Abstract
In this paper two closely related systems of paraconsistent belief change are presented. A paraconsistent system of belief change is one that allows for the non-trivial treatment of inconsistent belief sets. The aim in this paper is to provide theories that help us investigate responses to paradoxes. According to the strong system of paraconsistent belief change, if an agent accepts an inconsistent belief set, he or she has to accept the conjunctions of all the propositions in it. The weak system, on the other hand, allows people to have beliefs without believing their conjunctions. The weak system seems to deal better than with the lottery and preface paradoxes than does strong system, although the strong system has other virtues.
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Notes
- 1.
I use ‘systems’ of belief change throught this paper rather than the more usual ‘theories’ because I use ‘theory’ to refer to sets of formulas that are closed under conjunction and logical consequence. In the original draft of this paper I used ‘theory’ in both of these senses, producing a rather confusing mess (as a referee pointed out).
- 2.
In a recent paper Makinson (2012), Makinson returns to the preface paradox and compares it with the lottery paradox.
- 3.
Makinson has returned much more recently to work relevant logic. Recently, he is investigating the relationship between relevant and classical logic, in particular in his contribution to this volume.
- 4.
- 5.
For a survey of the various relevant logics, their properties, and their semantics, see Routley et al. (1983).
- 6.
Strictly speaking, a filter or ideal on the Lindenbaum algebra is not a set of formulas, but a set of equivalence classes of formulas.
- 7.
For a good survey of AGM-like theories based on non-classical logics, see Wassermann (2011).
- 8.
Restall and Slaney (1995) also try to expose more detail in how people do or should work with inconsistencies in belief revision, but they are concerned with the elimination of inconsistencies, not with when we should accept them.
- 9.
- 10.
- 11.
- 12.
As far as I know, Belnap never published his proof.
- 13.
I use ‘\(\sigma \)’ to connote ‘strong’ as in ‘strong theories’.
- 14.
Partial meets were first introduced into the AGM theory as a way of treating contraction Alchourón et al. (1985). Here I generalize this approach to define all the AGM operations.
- 15.
Although this solution might run afoul of Douven and Williamson’s generalizations of the lottery paradox Douven and Williamson (2006).
- 16.
Makinson develops Kyburg’s idea in a more formally percise way in terms of the notion of a “lossy rule”. Conjunction introduction is lossy; it is unreliable in certain situations. In Makinson (2012), Makinson formulates rules that allow conjunction introduction to be “applied sparingly”.
- 17.
In Kyburg (1997), Kyburg attacks Ray Jennings and Peter Schotch’s theory of forcing Jennings and Schotch (1984) as a means for dealing with the lottery paradox. Since my original idea in this paper was to adapt the forcing strategy to the AGM theory, it is appropriate for me to explain forcing briefly. In that theory, a (finite) premise set is partitioned into consistent subsets. The fewest number of sets such that their union is the original premise set is called the level of the premise set. Suppose that the level of a premise set \(X\) is \(n\). Then we say that \(X\) forces a formula \(A\) (written \(X[\vdash A\)) if and only if in every partition of \(X\) into \(n\) subsets there is a set \(Y\) in that partition such that \(Y\) classically entails \(A\). As a defender of forcing, Brown (1999) says that this method does not allow us to derive all the conjunctions that one would accept in lottery type cases. Rather, extra logical considerations usually come into play. This is true in the weak theory as well, since the formal requirements on the selection functions do not determine exactly which contents are selected, maximalized, and then intersected.
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Acknowledgments
David Makinson suggested to me that I write a paper for this volume and for this he is in part responsible for my rethinking my views on paraconsistency and belief revision. I am very grateful to David for his close reading of a draft of this paper and his extremely helpful suggestions. I am grateful to Rob Goldblatt, Patrick Girard, Jeremy Seligman, and Zach Weber for listening to and commenting on an earlier version of this paper. I am also indebted to Sven Ove Hansson and his two referees for very helpful comments. They saved me from some very embarrassing errors.
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Mares, E. (2014). Liars, Lotteries, and Prefaces: Two Paraconsistent Accounts of Belief Change. In: Hansson, S. (eds) David Makinson on Classical Methods for Non-Classical Problems. Outstanding Contributions to Logic, vol 3. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7759-0_7
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