Many-sorted logic

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Many-sorted logic can reflect formally our intention, not to handle the universe as a homogeneous collection of objects, but to partition it in a way that is similar to types in typeful programming. Both functional and assertive "parts of speech" in the language of the logic reflect this typeful partitioning of the universe, even on the syntax level: substitution and argument passing can be done only accordingly, respecting the “sorts”.

There are more ways to formalize the intention mentioned above; a many-sorted logic is any package of information which fulfills it. In most cases, the following are given:

a set of sorts, S
an appropriate generalization of the notion of signature to be able to handle the additional information that comes with the sorts.

The domain of discourse of any structure of that signature is then fragmented into disjoint subsets, one for every sort.

[edit] Algebraization

The algebraization of many-sorted logic is explained in On the Algebraization of Many-sorted Logics by Carlos Caleiro and Ricardo Gonçalves. The book generalizes abstract algebraic logic to the many-sorted case, but it can also be used as introductory material.