"A Sign is a Cognizable that, on the one hand, is so determined (i.e., specialized, bestimmt) by something other than itself, called its Object (or, in some cases, as if the Sign be the sentence "Cain killled Abel," in which Cain and Abel are equally Partial Objects, it may be more convenient to say that that which determines the Sign is the Complexus, or Totality, of Partial Objects. And in every case the Object is accurately the Universe of which the Special Object is member, or part), while, on the other hand, it so determines some actual or potential Mind, the determination whereof I term the Interpretant created by the Sign, that that Interpreting Mind is therein determined mediately by the Object." (A Letter to William James, EP 2:492, 1909)
"[A sign] must be determined to correspond, according to some principle, and by some species of causation, with something else, called its Object. In a word, whether physically, rationally, or otherwise directly or indirectly, its Object, as agent, acts upon the sign, as patient." ('The Basis of Pragmaticism', MS 283, 1905)
"A sign is intended to correspond to a real thing, or fact, or to something relatively real; and this object of the sign may be the very sign itself, as when a map is precisely superposed upon that which it maps. It is a perfection of the sign if it separately represents its object; in which case it becomes a proposition and is true or false." ([Foundations of Mathematics], MS 9:1, c. 1903?)
"By an object, I mean anything that we can think, i.e. anything we can talk about." ([Reflections on Real and Unreal Objects], MS 966, not dated)
"A sign stands for something to the idea which it produces, or modifies. Or, it is a vehicle conveying into the mind something from without. That for which it stands is called its object; that which it conveys, its meaning; and the idea to which it gives rise, its interpretant. The object of representation can be nothing but a representation of which the first representation is the interpretant. But an endless series of representations, each representing the one behind it, may be conceived to have an absolute object at its limit." (A Fragment, CP 1.339, not dated)
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Abbreviations (CP, EP, etc.) and sources; see here
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