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{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| <math>\Upsilon \langle p \rangle = 1 \quad \Leftrightarrow \quad p = 1.</math>
| <math>\Upsilon \langle u \rangle = 1 \quad \Leftrightarrow \quad u = 1.</math>
|}
|}
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| <math>\Upsilon \langle p, q \rangle = 1 \quad \Leftrightarrow \quad p \Rightarrow q.</math>
| <math>\Upsilon \langle u, v \rangle = 1 \quad \Leftrightarrow \quad u \Rightarrow v.</math>
|}
|}
|+ '''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 \langle f_i, f \rangle = \Upsilon \langle f_i \Rightarrow f \rangle</math>'''
|- style="background:ghostwhite"
|- style="background:ghostwhite"
| align="right" | <math>p:</math><br><math>q:</math>
| align="right" | <math>u:</math><br><math>v:</math>
| 1100<br>1010
| 1100<br>1010
| <math>f\!</math>
| <math>f\!</math>
| || || || || || || ||
| || || || || || || ||
|-
|-
| <math>f_1</math> || 0001 || <math>(p)(q)\!</math>
| <math>f_1</math> || 0001 || <math>(u)(v)\!</math>
| 1 || 1 || || || || || ||
| 1 || 1 || || || || || ||
| || || || || || || ||
| || || || || || || ||
|-
|-
| <math>f_2</math> || 0010 || <math>(p) q\!</math>
| <math>f_2</math> || 0010 || <math>(u) v\!</math>
| 1 || || 1 || || || || ||
| 1 || || 1 || || || || ||
| || || || || || || ||
| || || || || || || ||
|-
|-
| <math>f_3</math> || 0011 || <math>(p)\!</math>
| <math>f_3</math> || 0011 || <math>(u)\!</math>
| 1 || 1 || 1 || 1 || || || ||
| 1 || 1 || 1 || 1 || || || ||
| || || || || || || ||
| || || || || || || ||
|-
|-
| <math>f_4</math> || 0100 || <math>p (q)\!</math>
| <math>f_4</math> || 0100 || <math>u (v)\!</math>
| 1 || || || || 1 || || ||
| 1 || || || || 1 || || ||
| || || || || || || ||
| || || || || || || ||
|-
|-
| <math>f_5</math> || 0101 || <math>(q)\!</math>
| <math>f_5</math> || 0101 || <math>(v)\!</math>
| 1 || 1 || || || 1 || 1 || ||
| 1 || 1 || || || 1 || 1 || ||
| || || || || || || ||
| || || || || || || ||
|-
|-
| <math>f_6</math> || 0110 || <math>(p, q)\!</math>
| <math>f_6</math> || 0110 || <math>(u, v)\!</math>
| 1 || || 1 || || 1 || || 1 ||
| 1 || || 1 || || 1 || || 1 ||
| || || || || || || ||
| || || || || || || ||
|-
|-
| <math>f_7</math> || 0111 || <math>(p q)\!</math>
| <math>f_7</math> || 0111 || <math>(u v)\!</math>
| 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
| 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
| || || || || || || ||
| || || || || || || ||
|-
|-
| <math>f_8</math> || 1000 || <math>p q\!</math>
| <math>f_8</math> || 1000 || <math>u v\!</math>
| 1 || || || || || || ||
| 1 || || || || || || ||
| 1 || || || || || || ||
| 1 || || || || || || ||
|-
|-
| <math>f_9</math> || 1001 || <math>((p, q))\!</math>
| <math>f_9</math> || 1001 || <math>((u, v))\!</math>
| 1 || 1 || || || || || ||
| 1 || 1 || || || || || ||
| 1 || 1 || || || || || ||
| 1 || 1 || || || || || ||
|-
|-
| <math>f_{10}</math> || 1010 || <math>q\!</math>
| <math>f_{10}</math> || 1010 || <math>v\!</math>
| 1 || || 1 || || || || ||
| 1 || || 1 || || || || ||
| 1 || || 1 || || || || ||
| 1 || || 1 || || || || ||
|-
|-
| <math>f_{11}</math> || 1011 || <math>(p (q))\!</math>
| <math>f_{11}</math> || 1011 || <math>(u (v))\!</math>
| 1 || 1 || 1 || 1 || || || ||
| 1 || 1 || 1 || 1 || || || ||
