. (i) Any characteristic. (ii) A characteristic relevant for the indiscernibility of identicals (see LEIBNIZ’S LAW). Tully is the same as Cicero, but ‘Tully is hereby named by a five-letter name’ is true, while ‘Cicero is hereby named by a five-letter name’ is false. So being hereby named by a five-letter name is not a single property in this sense. It may, however, be doubted whether it is a characteristic at all, when the ‘hereby’ is unspecified (in which case we need not posit characteristics not subject to Leibniz’s law). (iii) A positive, as against negative, characteristic, e.g.
being red, but not being not red (which is too indefinite in its application: does it apply to abstract things?). (iv) A non-relational characteristic, e.g. being red, but not being a brother, (v) The Aristotelian and medieval proprium: a characteristic following from, and unique to, the essence of a species, but not part of its definition. Able to laugh is a proprium of a man, if only man can have it but man is not defined as a laughing animal. Other senses exist.
S.Shoemaker, Identity, Cause, and Mind, Cambridge UP, 1984, chapter 10. (Discusses sense near to (iv).)
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