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→‎Functional Quantifiers: bold definienda, tabular formating

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==Functional Quantifiers==

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The ''umpire measure'' of type <math>\Upsilon : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}</math> takes a single proposition of type <math>\mathbb{B}^2 \to \mathbb{B}</math> as argument, giving the constant proposition <math>1 : \mathbb{B}^2 \to \mathbb{B}</math> a value of 1 and everything else a value of 0. Expressed in symbolic form:

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The '''umpire measure''' of type <math>\Upsilon : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}</math> takes a single proposition of type <math>\mathbb{B}^2 \to \mathbb{B}</math> as argument, giving the constant proposition <math>1 : \mathbb{B}^2 \to \mathbb{B}</math> a value of 1 and everything else a value of 0. Expressed in symbolic form:

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The ''umpire operator'' of type <math>\Upsilon : (\mathbb{B}^2 \to \mathbb{B})^2 \to \mathbb{B}</math> takes two propositions of type <math>\mathbb{B}^2 \to \mathbb{B}</math> as arguments, giving pairs in which the first implies the second a value of 1 and everything else a value of 0. Expressed in symbolic form:

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The '''umpire operator''' of type <math>\Upsilon : (\mathbb{B}^2 \to \mathbb{B})^2 \to \mathbb{B}</math> takes two propositions of type <math>\mathbb{B}^2 \to \mathbb{B}</math> as arguments, giving pairs in which the first implies the second a value of 1 and everything else a value of 0. Expressed in symbolic form:

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===Tables===

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~~The auxiliary notations~~:

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Define two families of measures:

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: <math>\alpha_i ~~f =~~ \~~Upsilon~~ \~~langle f_i, f~~ \~~rangle =~~ \~~Upsilon~~ \~~langle f_i~~ \~~Rightarrow f~~ \~~rangle~~</math>

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| <math>\alpha_i, \beta_i : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}, i = 1 \ldots 15,</math>

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|}

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: ~~<math>\beta_i f = \Upsilon \langle f, f_i \rangle = \Upsilon \langle f \Rightarrow f_i \rangle</math>~~

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by means of the following formulas:

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~~define two series of measures:~~

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| <math>\alpha_i f = \Upsilon \langle f_i, f \rangle = \Upsilon \langle f_i \Rightarrow f \rangle,</math>

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|-

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| <math>\beta_i f = \Upsilon \langle f, f_i \rangle = \Upsilon \langle f \Rightarrow f_i \rangle.</math>

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|}

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: <math>\alpha_i, \~~beta_i~~ : (\mathbb{B}^2 \to \mathbb{B}~~) \to \mathbb{B},~~</math>

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The values of the sixteen <math>\alpha_i\!</math> on each of the sixteen boolean functions <math>f : \mathbb{B}^2 \to \mathbb{B}</math> are shown in Table 1.

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~~incidentally providing compact names for the column headings of the next two Tables~~.

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| 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1

|}<br>

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The values of the sixteen <math>\beta_i\!</math> on each of the sixteen boolean functions <math>f : \mathbb{B}^2 \to \mathbb{B}</math> are shown in Table 2.

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Jon Awbrey

12,122

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