Abstract
The present paper expounds a preferred models semantics of paraconsistent reasoning. The basic idea of this semantics is that we interpret the language L(V) of a theory T in such a way that the axioms of T are satisfied to a maximal extent. These preferred interpretations are described in terms of a network of partial structures. Upon this semantic analysis of paraconsistent reasoning we develop a corresponding proof theory using adaptive logics.
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Notes
- 1.
For a detailed investigation of the inconsistency of classical electrodynamics, see [8].
- 2.
The term consolidation is borrowed from belief revision theory (see, e.g., [9].).
- 3.
Cf. [17]. For a critical discussion of this doctrine, see H. Wansing et al.: On the methodology of paraconsistent logic (this volume).
- 4.
See [6] for a related strategy of dealing with inconsistencies.
- 5.
See [8, Chap. 2] for a detailed exposition of this inconsistency.
- 6.
This part is based on [1], which develops the network formalism as a paraconsistent semantics of theoretical terms.
- 7.
Unlike a simple pragmatic structure in [7], a partial structure does not contain a set P of sentences that are taken to be true in the correspondence sense.
- 8.
Partial structures are used here as a generalization of intended applications. An intended application of a theory-element \(\mathbf {T}\) leaves the \(\mathbf {T}\)-theoretical terms undetermined. This can be represented as follows: if \(R_k\) is \(\mathbf {T}\)-theoretical, \(R_k=\emptyset \). Partial structures, however, are a bit more flexible as they allow us to have a partial determination of the \(\mathbf {T}\)-theoretical terms for a given intended application.
- 9.
This understanding generalizes the standard definition of being a substructure. Unlike the standard definition, it is not required that a substructure and its superstructure share the same slots of relations.
- 10.
The question of whether \(<_s\) is smooth (in the set of all L(V) interpretations) for axiomatic theories T with an uncountable set of applications must be left for future research.
- 11.
We should note that the presently defined inference relation deviates from the one defined in [10, 14]. They define the relation
in such a way that \(\varphi \) must be verified by all classical models of A that are preferred on a given ordering <. This inference relation, however, is obviously not paraconsistent. - 12.
A strict partial order < is modular iff the relation R(x, y) defined by \(x \not<y \wedge y\not <x\) is an equivalence relation. There might be cases where the axioms are not ordered in a modular fashion, which would require some modifications of Definition 7. For simplicity, a modular ordering among the axioms is assumed.
- 13.
This is a simplification because the energy levels are only relatively stable. An electron may jump from one level to another.
- 14.
Where \(\mathbf {L}\) is a logic, a formula \(\varphi \) is \(\mathbf {L}\)-contingent iff there is an \(\mathbf {L}\)-model that falsifies \(\varphi \) and one that verifies \(\varphi \).
- 15.
We hope it is clear for the reader that we do not claim to define a new adaptive logic here, but rather apply existing logics to provide a new solution for an interesting problem, viz. providing a proof theory for the Modular Semantics inference relations.
in such a way that References
Andreas, H. (201x). Networks of partial structures. under review.
Balzer, W., Moulines, C. U., & Sneed, J. (1987). An architectonic for science. The structuralist program. Dordrecht: D. Reidel Publishing Company.
Batens, D. (2007). A universal logic approach to adaptive logics. Logica Universalis, 1, 221–242.
Benferhat, S., Dubois, D., & Prade, H. (1997). Some syntactic approaches to the handling of inconsistent knowledge bases: A comparative study part 1: the flat case. Studia Logica, 58(1), 17–45.
Brewka, G. (1991). Belief revision in a framework for default reasoning. In Proceedings of the Workshop on The Logic of Theory Change (pp. 602–622). London: Springer.
Brown, B., & Priest, G. (2004). Chunk and permeate: a paraconsistent inference strategy. part 1: the infinitesimal calculus. Journal of Philosophical Logic, 33, 379–388.
da Costa, N., & French, S. (2003). Science and partial truth. Oxford: Oxford University Press.
Frisch, M. (2005). Inconsistency, asymmetry, and non-locality. Oxford: Oxford University Press.
Hansson, S. O. (1999). A textbook of belief dynamics. Theory change and database updating. Dordrecht: Kluwer.
Kraus, S., Lehmann, D., & Madigor, M. (1990). Nonmonotonic reasoning, preferential models and cumulative logics. Artificial Intelligence, 44, 167–207.
Meheus, J., Straßer, C., & Verdée, P. (2016). Which style of reasoning to choose in the face of conflicting information? Journal of Logic and Computation, 26(1), 361–380. doi:10.1093/logcom/ext030.
Putte, F. V. D. & Straßer, C. (2012). Extending the standard format of adaptive logics to the prioritized case. Logique Et Analyse, 220, 601–641.
Rescher, N., & Manor, R. (1970). On inference from inconsistent premisses. Theory and Decision, 1(2), 179–217.
Shoham, Y. (1988). Reasoning about change: Time and causation from the standpoint of artificial intelligence. Cambridge, MA: MIT Press.
Sneed, J. (1979). The logical structure of mathematical physics. Dordrecht: D. Reidel Publishing Company.
Straßer, C. (2014). Adaptive logics for defeasible reasoning (Vol. 38). Trends in Logic Cham: Springer. Applications in Argumentation, Normative Reasoning and Default Reasoning, Heidelberg.
Wagner, G. (1991). Ex contradictione nihil sequitur. In Proceedings of the 12th IJCAI (pp. 538–543), Sidney, Australia.
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Andreas, H., Verdée, P. (2016). Adaptive Proofs for Networks of Partial Structures. In: Andreas, H., Verdée, P. (eds) Logical Studies of Paraconsistent Reasoning in Science and Mathematics. Trends in Logic, vol 45. Springer, Cham. https://doi.org/10.1007/978-3-319-40220-8_2
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