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The practical use of Peirce's categories is simply to organize our thoughts about what sorts of formal models are demanded by a material situation, for instance, a domain of phenomena from atoms to biology to culture.  To say that "k-ness" is involved in a phenomenon is simply to say that we need k-adic relations to model it adequately, and that the phenomenon itself appears to demand nothing less.  Aside from this, Peirce's realization that k-ness for k = 1, 2, 3 affords us with a sufficient basis for all that we need to model is a formal fact that depends on a particular theorem in the logic of relatives.  If it weren't for that, there would hardly be any reason to single out three.

The practical use of Peirce's categories is simply to organize our thoughts about what sorts of formal models are demanded by a material situation, for instance, a domain of phenomena from atoms to biology to culture.  To say that “''k''-ness” is involved in a phenomenon is simply to say that we need ''k''-adic relations to model it adequately, and that the phenomenon itself appears to demand nothing less.  Aside from this, Peirce's realization that ''k''-ness for ''k'' = 1, 2, 3 affords us with a sufficient basis for all that we need to model is a formal fact that depends on a particular theorem in the logic of relatives.  If it weren't for that, there would hardly be any reason to single out three.

In order to discuss the various forms of iconicity that might be involved in the application of Peirce's logical graphs and their kind to the object domain of logic itself, we will need to bring out two or three ''categories of structured individuals'' (COSIs), depending on how one counts.  These are called the ''object domain'', the ''sign domain'', and the ''interpretant sign domain'', which may be written <math>{O, S, I},\!</math> respectively, or <math>{X, Y, Z},\!</math> respectively, depending on the style that fits the current frame of discussion.

In order to discuss the various forms of iconicity that might be involved in the application of Peirce's logical graphs and their kind to the object domain of logic itself, we will need to bring out two or three ''categories of structured individuals'' (COSIs), depending on how one counts.  These are called the ''object domain'', the ''sign domain'', and the ''interpretant sign domain'', which may be written <math>{O, S, I},\!</math> respectively, or <math>{X, Y, Z},\!</math> respectively, depending on the style that fits the current frame of discussion.