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If the world of <math>\text{A}\!</math> and <math>\text{B},\!</math> the set <math>W = \{ \text{A}, \text{B}, {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}\!</math> is viewed abstractly, as an arbitrary set of six atomic points, then there are exactly <math>2^6 = 64\!</math> ''abstract properties'' or ''potential attributes'' that might be applied to or recognized in these points.  The elements of <math>W\!</math> that possess a given property form a subset of <math>W\!</math> called the ''extension'' of that property.  Thus the extensions of abstract properties are exactly the subsets of <math>W.\!</math>  The set of all subsets of <math>W\!</math> is called the ''power set'' of <math>W,\!</math> notated as <math>\operatorname{Pow}(W)\!</math> or <math>\mathcal{P}(W).\!</math> In order to make this way of talking about properties consistent with the previous definition of reality, it is necessary to say that one potential property is never realized, since no point has it, and its extension is the empty set <math>\varnothing = \{ \}.\!</math> All the ''natural'' properties of points that one observes in a concrete situation, properties whose extensions are known as ''natural kinds'', can be recognized among the ''abstract'', ''arbitrary'', or ''set-theoretic'' properties that are systematically generated in this way.  Typically, however, many of these abstract properties will not be recognized as falling among the more natural kinds.

If the world of A and B, the set W = {A, B, "A", "B", "i", "u"}, is viewed abstractly, as an arbitrary set of six atomic points, then there are exactly 26 = 64 "abstract properties" (APs) or "potential attributes" (PAs) that might be applied to or recognized in these points.  The extensions of these APs are the subsets of W, otherwise known as members of the "power set" Pow (W). In order to make this way of talking about properties consistent with the previous definition of reality, it is necessary to say that one potential property is never realized, since no point has it, and its extension is the empty set {}.  All the "natural" properties of points that one observes in a concrete situation, properties whose extensions are known as "natural kinds", can be recognized among the "abstract", "arbitrary", or "set theoretic" properties that are systematically generated in this way.  Typically, however, many of these abstract properties will not be recognized as falling among the more natural kinds.

 

Tables 54.1, 54.2, and 54.3 show three different ways of representing the elements of the world set W as vectors in the coordinate space W and as singular propositions in the universe of discourse W[].  Altogether, these Tables present the "literal" codes for the elements of W and W[] in their "mnemonic", "pragmatic", and "abstract" versions, respectively.  In each Table, Column 1 lists the element w C W, while Column 2 gives the corresponding coordinate vector w C W in the form of a bit string.  The next two Columns represent each w C W as a proposition in W[], in effect, reconstituting it as a function w : W >B.  Column 3 shows the propositional expression of each element in the form of a conjunct term, in other words, as a logical product of positive and negative features.  Column 4 gives the compact code for each element, using a conjunction of positive features in subscripted angle brackets to represent the singular proposition corresponding to each element.

Tables 54.1, 54.2, and 54.3 show three different ways of representing the elements of the world set W as vectors in the coordinate space W and as singular propositions in the universe of discourse W[].  Altogether, these Tables present the "literal" codes for the elements of W and W[] in their "mnemonic", "pragmatic", and "abstract" versions, respectively.  In each Table, Column 1 lists the element w C W, while Column 2 gives the corresponding coordinate vector w C W in the form of a bit string.  The next two Columns represent each w C W as a proposition in W[], in effect, reconstituting it as a function w : W >B.  Column 3 shows the propositional expression of each element in the form of a conjunct term, in other words, as a logical product of positive and negative features.  Column 4 gives the compact code for each element, using a conjunction of positive features in subscripted angle brackets to represent the singular proposition corresponding to each element.