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==Functional Quantifiers==
 
==Functional Quantifiers==
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The '''umpire measure''' of type <math>\Upsilon : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}</math> links the constant proposition <math>1 : \mathbb{B}^2 \to \mathbb{B}</math> to a value of 1 and every other proposition to a value of 0.  Expressed in symbolic form:
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===Table 1===
 
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{| align="center" cellpadding="8"
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| <math>\Upsilon \langle u \rangle = 1_\mathbb{B} \quad \Leftrightarrow \quad u = 1_{\mathbb{B}^2 \to \mathbb{B}}.</math>
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|}
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The '''umpire operator''' of type <math>\Upsilon : (\mathbb{B}^2 \to \mathbb{B})^2 \to \mathbb{B}</math> links pairs of propositions in which the first implies the second to a value of 1 and every other pair to a value of 0.  Expressed in symbolic form:
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{| align="center" cellpadding="8"
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| <math>\Upsilon \langle u, v \rangle = 1 \quad \Leftrightarrow \quad u \Rightarrow v.</math>
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|}
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===Tables===
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Define two families of measures:
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{| align="center" cellpadding="8"
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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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by means of the following formulas:
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{| align="center" cellpadding="8"
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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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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.  Expressed in terms of the implication ordering on the sixteen functions, <math>\alpha_i f = 1\!</math> says that <math>f\!</math> is ''above or identical to'' <math>f_i\!</math> in the implication lattice, that is, <math>\ge f_i\!</math> in the implication ordering.
      
{| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
 
{| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
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|}<br>
 
|}<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&nbsp;2.  Expressed in terms of the implication ordering on the sixteen functions, <math>\beta_i f = 1\!</math> says that <math>f\!</math> is ''below or identical to'' <math>f_i\!</math> in the implication lattice, that is, <math>\le f_i\!</math> in the implication ordering.
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===Table 2===
    
{| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
 
{| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
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|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
 
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
 
|}<br>
 
|}<br>
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===Table 3===
    
{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
 
{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
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| 1
 
| 1
 
|}<br>
 
|}<br>
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===Table 4===
    
{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
 
{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
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