Abstract
One diagnosis of Fitch’s paradox of knowability is that it hinges on the factivity of knowledge: that which is known is true. Yet the apparent role of factivity (in the paradox of knowability) and non-factive analogues in related paradoxes of justified belief can be shown to depend on familiar consistency and positive introspection principles. Rejecting arguments that the paradox hangs on an implausible consistency principle, this paper argues instead that the Fitch phenomenon is generated both in epistemic logic and logics of justification by the interaction of analogues of the knowability principle and positive introspection theses that are characteristic of, even if not entailed by, epistemic internalism.
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Notes
That is, regardless of whether one is committed to the knowability principle, the surprising result is still interesting and important. Hence, although there are some interesting arguments against accepting the principle in the first place, we do not address these in this paper. Rather, we take the view that the reasoning involved in the disappearing diamond is a phenomenon worth trying to understand, with interesting implications of its own for other general conceptual claims we may wish to make about truth, knowledge and justification. The introductory section of this paper also briefly notes other reasons for studying this phenomenon independent of any commitment to the knowability principle itself.
Although we won’t focus on this aspect of the issue, Stjernberg clearly owes us more by way of argument for the alleged uniqueness of knowledge here. Knowledge might well be the most general factive ‘stative’ attitude, but the category would appear to also include seeing, hearing, remembering and so forth (cf. Williamson 2000).
For discussion of similar issues concerning another weakening of the knowability principle, see MacIntosh (1984, p. 158).
Logics of this kind need not be normal, depending on the degree of idealisation J involves. For more discussion of this see Chase (2010).
Recall that we are talking about (internalist) evidentiary justification here, not simply justified belief—only if (accessible or internalist) evidence was possible could we be justified in believing a given claim. For example, we could not be justified in believing the number of stars in the universe was odd, absent any possibly accessible evidence to that effect, or any relevant connection between that belief and possible justificatory evidence. Thus we rule out purely doxastic justification such as a ‘set idea’ that the number of stars is odd, held without any clear connection to either actual or possible evidence either way (thanks to an anonymous referee for offering just this example to push this issue). Only if (ideal, possible) discoverable evidence (i.e. of the actual stars in the universe) is admitted and could somehow be relevantly connected to such a ‘set idea’, could the idea be considered justified. That is, we take it that whatever counts as evidence (on an internalist account) is what J-justifies a belief. A belief with no reference to evidence is not J-justified.
A well-known example from foundationalist internalism would be Chisholm’s account of knowledge in the original edition of Theory of Knowledge in terms of the notion of concurrence, which requires reasonable propositions to be mutually consistent (Chisholm 1966, p. 53); Chisholm there acknowledges the links between this idea, Price’s notion of coherence, and C.I. Lewis’s notion of congruence. Traditional coherentist accounts of knowledge or justification also standardly impose a logical consistency requirement as a partial explication of that notion (eg., cf. BonJour 1985, pp. 94–95). A logical consistency condition is also natural within Bayesian epistemologies (eg., Kaplan 1996, p. 125) and Quinean best explanation accounts (Quine and Ullian 1978, pp. 14–18).
For instance: let sufJ be a J operator that is (one way or another) sufficient for the job. One way of spelling out the minimum effectiveness for the justification operator in (JP) would be to stipulate that sufJ must transmit across entailment/inference. Where we are sufJ-justified in believing the truth of a proposition p, we ought to be sufJ-justified in believing the truth of any propositions entailed by p.
Consider Wright’s disjunctive template:
“According to Wright (2000), for any propositions p, q and r and subject s, the disjunctive template is instantiated whenever:
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(a)
p entails q;
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(b)
s’s justification for p consists in s’s being in a state subjectively indistinguishable from a state in which r would be true;
-
(c)
r is incompatible with p;
-
(d)
r would be true if q were false.
(or, for a stronger version: replace c above with (c*) r is incompatible (not necessarily with p but) with some presupposition of the cognitive project of obtaining a justification for p in the relevant fashion).”, in Moretti and Piazza (2013).
