 Mceldowney (2020) argues that the concept of bi-interpretability is the best measure of theoretical equivalence in MSL. The oldest criterion for theoretical equivalence between first-order theories with the same signature is the notion of definitional equivalence (see Eilenberg &amp; Mac Lane 1942, 1945 and Glymour 1971). Therefore, the necessity of a many-sorted counterpart was apparent. As a result, a debate on the nature of equivalence in MSL has taken place since 2016. Barrett and Halvorson (2016) generalizes the idea of definitional extension to the concept of Morita extension: intuitively, \(T^+\) is a Morita extension of \(T\) iff both \(T\) and \(T^+\) are MSL theories and \(T^+\) “says no more” than the original theory (see 2016: 565 for formal details).