This is obvious enough, and yet the expression inverse variation is open to objection. For the denotation may be increased in a sense without affecting the connotation. The birth of an animal may be said to increase the denotation: every year thousands of new houses are built: there are swarms of flies in a hot summer and few in a cold. But all the time the connotation of animal, house, or fly remains the same: the word does not change its meaning.
It is obviously wrong to say that they vary in inverse proportion. Double or treble the number of attributes, and you do not necessarily reduce the denotation by one-half or one-third.
It is, in short, the meaning or connotation that is the main thing. This determines the application of a word. As a rule if you increase meaning, you restrict scope. Let your idea, notion, or concept of culture be a knowledge of Mathematics, Latin and Greek: your men of culture will be more numerous than if you require from each of them these qualifications plus a modern language, an acquaintance with the Fine Arts, urbanity of manners, etc.
It is just possible to increase the connotation without decreasing the denotation, to thicken or deepen the concept without diminishing the class. This is possible only when two properties are exactly co-extensive, as equilaterality and equiangularity in triangles.
Singular and Proper Names. A Proper or Singular name is a name used to designate an individual. Its function, as distinguished from that of the general name, is to be used purely for the purpose of distinctive reference.
A man is not called Tom or Dick because he is like in certain respects to other Toms or other Dicks. The Toms or the Dicks do not form a logical class. The names are given purely for purposes of distinction, to single out an individual subject. The Arabic equivalent for a Proper name, alam, "a mark," "a sign-post," is a recognition of this.
In the expressions "a Napoleon," "a Hotspur," "a Harry," the names are not singular names logically, but general names logically, used to signify the possession of certain attributes.
A man may be nicknamed on a ground, but if the name sticks and is often used, the original meaning is forgotten. If it suggests the individual in any one of his qualities, any point in which he resembles other individuals, it is no longer a Proper or Singular name logically, that is, in logical function. That function is fulfilled when it has called to mind the individual intended.
To ask, as is sometimes done, whether Proper names are connotative or denotative, is merely a confusion of language. The distinction between connotation and denotation, extension and intension, applies only to general names. Unless a name is general, it has neither extension nor intension:[3] a Proper or Singular name is essentially the opposite of a general name and has neither the one nor the other.
A nice distinction may be drawn between Proper and Singular names, though they are strict synonyms for the same logical function. It is not essential to the discharge of that function that the name should be strictly appropriated to one object. There are many Toms and many Dicks. It is enough that the word indicates the individual without confusion in the particular circumstances.