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Is There a Viable Account of Well-Founded Belief?

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Abstract

My starting point is some widely accepted and intuitive ideas about justified, well-founded belief. By drawing on John Pollock’s work, I sketch a formal framework for making these ideas precise. Central to this framework is the notion of an inference graph. An inference graph represents everything that is relevant about a subject for determining which of her beliefs are justified, such as what the subject believes based on what. The strengths of the nodes of the graph represent the degrees of justification of the corresponding beliefs. There are two ways in which degrees of justification can be computed within this framework. I argue that there is not any way of doing the calculations in a broadly probabilistic manner. The only alternative looks to be a thoroughly non-probabilistic way of thinking wedded to the thought that justification is closed under competent deduction. However, I argue that such a view is unable to capture the intuitive notion of justification, for it leads to an uncomfortable dilemma: either a widespread scepticism about justification, or drawing epistemically spurious distinctions between different types of lotteries. This should worry anyone interested in well-founded belief.

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Notes

  1. We might term a reason that is adequate for believing p a prima facie reason to believe p. Pollock (1974) discusses the idea of prima facie justification, and Pryor (2004), for instance, make use of the notion. However, note that, unlike Pollock, I am not intending to restrict prima facie reasons to non-deductive reasons.

  2. For instance, some of the things Earl Conee and Richard Feldman say could be interpreted as an endorsement of such a view. As an example, they claim that as long as believing a proposition fits the evidence a subject has, the subject’s belief in that proposition is justified, despite failing to be founded or based on adequate evidence (see Feldman and Conee 2004, pp. 92–93).

  3. Very many epistemologists flag ideas along these lines. For instance, in the contemporary debate the ideas about justification sketched by Jim Pryor (2004) come at least close. What complicates things a bit is that Pryor allows irrational, unjustified doubts to turn justified beliefs into unjustified ones (see 2004, p. 365). Perhaps such doubts could just be viewed as reasons the subject has (after all, Pryor restricts his claim to doubts that the subject does not recognise as unjustified), in which case the view is perfectly compatible with Structuralism.

  4. I will talk about beliefs corresponding to nodes in an inference graph, intending to remain neutral on this choice. If the occupiers of nodes are propositions, then a belief corresponds to a node just in case the occupier of the node is a proposition p, the belief has p as its content, and the belief has the right sort of inferential history, a history encoded by the relevant part of the inference graph. If the occupiers of nodes are mental state types, then a belief corresponds to a node just in case it falls under the type assigned to that node, and the belief has the right sort of inferential history.

  5. Pollock (personal correspondence) draws a contrast between the actual historical basis of a belief and the basis recorded by the reasoning system.

  6. See Pollock and Cruz (1999, p. 195). However, the basic idea goes back much further. For instance, Chisholm (1964) defends a similar thought. Instead of prima facie reasons, Chisholm talks about a proposition completely justifying a belief in another proposition. Assume that a proposition p completely justifies a belief in a proposition q. The idea is that d defeats this justification if and only if d is a true proposition and the conjunction d & q does not completely justify the subject in believing q. In effect, this criterion for being a defeater is assumed pretty much by anyone defending the sort of defeasibilist analysis of knowledge popular in the post-Gettier debate. See, for instance, Lehrer and Paxson (1978) and Annis (1978).

  7. There is also another reason for adding D to the definition of an inference graph, which has to do with the function σ. In particular, one might worry that the strengths of the members of the defeat-relation cannot be read from the rest of the inference graph, but must be specified separately through σ.

  8. This definition allows the roots of defeat-links to be sets consisting of more than one node. Someone who thought that a defeater was always the occupier of a single node (such as Pollock 1995), could require that J always contains only a single node.

  9. Note that I am not here defining the reflexive ancestral relation: a node k is not automatically a support-ancestor of itself.

  10. See, for instance, Enderton (1977) for the definition of a well-founded relation.

  11. One might think that in addition to the conditions given, every node should either be basic or the target of a support-link. However, it’s not clear what work ruling out isolated nodes would do, as long as beliefs corresponding to such nodes would not be justified, and as long as such nodes could not be roots of defeat-links.

