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| <blockquote> | | <blockquote> |
− | <math>(\forall x \in X)(Px \Rightarrow Qx)</math> | + | <math>(\forall x \in X)(p(x) \Rightarrow q(x))</math> |
| </blockquote> | | </blockquote> |
| | | |
− | This is just the form <math>\operatorname{All}\ P\ \operatorname{are}\ Q,</math> already covered here: | + | <blockquote> |
| + | <math>\prod_{x \in X} (p_x (q_x)) = 1</math> |
| + | </blockquote> |
| + | |
| + | This is just the form <math>\operatorname{All}\ p\ \operatorname{are}\ q,</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>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> |
| + | |
| + | {| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%" |
| + | |+ '''Table 1. Simple Qualifiers of Propositions (''n'' = 2)''' |
| + | |- style="background:ghostwhite" |
| + | | align="right" | <math>p:</math><br><math>q:</math> |
| + | | 1100<br>1010 |
| + | | <math>f\!</math> |
| + | | <math>(\ell_{11})</math><br><math>\text{No } p </math><br><math>\text{is } q </math> |
| + | | <math>(\ell_{10})</math><br><math>\text{No } p </math><br><math>\text{is }(q)</math> |
| + | | <math>(\ell_{01})</math><br><math>\text{No }(p)</math><br><math>\text{is } q </math> |
| + | | <math>(\ell_{00})</math><br><math>\text{No }(p)</math><br><math>\text{is }(q)</math> |
| + | | <math> \ell_{00} </math><br><math>\text{Some }(p)</math><br><math>\text{is }(q)</math> |
| + | | <math> \ell_{01} </math><br><math>\text{Some }(p)</math><br><math>\text{is } q </math> |
| + | | <math> \ell_{10} </math><br><math>\text{Some } p </math><br><math>\text{is }(q)</math> |
| + | | <math> \ell_{11} </math><br><math>\text{Some } p </math><br><math>\text{is } q </math> |
| + | |- |
| + | | <math>f_0</math> || 0000 || <math>(~)</math> |
| + | | 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0 |
| + | |- |
| + | | <math>f_1</math> || 0001 || <math>(p)(q)\!</math> |
| + | | 1 || 1 || 1 || 0 || 1 || 0 || 0 || 0 |
| + | |- |
| + | | <math>f_2</math> || 0010 || <math>(p) q\!</math> |
| + | | 1 || 1 || 0 || 1 || 0 || 1 || 0 || 0 |
| + | |- |
| + | | <math>f_3</math> || 0011 || <math>(p)\!</math> |
| + | | 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0 |
| + | |- |
| + | | <math>f_4</math> || 0100 || <math>p (q)\!</math> |
| + | | 1 || 0 || 1 || 1 || 0 || 0 || 1 || 0 |
| + | |- |
| + | | <math>f_5</math> || 0101 || <math>(q)\!</math> |
| + | | 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 |
| + | |- |
| + | | <math>f_6</math> || 0110 || <math>(p, q)\!</math> |
| + | | 1 || 0 || 0 || 1 || 0 || 1 || 1 || 0 |
| + | |- |
| + | | <math>f_7</math> || 0111 || <math>(p q)\!</math> |
| + | | 1 || 0 || 0 || 0 || 1 || 1 || 1 || 0 |
| + | |- |
| + | | <math>f_8</math> || 1000 || <math>p q\!</math> |
| + | | 0 || 1 || 1 || 1 || 0 || 0 || 0 || 1 |
| + | |- |
| + | | <math>f_9</math> || 1001 || <math>((p, q))\!</math> |
| + | | 0 || 1 || 1 || 0 || 1 || 0 || 0 || 1 |
| + | |- |
| + | | <math>f_{10}</math> || 1010 || <math>q\!</math> |
| + | | 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 |
| + | |- |
| + | | <math>f_{11}</math> || 1011 || <math>(p (q))\!</math> |
| + | | 0 || 1 || 0 || 0 || 1 || 1 || 0 || 1 |
| + | |- |
| + | | <math>f_{12}</math> || 1100 || <math>p\!</math> |
| + | | 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 |
| + | |- |
| + | | <math>f_{13}</math> || 1101 || <math>((p) q)\!</math> |
| + | | 0 || 0 || 1 || 0 || 1 || 0 || 1 || 1 |
| + | |- |
| + | | <math>f_{14}</math> || 1110 || <math>((p)(q))\!</math> |
| + | | 0 || 0 || 0 || 1 || 0 || 1 || 1 || 1 |
| + | |- |
| + | | <math>f_{15}</math> || 1111 || <math>((~))</math> |
| + | | 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 |
| + | |}<br> |
| + | |
| + | {| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%" |
| + | |+ '''Table 2. 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}\ p\ \text{is}\ (q)</math> |
| + | | |
| + | | <math>\text{No}\ p\ \text{is}\ q </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}\ p\ \text{is}\ q </math> |
| + | | |
| + | | <math>\text{No}\ p\ \text{is}\ (q)</math> |
| + | | <math>(\ell_{10})</math> |
| + | |- |
| + | | |
| + | | |
| + | | <math>\text{All}\ q\ \text{is}\ p </math> |
| + | | <math>\text{No}\ q\ \text{is}\ (p)</math> |
| + | | <math>\text{No}\ (p)\ \text{is}\ q </math> |
| + | | <math>(\ell_{01})</math> |
| + | |- |
| + | | |
| + | | |
| + | | <math>\text{All}\ (q)\ \text{is}\ p </math> |
| + | | <math>\text{No}\ (q)\ \text{is}\ (p)</math> |
| + | | <math>\text{No}\ (p)\ \text{is}\ (q)</math> |
| + | | <math>(\ell_{00})</math> |
| + | |- |
| + | | |
| + | | |
| + | | <math>\text{Some}\ (p)\ \text{is}\ (q)</math> |
| + | | |
| + | | <math>\text{Some}\ (p)\ \text{is}\ (q)</math> |
| + | | <math>\ell_{00}\!</math> |
| + | |- |
| + | | |
| + | | |
| + | | <math>\text{Some}\ (p)\ \text{is}\ q</math> |
| + | | |
| + | | <math>\text{Some}\ (p)\ \text{is}\ q</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}\ p\ \text{is}\ (q)</math> |
| + | | |
| + | | <math>\text{Some}\ p\ \text{is}\ (q)</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}\ p\ \text{is}\ q</math> |
| + | | |
| + | | <math>\text{Some}\ p\ \text{is}\ y</math> |
| + | | <math>\ell_{11}\!</math> |
| + | |}<br> |
| | | |
| ====Exercise 2==== | | ====Exercise 2==== |