2. The Degree of Epistemic Justification

1. 2. The Degree of Epistemic Justification

2. Question • Suppose you are justified in believing that the Roman Empire declined because of the Germanization of its military • Suppose also that you are justified in believing that salmon’s immune system has its base in its gills • Are you then justified in believing that the Roman Empire declined because of the Germanization of its military and salmon’s immune system has its base in its gills?

3. Overview • Subject: Formal Measure of (epistemic) Justification • Common View: Degree of justification is the conditional probability of the proposition given the evidence, P(h|e) • Proposal:The degree of justification is a function of the conditional probabilityP(h|e) and the prior probabilityP(h) • Motivation: The proposal is motivated by the dual goal of cognition.

4. Dual Goal of Cognition • To increase true beliefs and reduce false beliefs • It is easy to increase true beliefs: Believe everything you can think of!! • It is easy to reduce false beliefs: Abandon all beliefs you have!! • The challenge is to achieve both of these goals, or balance the two goals.

5. Dual Goal In Adding Beliefs • Addition and Removal • . • When we add propositions: We want to increase true beliefs but not increase false beliefs . • When we remove propositions: We want to reduce false beliefs but not reduce true beliefs • We will focus on cases of adding propositions • . • We are more justifiedin accepting propositions: We are more likely to achieve the dual goal of increasing true beliefs while not increasing false beliefs • “accept” here means adding to your body of beliefs.

6. Common View and the Dual Goal • Common View: The degree of justification is the conditional probability, J(h, e) = P(h|e) • Accommodating the Dual Goal: To balance the two demands,weadjustthe level of the probabilistic threshold tof justification • . • If we place more emphasis on increasing true beliefs, set the threshold low • If we place more emphasis on not increasing false beliefs, set the threshold high.

7. The Problem of the Conjunction • Intuitive Reasoning • Suppose we are justified in accepting h1 • Suppose we are justified in accepting h2 . • Then, we are justified in accepting h1h2 (Recall the initial question for this segment). • Formal Counterexample • . • P(h1|e) t • P(h2|e) t where e is the total evidence . • But often P(h1h2|e) < t Related puzzles:Lottery Paradox, Preface Paradox

8. Case of Independent Propositions • Consider probabilistically independent (mutually irrelevant) propositions h1, …, hn (as in the initial question) • Assume we are justified in accepting each of the propositionsh1, …, hn • We are then justified in accepting all of h1, …, hn • . • We are justified in accepting h1, i.e. J(h1, e)  t • Since h1 and h2 are mutually irrelevant, the prior acceptance of h1 does not affect the evaluation of h2 • So, we are justified in also accepting h2 • Repeat the same reasoning for h3, …, hn • . • We are then justified in accepting their conjunction, i.e. J(h1…hn, e) t (accepting all of them is no different from accepting their conjunction).

9. Diagnosis of the Conjunction Problem • Adding the conjunction is riskier (more likely to add a false belief) than adding a conjunct • But adding the conjunction can produce higher gain (can increase more true beliefs) than adding a conjunct • The higher risk is counterbalanced by the higher potential gain • The degree of justification for the conjunction need not be lower than that for a conjunct • The degree of confidence for the conjunction should be lower (because of the lower risk) than the degree of confidence for a conjunct.

10. Formalizing Two Factors • Two determinants of the degree of justification • .Risk of adding false beliefs • .Potential gain in true beliefs • Risk • The higher the conditional probability, the lower the risk • J(h, e) is an increasingfunction of P(h|e) • Potential Gain • The number of beliefs is irrelevant • The more information is added, the higher the potential gain • The amount of information: I(h) = – logP(h) • J(h, e) is an increasing function of I(h) = – logP(h) • J(h, e) is a decreasing function of P(h) More information  More specific  Lower prior probability • We need to balancethe risk and the potential gain.

11. A Third Factor? • IfP(h|e) and P(h) are the only determinants of the degree of justification, then J(h, e) = F(P(h|e), P(e)) for some function F • Is there a third determinant that affects the degree of justification? • The most (only?) plausible candidate is P(e) • P(e) is often an easily overlooked determinant, e.g. P(e|h) is determined by P(h), P(h|e), and P(e) • There is additional reason to consider this possibility seriously.

