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Many-sorted logic can reflect formally our intention not to handle the universe as a
homogeneous Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
collection of objects, but to
partition Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of a ...
it in a way that is similar to types in typeful programming. Both functional and assertive "
parts of speech In grammar, a part of speech or part-of-speech (abbreviated as POS or PoS, also known as word class or grammatical category) is a category of words (or, more generally, of lexical items) that have similar grammatical properties. Words that are ass ...
" 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 various ways to formalize the intention mentioned above; a ''many-sorted logic'' is any package of information which fulfils it. In most cases, the following are given: * a set of sorts, ''S'' * an appropriate generalization of the notion of ''
signature A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...
'' to be able to handle the additional information that comes with the sorts. The
domain of discourse In the formal sciences, the domain of discourse, also called the universe of discourse, universal set, or simply universe, is the set of entities over which certain variables of interest in some formal treatment may range. Overview The domain ...
of any
structure A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...
of that signature is then fragmented into disjoint subsets, one for every sort.


Example

When reasoning about biological organisms, it is useful to distinguish two sorts: \mathrm and \mathrm. While a function \mathrm\colon \mathrm \to \mathrm makes sense, a similar function \mathrm\colon \mathrm \to \mathrm usually does not. Many-sorted logic allows one to have terms like \mathrm(\mathrm), but to discard terms like \mathrm(\mathrm) as syntactically ill-formed.


Algebraization

The algebraization of many-sorted logic is explained in an article by Caleiro and Gonçalves, which generalizes
abstract algebraic logic In mathematical logic, abstract algebraic logic is the study of the algebraization of deductive systems arising as an abstraction of the well-known Lindenbaum–Tarski algebra, and how the resulting algebras are related to logical systems.Font, 20 ...
to the many-sorted case, but can also be used as introductory material.


Order-sorted logic

While ''many-sorted'' logic requires two distinct sorts to have disjoint universe sets, ''order-sorted'' logic allows one sort s_1 to be declared a subsort of another sort s_2, usually by writing s_1 \subseteq s_2 or similar syntax. In the above biology example, it is desirable to declare :\text \subseteq \text, :\text \subseteq \text, :\text \subseteq \text, :\text \subseteq \text, :\text \subseteq \text, :\text \subseteq \text, and so on; cf. picture. Wherever a term of some sort s is required, a term of any subsort of s may be supplied instead (''
Liskov substitution principle The Liskov substitution principle (LSP) is a particular definition of a subtyping relation, called strong behavioral subtyping, that was initially introduced by Barbara Liskov in a 1988 conference keynote address titled ''Data abstraction and h ...
''). For example, assuming a function declaration \text: \text \longrightarrow \text, and a constant declaration \text: \text, the term \text(\text) is perfectly valid and has the sort \text. In order to supply the information that the mother of a dog is a dog in turn, another declaration \text: \text \longrightarrow \text may be issued; this is called ''function overloading'', similar to
overloading in programming languages In programming language theory and type theory, polymorphism is the provision of a single interface to entities of different types or the use of a single symbol to represent multiple different types.: "Polymorphic types are types whose operatio ...
. Order-sorted logic can be translated into unsorted logic, using a unary predicate p_i(x) for each sort s_i, and an axiom \forall x (p_i(x) \rightarrow p_j(x)) for each subsort declaration s_i \subseteq s_j. The reverse approach was successful in automated theorem proving: in 1985,
Christoph Walther Christoph Walther (born 9 August 1950) is a German computer scientist, known for his contributions to automated theorem proving. He is Professor emeritus ''Emeritus'' (; female: ''emerita'') is an adjective used to designate a retired chair ...
could solve a then benchmark problem by translating it into order-sorted logic, thereby boiling it down an order of magnitude, as many unary predicates turned into sorts. In order to incorporate order-sorted logic into a clause-based automated theorem prover, a corresponding ''
order-sorted unification In logic and computer science, unification is an algorithmic process of solving equations between symbolic expressions. Depending on which expressions (also called ''terms'') are allowed to occur in an equation set (also called ''unification prob ...
'' algorithm is necessary, which requires for any two declared sorts s_1, s_2 their intersection s_1 \cap s_2 to be declared, too: if x_1 and x_2 are variables of sort s_1 and s_2, respectively, the equation x_1 \stackrel\,x_2 has the solution \, where x: s_1 \cap s_2. Smolka generalized order-sorted logic to allow for
parametric polymorphism In programming languages and type theory, parametric polymorphism allows a single piece of code to be given a "generic" type, using variables in place of actual types, and then instantiated with particular types as needed. Parametrically polymorph ...
. In his framework, subsort declarations are propagated to complex type expressions. As a programming example, a parametric sort \text(X) may be declared (with X being a type parameter as in a
C++ template Templates are a feature of the C++ programming language that allows functions and classes to operate with generic types. This allows a function or class to work on many different data types without being rewritten for each one. The C++ Standa ...
), and from a subsort declaration \text \subseteq \text the relation \text(\text) \subseteq \text(\text) is automatically inferred, meaning that each list of integers is also a list of floats. Schmidt-Schauß generalized order-sorted logic to allow for term declarations. As an example, assuming subsort declarations \text \subseteq \text and \text \subseteq \text, a term declaration like \forall i:\text. \; (i+i):\text allows to declare a property of integer addition that could not be expressed by ordinary overloading.


See also

*
Categorical logic __NOTOC__ Categorical logic is the branch of mathematics in which tools and concepts from category theory are applied to the study of mathematical logic. It is also notable for its connections to theoretical computer science. In broad terms, categ ...
* Many-sorted first-order logic#Many-sorted logic


References

Early papers on many-sorted logic include: * , collected in the author's ''Computation, Logic, Philosophy. A Collection of Essays'', Beijing: Science Press; Dordrecht: Kluwer Academic, 1990. * * * {{cite journal, author=F. Jeffry Pelletier, title=Sortal Quantification and Restricted Quantification, journal=Philosophical Studies, year=1972, volume=23, issue=6, pages=400–404, url=https://www.sfu.ca/~jeffpell/papers/SortalRestrQuant.pdf, doi=10.1007/bf00355532, s2cid=170303654


External links

*"Many-sorted Logic", the first chapter i
''Lecture Notes on Decision Procedures''
b
Calogero G. Zarba
Systems of formal logic