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Symmetric and dual paraconsistent logics

100%
Kamide N., Wansing H.
Logic and Logical Philosophy
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2010
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vol. 19
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issue 1-2
7–30
EN
Two new first-order paraconsistent logics with De Morgan-type negations and co-implication, called symmetric paraconsistent logic (SPL) and dual paraconsistent logic (DPL), are introduced as Gentzen-type sequent calculi. The logic SPL is symmetric in the sense that the rule of contraposition is admissible in cut-free SPL. By using this symmetry property, a simpler cut-free sequent calculus for SPL is obtained. The logic DPL is not symmetric, but it has the duality principle. Simple semantics for SPL and DPL are introduced, and the completeness theorems with respect to these semantics are proved. The cut-elimination theorems for SPL and DPL are proved in two ways: One is a syntactical way which is based on the embedding theorems of SPL and DPL into Gentzen’s LK, and the other is a semantical way which is based on the completeness theorems.
2

On the proof-theory of a first-order extension of GL

58%
Schwartz Y., Tourlakis G.
Logic and Logical Philosophy
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2014
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vol. 23
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issue 3
329–363
EN
We introduce a first order extension of GL, called ML3, and develop its proof theory via a proxy cut-free sequent calculus GLTS. We prove the highly nontrivial result that cut is a derived rule in GLTS, a result that is unavailable in other known first-order extensions of GL. This leads to proofs of weak reflection and the related conservation result for ML3, as well as proofs for Craig’s interpolation theorem for GLTS. Turning to semantics we prove that ML3 is sound with respect to arithmetical interpretations and that it is also sound and complete with respect to converse well-founded and transitive finite Kripke models. This leads us to expect that a Solovay-like proof of arithmetical completeness of ML3 is possible.
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