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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
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 formal languages, truth functions are represented by unambiguous symbols.This allows logical statements to not be understood in an ambiguous way. These symbols are called logical connectives, logical operators, propositional operators, or, in classical logic, truth-functional connectives.
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.
Various extensions of logic programming have been developed to provide a logical framework for such destructive change of state. [53] [54] [55] The broad range of Prolog applications, both in isolation and in combination with other languages is highlighted in the Year of Prolog Book, [21] celebrating the 50 year anniversary of Prolog in 2022.
(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 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.
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.