Abstract
In this paper, I’ll survey a number of closure principles of epistemic justification and find them all wanting. However, it’ll be my contention that there’s a novel closure principle of epistemic justification that has the virtues of its close cousin closure principles, without their vices. This closure principle of epistemic justification can be happily thought of as a multi-premise closure principle and it cannot be used in Cartesian skeptical arguments that employ a closure principle of epistemic justification. In this way, then, it represents marked improvement over other contemporary closure principles of epistemic justification that require both sacrificing multi-premise closure and forcing anti-Cartesian skeptics who reject the closure principle employed in certain Cartesian skeptical arguments to cast aside justification closure.
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Notes
I treat “implication” and “entailment” as synonymous in this paper.
In this paper, I’m taking justification to be fallibilist in character, that is, I’m taking it for granted that one can be justified in believing that p even though the probability of p on her evidence is less than 1.
E.g., for all S, p, and q: If S is justified in believing that p, S knows that p entails q, then S is justified in believing that q.
In this paper, I’m only interested in deductive versions of the closure principle, not probabilistic or inductive versions.
Some Cartesian skeptical arguments, of course, employ an underdetermination principle, which is, standardly, treated as logically distinct from the closure principle, although some philosophers—e.g., Anthony Brueckner (1994)—disagree that these principles are logically distinct.
I take S to have knowledge-grade doxastic epistemic justification for believing that p only if (i) S has whatever degree of propositional justification for believing that p that’s required for S knowing that p, (ii) S believes that p, and (iii) S’s belief in p is appropriately connected to S’s propositional justification for believing that p, where S has the degree of propositional justification for believing that p that’s required for knowledge that p only if S has sufficient epistemic reason to believe that p or adequate evidence for believing that p and S’s epistemic grounds for believing that p aren’t subject to a relevant defeater.
Take note that the phrase “justifiably believes” is synonymous with “has doxastic justification for believing”.
For reasons of style and convenience, I’ll omit “knowledge-grade” from “knowledge-grade justification for believing that p,” “or S knowledge-grade justifiably believes that p” from “S has knowledge-grade justification for believing that p or S knowledge-grade justifiably believes that p,” and “knowledge-grade” from “S knowledge-grade justifiably believes that q” for the remainder of this paper, but it should be supplied by the reader unless otherwise noted by me.
See, e.g., Andrew Wohlgemuth (1990: p. 234).
See Timothy Williamson (2000: p. 117) for a similar competent deduction condition on knowledge closure.
If (a) S is justified in believing that p, (b) S competently deduces p from q (or S knows that p entails q), then S is justified in believing that q in virtue of (a) and (b). (cf. Moretti and Tommaso 2013: p. 7).
It’s worth noting that any transmission principle is a closure principle, but the converse doesn’t hold.
Evidence, E, for q is semi-independent iff E is an evidence set of S’s that includes some evidence in favor of q that’s independent of either (x) or her competent deduction of q from p (or her knowledge that p entails q) and E includes (x) and S’s competent deduction of q from p (or her knowledge that p entails q).
To get a clearer picture of how this is supposed to work, consider the following scenario. Imagine Wayne, an undergraduate student, is taking a low-level mathematics class. Suppose Wayne’s professor and the class TA tell him that q, some complicated-looking mathematical proposition, logically follows from p, some rather innocent looking mathematical proposition. Intuitively, Wayne knows (by way of testimony) that p entails q, but he has no idea how to deduce q from p.
See Christopher Hookway (2006: p. 107) for a different, but very interesting proposal for defanging the so-called lottery paradox.
There are, of course, some complicated issues involving both normative and mental state defeaters—see, e.g., Michael Sudduth (2017: ch. 3)—but we can safely set those issues aside for the purposes of this paper. I’m also setting to the side recent challenges to the very idea of defeat from, e.g., Lasonen-Aarnio (2010) and Bob Beddor (2015). Although it’s my view that these challenges to defeat can be overcome.
A referee indicated that for a single premise closure principle a no-defeaters condition like in (6) might be redundant. To see why one might think this, consider an example. Suppose Ed believes that Harry is in London, and so Ed competently deduces that he’s not in Atlanta, but Ed acquires evidence, D, that Harry is in Atlanta. D not only defeats Ed’s justification for the entailed proposition, but also for the entailing proposition as well. In which case, one might be tempted by the thought that if there’s a defeater for the entailed proposition, then there will be a defeater for the single entailing proposition, and so single premise closure principle isn’t violated in cases involving defeat.
