 Further motivation for systems of connexive logic comes from more instrumental studies. In McCall 1967, connexive implication is motivated by reproducing in a first-order language all valid moods of Aristotle’s syllogistic (see the entry on  Aristotle’s logic).  In particular, the classically invalid inference from ‘All A is B’ to ‘Some A is B’ is obtained by translating ‘Some A is B’ as ∃x(~(A(x) → ~B(x))), where → is a connexive implication. In Wansing 2007, connexive implication is motivated by introducing a negation connective into Categorial Grammar in order to express negative information about membership in syntactic categories (see the entry on  typelogical grammar).  Consider, for example, the syntactic category (type) (n → s) of intransitive verbs, i.e., of expressions that in combination with a name (an expression of type n) result in a sentence (an expression of type s). The idea is that an expression is of type ~(n → s) iff in combination with a name it results in an expression that is not a sentence. In other words, an expression belongs to type ~(n → s) iff it is of type (n → ~s). In the short note Besnard 2011, Aristotle’s thesis AT′ is motivated as expressing a notion of rule consistency for rule-based systems in knowledge representation. A further motivation arises from the problem of modelling conditional obligations in deontic logic. Weiss (2019) suggests to understand a certain implication that validates Aristotle’s theses and weak versions of Boethius’ theses (cf. sections 2 and 3.2) as expressing a conditional obligation operator. Yet another motivation arises from the problem of modelling conditional obligations in deontic logic. Another motivation in terms of applications comes from non-classical mathematics. There is an extended literature on mathematical theories based on non-classical logics, including intutionistic, fuzzy, relevant, and linear arithmetic and paraconsistent set theory. Early contributions in the context of connexive logic are McCall’s 1967 connexive class theory, and Wiredu’s 1974 paper on connexive set theory. Ferguson (2016, 2019) takes up the challenge of investigating the prospects for a connexive mathematics and explores the feasibility of a connexive arithmetic.