Changes

Suppose we have a sign relation <math>L \subseteq O \times S \times I,</math> where <math>O\!</math> is the object domain, <math>S\!</math> is the sign domain, and <math>I\!</math> is the interpretant domain of the sign relation <math>L.\!</math>

Suppose we have a sign relation <math>L \subseteq O \times S \times I,</math> where <math>O\!</math> is the object domain, <math>S\!</math> is the sign domain, and <math>I\!</math> is the interpretant domain of the sign relation <math>L.\!</math>

Now suppose that the situation with respect to the ''denotative component'' of <math>L,\!</math> in other words, the projection of <math>L\!</math> on the subspace <math>O \times S,</math> can be pictured in the following manner, where equal signs written between ostensible nodes identify them into a single actual node.

Now suppose that the situation with respect to the ''denotative component'' of <math>L,\!</math> in other words, the projection of <math>L\!</math> on the subspace <math>O \times S,</math> can be pictured in the following manner:

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The Figure depicts a situation where each of the three objects, <math>o_1, o_2, o_3,\!</math> has a ''proper name'' that denotes it alone, namely, the three proper names <math>s_1, s_2, s_3,\!</math> respectively.  Over and above the objects denoted by their proper names, there is the general sign <math>s,\!</math> which denotes any and all of the objects <math>o_1, o_2, o_3.\!</math>  This kind of sign is described as a ''general name'' or a ''plural term'', and its relation to its objects is a ''general reference'' or a ''plural denotation''.

The Figure depicts a situation where each of the three objects, <math>o_1, o_2, o_3,\!</math> has a ''proper name'' that denotes it alone, namely, the three proper names <math>s_1, s_2, s_3,\!</math> respectively.  Over and above the objects denoted by their proper names, there is the general sign <math>s,\!</math> which denotes any and all of the objects <math>o_1, o_2, o_3.\!</math>  This kind of sign is described as a ''general name'' or a ''plural term'', and its relation to its objects is a ''general reference'' or a ''plural denotation''.

Now, at this stage of the game, if you ask:  ''Is the object of the sign <math>s\!</math> one or many?'', the answer has to be:  ''Not one, but many''.  That is, there is not one <math>o\!</math> that <math>s\!</math> denotes, but only the three <math>o\!</math>'s in the object space.  Nominal thinkers would ask:  ''Granted this, what need do we have really of more excess?''  The maxim of the nominal thinker is ''never read a general name as a name of a general'', meaning that we should never jump from the accidental circumstance of a plural sign <math>s\!</math> to the abnominal fact that a unit <math>o\!</math> exists.

If you now ask, ''Is the object of the sign <math>s\!</math> one or many?'', the answer has to be ''many''.  That is, there is not one <math>o\!</math> that <math>s\!</math> denotes, but only the three <math>o\!</math>'s in the object space.  Nominal thinkers would ask:  ''Granted this, what need do we have really of more excess?''  The maxim of the nominal thinker is ''never read a general name as a name of a general'', meaning that we should never jump from the accidental circumstance of a plural sign <math>s\!</math> to the abnominal fact that a unit <math>o\!</math> exists.

In actual practice this would be just one segment of a much larger sign relation, but let us continue to focus on just this one piece.  The association of objects with signs is not in general a function, no matter which way, from <math>O\!</math> to <math>S\!</math> or from <math>S\!</math> to <math>O,\!</math> that we might try to read it, but very often one will choose to focus on a selection of links that do make up a function in one direction or the other.

In actual practice this would be just one segment of a much larger sign relation, but let us continue to focus on just this one piece.  The association of objects with signs is not in general a function, no matter which way, from <math>O\!</math> to <math>S\!</math> or from <math>S\!</math> to <math>O,\!</math> that we might try to read it, but very often one will choose to focus on a selection of links that do make up a function in one direction or the other.