Changes

For brevity in the present case, and to serve as a generic device in any similar array of situations, let <math>S\!</math> be the type of an arbitrary sentence, possibly empty, and let <math>S'\!</math> be the type of a specifically non-empty sentence.  In addition, let <math>\underline\varepsilon</math> be the type of the empty sentence, in effect, the language

For brevity in the present case, and to serve as a generic device in any similar array of situations, let <math>S\!</math> be the type of an arbitrary sentence, possibly empty, and let <math>S'\!</math> be the type of a specifically non-empty sentence.  In addition, let <math>\underline\varepsilon</math> be the type of the empty sentence, in effect, the language

<math>\underline\varepsilon = \{ \varepsilon \}</math> that contains a single empty string, and let a plus sign "+" signify a disjoint union of types.  In the most general type of situation, where the type <math>S\!</math> is permitted to include the empty string, one notes the following relation among types:

<math>\underline\varepsilon = \{ \varepsilon \}</math> that contains a single empty string, and let a plus sign <math>^{\backprime\backprime} + ^{\prime\prime}</math> signify a disjoint union of types.  In the most general type of situation, where the type <math>S\!</math> is permitted to include the empty string, one notes the following relation among types:

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