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==Functional Quantifiers==
==Functional Quantifiers==
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:
The '''umpire measure''' of type <math>\Upsilon : (X \to \mathbb{B}) \to \mathbb{B}</math> links the constant proposition <math>1 : X \to \mathbb{B}</math> to a value of 1 and every other proposition to a value of 0. Expressed in symbolic form:
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| <math>\Upsilon \langle u \rangle = 1_\mathbb{B} \quad \Leftrightarrow \quad u = 1_{\mathbb{B}^2 \to \mathbb{B}}.</math>
| <math>\Upsilon (u) = 1_\mathbb{B} \quad \Leftrightarrow \quad u = 1_{X \to \mathbb{B}}.</math>
|}
|}
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:
The '''umpire operator''' of type <math>\Upsilon : (X \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:
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| <math>\Upsilon \langle u, v \rangle = 1 \quad \Leftrightarrow \quad u \Rightarrow v.</math>
| <math>\Upsilon (u, v) = 1 \quad \Leftrightarrow \quad u \Rightarrow v.</math>
|}
|}
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| <math>\alpha_i f = \Upsilon \langle f_i, f \rangle = \Upsilon \langle f_i \Rightarrow f \rangle,</math>
| <math>\alpha_i f = \Upsilon (f_i, f) = \Upsilon (f_i \Rightarrow f),</math>
|-
|-
| <math>\beta_i f = \Upsilon \langle f, f_i \rangle = \Upsilon \langle f \Rightarrow f_i \rangle.</math>
| <math>\beta_i f = \Upsilon (f, f_i) = \Upsilon (f \Rightarrow f_i).</math>
|}
|}
{| 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%"
|+ '''Table 1. Qualifiers of Implication Ordering: <math>\alpha_i f = \Upsilon \langle f_i, f \rangle = \Upsilon \langle f_i \Rightarrow f \rangle</math>'''
|+ '''Table 1. Qualifiers of Implication Ordering: <math>\alpha_i f = \Upsilon (f_i, f) = \Upsilon (f_i \Rightarrow f)</math>'''
|- style="background:ghostwhite"
|- style="background:ghostwhite"
| align="right" | <math>u:</math><br><math>v:</math>
| align="right" | <math>u:</math><br><math>v:</math>
{| 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%"
|+ '''Table 2. Qualifiers of Implication Ordering: <math>\beta_i f = \Upsilon \langle f, f_i \rangle = \Upsilon \langle f \Rightarrow f_i \rangle</math>'''
|+ '''Table 2. Qualifiers of Implication Ordering: <math>\beta_i f = \Upsilon (f, f_i) = \Upsilon (f \Rightarrow f_i)</math>'''
|- style="background:ghostwhite"
|- style="background:ghostwhite"
| align="right" | <math>u:</math><br><math>v:</math>
| align="right" | <math>u:</math><br><math>v:</math>
| <math>f_{14}</math>
| <math>f_{14}</math>
| 1110
| 1110
| <math>((u)(y))\!</math>
| <math>((u)(v))\!</math>
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| <math>\text{Some}\ u\ \text{is}\ v</math>
| <math>\text{Some}\ u\ \text{is}\ v</math>
|
|
| <math>\text{Some}\ u\ \text{is}\ y</math>
| <math>\text{Some}\ u\ \text{is}\ v</math>
| <math>\ell_{11}\!</math>
| <math>\ell_{11}\!</math>
|}<br>
|}<br>