  12. Hence, thought I talk about degrees of justification, the picture being sketched is compatible with the thought that being justified in believing a proposition is an all-or-nothing affair. As an analogy, one could talk about different degrees of tallness, while holding that only people above, say, 180 cm are tall.

  13. Pollock (2001) attempted to formulate set of rules, or what is referred to within AI as a “semantics”, that takes into account degrees of justification, instead of just assigning one of two defeat-statuses to each node, but retracted this semantics. The semantics he proposed in (1995) and (1999) involving just two defeat-statuses have also turned out to have counterexamples. He subsequently worked on a new semantics.

  14. I am here simply ignoring possible cases in which there is a maximal degree of support without an entailment.

  15. To be more precise, the strength of j in inference graph G in which q is inferred from p 1, …, p n in one step equals the strength of a node j* in a different inference graph G* that differs from G only in that q is inferred through two steps, first one in which a subject infers the conjunction p 1 & … & p n from p 1, …, p n and second a step in which she infers q from p 1 & … & p n .

  16. No doubt numerous philosophers would disagree, arguing that WLP is false for the reason that risks can pile up over multi-premise deductions in a way that it does not allow for.

  17. When I talk about the degree to which a set of premises supports a conclusion, it is worth underlining that this notion of support is different from the notion of degree of evidential support that arises in connection with different measures of confirmation. The latter is a comparative notion, having to do with what impact acquiring certain evidence has on the probability of a proposition. For instance, a well-known idea is one on which the degree of confirmation of a proposition h by a body of evidence E is the probability of h conditional on E minus the unconditional probability of h. If we translate this talk of probabilities to talk of degrees of justification, the question is how much more justified acquiring a piece of evidence makes one in believing a given proposition than one previously was.

  18. The definition is the following: Pr(p|q) = Pr(p & q)/Pr(q). Some prefer to treat this as a probability axiom instead of a definition.

  19. Of course, this idea would have to be qualified in light of non-propositional items of evidence.

  20. I am assuming here that none of the conjuncts logically entail others.

  21. For instance, assume that p is the proposition that it will snow, and q is the proposition that the temperature will be freezing. These propositions are positively probabilistically relevant, and intuitively, the degree to which one is justified in believing their conjunction ought to be higher than the degree of justification that the rule mentioned would yield.

  22. The idea that one should proportion one’s belief to the evidence has been championed by Hume, Quine, and many others. In so far as our concern is a finer-grained notion of belief, this leads very naturally to the thought that one’s degree of belief in a proposition should mirror how likely it is on one’s evidence. The present framework replaces talk of evidence by talk of reasons. Then, proportioning one’s beliefs to one’s reasons will require believing a proposition to a degree that reflects the strength of one’s reasons for that proposition—or, more precisely, to a degree that reflects the strength of the node corresponding to that belief, this strength just being the degree to which one is justified in believing that proposition.

  23. The idea is that defeat will be an all-or-nothing affair, and that defeated nodes will have minimal strengths.

  24. Ultimately, the structuralist wedded to WLP will have to give the same type of sceptical treatment of lotteries in which at least one ticket will win.

  25. The defeat-links <j 1, i 1>, <j 2, i 2>, and <j 3, i 3> do not go both ways: for instance, node j 1 is considered to be a defeater for node i 1, but not vice versa. First, for present purposes I am assuming a criterion for being a defeater on which deductive inferences are not defeasible, and the inferences to t 1, t 2, and t 3 are deductive. Second, even if an alternative criterion for being a defeater was adopted on which deductive inferences were defeasible, I do not think a case could be made for thinking that i 1, for instance, was a defeater for j 1. We do not seem to be dealing with a typical case of defeat of a deductive inference: the subject does not, for instance, have any reason to think that she has made a mistake in inferring any of the propositions t 1, t 2, and t 3 from the premises that logically entail them. Finally, even if the defeat-links did go both ways, I think the right verdict on the graph would be the one I reach below.