12. Stability (Resilience) 1. Suppose you have a die from a factory known to produce biased dice, but you don’t know which way the die you have is biased, so that the probability of h that you will not get one is 5/6 2. You roll the die 600 times and you get one 100 times, so that the probability for h is still 5/6 • Some people (e.g. Joyce 2005) think you are more justified in believing h in (2) than in (1) because of added stability (added resilience to revision) • Should we consider P(e) as a third determinant of the degree of justification? . P(h) = P(h|e) = 5/6 in both cases P(e) = 1 (tautology) in (1) P(e) is very small in (2)

13. P(e) Is Not a Third Factor • Suppose P(e) is a third factor, so that J(h, e) = F(P(h|e), P(h), P(e)) F(x, y, z1) ≠ F(x, y, z2) for some x, y, z1 and z2 such that z1 > z2 • Let P(h|e1) = x, P(h) = y, P(e1) = z1 • Let e* be irrelevant to h, e1and he1, and P(e*) = z2/z1 • So, P(h|e1  e*) = x, P(h) = y, P(e1  e*) = z1 [z2/z1] = z2 • This means that if P(e) is a third determinant, adding irrelevant evidence could change the degree of justification, which should not be the case • Caution: P(e) can still influenceJ(h, e) indirectly through its influence on P(h) and P(h|e), but it is not a third factor.

14. The Place of Stability • OK, P(e) is not a third factor, but shouldn’t we take stability into account somehow? • It should be part of the analysis of knowledge . • Justified (for the dual goal of cognition) • True • Belief • Gettier Condition (Stability) . • If you insist, you may call the combination of (what I call) justification and the Gettier condition “justification” (or give any other name), but our focus is justification in the less demanding sense • We now return to the challenge of balancing risk and potential gain.

15. General Conjunction Requirement • In order to narrow down the way we balance the two factors, we introduce the General Conjunction Requirement (GCR) • Idea:Conjunction operation on probabilistically independent (mutually irrelevant) propositions preserves the justification status • Formal Expression: Suppose h1, …, hn are probabilistically independent both unconditionally and conditionally on evidence e Then, for any threshold t, 1)If J(h1, e), …, J(hn, e) t, then J(h1  … hn, e) t 2)If J(h1, e), …, J(hn, e) <t, then J(h1  … hn, e) <t.

16. Formal Reasoning for GCR • GCR is intuitively plausible as it stands, but we can formally argue for it--we argue for the positive half by showing: . • the additional risk we take in adding hm+1 to {h1, …, hm} is the same as the risk we take in accepting hm+1 (if hm+1 is probabilistically independent of h1…hm on condition of e) . • the additional gain we may make in adding hm+1 to {h1, …, hm} is the same as the additional gain we may make in accepting hm+1 (if hm+1 is probabilistically independent of h1…hm)

17. Determining the Additional Risk • P(h1…hm+1|e) = P(hm+1|h1…hm  e)P(h1…hm|e) . • P(h1…hm+1|e) [on the left] determines the total risk we take in accepting h1…hm+1 • P(h1…hm|e) [second factor on the right] determines the original risk we have taken in accepting h1…hm • So, P(hm+1|h1…hm  e) [first factor on the right] determines the additionalrisk we take in adding hm+1 to {h1, …, hm} • P(hm+1|h1…hm  e) = P(hm+1|e) by probabilistic independence on condition of e . • P(hm+1|e) determines the independent risk we take in accepting hm+1 • So, the additionalrisk is the same as the independent risk.

18. Determining the Additional Potential Gain • The additionalinformationI(hm+1|h1…hm) we may gain in adding hm+1 to {h1, …, hm} is the difference between the totalinformationI(h1…hm+1) we may gain in accepting h1…hm+1 and the original informationI(h1…hm+1) we may have gained in accepting h1…hm+1 by Independence The additional information is the same as the independent informationI(hm+1) we may gain in accepting hm+1.

