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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:
| + | ===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 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.
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| {| 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 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.
| + | ===Table 2=== |
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| {| 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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| | || || || || || || || 1 | | | || || || || || || || 1 |
| |}<br> | | |}<br> |
| + | |
| + | ===Table 3=== |
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| {| 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> |
| + | |
| + | ===Table 4=== |
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| {| 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%" |