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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 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.
(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.
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
Recently, conservative extensions have been used for defining a notion of module for ontologies [citation needed]: if an ontology is formalized as a logical theory, a subtheory is a module if the whole ontology is a conservative extension of the subtheory. An extension which is not conservative may be called a proper extension.
Logic is the study of proof and deduction as manifested in language (abstracting from any underlying psychological or biological processes). [1] Logic is not a closed, completed science, and presumably, it will never stop developing: the logical analysis can penetrate into varying depths of the language [2] (sentences regarded as atomic, or splitting them to predicates applied to individual ...
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.