It is known that a number of inference principles can be used to trivialise the axioms of naïve comprehension - the axioms underlying the naïve theory of sets. In this paper we systematise and extend these known results, to provide a number of general classes of axioms responsible for trivialising naïve comprehension.

Curry's paradox, sometimes described as a general version of the better known Russell's paradox, has intrigued logicians for some time. This paper examines the paradox in a natural deduction setting and critically examines some proposed restrictions to the logic by Fitch and Prawitz. We then offer a tentative counterexample to a conjecture by Tennant proposing a criterion for what is to count as a genuine paradox.

In his paper [6], Greg Restall conjectured that a logic supports a naïve comprehension scheme if and only if it is robustly contraction free, that is, if and only if no contracting connective is definable in terms of the primitive connectives of the logic. In this paper, we present infinitely many counterexamples to Restall''s conjecture, in the form of purely implicational logics which are robustly contraction free, but which trivialize naïve comprehension.

In this paper we consider the implicational fragment of Abelian logic \ . We show that although the Abelian groups provide an semantics for the set of theorems of \ they do not for the associated consequence relation. We then show that the consequence relation is not algebraizable in the sense of Blok and Pigozzi . In the second part of the paper, we investigate an extension of \ in the same language and having the same set of theorems and (...) show that this new consequence relation is algebraizable with the Abelian groups as its equivalent algebraic semantics. Finally, we show that although \ is not algebraizable, it is order-algebraizable in the sense of Raftery. (shrink)

Abelian Logic is a paraconsistent logic discovered independently by Meyer and Slaney [10] and Casari [2]. This logic is also referred to as Abelian Group Logic (AGL) [12] since its set of theorems is sound and complete with respect to the class of Abelian groups. In this paper we investigate the pure implication fragment A→ of Abelian logic. This is an extension of the implication fragment of linear logic, BCI. A Hilbert style axiomatic system for A→ can obtained by adding (...) the axiom A (dubbed the ‘axiom of relativity’ by Meyer and Slaney) to BCI, as follows: B (α → β) → ((γ → α) → (γ → β)) C (α → (β → γ)) → ((β → (α → γ)) I α → α A ((α → β) → β) → α MP α, α → β ⇒ β. (shrink)

In his paper, ìRight Weakening and Right Contraction in LK î, Hirokawa investigates the properties of the structural rules of contraction and weak- ening as they appear in a certain sequent calculus formulation of Örst order classical logic. In what follows we explore the notion of correspondence, in particular with reference to the structural rules in the succedent, and in doing so critically examine the sensitivity of Hirokawaís results to the formulation of the calculus, both with respect to the formulations (...) of the rules governing the connectives present and to the choice of the connectives themselves. (shrink)