This paper presents a range of new triviality proofs pertaining to naïve truth theory formulated in paraconsistent relevant logics. It is shown that excluded middle together with various permutation principles such as A → (B → C)⊩B → (A → C) trivialize naïve truth theory. The paper also provides some new triviality proofs which utilize the axioms ((A → B)∧ (B → C)) → (A → C) and (A → ¬A) → ¬A, the fusion connective and the Ackermann constant. An (...) overview over various ways to formulate Leibniz’s law in non-classical logics and two new triviality proofs for naïve set theory are also provided. (shrink)

This essay discusses rules and semantic clauses relating to Substitution—Leibniz’s law in the conjunctive-implicational form s=t ∧ A(s) → A(t)—as these are put forward in Priest’s books "In Contradiction" and "An Introduction to Non-Classical Logic: From If to Is." The stated rules and clauses are shown to be too weak in some cases and too strong in others. New ones are presented and shown to be correct. Justification for the various rules are probed and it is argued that Substitution ought (...) to fail. (shrink)

Restall set forth a "consecution" calculus in his "An Introduction to Substructural Logics." This is a natural deduction type sequent calculus where the structural rules play an important role. This paper looks at different ways of extending Restall's calculus. It is shown that Restall's weak soundness and completeness result with regards to a Hilbert calculus can be extended to a strong one so as to encompass what Restall calls proofs from assumptions. It is also shown how to extend the calculus (...) so as to validate the metainferential rule of reasoning by cases, as well as certain theory-dependent rules. (shrink)

Within the framework of Meinongianism, nothingness turns out to have contradictory features—it seems to be an object and not. In this paper, we explore two different kinds of Meinongian accounts of nothingness. The first one is the consistent account, which rejects the contradiction of nothingness, while the second one is the inconsistent account, which accepts the contradiction of nothingness. First of all, after showing that the consistent account of nothingness defended by Jacquette fails, we express some concerns on the general (...) possibility of consistently characterizing nothingness. Secondly, starting from Priest’s inconsistent characterization of nothingness :146–158, 2014b), we will introduce our own inconsistent account. The key idea of our account is to take nothingness as the complement of the totality. Finally, we will make formal sense of it by constructing an inconsistent mereological system, which is the development of the paraconsistent mereology proposed by Weber and Cotnoir. (shrink)

This paper is a step toward showing what is achievable using non-classical metatheory—particularly, a substructural paraconsistent framework. What standard results, or analogues thereof, from the classical metatheory of first order logic can be obtained? We reconstruct some of the originals proofs for Completeness, Löwenheim-Skolem and Compactness theorems in the context of a substructural logic with the naive comprehension schema. The main result is that paraconsistent metatheory can ‘re-capture’ versions of standard theorems, given suitable restrictions and background assumptions; but the shift (...) to non-classical logic may recast the meanings of these apparently ‘absolute’ theorems. (shrink)