Changes

==Functional Quantifiers==

==Functional Quantifiers==

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:

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:

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===

===Tables===

The auxiliary notations:

Define two families of measures:

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

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

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

by means of the following formulas:

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

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&nbsp;1.

 

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

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

|}<br>

|}<br>

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