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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 ...
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
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.
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.
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
An empty set exists. This formula is a theorem and considered true in every version of set theory. The only controversy is over how it should be justified: by making it an axiom; by deriving it from a set-existence axiom (or logic) and the axiom of separation; by deriving it from the axiom of infinity; or some other method.
Karel Lambert, who coined the term "free logic", has suggested that free logic can be understood as a generalization of classical predicate logic just as predicate logic is a generalization of Aristotelian logic. On this view, classical predicate logic introduces predicates with an empty extension while free logic introduces singular terms of ...