A Uniform Proof-theoretic Foundation for Abstract Paraconsistent Logic Programming

It is known that paraconsistent logic programming, which is usually based upon a paraconsistent logic, is important in dealing with inconsistency-tolerant and non-monotonic reasoning appropriately. Firstly in this paper, a cut-free single succedent sequent system SN4 and a cut-free multiple succedent sequent system MN4 are introduced for Nelson’s paraconsistent 4-valued logic N4 [3], and the uniformity theorem (with respect to the notion of uniform proofs) which is established by Miller et al. [34] is shown for SN4 and MN4. The framework using SN4 provides us with an abstract paraconsistent logic programming language that can express inconsistency-tolerant reasoning and inexact information by using the properties of paraconsistency and constructible falsity. The framework using MN4 gives an abstract paraconsistent disjunctive logic programming language that can allow to express disjunctive (indefinite) information in the program clauses. Secondly, a cut-free single succedent sequent system SN16 is introduced for an extension N16 of N4, which is a variant of Shramko and Wansing’s 16-valued logics [42], and the uniformity theorem for SN16 is shown. The framework using SN16 produces an abstract (extended) paraconsistent logic programming language that can also express a certain kind of synonymous information. Thirdly, a cut-free single succedent sequent system SC is introduced for a fragment of Wansing’s non-commutative logic COSPL [46], which is a non-commutative version of N4, and the uniformity theorem for SC is shown. The framework using SC provides us with an abstract paraconsistent ordered linear logic programming language that can represent both ordered and hierarchical information. The results of this paper are regarded as natural extensions of the results by Miller et al. [34] and by Harland et al. [16].