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 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 of any
structure 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:
and
. While a function
makes sense, a similar function
usually does not. Many-sorted logic allows one to have terms like
, but to discard terms like
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 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
to be declared a subsort of another sort
, usually by writing
or similar syntax. In the
above biology example, it is desirable to declare
:
,
:
,
:
,
:
,
:
,
:
,
and so on; cf. picture.
Wherever a term of some sort
is required, a term of any subsort of
may be supplied instead (''
Liskov substitution principle''). For example, assuming a function declaration
, and a constant declaration
, the term
is perfectly valid and has the sort
. In order to supply the information that the mother of a dog is a dog in turn, another declaration
may be issued; this is called ''function overloading'', similar to
overloading in programming languages.
Order-sorted logic can be translated into unsorted logic, using a unary predicate
for each sort
, and an axiom
for each subsort declaration
. The reverse approach was successful in automated theorem proving: in 1985,
Christoph Walther 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'' algorithm is necessary, which requires for any two declared sorts
their intersection
to be declared, too: if
and
are variables of sort
and
, respectively, the equation
has the solution
, where
.
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
may be declared (with
being a type parameter as in a
C++ template), and from a subsort declaration
the relation
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
and
, a term declaration like
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