| 1 || 1 || 1 || 1 || || || ||
| 1 || 1 || 1 || 1 || || || ||
|-
|-
| <math>f_{12}</math> || 1100 || <math>p\!</math>
| <math>f_{12}</math> || 1100 || <math>u\!</math>
| 1 || || || || 1 || || ||
| 1 || || || || 1 || || ||
| 1 || || || || 1 || || ||
| 1 || || || || 1 || || ||
|-
|-
| <math>f_{13}</math> || 1101 || <math>((p) q)\!</math>
| <math>f_{13}</math> || 1101 || <math>((u) v)\!</math>
| 1 || 1 || || || 1 || 1 || ||
| 1 || 1 || || || 1 || 1 || ||
| 1 || 1 || || || 1 || 1 || ||
| 1 || 1 || || || 1 || 1 || ||
|-
|-
| <math>f_{14}</math> || 1110 || <math>((p)(q))\!</math>
| <math>f_{14}</math> || 1110 || <math>((u)(v))\!</math>
| 1 || || 1 || || 1 || || 1 ||
| 1 || || 1 || || 1 || || 1 ||
| 1 || || 1 || || 1 || || 1 ||
| 1 || || 1 || || 1 || || 1 ||
|+ '''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 \langle f, f_i \rangle = \Upsilon \langle f \Rightarrow f_i \rangle</math>'''
|- style="background:ghostwhite"
|- style="background:ghostwhite"
| align="right" | <math>p:</math><br><math>q:</math>
| align="right" | <math>u:</math><br><math>v:</math>
| 1100<br>1010
| 1100<br>1010
| <math>f\!</math>
| <math>f\!</math>
| 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
| 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
|-
|-
| <math>f_1</math> || 0001 || <math>(p)(q)\!</math>
| <math>f_1</math> || 0001 || <math>(u)(v)\!</math>
| || 1 || || 1 || || 1 || || 1
| || 1 || || 1 || || 1 || || 1
| || 1 || || 1 || || 1 || || 1
| || 1 || || 1 || || 1 || || 1
|-
|-
| <math>f_2</math> || 0010 || <math>(p) q\!</math>
| <math>f_2</math> || 0010 || <math>(u) v\!</math>
| || || 1 || 1 || || || 1 || 1
| || || 1 || 1 || || || 1 || 1
| || || 1 || 1 || || || 1 || 1
| || || 1 || 1 || || || 1 || 1
|-
|-
| <math>f_3</math> || 0011 || <math>(p)\!</math>
| <math>f_3</math> || 0011 || <math>(u)\!</math>
| || || || 1 || || || || 1
| || || || 1 || || || || 1
| || || || 1 || || || || 1
| || || || 1 || || || || 1
|-
|-
| <math>f_4</math> || 0100 || <math>p (q)\!</math>
| <math>f_4</math> || 0100 || <math>u (v)\!</math>
| || || || || 1 || 1 || 1 || 1
| || || || || 1 || 1 || 1 || 1
| || || || || 1 || 1 || 1 || 1
| || || || || 1 || 1 || 1 || 1
|-
|-
| <math>f_5</math> || 0101 || <math>(q)\!</math>
| <math>f_5</math> || 0101 || <math>(v)\!</math>
| || || || || || 1 || || 1
| || || || || || 1 || || 1
| || || || || || 1 || || 1
| || || || || || 1 || || 1
|-
|-
| <math>f_6</math> || 0110 || <math>(p, q)\!</math>
| <math>f_6</math> || 0110 || <math>(u, v)\!</math>
| || || || || || || 1 || 1
| || || || || || || 1 || 1
| || || || || || || 1 || 1
| || || || || || || 1 || 1
|-
|-
| <math>f_7</math> || 0111 || <math>(p q)\!</math>
| <math>f_7</math> || 0111 || <math>(u v)\!</math>
| || || || || || || || 1
| || || || || || || || 1
| || || || || || || || 1
| || || || || || || || 1
|-
|-
| <math>f_8</math> || 1000 || <math>p q\!</math>
| <math>f_8</math> || 1000 || <math>u v\!</math>
| || || || || || || ||
| || || || || || || ||
| 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
| 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
|-
|-
| <math>f_9</math> || 1001 || <math>((p, q))\!</math>
| <math>f_9</math> || 1001 || <math>((u, v))\!</math>
| || || || || || || ||
| || || || || || || ||
| || 1 || || 1 || || 1 || || 1
| || 1 || || 1 || || 1 || || 1
|-
|-
