Abstract
We uncover a ‘naturally occurring’ first degree system, AL of Articular Logic that is both relevant and paraconsistent. The principal semantic innovation is an informationally articulated, but nevertheless entirely classical representation of wffs as clutters (Clutters are also referred to as simple hypergraphs in some contexts.) on the power set of a set of possible states. The principal methodological novelty is the general observation that distinct classical representations of wffs can be selected and combined with redeployments of classical inference to accommodate particular inferential requirements such as paraconsistency and relevance.
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Notes
- 1.
A hypergraph on a set is a collection of collections of objects from that set whose elements are called the vertices of the hypergraph. The 3-harmonic hypergraph is a hypergraph on the premise set. Members of the hypergraph are called edges. A hypergraph is n-harmonic for any natural number n if n + 1 is the least number of edges whose intersection is non-empty.
- 2.
Every n − tuple of edges has a common vertex.
- 3.
A hypergraph H is n-uncolourable if the width of the narrowest colouring of the set which will not result in monochromatic cells is greater that n; else it is n − colourable. The least n such that H is n − colourable is called the chromatic number of the set.
- 4.
By \(\overline{a}\) we mean the set theoretic complement of a.
- 5.
H so far has been used to denote clutters. However, when there is no danger of confusion, we refer to hypergraph by it as well.
- 6.
The subscript suggests that the binary axioms belong to AL 1.
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Acknowledgements
The research presented in this paper is supported by SSHRC research grant # 410-2008-2330. We would like to express our thanks to Professor J. Michael Dunn for his stimulating correspondence and to Bryson Brown, Julian Sahasrabudhe and Andrew Withy for their valuable observations.
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Jennings, R.E., Chen, Y. (2013). FDE: A Logic of Clutters. In: Tanaka, K., Berto, F., Mares, E., Paoli, F. (eds) Paraconsistency: Logic and Applications. Logic, Epistemology, and the Unity of Science, vol 26. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4438-7_10
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