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

Typeful programming

In computer science, typeful programming is a programming style identified by widespread use of type information handled through mechanical typechecking techniques. The concept was introduced in a paper of the same name by Luca Cardelli in 1991....

. Both functional and assertive "parts of speech

Lexical category

In grammar, a part of speech is a linguistic category of words , which is generally defined by the syntactic or morphological behaviour of the lexical item in question. Common linguistic categories include noun and verb, among others...

" 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
  Signature (logic)
  In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an algebraic structure. In model theory, signatures are used for both purposes.Signatures play the same...
  
  to be able to handle the additional information that comes with the sorts.

The domain of discourse

Domain of discourse

In the formal sciences, the domain of discourse, also called the universe of discourse , is the set of entities over which certain variables of interest in some formal treatment may range...

 of any structure

Structure (mathematical logic)

In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations which are defined on it....

 of that signature is then fragmented into disjoint subsets, one for every sort.

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

Abstract Algebraic Logic

In mathematical logic, abstract algebraic logic is the study of the algebraization of deductive systemsarising as an abstraction of the well-known Lindenbaum-Tarski algebra, and how the resulting algebras are related to logical systems.-Overview:...

to the many-sorted case, but it can also be used as introductory material.

External links

• "Many-sorted Logic", the first chapter in Lecture Notes on Decision Procedures by Calogero G. Zarba