Likelihood Ratio Measure: a Reply to Glass and McCartney. Acta Anal 32, 491–513 (2017). https://doi.org/10.1007/s12136-017-0322-7
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DOI: https://doi.org/10.1007/s12136-017-0322-7