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Denotation as in poetry is the literal meaning of a word, and connotation is the suggestive meaning of a word.
For example, the word "city" connotes the attributes of largeness, populousness. It denotes individual objects such as London, New York, Paris.
It should not to be confused (though it often is) with Gottlob Frege's distinction between sense and reference, though it has some affinity with his distinction between concept and object. Contemporary philosophers employ the terms intension and extension for connotation and denotation respectively.
Mill's definition of the term "connotation" is altogether different from that used by scholastic logicians. In scholastic logic, a "connotative" term was originally what would now be called an adjective, "signifying an attribute as qualifying a subject". For example, "brave", as used to say or imply of some particular person that they are brave. By contrast, the abstract noun "bravery" was thought to signify something independent of the subject, an "independent entity", thus is non-connotative. The distinction is connected with the metaphysical one between substance and attribute.
There is a related distinction in linguistics between the objective meaning or denotation of a word such as "vulgar", and the positive or negative association or connotation we attach to such a word. "Vulgar" derives from the Latin word for "common" and literally means ubiquitous, found everywhere, and was its original meaning. The word has now acquired the negative connotation of "gross" or "crudely obscene" (also of showy ostentatiousness). The process of acquiring a negative connotation is known as pejoration.
Connotations often give insight into the associations of the real usage of a word.
This distinction is a useful one in psychotherapy, helping to understand the weight of words