However, while it is certainly true in some cases that a defeater for the entailed proposition is a defeater for the single entailing proposition, this isn’t true in all cases. Imagine, for instance, that Arwen, a graduate student in mathematics, has excellent evidence for the proposition the number 2 is the only even prime number. Since any logically true proposition entails any other logically true proposition, the proposition the number 2 is the only even prime number entails that any bounded nonempty subset of ℝ has a least upper bound in ℝ. Suppose that Arwen, in fact, competently deduces the latter proposition from the former. It’s a complicated deduction, but Arwen carries it out competently. Suppose, though, that Arwen’s teachers and peers in the mathematics department tell her that it’s false that any bounded nonempty subset of ℝ has a least upper bound in ℝ. Intuitively, Arwen isn’t justified in believing that any bounded nonempty subset of ℝ has a least upper bound in ℝ, since the testimony of her teachers and peers undercuts her epistemic grounds for believing the entailed proposition in this connection, but that defeater does nothing to negatively impact her justification for believing that the number 2 is the only even prime number. Her epistemic grounds with respect to that proposition are left intact in the face of her colleagues’ testimony. So, it’s just not the case that when an entailed proposition is subject to a defeater that, necessarily, the single entailing proposition is subject to a defeater as well.
A propositional defeater is a proposition that, were S to become aware of its truth, would negatively impact S’s justification. See Moretti and Piazza (2018: p. 2846).
For example, if X is told by four experts that the TV is broken, then, if one of the four experts just made up his testimony that the TV is broken, that would negatively impact X’s justification, but X would still have knowledge-grade justification for believing that the TV is broken.
Consider also a further example. Imagine X has excellent evidence that p and no one is aware of any counterevidence for p, but there’s some undefeated incredibly complex, highly advanced true scientific proposition that’s not even related to p and that it’s metaphysically impossible for X to become aware of its truth. Now, even though, the true scientific proposition is, by definition, a propositional defeater for p, since it’s true by way of the relevant antecedent being false—i.e., it’s false that X could become aware of the scientific proposition’s truth—X is still justified (to whatever degree is required for knowledge) in believing that p.
Consider one last example. Suppose that X has an overwhelming amount of evidence in favor of some true scientific proposition, p, at t and no one is aware of any counterevidence to p at t. But one hundred years from t a scientific discovery will be made and, at that distant point in the future, there will be a proposition, q, that if X were aware of its truth at t, would negatively impact his justification for p at t. Despite there being a propositional defeater for p, it’s very plausible to think that X still has knowledge-grade justification for p at t.
A similar verdict would be reached in R.E. Fraser and Hawthorne’s (2015: p. 166) Nostradamus case against justification closure.
Similar considerations would, of course, apply to Vann McGee’s (1985) purported counterexample to modus ponens.
Keep in mind that I’m stipulating that there’s neither a normative defeater nor a mental state defeater for Darian’s perceptual belief in dog.
Some philosophers have followed Klein here, e.g., Rodrigo Borges (2015).
A view that fits this mold was suggested to me by a referee.
An extraordinary or heavyweight proposition is a proposition that our everyday beliefs presuppose and that’s an implication of an everyday proposition, but one that we typically take to go beyond our powers to know—e.g., we typically take it to go beyond our powers to know that we aren’t handless-brains-in-vats. For some discussion of propositions of this kind, see Dretske (2003, 2005), Crispin Wright (2004), and Avnur (2012).
Everyday beliefs are beliefs that we straightforwardly take ourselves to know.
And, as we saw with Klein’s Suggestion, appealing to the antecedent of P1 itself as doing the justifying of the consequent of P1 has implausible consequences.
For Cartesian skeptical arguments to get off the ground, the evidence that the subject has for either the entailing proposition or the entailed proposition must be skeptic-friendly evidence. For example, the skeptic isn’t going to give the subject testimony from God that he has hands, since that’s excellent evidence for thinking that one isn’t a handless-brain-in-a-vat, which would undermine the skeptic’s argument.
Note well that condition (II) of (9) isn’t a necessary antecedent condition to the effect that the subject is justified in believing the entailed proposition.
This question was posed to me by a referee.
Thanks to referees for pressing these questions.
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Simpson, J. Getting a little closure for closure. Synthese 199, 12331–12361 (2021). https://doi.org/10.1007/s11229-021-03335-w
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DOI: https://doi.org/10.1007/s11229-021-03335-w