Many-sorted logic

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:

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

When reasoning about biological organisms, it is useful to distinguish two sorts: ${\displaystyle {\textit {plant}}}$ and ${\displaystyle {\textit {animal}}}$. While a function ${\displaystyle {\textit {mother}}:{\textit {animal}}\longrightarrow {\textit {animal}}}$ makes sense, a similar function ${\displaystyle {\textit {mother}}:{\textit {plant}}\longrightarrow {\textit {plant}}}$ usually does not. Many-sorted logic allows one to have terms like ${\displaystyle {\textit {mother}}({\textit {lassie}})}$, but to discard terms like ${\displaystyle {\textit {mother}}({\textit {my}}\_{\textit {favorite}}\_{\textit {oak}})}$ as syntactically ill-formed.

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