 It is important to be clear on what the equi-interpretability of PFO and PFO+ with respectively MSO and MSO+ does and does not show (Florio and Linnebo 2021: sect. 4.2). It shows that these two pairs of theories are equivalent for most technical purposes. But by itself it does not show anything about these two pairs of theories’ being equivalent in any of the more demanding senses that philosophers often care about (such as having the same epistemic status, ontological commitments, or degree of analyticity). (For instance PFO is equi-interpretable with atomic extensional mereology, which philosophers tend to find much more problematic than PFO.) In order to show that the two pairs of theories are equivalent in some philosophically important respect \(F\), we would need to show that the above translations preserve \(F\)-ness.