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In mathematical logic, a theory can be extended with new constants or function names under certain conditions with assurance that the extension will introduce no contradiction. Extension by definitions is perhaps the best-known approach, but it requires unique existence of an object with the desired property. Addition of new names can also be ...
An extensional definition gives meaning to a term by specifying its extension, that is, every object that falls under the definition of the term in question.. For example, an extensional definition of the term "nation of the world" might be given by listing all of the nations of the world, or by giving some other means of recognizing the members of the corresponding class.
(That set might be empty, currently.) For example, the extension of a function is a set of ordered pairs that pair up the arguments and values of the function; in other words, the function's graph. The extension of an object in abstract algebra, such as a group, is the underlying set of the object. The extension of a set is the set itself.
In mathematical logic, more specifically in the proof theory of first-order theories, extensions by definitions formalize the introduction of new symbols by means of a definition. For example, it is common in naive set theory to introduce a symbol ∅ {\displaystyle \emptyset } for the set that has no member.
While the name "logic programming" is used to refer to the entire paradigm of programming languages including Datalog and Prolog, when discussing formal semantics, it generally refers to an extension of Datalog with function symbols. Logic programs are also called Horn clause programs.
Some of these languages, such as miniKanren [28] and relational linear programming [30] are logic programming languages in the sense of this article. However, the relational language RML is an imperative programming language [ 31 ] whose core construct is a relational expression, which is similar to an expression in first-order predicate logic.
In logic, extensionality, or extensional equality, refers to principles that judge objects to be equal if they have the same external properties. It stands in contrast to the concept of intensionality, which is concerned with whether the internal definitions of objects are the same.
The extension of a predicate – a truth-valued function – is the set of tuples of values that, used as arguments, satisfy the predicate. Such a set of tuples is a relation . Examples