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(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, 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 ...
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.
In logic, a set of symbols is commonly used to express logical representation. The following table lists many common symbols, together with their name, how they should be read out loud, and the related field of mathematics.
meaning "If each is true, then is true". Logic programs compute the set of facts that are implied by their rules. Many implementations of Datalog, Prolog, and related languages add procedural features such as Prolog's cut operator or extra-logical features such as a foreign function interface. The formal semantics of such extensions are beyond ...
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
anti-extension In set theory and logic, the complement of the extension of a concept or predicate, consisting of all objects that do not fall under the concept. [11] [12] [13] antilogism A syllogism with three premises leading to a contradiction, showing the inconsistency of the premises. [14] [15] [16] antinomy
Prolog is a logic programming language that has its origins in artificial intelligence, automated theorem proving and computational linguistics. [1] [2] [3]Prolog has its roots in first-order logic, a formal logic, and unlike many other programming languages, Prolog is intended primarily as a declarative programming language: the program is a set of facts and rules, which define relations.