<p>A logic is <i>relevant</i> if it is a set of formulas closed under adjunction, modus ponens and enjoys the variable sharing property. It is <i>connexive</i> if it has some or all of the versions of Aristotle’s thesis and Boethius’ thesis. It is <i>connexively acceptable</i> if it does not sanction such formulas as the negation of the self-identity axiom. Finally, it is <i>strictly connexive</i> if it does not validate some formula implying (or being implied by) its own negation. Now, it is known that addition of Aristotle’s thesis to a relevant logic does not necessarily result in an acceptable connexive logic. Then, the aim of this paper is to examine the prospects of defining relevant acceptable strictly connexive logics based upon weak relevant logics.</p>