-
(a)
This is a common feature of characterisations of internalism in the period when it first because a subject for sustained investigation; cf. the accounts in Pollock (1986, p. 22), Chisholm (1988, pp. 285–286), Taylor (1990, p. 210), Sosa (1991, p. 193), Audi (1998, pp. 231–232), and Zagzebski (1996, p. 21).
Neither Audi nor Williamson argue that justification is ‘luminous’ (i.e. \(\hbox {J}\alpha \rightarrow \, \hbox {KJ}\alpha \): that we are always able to know when we are justified), and Williamson’s argument (2000) that this applies to all non-trivial concepts is perhaps the strongest contender undermining (4) for knowledge. But that a pure-modal (4) axiom ought to hold for (an effective, internalist) J seems far less contentious.
And just such a J (with the upper and lower bounds outlined) is useful for, and in fact utilised by, key internalists and anti-realists. Recall that the heart of the matter is that anti-realists want to combat metaphysical realism, specifically its “concept of truth which is evidentially unconstrained—a concept which would permit an empirical theory that was by ideal by all internal and operational criterion to be false” (Wright 2003, p. 288). The precise problem is how to do this while still denying factivity for (a suitably idealisable) J operator.
As Putnam points out, anti-realists have tried to combat this metaphysical notion by “propos[ing] a notion whereby truth would coincide with some kind of idealisation of rational acceptability” (335). It is for this reason that Wright (in Wright 2003, p. 356, footnote 26) appeals explicitly to (JO*) in showing why Putnam’s internalism falls foul of Fitch: his explanation appeals to the following two principles: (1) \(\hbox {JB}\alpha \rightarrow \lnot \lnot \hbox {JB}\alpha \), and (2) \(\hbox {JB}\lnot \hbox {JB}\alpha \rightarrow \lnot \hbox {JB}\alpha \) (though without formally stating them as such). For Wright, a J consonant with Putnam’s (and any good) internalism, both (1) and (2) hold. His defence of (2) is as follows: “on the other hand, if under C the second conjunct is rationally believed—namely that no thinker will ever rationally believe that Q—then indeed no thinker ever will; so, in particular, there will be no rational belief that Q under conditions C”. The second conjunct Wright is talking about here is the second conjunct of assumption 2 in the standard Fitch proof. Wright is claiming that if we have a justified belief that this second conjunct is true (i.e., if \(\hbox {J}\lnot \hbox {J}\alpha )\) then in fact (under Putnam’s account) that conjunct holds true (ie \(\lnot \hbox {J}\alpha \) is true). Thus Wright endorses \(\hbox {J}\lnot \hbox {J}\alpha \rightarrow \lnot \hbox {J}\alpha \), which is (JO*).
Gettier cases have often been put to work in this way; cf., for instance, Sturgeon (1993), a reflection on the first thirty years of the Gettier literature.
This is not Kripke’s own conclusion regarding them; rather, his example is used to defend a Millian theory of names, by showing that similar difficulties arise for Fregean accounts as well (Kripke 2011, pp. 136, 158).
The Pierre-like example we refer to in note 5 above, wherein an agent justifiably believes that the number of stars is odd without reference to any kind of (even possible) evidence, also draws attention to the issue of ‘slippage’—the question it raises is whether justified belief can come apart from evidence that justifies. But, as we argued when we first introduced our notion of J, we take it that any notion of justification allowing such examples is going to be too weak to do the work (JP) is supposed to do. We are specifically interested (as Kelp and Pritchard are) in a concept of justification which enables (JP) to do the work that (KP) was supposed to do.
See Priest (2006), chapter 13, for discussion of contradictory obligations in legal systems, and analysis of them in a deontic logic built on LP.
It could even be argued that, contra Kelp and Pritchard, allowing (JO) to hold and thus to carry a contradiction in Pierre’s internal beliefs to a contradiction in the world is a more natural move than Kelp and Pritchard’s denial of (JO), just so long as we hold (JP) as an effective truth-tracking principle in the first place. It could, in this case, seem quite right that an inconsistency in justified beliefs should entail an inconsistency in the world.