| <math>f_{10}</math> || 1010 || <math>q\!</math>
| <math>f_{10}</math> || 1010 || <math>v\!</math>
| || || || || || || ||
| || || || || || || ||
| || || 1 || 1 || || || 1 || 1
| || || 1 || 1 || || || 1 || 1
|-
|-
| <math>f_{11}</math> || 1011 || <math>(p (q))\!</math>
| <math>f_{11}</math> || 1011 || <math>(u (v))\!</math>
| || || || || || || ||
| || || || || || || ||
| || || || 1 || || || || 1
| || || || 1 || || || || 1
|-
|-
| <math>f_{12}</math> || 1100 || <math>p\!</math>
| <math>f_{12}</math> || 1100 || <math>u\!</math>
| || || || || || || ||
| || || || || || || ||
| || || || || 1 || 1 || 1 || 1
| || || || || 1 || 1 || 1 || 1
|-
|-
| <math>f_{13}</math> || 1101 || <math>((p) q)\!</math>
| <math>f_{13}</math> || 1101 || <math>((u) v)\!</math>
| || || || || || || ||
| || || || || || || ||
| || || || || || 1 || || 1
| || || || || || 1 || || 1
|-
|-
| <math>f_{14}</math> || 1110 || <math>((p)(y))\!</math>
| <math>f_{14}</math> || 1110 || <math>((u)(y))\!</math>
| || || || || || || ||
| || || || || || || ||
| || || || || || || 1 || 1
| || || || || || || 1 || 1
|+ '''Table 3. Simple Qualifiers of Propositions (''n'' = 2)'''
|+ '''Table 3. Simple Qualifiers of Propositions (''n'' = 2)'''
|- style="background:ghostwhite"
|- style="background:ghostwhite"
| align="right" | <math>p:</math><br><math>q:</math>
| align="right" | <math>u:</math><br><math>v:</math>
| 1100<br>1010
| 1100<br>1010
| <math>f\!</math>
| <math>f\!</math>
| <math>(\ell_{11})</math><br><math>\text{No } p </math><br><math>\text{is } q </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 } p </math><br><math>\text{is }(q)</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 }(p)</math><br><math>\text{is } q </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 }(p)</math><br><math>\text{is }(q)</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 }(p)</math><br><math>\text{is }(q)</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 }(p)</math><br><math>\text{is } q </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 } p </math><br><math>\text{is }(q)</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 } p </math><br><math>\text{is } q </math>
| <math> \ell_{11} </math><br><math>\text{Some } u </math><br><math>\text{is } v </math>
|-
|-
| <math>f_0</math> || 0000 || <math>(~)</math>
| <math>f_0</math> || 0000 || <math>(~)</math>
| 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0
| 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0
|-
|-
| <math>f_1</math> || 0001 || <math>(p)(q)\!</math>
| <math>f_1</math> || 0001 || <math>(u)(v)\!</math>
| 1 || 1 || 1 || 0 || 1 || 0 || 0 || 0
| 1 || 1 || 1 || 0 || 1 || 0 || 0 || 0
|-
|-
| <math>f_2</math> || 0010 || <math>(p) q\!</math>
| <math>f_2</math> || 0010 || <math>(u) v\!</math>
| 1 || 1 || 0 || 1 || 0 || 1 || 0 || 0
| 1 || 1 || 0 || 1 || 0 || 1 || 0 || 0
|-
|-
| <math>f_3</math> || 0011 || <math>(p)\!</math>
| <math>f_3</math> || 0011 || <math>(u)\!</math>
| 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0
| 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0
|-
|-
| <math>f_4</math> || 0100 || <math>p (q)\!</math>
| <math>f_4</math> || 0100 || <math>u (v)\!</math>
| 1 || 0 || 1 || 1 || 0 || 0 || 1 || 0
| 1 || 0 || 1 || 1 || 0 || 0 || 1 || 0
|-
|-
| <math>f_5</math> || 0101 || <math>(q)\!</math>
| <math>f_5</math> || 0101 || <math>(v)\!</math>
