Abstract
Transmission arguments against closure of knowledge base the case against closure on the premise that a necessary condition for knowledge is not closed. Warfield argues that this kind of argument is fallacious whereas Brueckner, Murphy and Yan try to rescue it. According to them, the transmission argument is no longer fallacious once an implicit assumption is made explicit. I defend Warfield’s objection by arguing that the various proposals for the unstated assumption either do not avoid the fallacy or turn the central premise of the transmission argument, namely that a necessary condition is not closed, into a redundant and superfluous premise. I conclude that Warfield’s advice is still to be heeded: Arguments against closure must not rely essentially on the premise that a necessary condition for knowledge is not closed.
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Notes
In this paper I only discuss the principle of knowledge closure, not other epistemic closure principles (for example closure of justification). Henceforth, “closure” refers to closure of knowledge unless noted otherwise.
The argument without the formulas is taken verbatim from (Warfield (2004), p. 38).
Precursors or contemporaries of Warfield are Vogel who gives an example of an analysis of knowledge that contains a non-closed condition but entails that knowledge is closed (so-called “contracking”, 1987, p. 201), Hales who argues that the transmission argument commits the “fallacy of division” (Hales 1995, p. 188), Leite who points out that the conclusion entails the possibility of knowledge but the premises do not (2004, p. 339) and Huemer who gives a set-theoretic counterexample (2005, p. 178).
Warfield comes dangerously close to committing the fallacy fallacy. Although he offers everything that is needed to avoid this fallacy, he does so only “for concreteness” and as an “illustration” (2004, p. 39). Moreover, he writes: “This argument form (of which the two previously displayed arguments are instances) is invalid” (2004, p. 38). With the parenthesis Warfield invites the mistaken belief that being an instance of the transmission argument is sufficient for being invalid.
Knowledge is an example: It is both necessary and sufficient for knowledge. Trivially, if knowledge is not closed, knowledge is not closed. Other valid instances (besides the uninteresting ones with contradictory premises) can be generated by choosing a relation \(R\) such that \(R(p, q)\) entails \(Kp\).
Warfield argues for two instances of (A)—cf. his (7) and (10)—that they are consistent with the other premises (cf. 2004, p. 39). (A) is equivalent to \(\square \,\forall p \,\forall q \, (Kp \wedge R(p, q) \rightarrow Xq)\). I will rely on both versions in the following.
Warfield does not offer a displayed version of his proposal. This might explain why his alternative to the transmission argument went largely unnoticed.
For proofs of (T*) cf. (Huemer (2005), p. 180) and (Murphy (2006), p. 367). However, Huemer and Murphy assume without argument the existence of necessary but insufficient conditions that are jointly sufficient. The existence of such conditions is a controversial issue (cf. Williamson 2000). But note that (T*) only requires the existence of some set of such conditions; it does not require the existence of a set of informative or conceptually more basic conditions. If no specific further requirements are in play for being a necessary condition, it is not difficult to construct a set of necessary but insufficient conditions that are jointly sufficient. A trivial example can be constructed by quantifying over times. Let \(T\) be the set of all \(t\) such that \(S\) knows that \(p\) at \(t\). Then \(S\) knows that \(p\) at \(t_{0}\) iff (i) \(t_{0}\) is not after the last \(t \in T\), (ii) \(t_{0}\) is not before the earliest \(t \in T\) and (iii) \(t_{0}\) does not fall into a gap within \(T\). These conditions are necessary but insufficient conditions that are jointly sufficient. Everything that is known by someone is known at some time interval. Since one may lose and reacquire knowledge, the time interval need not be continuous. Hence, it is necessary but not sufficient for knowledge to be within the boundaries of the knowledge time interval. It is also necessary but not sufficient to not be in a gap of that interval (all gaps are inside the outmost boundaries of an interval, for example, times before the earliest point of a time interval do not fall into a gap within that interval).
The example rests on the assumption that (T*) requires only the existence of some unclosed necessary condition whatsoever. Hence, (T*) does not presuppose the rejection of (Williamsonian) primitivism about knowledge; my three conditions are obviously not conceptually more basic than the analysandum and, hence, not an analysis of knowledge. However, if more specific requirements are in play for being a necessary condition for knowledge, (T*) must be weakened so that it no longer presupposes the existence of necessary and jointly sufficient conditions: If knowledge is not closed and \(M\) is a set of necessary but insufficient conditions that are jointly sufficient (and meet the imposed requirements), at least one member of \(M\) is not closed.
In this section I argue that independence in Brueckner’s sense is not self-evident. In Sect. 4.1 I argue that it is even false by offering a counterexample, that is, a (type of) analysis of knowledge whose conditions are not independent in Brueckner’s sense.
Step by step derivation of (CP*\('\)):
(CP*) \(\lnot \square \, \forall p \, \forall q \, [Cp \wedge R(p,q) \wedge \lnot Cq \rightarrow (Dp \wedge R(p,q) \rightarrow Cq)]\)
\(\equiv \lozenge \, \exists p \, \exists q \, (Cp \wedge R(p,q) \wedge \lnot Cq \wedge Dp \wedge R(p,q) \wedge \lnot Cq)\)
\(\equiv \lozenge \, \exists p \, \exists q \, (Cp \wedge R(p,q) \wedge Dp \wedge \lnot Cq)\)
\(\equiv \lnot \square \, \forall p \, \forall q \, (Cp \wedge Dp \wedge R(p,q) \rightarrow Cq).\)
Both (P2) and (CP*\('\)) are equivalent to existence statements. Applying conjunction elimination to (CP*\('\)) yields (P2).
This assessment also applies to Bernecker’s “argument from knowledge conditions” (2011, p. 369). According to Bernecker, who explicitly endorses Brueckner’s and Murphy’s replies to Warfield:
to validly argue from belief in the entailed q not satisfying X to knowledge not being closed it must be shown that one meets all necessary conditions for knowledge—including X—with respect to both p and p entails q, and that the only thing that prevents belief in the entailed q from qualifying as knowledge is the failure to satisfy X. (Bernecker 2011, p. 369)
The strategy described by Bernecker is not a transmission argument. Necessary conditions are introduced not in order to determine whether they are closed and to infer the non-closure of knowledge, but in order to determine whether a particular case is a case of knowledge. This “argument from knowledge conditions” is completely in line with Warfield’s argument against closure.
(P3) is equivalent to an existence statement and (P1) allows weakening the first conjunct \(Kp\) to \(Xp\). The result is equivalent to (P2). Even Yan herself does not use (P2) at any stage of her argument (cf. 2013, p. 262).
This reason for distinguishing transmission arguments from other arguments against closure does not assume that anti-sceptical strategies should or even must be neutral. To the contrary, relying on the transmission argument is an example of a neutral anti-sceptical strategy that must fail and, hence, its failure is an additional reason for giving up the requirement that anti-sceptical strategies must be neutral.
This seems to be Holliday’s view: “It is up to defenders of these theories to explain why knowledge is closed in ways that their conditions on knowledge are not” (Holliday forthcoming, p. 17).
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Acknowledgments
I am grateful to Hans Rott, Stefan Ruhland and Verena Wagner for helpful discussions of earlier versions and two anonymous referees for helping me to improve this paper.
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Kraft, T. Transmission arguments against knowledge closure are still fallacious. Synthese 191, 2617–2632 (2014). https://doi.org/10.1007/s11229-014-0403-8
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DOI: https://doi.org/10.1007/s11229-014-0403-8