Many-sorted, or multi-typed, logic allows multiple sorts. Normally these are disjoint, but there is also order-sorted logic. These may be translated into unsorted logic if understood as a carved from a single domain.

But also they may be seen as simple instances of a category (or higher) of types, a set or a poset.

References

F. Jeffry Pelletier (1972). “Sortal Quantification and Restricted Quantification” (PDF). Philosophical Studies. 23: 400–404. (pdf)

Oberschelp A. (1990) Order sorted predicate logic. In: Bläsius K.H., Hedtstück U., Rollinger CR. (eds) Sorts and Types in Artificial Intelligence. Lecture Notes in Computer Science (Lecture Notes in Artificial Intelligence), vol 418. Springer, Berlin, Heidelberg . (doi)

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