  26. I think it is also extremely plausible that given these assignments, k will be undefeated, but do not really need this assumption, and will not complicate things by including more principles.

  27. I am assuming that subjects who instantiate inference graph GL hold the same attitude towards each ticket in the lottery, and perform the same inferences regarding each ticket.

  28. One could argue that in cases in which a node j is based on a defeated node i that is completely redundant for the inference, j is not defeated.

  29. This is a problem that Pollock (1995, pp. 64–65) raises for his own account.

  30. See Pollock and Cruz (1999, pp. 231–233), Pollock (1995, pp. 64–67).

  31. See Makinson (1965).

  32. Throughout the discussion of the statistical syllogism and its relevance for the Paradox of the Preface, I am using notation that is essentially the same that Pollock uses. See, for instance, Pollock (1995, p. 125, § 9).

  33. See, for instance, Pollock and Cruz (1999, pp. 229–234).

  34. In the lottery case discussed, r, a description of the structure of the lottery, provides a subject with both a reason to believe each of ~t 1, ~t 2, and ~t 3 stating of individual tickets that they will lose, and a deductive reason to believe that not all of ~t 1, ~t 2, and ~t 3 are true. The kind of Preface Paradox situation I will describe is one in which a subject has independent reasons for believing each of p 1, …, p n , and a yet separate reason of the kind just discussed for believing that not all of p 1, …, p n are true, simply because such situations are more typical. However, this is not essential to Preface Paradox cases.

  35. Generally, \( \left\lceil {r\& \Pr (p|q\& r) \ne \Pr (\Pr (p|q))} \right\rceil \) is an undercutting defeater for the reason for believing p provided by \( \left\lceil {q\& \Pr (p|q) \ge r} \right\rceil \). See Pollock (1995, pp. 66–67).

  36. Note that this is also true of lotteries in which there is guaranteed to be at least (but maybe not exactly) one winning ticket.

  37. I am bracketing the issue of what the inference parents of this node are, just as I bracketed the issue of the inferential history of the node supporting the belief in the following proposition: \( \Pr (\exists p_{i}(p_{i} \in X\;\&\; \sim p_{i} )\;|\;{\mathbf{F}}(X)) \ge 2/3. \)

  38. There are numerous other possible forms of lotteries where there is negative relevance, but of a degree lesser than in standard lotteries.

  39. However, there are such lotteries, since the equation 1 − r n = r has real solutions.

  40. To be more precise, we might assume p to state the following facts: There are three standard lotteries, and each of these has exactly three tickets. t 1 is true just in case ticket #1 wins the first standard lottery, t 2 is true just in case ticket #2 wins the second standard lottery, and t 3 is true just in case ticket #3 wins the third standard lottery. The outcomes of the standard lotteries are independent. Then, for each ~t i , p makes ~t i probable to degree 2/3 (≈0.67). This is the strength of the relevant inference link. The probability that at least one of the tickets will win is given by 1 − (2/3)3 (≈0.70). This would be a case in which p supports the proposition that at least one of the tickets will win to a higher degree than it supports any of the propositions stating of the individual tickets that they will lose. Again, set the minimal threshold level for justification at 2/3.

  41. This inference graph is identical to a graph representing a Preface situation in which the subject’s reason for believing each of three propositions p 1, p 2, and p 3 and her reason for believing ~(p 1 & p 2 & p 3) is the same. Assume that this reason consists of believing that the set of propositions {p 1, …, p n } has the following feature F: each proposition is supported to degree 2/3 by the evidence and the propositions are probabilistically independent. Then, the set having feature F is a reason to believe each of p 1, p 2, and p 3, and it is a reason to believe ~(p 1 & p 2 & p 3).

  42. See Pollock (1995, pp. 61–62).

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Lasonen-Aarnio, M. Is There a Viable Account of Well-Founded Belief?. Erkenn 72, 205–231 (2010). https://doi.org/10.1007/s10670-009-9200-z

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