| 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0
| 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0
|-
|-
| <math>f_6</math> || 0110 || <math>(p, q)\!</math>
| <math>f_6</math> || 0110 || <math>(u, v)\!</math>
| 1 || 0 || 0 || 1 || 0 || 1 || 1 || 0
| 1 || 0 || 0 || 1 || 0 || 1 || 1 || 0
|-
|-
| <math>f_7</math> || 0111 || <math>(p q)\!</math>
| <math>f_7</math> || 0111 || <math>(u v)\!</math>
| 1 || 0 || 0 || 0 || 1 || 1 || 1 || 0
| 1 || 0 || 0 || 0 || 1 || 1 || 1 || 0
|-
|-
| <math>f_8</math> || 1000 || <math>p q\!</math>
| <math>f_8</math> || 1000 || <math>u v\!</math>
| 0 || 1 || 1 || 1 || 0 || 0 || 0 || 1
| 0 || 1 || 1 || 1 || 0 || 0 || 0 || 1
|-
|-
| <math>f_9</math> || 1001 || <math>((p, q))\!</math>
| <math>f_9</math> || 1001 || <math>((u, v))\!</math>
| 0 || 1 || 1 || 0 || 1 || 0 || 0 || 1
| 0 || 1 || 1 || 0 || 1 || 0 || 0 || 1
|-
|-
| <math>f_{10}</math> || 1010 || <math>q\!</math>
| <math>f_{10}</math> || 1010 || <math>v\!</math>
| 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1
| 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1
|-
|-
| <math>f_{11}</math> || 1011 || <math>(p (q))\!</math>
| <math>f_{11}</math> || 1011 || <math>(u (v))\!</math>
| 0 || 1 || 0 || 0 || 1 || 1 || 0 || 1
| 0 || 1 || 0 || 0 || 1 || 1 || 0 || 1
|-
|-
| <math>f_{12}</math> || 1100 || <math>p\!</math>
| <math>f_{12}</math> || 1100 || <math>u\!</math>
| 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1
| 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1
|-
|-
| <math>f_{13}</math> || 1101 || <math>((p) q)\!</math>
| <math>f_{13}</math> || 1101 || <math>((u) v)\!</math>
| 0 || 0 || 1 || 0 || 1 || 0 || 1 || 1
| 0 || 0 || 1 || 0 || 1 || 0 || 1 || 1
|-
|-
| <math>f_{14}</math> || 1110 || <math>((p)(q))\!</math>
| <math>f_{14}</math> || 1110 || <math>((u)(v))\!</math>
| 0 || 0 || 0 || 1 || 0 || 1 || 1 || 1
| 0 || 0 || 0 || 1 || 0 || 1 || 1 || 1
|-
|-
| <math>\text{E}\!</math><br><math>\text{Exclusive}</math>
| <math>\text{E}\!</math><br><math>\text{Exclusive}</math>
| <math>\text{Universal}</math><br><math>\text{Negative}</math>
| <math>\text{Universal}</math><br><math>\text{Negative}</math>
| <math>\text{All}\ p\ \text{is}\ (q)</math>
| <math>\text{All}\ u\ \text{is}\ (v)</math>
|
|
| <math>\text{No}\ p\ \text{is}\ q </math>
| <math>\text{No}\ u\ \text{is}\ v </math>
| <math>(\ell_{11})</math>
| <math>(\ell_{11})</math>
|-
|-
| <math>\text{A}\!</math><br><math>\text{Absolute}</math>
| <math>\text{A}\!</math><br><math>\text{Absolute}</math>
| <math>\text{Universal}</math><br><math>\text{Affirmative}</math>
| <math>\text{Universal}</math><br><math>\text{Affirmative}</math>
| <math>\text{All}\ p\ \text{is}\ q </math>
| <math>\text{All}\ u\ \text{is}\ v </math>
|
|
| <math>\text{No}\ p\ \text{is}\ (q)</math>
| <math>\text{No}\ u\ \text{is}\ (v)</math>
| <math>(\ell_{10})</math>
| <math>(\ell_{10})</math>
|-
|-
|
|
|
|
| <math>\text{All}\ q\ \text{is}\ p </math>
| <math>\text{All}\ v\ \text{is}\ u </math>
| <math>\text{No}\ q\ \text{is}\ (p)</math>
| <math>\text{No}\ v\ \text{is}\ (u)</math>
| <math>\text{No}\ (p)\ \text{is}\ q </math>
| <math>\text{No}\ (u)\ \text{is}\ v </math>
| <math>(\ell_{01})</math>
| <math>(\ell_{01})</math>
|-
|-
|
|
|
|
| <math>\text{All}\ (q)\ \text{is}\ p </math>
