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| By way of equipping this inquiry with a bit of concrete material, I begin with a consideration of ''higher order propositional expressions'' (HOPE's), in particular, those that stem from the propositions on 1 and 2 variables. | | By way of equipping this inquiry with a bit of concrete material, I begin with a consideration of ''higher order propositional expressions'' (HOPE's), in particular, those that stem from the propositions on 1 and 2 variables. |
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− | ====2.1.1. Higher Order Propositions, Higher Order Logical Operators (''n'' = 1)==== | + | ====Higher Order Propositions and Higher Order Logical Operators (''n'' = 1)==== |
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| A ''higher order proposition'' is, very roughly speaking, a proposition about propositions. If the original order of propositions is a class of indicator functions <math>F : X \to \mathbb{B},</math> then the next higher order of propositions consists of maps of the type <math>m : (X \to \mathbb{B}) \to \mathbb{B}.</math> | | A ''higher order proposition'' is, very roughly speaking, a proposition about propositions. If the original order of propositions is a class of indicator functions <math>F : X \to \mathbb{B},</math> then the next higher order of propositions consists of maps of the type <math>m : (X \to \mathbb{B}) \to \mathbb{B}.</math> |
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| For example, consider the case where <math>X = B.\!</math> Then there are exactly four propositions <math>F : \mathbb{B} \to \mathbb{B},</math> and exactly sixteen higher order propositions that are based on this set, all bearing the type <math>m : (\mathbb{B} \to \mathbb{B}) \to \mathbb{B}.</math> | | For example, consider the case where <math>X = B.\!</math> Then there are exactly four propositions <math>F : \mathbb{B} \to \mathbb{B},</math> and exactly sixteen higher order propositions that are based on this set, all bearing the type <math>m : (\mathbb{B} \to \mathbb{B}) \to \mathbb{B}.</math> |
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− | Table 10 lists the sixteen higher order propositions about propositions on one boolean variable, organized in the following fashion: Columns 1 and 2 form a truth table for the four <math>F : \mathbb{B} \to \mathbb{B},</math> turned on its side from the way that one is most likely accustomed to see truth tables, with the row leaders in Column 1 displaying the names of the functions <math>F_i,\!</math> for <math>i = 1\ \operatorname{to}\ 4,\!</math> while the entries in Column 2 give the values of each function for the argument values that are listed in the corresponding column head. Column 3 displays one of the more usual expressions for the proposition in question. The last sixteen columns are topped by a collection of conventional names for the higher order propositions, also known as the ''measures'' <math>m_j,\!</math> for <math>j = 0\ \operatorname{to}\ 15,\!</math> where the entries in the body of the Table record the values that each <math>m_j\!</math> assigns to each <math>F_i.\!</math> | + | Table 10 lists the sixteen higher order propositions about propositions on one boolean variable, organized in the following fashion: Columns 1 and 2 form a truth table for the four <math>F : \mathbb{B} \to \mathbb{B},</math> turned on its side from the way that one is most likely accustomed to see truth tables, with the row leaders in Column 1 displaying the names of the functions <math>F_i, i = 1\ \operatorname{to}\ 4,</math> while the entries in Column 2 give the values of each function for the argument values that are listed in the corresponding column head. Column 3 displays one of the more usual expressions for the proposition in question. The last sixteen columns are topped by a collection of conventional names for the higher order propositions, also known as the ''measures'' <math>m_j, j = 0\ \operatorname{to}\ 15,</math> where the entries in the body of the Table record the values that each <math>m_j\!</math> assigns to each <math>F_i.\!</math> |
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| {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" | | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |