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Revision as of 20:16, 12 December 2008
, 20:16, 12 December 2008+ § [functional quantifiers]
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==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:
{| align="center" cellpadding="8"
| <math>\Upsilon \langle u \rangle = 1_\mathbb{B} \quad \Leftrightarrow \quad u = 1_{\mathbb{B}^2 \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:
{| align="center" cellpadding="8"
| <math>\Upsilon \langle u, v \rangle = 1 \quad \Leftrightarrow \quad u \Rightarrow v.</math>
|}
===Tables===
Define two families of measures:
{| align="center" cellpadding="8"
| <math>\alpha_i, \beta_i : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}, i = 1 \ldots 15,</math>
|}
by means of the following formulas:
{| align="center" cellpadding="8"
| <math>\alpha_i f = \Upsilon \langle f_i, f \rangle = \Upsilon \langle f_i \Rightarrow f \rangle,</math>
|-
| <math>\beta_i f = \Upsilon \langle f, f_i \rangle = \Upsilon \langle f \Rightarrow f_i \rangle.</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 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%"
|+ '''Table 1. Qualifiers of Implication Ordering: <math>\alpha_i f = \Upsilon \langle f_i, f \rangle = \Upsilon \langle f_i \Rightarrow f \rangle</math>'''
|- style="background:ghostwhite"
| align="right" | <math>u:</math><br><math>v:</math>
| 1100<br>1010
| <math>f\!</math>
| <math>\alpha_0</math>
| <math>\alpha_1</math>
| <math>\alpha_2</math>
| <math>\alpha_3</math>
| <math>\alpha_4</math>
| <math>\alpha_5</math>
| <math>\alpha_6</math>
| <math>\alpha_7</math>
| <math>\alpha_8</math>
| <math>\alpha_9</math>
| <math>\alpha_{10}</math>
| <math>\alpha_{11}</math>
| <math>\alpha_{12}</math>
| <math>\alpha_{13}</math>
| <math>\alpha_{14}</math>
| <math>\alpha_{15}</math>
|-
| <math>f_0</math> || 0000 || <math>(~)</math>
| 1 || || || || || || ||
| || || || || || || ||
|-
| <math>f_1</math> || 0001 || <math>(u)(v)\!</math>
| 1 || 1 || || || || || ||
| || || || || || || ||
|-
| <math>f_2</math> || 0010 || <math>(u) v\!</math>
| 1 || || 1 || || || || ||
| || || || || || || ||
|-
| <math>f_3</math> || 0011 || <math>(u)\!</math>
| 1 || 1 || 1 || 1 || || || ||
| || || || || || || ||
|-
| <math>f_4</math> || 0100 || <math>u (v)\!</math>
| 1 || || || || 1 || || ||
| || || || || || || ||
|-
| <math>f_5</math> || 0101 || <math>(v)\!</math>
| 1 || 1 || || || 1 || 1 || ||
| || || || || || || ||
|-
| <math>f_6</math> || 0110 || <math>(u, v)\!</math>
| 1 || || 1 || || 1 || || 1 ||
| || || || || || || ||
|-
| <math>f_7</math> || 0111 || <math>(u v)\!</math>
| 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
| || || || || || || ||
|-
| <math>f_8</math> || 1000 || <math>u v\!</math>
| 1 || || || || || || ||
| 1 || || || || || || ||
|-
| <math>f_9</math> || 1001 || <math>((u, v))\!</math>
| 1 || 1 || || || || || ||
| 1 || 1 || || || || || ||
|-
| <math>f_{10}</math> || 1010 || <math>v\!</math>
| 1 || || 1 || || || || ||
| 1 || || 1 || || || || ||
|-
| <math>f_{11}</math> || 1011 || <math>(u (v))\!</math>
| 1 || 1 || 1 || 1 || || || ||
| 1 || 1 || 1 || 1 || || || ||
|-
| <math>f_{12}</math> || 1100 || <math>u\!</math>
| 1 || || || || 1 || || ||
| 1 || || || || 1 || || ||
|-
| <math>f_{13}</math> || 1101 || <math>((u) v)\!</math>
| 1 || 1 || || || 1 || 1 || ||
| 1 || 1 || || || 1 || 1 || ||
|-
| <math>f_{14}</math> || 1110 || <math>((u)(v))\!</math>
| 1 || || 1 || || 1 || || 1 ||
| 1 || || 1 || || 1 || || 1 ||
|-
| <math>f_{15}</math> || 1111 || <math>((~))</math>
| 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
| 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
|}<br>
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.
{| 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>'''
|- style="background:ghostwhite"
| align="right" | <math>u:</math><br><math>v:</math>
| 1100<br>1010
| <math>f\!</math>
| <math>\beta_0</math>
| <math>\beta_1</math>
| <math>\beta_2</math>
| <math>\beta_3</math>
| <math>\beta_4</math>
| <math>\beta_5</math>
| <math>\beta_6</math>
| <math>\beta_7</math>
| <math>\beta_8</math>
| <math>\beta_9</math>
| <math>\beta_{10}</math>
| <math>\beta_{11}</math>
| <math>\beta_{12}</math>
| <math>\beta_{13}</math>
| <math>\beta_{14}</math>
| <math>\beta_{15}</math>
|-
| <math>f_0</math> || 0000 || <math>(~)</math>
| 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
| 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
|-
| <math>f_1</math> || 0001 || <math>(u)(v)\!</math>
| || 1 || || 1 || || 1 || || 1
| || 1 || || 1 || || 1 || || 1
|-
| <math>f_2</math> || 0010 || <math>(u) v\!</math>
| || || 1 || 1 || || || 1 || 1
| || || 1 || 1 || || || 1 || 1
|-
| <math>f_3</math> || 0011 || <math>(u)\!</math>
| || || || 1 || || || || 1
| || || || 1 || || || || 1
|-
| <math>f_4</math> || 0100 || <math>u (v)\!</math>
| || || || || 1 || 1 || 1 || 1
| || || || || 1 || 1 || 1 || 1
|-
| <math>f_5</math> || 0101 || <math>(v)\!</math>
| || || || || || 1 || || 1
| || || || || || 1 || || 1
|-
| <math>f_6</math> || 0110 || <math>(u, v)\!</math>
| || || || || || || 1 || 1
| || || || || || || 1 || 1
|-
| <math>f_7</math> || 0111 || <math>(u v)\!</math>
| || || || || || || || 1
| || || || || || || || 1
|-
| <math>f_8</math> || 1000 || <math>u v\!</math>
| || || || || || || ||
| 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
|-
| <math>f_9</math> || 1001 || <math>((u, v))\!</math>
| || || || || || || ||
| || 1 || || 1 || || 1 || || 1
|-
| <math>f_{10}</math> || 1010 || <math>v\!</math>
| || || || || || || ||
| || || 1 || 1 || || || 1 || 1
|-
| <math>f_{11}</math> || 1011 || <math>(u (v))\!</math>
| || || || || || || ||
| || || || 1 || || || || 1
|-
| <math>f_{12}</math> || 1100 || <math>u\!</math>
| || || || || || || ||
| || || || || 1 || 1 || 1 || 1
|-
| <math>f_{13}</math> || 1101 || <math>((u) v)\!</math>
| || || || || || || ||
| || || || || || 1 || || 1
|-
| <math>f_{14}</math> || 1110 || <math>((u)(y))\!</math>
| || || || || || || ||
| || || || || || || 1 || 1
|-
| <math>f_{15}</math> || 1111 || <math>((~))\!</math>
| || || || || || || ||
| || || || || || || || 1
|}<br>
{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
|+ '''Table 3. Simple Qualifiers of Propositions (''n'' = 2)'''
|- style="background:ghostwhite"
| align="right" | <math>u:</math><br><math>v:</math>
| 1100<br>1010
| <math>f\!</math>
| <math>(\ell_{11})</math><br><math>\text{No } u </math><br><math>\text{is } v </math>
| <math>(\ell_{10})</math><br><math>\text{No } u </math><br><math>\text{is }(v)</math>
| <math>(\ell_{01})</math><br><math>\text{No }(u)</math><br><math>\text{is } v </math>
| <math>(\ell_{00})</math><br><math>\text{No }(u)</math><br><math>\text{is }(v)</math>
| <math> \ell_{00} </math><br><math>\text{Some }(u)</math><br><math>\text{is }(v)</math>
| <math> \ell_{01} </math><br><math>\text{Some }(u)</math><br><math>\text{is } v </math>
| <math> \ell_{10} </math><br><math>\text{Some } u </math><br><math>\text{is }(v)</math>
| <math> \ell_{11} </math><br><math>\text{Some } u </math><br><math>\text{is } v </math>
|-
| <math>f_0</math>
| 0000
| <math>(~)</math>
| 1
| 1
| 1
| 1
|
|
|
|
|-
| <math>f_1</math>
| 0001
| <math>(u)(v)\!</math>
| 1
| 1
| 1
|
| 1
|
|
|
|-
| <math>f_2</math>
| 0010
| <math>(u) v\!</math>
| 1
| 1
|
| 1
|
| 1
|
|
|-
| <math>f_3</math>
| 0011
| <math>(u)\!</math>
| 1
| 1
|
|
| 1
| 1
|
|
|-
| <math>f_4</math>
| 0100
| <math>u (v)\!</math>
| 1
|
| 1
| 1
|
|
| 1
|
|-
| <math>f_5</math>
| 0101
| <math>(v)\!</math>
| 1
|
| 1
|
| 1
|
| 1
|
|-
| <math>f_6</math>
| 0110
| <math>(u, v)\!</math>
| 1
|
|
| 1
|
| 1
| 1
|
|-
| <math>f_7</math>
| 0111
| <math>(u v)\!</math>
| 1
|
|
|
| 1
| 1
| 1
|
|-
| <math>f_8</math>
| 1000
| <math>u v\!</math>
|
| 1
| 1
| 1
|
|
|
| 1
|-
| <math>f_9</math>
| 1001
| <math>((u, v))\!</math>
|
| 1
| 1
|
| 1
|
|
| 1
|-
| <math>f_{10}</math>
| 1010
| <math>v\!</math>
|
| 1
|
| 1
|
| 1
|
| 1
|-
| <math>f_{11}</math>
| 1011
| <math>(u (v))\!</math>
|
| 1
|
|
| 1
| 1
|
| 1
|-
| <math>f_{12}</math>
| 1100
| <math>u\!</math>
|
|
| 1
| 1
|
|
| 1
| 1
|-
| <math>f_{13}</math>
| 1101
| <math>((u) v)\!</math>
|
|
| 1
|
| 1
|
| 1
| 1
|-
| <math>f_{14}</math>
| 1110
| <math>((u)(v))\!</math>
|
|
|
| 1
|
| 1
| 1
| 1
|-
| <math>f_{15}</math>
| 1111
| <math>((~))</math>
|
|
|
|
| 1
| 1
| 1
| 1
|}<br>
{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
|+ '''Table 4. Simple Qualifiers of Propositions (''n'' = 2)'''
|- style="background:ghostwhite"
| align="right" | <math>u:</math><br><math>v:</math>
| 1100<br>1010
| <math>f\!</math>
| <math>(\ell_{11})</math><br><math>\text{No } u </math><br><math>\text{is } v </math>
| <math>(\ell_{10})</math><br><math>\text{No } u </math><br><math>\text{is }(v)</math>
| <math>(\ell_{01})</math><br><math>\text{No }(u)</math><br><math>\text{is } v </math>
| <math>(\ell_{00})</math><br><math>\text{No }(u)</math><br><math>\text{is }(v)</math>
| <math> \ell_{00} </math><br><math>\text{Some }(u)</math><br><math>\text{is }(v)</math>
| <math> \ell_{01} </math><br><math>\text{Some }(u)</math><br><math>\text{is } v </math>
| <math> \ell_{10} </math><br><math>\text{Some } u </math><br><math>\text{is }(v)</math>
| <math> \ell_{11} </math><br><math>\text{Some } u </math><br><math>\text{is } v </math>
|-
| <math>f_0</math>
| 0000
| <math>(~)</math>
| 1
| 1
| 1
| 1
|
|
|
|
|-
| <math>f_1</math>
| 0001
| <math>(u)(v)\!</math>
| 1
| 1
| 1
|
| 1
|
|
|
|-
| <math>f_2</math>
| 0010
| <math>(u) v\!</math>
| 1
| 1
|
| 1
|
| 1
|
|
|-
| <math>f_4</math>
| 0100
| <math>u (v)\!</math>
| 1
|
| 1
| 1
|
|
| 1
|
|-
| <math>f_8</math>
| 1000
| <math>u v\!</math>
|
| 1
| 1
| 1
|
|
|
| 1
|-
| <math>f_3</math>
| 0011
| <math>(u)\!</math>
| 1
| 1
|
|
| 1
| 1
|
|
|-
| <math>f_{12}</math>
| 1100
| <math>u\!</math>
|
|
| 1
| 1
|
|
| 1
| 1
|-
| <math>f_6</math>
| 0110
| <math>(u, v)\!</math>
| 1
|
|
| 1
|
| 1
| 1
|
|-
| <math>f_9</math>
| 1001
| <math>((u, v))\!</math>
|
| 1
| 1
|
| 1
|
|
| 1
|-
| <math>f_5</math>
| 0101
| <math>(v)\!</math>
| 1
|
| 1
|
| 1
|
| 1
|
|-
| <math>f_{10}</math>
| 1010
| <math>v\!</math>
|
| 1
|
| 1
|
| 1
|
| 1
|-
| <math>f_7</math>
| 0111
| <math>(u v)\!</math>
| 1
|
|
|
| 1
| 1
| 1
|
|-
| <math>f_{11}</math>
| 1011
| <math>(u (v))\!</math>
|
| 1
|
|
| 1
| 1
|
| 1
|-
| <math>f_{13}</math>
| 1101
| <math>((u) v)\!</math>
|
|
| 1
|
| 1
|
| 1
| 1
|-
| <math>f_{14}</math>
| 1110
| <math>((u)(v))\!</math>
|
|
|
| 1
|
| 1
| 1
| 1
|-
| <math>f_{15}</math>
| 1111
| <math>((~))</math>
|
|
|
|
| 1
| 1
| 1
| 1
|}<br>
{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
|+ '''Table 5. Relation of Quantifiers to Higher Order Propositions'''
|- style="background:ghostwhite"
| <math>\text{Mnemonic}</math>
| <math>\text{Category}</math>
| <math>\text{Classical Form}</math>
| <math>\text{Alternate Form}</math>
| <math>\text{Symmetric Form}</math>
| <math>\text{Operator}</math>
|-
| <math>\text{E}\!</math><br><math>\text{Exclusive}</math>
| <math>\text{Universal}</math><br><math>\text{Negative}</math>
| <math>\text{All}\ u\ \text{is}\ (v)</math>
|
| <math>\text{No}\ u\ \text{is}\ v </math>
| <math>(\ell_{11})</math>
|-
| <math>\text{A}\!</math><br><math>\text{Absolute}</math>
| <math>\text{Universal}</math><br><math>\text{Affirmative}</math>
| <math>\text{All}\ u\ \text{is}\ v </math>
|
| <math>\text{No}\ u\ \text{is}\ (v)</math>
| <math>(\ell_{10})</math>
|-
|
|
| <math>\text{All}\ v\ \text{is}\ u </math>
| <math>\text{No}\ v\ \text{is}\ (u)</math>
| <math>\text{No}\ (u)\ \text{is}\ v </math>
| <math>(\ell_{01})</math>
|-
|
|
| <math>\text{All}\ (v)\ \text{is}\ u </math>
| <math>\text{No}\ (v)\ \text{is}\ (u)</math>
| <math>\text{No}\ (u)\ \text{is}\ (v)</math>
| <math>(\ell_{00})</math>
|-
|
|
| <math>\text{Some}\ (u)\ \text{is}\ (v)</math>
|
| <math>\text{Some}\ (u)\ \text{is}\ (v)</math>
| <math>\ell_{00}\!</math>
|-
|
|
| <math>\text{Some}\ (u)\ \text{is}\ v</math>
|
| <math>\text{Some}\ (u)\ \text{is}\ v</math>
| <math>\ell_{01}\!</math>
|-
| <math>\text{O}\!</math><br><math>\text{Obtrusive}</math>
| <math>\text{Particular}</math><br><math>\text{Negative}</math>
| <math>\text{Some}\ u\ \text{is}\ (v)</math>
|
| <math>\text{Some}\ u\ \text{is}\ (v)</math>
| <math>\ell_{10}\!</math>
|-
| <math>\text{I}\!</math><br><math>\text{Indefinite}</math>
| <math>\text{Particular}</math><br><math>\text{Affirmative}</math>
| <math>\text{Some}\ u\ \text{is}\ v</math>
|
| <math>\text{Some}\ u\ \text{is}\ y</math>
| <math>\ell_{11}\!</math>
|}<br>