| <math>\text{All}\ (v)\ \text{is}\ u </math>
| <math>\text{No}\ (q)\ \text{is}\ (p)</math>
| <math>\text{No}\ (v)\ \text{is}\ (u)</math>
| <math>\text{No}\ (p)\ \text{is}\ (q)</math>
| <math>\text{No}\ (u)\ \text{is}\ (v)</math>
| <math>(\ell_{00})</math>
| <math>(\ell_{00})</math>
|-
|-
|
|
|
|
| <math>\text{Some}\ (p)\ \text{is}\ (q)</math>
| <math>\text{Some}\ (u)\ \text{is}\ (v)</math>
|
|
| <math>\text{Some}\ (p)\ \text{is}\ (q)</math>
| <math>\text{Some}\ (u)\ \text{is}\ (v)</math>
| <math>\ell_{00}\!</math>
| <math>\ell_{00}\!</math>
|-
|-
|
|
|
|
| <math>\text{Some}\ (p)\ \text{is}\ q</math>
| <math>\text{Some}\ (u)\ \text{is}\ v</math>
|
|
| <math>\text{Some}\ (p)\ \text{is}\ q</math>
| <math>\text{Some}\ (u)\ \text{is}\ v</math>
| <math>\ell_{01}\!</math>
| <math>\ell_{01}\!</math>
|-
|-
| <math>\text{O}\!</math><br><math>\text{Obtrusive}</math>
| <math>\text{O}\!</math><br><math>\text{Obtrusive}</math>
| <math>\text{Particular}</math><br><math>\text{Negative}</math>
| <math>\text{Particular}</math><br><math>\text{Negative}</math>
| <math>\text{Some}\ p\ \text{is}\ (q)</math>
| <math>\text{Some}\ u\ \text{is}\ (v)</math>
|
|
| <math>\text{Some}\ p\ \text{is}\ (q)</math>
| <math>\text{Some}\ u\ \text{is}\ (v)</math>
| <math>\ell_{10}\!</math>
| <math>\ell_{10}\!</math>
|-
|-
| <math>\text{I}\!</math><br><math>\text{Indefinite}</math>
| <math>\text{I}\!</math><br><math>\text{Indefinite}</math>
| <math>\text{Particular}</math><br><math>\text{Affirmative}</math>
| <math>\text{Particular}</math><br><math>\text{Affirmative}</math>
| <math>\text{Some}\ p\ \text{is}\ q</math>
| <math>\text{Some}\ u\ \text{is}\ v</math>
|
|
| <math>\text{Some}\ p\ \text{is}\ y</math>
| <math>\text{Some}\ u\ \text{is}\ y</math>
| <math>\ell_{11}\!</math>
| <math>\ell_{11}\!</math>
|}<br>
|}<br>
<blockquote>
<blockquote>
<math>(\forall x \in X)(p(x) \Rightarrow q(x))</math>
<math>(\forall x \in X)(u(x) \Rightarrow v(x))</math>
</blockquote>
</blockquote>
<blockquote>
<blockquote>
<math>\prod_{x \in X} (p_x (q_x)) = 1</math>
<math>\prod_{x \in X} (u_x (v_x)) = 1</math>
</blockquote>
</blockquote>
This is just the form <math>\operatorname{All}\ p\ \operatorname{are}\ q,</math> already covered here:
This is just the form <math>\operatorname{All}\ u\ \operatorname{are}\ v,</math> already covered here:
: [[Directory:Jon_Awbrey/Papers/Functional_Logic_:_Quantification_Theory#Application_of_Higher_Order_Propositions_to_Quantification_Theory|Application of Higher Order Propositions to Quantification Theory]]
: [[Directory:Jon_Awbrey/Papers/Functional_Logic_:_Quantification_Theory#Application_of_Higher_Order_Propositions_to_Quantification_Theory|Application of Higher Order Propositions to Quantification Theory]]
Need to think a little more about the proposition <math>p \Rightarrow q</math> as a boolean function of type <math>\mathbb{B}^2 \to \mathbb{B}</math> and the corresponding higher order proposition of type <math>(\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}.</math>
Need to think a little more about the proposition <math>u \Rightarrow v</math> as a boolean function of type <math>\mathbb{B}^2 \to \mathbb{B}</math> and the corresponding higher order proposition of type <math>(\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}.</math>
====Exercise 2====
====Exercise 2====
