Abstract
The paper explains how Gettier’s conclusion can be reached on general theoretical grounds within the framework of epistemic logic, without reliance on thought experiments. It extends the argument to permissive conceptions of justification that invalidate principles of multi-premise closure and require neighbourhood semantics rather than semantics of a more standard type. The paper concludes by recommending a robust methodology that aims at convergence in results between thought experimentation and more formal methods. It also warns against conjunctive definitions as sharing several of the drawbacks of disjunctive definitions.
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Notes
Much of the debate concerns alleged experimental findings of ethnic or gender variation in judgments about Gettier cases, following Weinberg et al. (2001). For a recent defence of the method of cases see Nagel (2012), and for a recent experimental study that did not find such bias see Nagel et al. (2013). For a different kind of scepticism about Gettier cases see Weatherson (2003). My discussion of Gettier cases in Williamson (2007) also concentrated on case-specific judgments.
Proof Suppose Kp ⊆ Jp for all propositions p; but for any world w, w ∊ KR(w), so w ∊ JR(w), so S(w) ⊆ R(w). Conversely, suppose S(w) ⊆ R(w) for all worlds w; but if w ∊ Kp, then R(w) ⊆ p, so S(w) ⊆ p, so w ∊ Jp; thus Kp ⊆ Jp.
The case was first published in Goldman (1976), which acknowledges Carl Ginet for the example.
See Kroedel (2012).
See Hughes and Cresswell (1996), pp. 221–223.
Chellas (1980, p. 234), calls modal logics with this rule monotonic.
Of course, if p is true at x then p ∪ {x} = p.
See Weatherson (2003).
Artemov (2008) analyses Gettier’s arguments and related considerations in the framework of justification logic, a refinement of epistemic logic in which the structure of justifications can be explicitly represented in the formal language. For present purposes, the austere framework of unrefined epistemic logic is preferable, because it assumes less and makes the comparison between knowledge and justified true belief more perspicuous. Nevertheless, justification logic is an intriguing resource for epistemologists to exploit.
Another problem for strictly conjunctive analyses is that they disallow compensation between how a putative instance scores on the various dimension relevant to the conjuncts. To put the point schematically, let being F depend on doing well on n dimensions, with compensation between dimensions. Suppose that we analyse what it is for x to be F as a conjunction of n conjuncts, where the ith conjunct is that t i < x i , where x i is how well x does on the ith dimension, t i is the required threshold for that dimension, and < is the relevant ordering relation. Given compensation between dimensions, we should have cases like this: a is F and b is F, where b i < a i but a j < b j (b compensates for doing worse than a on dimension i by doing better than a on dimension j), but c is not F, where c i = b i and c j = a j (c does not compensate for doing worse than a on dimension i by doing better than a on dimension j; for simplicity, assume that on each other dimension a, b, and c are equal). But this cannot happen on the conjunctive model. For since b is F, it satisfies the ith conjunct, so t i < b i = c i , so c satisfies the ith conjunct too; since a is F, it satisfies the jth conjunct, so t j < a j = c j , so c satisfies the jth conjunct too; since c equals a and b on all the other dimensions, it also satisfies all the other conjuncts; thus c is F on the conjunctive analysis.
References
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Acknowledgments
This article develops half of my talk at the 2013 ‘Gettier Problem at 50’ conference in Edinburgh; Williamson 201X develops the other half. I thank Allan Hazlett and audiences there and at the Universities of Michigan, Oxford, and Virginia for helpful comments.
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Williamson, T. A note on Gettier cases in epistemic logic. Philos Stud 172, 129–140 (2015). https://doi.org/10.1007/s11098-014-0357-1
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DOI: https://doi.org/10.1007/s11098-014-0357-1