This is not to say that a contradiction-tolerant internalism is irrelevant. Indeed an interesting question raised by the above arguments is whether an internalist is ultimately faced with the dilemma of having to weaken justification (undermining the effectiveness of (JP)) or to accept (JO) and so contradictions in the world (potentially the cost of retaining the conception of (JP) as an effective truth-tracking principle).
E.g. Williamson’s comments (2013) suggest that the problem with Pierre-cases (or, as he dubs them: ‘Frege cases’) is a problem with understanding what the propositions actually say in each case, rather than with the concept J (or K), consistency, etc. On this view, rather than indictments on the consistency or otherwise of justification, we should see such cases as coarse-grained propositions in need of refinement (while Williamson does not spell this out, he implicitly suggests that if we fine tune the propositions in Pierre’s case, giving them more internal structure etc, their apparent inconsistency will be resolved).
Richard (2011) makes a related point, arguing that from his own point of view, Pierre does not contradict himself, and nor is it ‘unequivocal’ that he contradicts himself from a third person (or ‘external’) perspective. Indeed, Richard makes a compelling case that while we can give a linguistic interpretation of Pierre’s utterances, we can’t so easily give an accurate ‘individual’ interpretation of those utterances “as expressive of what [Pierre] believes” (24). That is, we can focus on Pierre’s ‘French beliefs’ or his ‘English belief’ but not both at once, at least not without “making use of an idiom whose syntactic resources reflect the structure of Pierre’s conceptual system” (25).
Williamson (2013) also offers a direct argument for (DJ) from modelling: namely that it is a very reasonable and minimal assumption that “from each world at least one world in the model is accessible”. And this assumption yields the result that there are no inconsistencies in JB as well (if neither \(\hbox {B}\alpha \wedge \lnot \hbox {B}\alpha \) nor \(\hbox {B}(\alpha \wedge \lnot \alpha )\) is allowed, then surely neither (JCB) nor (CB) is either).
We could also appeal to Wright’s transmission failure, outlined in note 8. In terms of the disjunctive template, the Pierre case satisfies (a)—the city Pierre experiences as ugly and his Parisian friend’s statement together justify the claim p (that London is ugly and Londres is pretty). This claim p entails q (that London is not Londres). Pierre’s justification for q satisfies (b) in that his justification for p consists in a state subjectively indistinguishable from a state in which it is true that r (that Londres is ugly). So the situation also satisfies (c)—r is incompatible with p. It also satisfies (d)—it would be true that Londres is ugly if London was Londres (i.e. if it were false that q). If we agree with Wright that when an argument fails to transmit warrant, it is not an argument “whereby someone could be moved to rational [or justified] conviction of its conclusion” (2000, p. 140, in Moretti and Piazza 2013), then we have grounds to doubt that Pierre is sufJ-justified at all. Whatever he has for his claim q, it is not sufJ-justification, so the thought experiment does not undermine (JO*) for the kind of J the anti-realist needs in (JP) in the first place.
This objection was suggested to us by an anonymous referee.
(4K) also features in other Fitch-like derivations of trouble from the knowability principle (see Brogaard and Salerno 2006 for an example).
Of course, the plausibility of such axioms is already the subject of debate; for instance, (as we also note in various places throughout the paper) Williamson’s well-known anti-luminosity argument is deployed against (4K) among many other targets (cf. Williamson 2000, Chap. 5).
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Acknowledgements
Our thanks for comments to Dirk Baltzly, JC Beall, David Coady, Richard Corry, Lucy Tatman, two anonymous reviewers and an audience at the University of Tasmania.
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Chase, J., Rush, P. Factivity, consistency and knowability. Synthese 195, 899–918 (2018). https://doi.org/10.1007/s11229-016-1253-3
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DOI: https://doi.org/10.1007/s11229-016-1253-3