Describes a collection of related deductive systems that intend to capture the position that no proposition should entail its own negation. Frequently, logics described as connexive enjoy theorems such as Aristotle’s thesis:

• •$\neg (\phi \to \neg \phi )$

or Boethius’ thesis:

• •$(\phi \to \psi )\to \neg (\phi \to \neg \psi )$

Formal accounts of these theses have generally been accompanied by severe restrictions. For example, because axiomatic representations of the inference rule of simplification entail that both $(\phi \wedge \neg \phi )\to \phi$ and $(\phi \wedge \neg \phi )\to \neg \phi$, systems containing Boethius’ thesis are inconsistent with axiomatic forms of this rule. Likewise, contraposition is not admissible in many connexive logics.