Abstract
In this paper we look at two classic methods of deriving consequences from inconsistent premises: Rescher-Manor and Schotch-Jennings. The overall goal of the project is to confine the method of drawing consequences from inconsistent sets to those that do not require reference to any information outside of very general facts about the set of premises. Methods in belief revision often require imposing assumptions on premises, e.g., which are the important premises, how the premises relate in non-logical ways. Such assumptions enable one to select a reasonable collection of formulas from all the formulas of the language (the set of consequences of an inconsistent set). Basic versions of the classic methods only use logical relations between the premises. We compare and criticize each of the classic views with an eye to combining the views to get the most out of inconsistent premises. We do this in a way that will respect the theoretical grounding of each view while meeting our other restrictions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
This would be the suggestion of [6].
For more discussion and motivation for forcing see [5].
Recall that we will usually omit the subscript X referring to the logic.
ω is the set of natural numbers 0,1,2,….
For more on this see [4].
The original notion of cover was given in terms of partitions, and although the generalization of cover that is prominent now has certain nice mathematical properties, partitions will play an important role in the sequel.
See [4].
See [8, Ch. 9].
See [3].
The case where ℓ(Γ)=0 doesn’t pose any real problems, it is simply that every partition cover of Γ would have width 1, which isn’t the level of Γ.
This new notion of cover can be seen as a special case of an A-cover where A(Γ)=N(Γ), but N(Γ) is contained in the closure of every non-empty cell of the cover. This means that, in general, N-forcing would extend the corresponding version of A-forcing since there are fewer special A-covers.
Thank you to an anonymous referee for this journal for suggesting generalizing the notion of N-forcing.
A similar situation arises in the case of AGM belief revision [1]. They have the two extremes: all of the MCS or a single MCS. They offer a way to characterize any method of belief revision based on selecting some proper sub-collection of MCSs.
In [2], the authors phrase argued consequence in terms of there being no MCS that proves ¬α, but that is not available in our abstract setting.
Note that we could also characterize the Neg-preferred MCSs by those in \( \{\Gamma^{\prime}: \textsc{MCS}(\Gamma^{\prime},\Gamma)\& \ell(\Gamma\setminus\Gamma^{\prime})=\ell(\Gamma)\}\).
References
Alchourrón, C.E., Gärdenfors, P., Makinson, D. (1985). On the logic of theory change: Partial meet contraction and revision functions. Journal of Symbolic Logic, 50(2), 510–530.
Benferhat, S., Dubois, D., Prade, H. (1997). Some syntactic approaches to the handling of inconsistent knowledge bases: A comparative study, part i. Studia Logica, 58(1), 17–45.
Brown, B., & Priest, G. (2004). Chunk and permeate, a paraconsistent inference strategy. part i: The infinitesimal calculus. Journal of Philosophical Logic, 33, 379388.
Payette, G. (2009). Preserving logical structure In Brown, B., Jennings, R.E., Schotch, P.K. (Eds.), On Preserving: Essays on Preservationism and Paraconsistent Logic, chap. 7, (pp. 105–43). Toronto: University of Toronto Press.
Payette, G., & Schotch, P.K. (2007). On preserving. Logica Universalis, 1(2), 295–310.
Quine, W.V.O., & Ullian, J.S. (1978). The Web of Belief, 2nd edn. Random House, New York (1970).
Rescher, N., & Manor, R. (1970). On inference from inconsistent premises. Theory and Decision, 1, 179–217.
Schotch, P., Brown, B., Jennings, R. (Eds.) (2009). On Preserving: Essays on Preservationism and Paraconsistency. Toronto Studies in Philosophy. Toronto: Toronto University Press.
Schotch, P.K., & Jennings, R.E. (1979). Multiple-valued consistency. In: Proceedings of the 9th international symposium on many-valued logic. Bath: IEEE.
Schotch, P.K., & Jennings, R.E. (1980). Inference and necessity. Journal of Philosophical Logic, 9(3), 327–40.
Wong, P. (2000). Inconsistency and preservation. In:PRICAI pp. 50–60.
Acknowledgements
The author would like the thank the Killam Trusts at Dalhousie University for funding this research, and for an anonymous referee for this journal for comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Appendix
Appendix
Observation 7
\(\vdash ^{I}\not \subseteq \Vdash^N \).
Proof
The following is a counterexample to \(\vdash ^{I}\subseteq \Vdash^N \).
Let Γ be
Then N(Γ)=∅, and there are three MCSs:
-
\(A=\{a\wedge b\wedge d,(b\supset c)\wedge \neg f,(\neg b\supset c)\wedge \neg f\}\),
-
\(B=\{(\neg b\supset c)\wedge \neg f, \neg a\wedge \neg b, (b\supset c)\wedge \neg f, \neg e\wedge \neg b\}\), and
-
Φ={a∧¬d∧(e∨c)∧f,¬e∧¬b}.
So we have \(A\vdash c\), \(B\vdash c\) and \({\Phi }\vdash c\). Thus \(M({\Gamma })\vdash c\). However, Γ has level 3 as shown by the following partition cover
But no member of this partition cover proves c, thus neither Γℓ c nor ΓN c. Therefore \({\Gamma }\vdash ^{I} c\) while ΓN c. □
Corollary 1
From observation 7, ⊩M ⊆ ⊩ N , and ⊩ ⫋ ⊩M.
Rights and permissions
About this article
Cite this article
Payette, G. Getting the Most Out of Inconsistency. J Philos Logic 44, 573–592 (2015). https://doi.org/10.1007/s10992-014-9343-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10992-014-9343-5