Changes

This discussion draws on concepts from two previous papers (Awbrey, 1989 and 1994), changing notations as needed to fit the current context.  Except for a number of special sets like <math>\mathbb{B},\!</math> <math>\mathbb{N},\!</math> <math>\mathbb{Z},\!</math> and <math>\mathbb{R},\!</math> I use plain capital letters for ordinary sets, singly underlined capitals for coordinate spaces and vector spaces, and doubly underlined capitals for the ''alphabets'' and ''lexicons'' that generate formal languages and logical universes of discourse.

This discussion draws on concepts from two previous papers (Awbrey, 1989 and 1994), changing notations as needed to fit the current context.  Except for a number of special sets like <math>\mathbb{B},\!</math> <math>\mathbb{N},\!</math> <math>\mathbb{Z},\!</math> and <math>\mathbb{R},\!</math> I use plain capital letters for ordinary sets, singly underlined capitals for coordinate spaces and vector spaces, and doubly underlined capitals for the ''alphabets'' and ''lexicons'' that generate formal languages and logical universes of discourse.

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If <math>X = \{ x_1, \ldots, x_n \}\!</math> is a set of <math>n\!</math> elements, it is possible to construct a ''formal alphabet'' of <math>n\!</math> ''letters'' or a ''formal lexicon'' of <math>n\!</math> ''words'' corresponding to the elements of <math>X\!</math> and notated as follows:

If X = {x1, ... , xn} is a set of n elements, it is possible to construct a "formal alphabet" of n "letters" or a "formal lexicon" of n "words" that exists in a one to one correspondence with the elements of X and can be notated as follows:

X = Lit (X) = {x1, ... , xn}.

| <math>\underline{\underline{X}} = \operatorname{Lit}(X) = \{ \underline{\underline{x_1}}, \ldots, \underline{\underline{x_n}} \}.\!</math>

The set X is known in formal settings as the "literal alphabet" or the "literal lexicon" associated with X, but on more familiar grounds it can be called the "double" of X.  Under conditions of careful interpretation, any finite set X can be construed as its own double, but for now it is safest to preserve the apparent distinction in roles until the sense of this double usage has become second nature.

The set <math>\underline{\underline{X}}\!</math> is known in formal settings as the ''literal alphabet'' or the ''literal lexicon'' associated with <math>X,\!</math> but on more familiar grounds it can be called the ''double'' of <math>X.\!</math> Under conditions of careful interpretation, any finite set <math>X\!</math> can be construed as its own double, but for now it is safest to preserve the apparent distinction in roles until the sense of this double usage has become second nature.

This construction is often useful in situations where has to deal with a set of signs {"s1", ... , "sn"} with a fixed or a faulty interpretation.  Here, one needs a fresh set of signs {x1, ... , xn} that can be used in analogous ways to the original, but free enough to be controlled and flexible enough to be repaired.  In other words, the interpretation of the new list is subject to experimental variation, freely controllable in such a way that it can follow or assimilate the original interpretation whenever it makes sense to do so, but critically reflected and flexible enough to have its interpretation amended whenever necessary.

This construction is often useful in situations where has to deal with a set of signs {"s1", ... , "sn"} with a fixed or a faulty interpretation.  Here, one needs a fresh set of signs {x1, ... , xn} that can be used in analogous ways to the original, but free enough to be controlled and flexible enough to be repaired.  In other words, the interpretation of the new list is subject to experimental variation, freely controllable in such a way that it can follow or assimilate the original interpretation whenever it makes sense to do so, but critically reflected and flexible enough to have its interpretation amended whenever necessary.