19. Justification for Addition • The degree of justification J(hm+1, e|h1 … hm) for adding hm+1 to {h1, …, hm} is the same as the degree of justification J(hm+1, e) for independently accepting hm+1 . • The additional risk is the same as the independent risk • The additional potential gain is the same as the independent potential gain • So, we are as justified in adding hm+1 to {h1, …, hm} as we are in accepting hm+1 Caution: h1 … hm in J(hm+1, e|h1 … hm) is not the background assumption • Since we are justified (by hypothesis) in accepting hm+1 independently, we are also justified in adding hm+1 to{h1, …, hm}. .

20. Finding J(h, e) • The measure of justificationJ(h, e) should: (a) be an increasing function of P(h|e) (b) be a decreasing function of P(h) (c) satisfy GCR • Any measure of confirmation meets (a) and (b) • Among many measures of confirmation, the logarithmic version of the rateof confirmation meets (c) [proof in Shogenji (forthcoming)] (We assume P(h) is neither 0 nor 1.)

21. Ordinal Equivalence • Uniqueness: Is J(h, e) theonly measure of justification (the only measure that meets GCR)? • . • No, it is not • Ordinal Equivalence: However, all measures of justification (GCR-satisfying measures) are ordinally equivalent [proof in Shogenji (forthcoming); Atkinson (forthcoming)] • . Recall: J1(h, e) and J2(h, e) are ordinally equivalent if and only if for any two pairs <hi, ei> and <hj, ej>, J1(hi, ei) < (=, >) J1(hj, ej) if and only if J2(hi, ei) < (=, >) J2(hj, ej) • I(h, e), R*(h, e), Z+(h, e) are not ordinally equivalent to J(h, e), so they are not measures of justification • J(h, e) is the measure of justification up to ordinal equivalence.

22. Notable Features of J(h, e) • The degree of justification (the rate of confirmation) has the following features: • It is equi-neutral • It is equi-maximal • It is not symmetric (It is not the degree of coherence).

23. Implication of Equi-Neutrality • Suppose P(h1|e1) = P(h1) is very highwhile P(h2|e2) = P(h2) is very low • In the absence of relevant evidence, we are no more justified in accepting highly probable h1 than we are in accepting highly improbable h2!! • Reason • Accepting h1 is low risk-low potential gain • Accepting h1 is high risk-high potential gain • The risk and the potential gain exactly offset each other • Justification vs. Confidence • We should be more confident about h1 than we are about h2 • We should use the degree of confidence, P(h|e), in some contexts (e.g. for calculating the expected utility).

24. Equi-Maximality • Beyond Conditional Maximality • For any given P(h), J(h, e) is maximal when P(h|e) = 1 because J(h, e) is an increasing function of P(h|e) • Equi-maximality says that this maximal value is constant regardless of P(h) • Intuitive Meaning • Whether P(h) is high or low, P(h|e) = 1 makes hcompletely justified by e • We need P(h) ≠ 1 to avoid P(h|e) = P(h) = 1 (avoid being neural and maximal at the same time).

25. Review: Measures of Information • Self Information: • Amount of information on h we gain when we come to know h • .I(h) = – log2P(h) • . • Example: If P(h) = 1/8, then I(h) = – log22–3 = 3 • Mutual Information: • Amount of information on h we gain when we come to know e • .I(h, e) = log2P(h|e) – log2P(h) • . • Example: If P(h) = 1/8 and P(h|e) = 1/2, then I(h, e) = log22–1 – log22–3 = 2.

26. Justification and Information • . • The degree of justification is the ratio of mutual information to self information • Mutual information is earnedinformation • Self information registeredinformation • The degree of justification is the ratio of the earned information to the registered information • The higher the ratio, the more likely the acceptance servesthe dual goal of cognition.

27. The Logarithmic System of Measures • Earlier we considered the logarithmic and the non-logarithmic systems of the amount/rate of confirmation/mutual confirmation, e.g. • The logarithmic rate of confirmation: J(h, e) • The non-logarithmic rate of confirmation: Z+(h, e) • Since J(h, e) is the measure of justification (up to ordinal equivalence, while Z+(h, e) is not, the logarithmic system is preferable: