Abstract
In order to reply to the contemporary skeptic’s argument for the conclusion that we don’t have any empirical knowledge about the external world, several authors have suggested different fallibilist theories of knowledge that reject the epistemic closure principle. Holliday [8], however, shows that almost all of them suffer from either the problem of containment or the problem of vacuous knowledge. Furthermore, Holliday [9] suggests that the fallibilist should allow a proposition to have multiple sets of relevant alternatives, each of which is sufficient while none is necessary, if all its members are eliminated, for knowing that proposition. Not completely satisfied with Holliday’s multi-path reply to the skeptic, the author suggests a new single-path relevant alternative theory of knowledge and argues that it can avoid both the problem of containment and the problem of vacuous knowledge while rejecting skepticism.
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Notes
- 1.
This is, in essence but not in form, Unger’s argument in Chap. 1 of [12].
- 2.
Or rule out, or eliminate. In what follows, ‘eliminate’, ‘rule out’, and ‘exclude’ will be used as synonyms.
- 3.
Following Holliday, we call a sentence of the form ‘an epistemic closure principle’:
$$\phi _0 \wedge \text { K} \phi _1 \wedge \ldots \wedge \text { K} \phi _n \rightarrow \text { K}\psi _1 \vee \ldots \vee \text { K}\psi _m,$$where ‘\(\phi _0\)’ is a propositional conjunction, i.e., a conjunction none of whose conjuncts contains an occurrence of ‘K’. When n is equal to 1, we call such an epistemic closure principle ‘a single-premise epistemic closure principle’. When n is greater than 1, we call such an epistemic closure principle ‘a multiple-premise epistemic closure principle’.
- 4.
For example, in Holliday’s formalization of Dretske’s relevant alternative theory, the function r is such that, for every model \(\mathfrak {M}\), world w, and proposition \(\phi \), r\(_\mathfrak {M}\)(w, \(\phi \)) = Min\(_{\le _\mathfrak {M}^w}\)[\(\lnot \phi \)]\(_\mathfrak {M}\) = {\(v\in [\lnot \phi ]\) \(_\mathfrak {M}\) \(\cap \)W\(^w_\mathfrak {M}\) | \(\lnot \exists u(u\in [\lnot \phi ]\) \(_\mathfrak {M}\) \(\wedge \) \(u\le _\mathfrak {M}^{w}v\) \(\wedge \) \(\lnot v\le _\mathfrak {M}^{w}u\))}.
- 5.
An area \(\Sigma \) is a set of sentences such that if \(\phi \in \Sigma \) and \(\psi \) is truth-functional consequence of \(\phi \), then \(\psi \in \Sigma \). Note that where I talk about a model \(\mathfrak {M}\) and a world w, Holliday [9] talks about a context C and a scenario w; the terminological difference here is unimportant.
- 6.
Where I use the phrase ‘standard fallibilist picture’ Holliday [9] uses the phrase ‘standard relevant picture’. Again, I think the terminological difference here is unimportant.
- 7.
If a clause C\('\) can be obtained by adding zero or more disjuncts to C, then C\('\) is a superclause of C, and C is a subclause of C\('\): e.g., ‘(p \(\vee \) \(\lnot \)q \(\vee \) r)’ is a superclause of ‘(p \(\vee \) \(\lnot \)q)’ and a subclause of ‘(p \(\vee \) \(\lnot \)q \(\vee \lnot \)s \(\vee \) r)’.
- 8.
What follows is not the only way that multiple sets of relevant alternatives can be assigned to a proposition at a world, but it seems to be a quite natural way to do so.
- 9.
See footnote 5 for how it is to be defined.
- 10.
If \(\phi \) is a truth-functional tautology, we define r \(_\mathfrak {M}^r\)(\(\phi \), w) = r \(_\mathfrak {M}^r\)((p \(\vee \) \(\lnot \)p), w).
- 11.
More correctly, the following conditions and several others are jointly consistent. But these extra ones are not important for my purpose in this paper.
- 12.
- 13.
- 14.
- 15.
When n = 1, this case reduces to case (r\(_1\)).
- 16.
When n = 1, this case also reduces to case (r\(_1\)).
- 17.
Heller takes (ERA*) to be merely a necessary condition, yet, for the sake of simplicity, I take it to be both a necessary and a sufficient condition. As far as I can tell, nothing important hinges on this difference for the purpose of this paper.
- 18.
Compare Holliday’s noVK \(^{multi}\) with my noVK \(^{H* }\). If we identify W\(^w_\mathfrak {M}\) in noVK \(^{multi}\) with \(\cup \$_\mathfrak {M}^w\) in noVK \(^{H* }\), it can be seen vividly that the two results are essentially the same.
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Wang, Wf. (2017). Multi-Path vs. Single-Path Replies to Skepticism. In: Baltag, A., Seligman, J., Yamada, T. (eds) Logic, Rationality, and Interaction. LORI 2017. Lecture Notes in Computer Science(), vol 10455. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55665-8_5
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