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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.

====Higher Order Propositions and Logical Operators (''n'' = 1)====

====2.1.1.  Higher Order Propositions, Higher Order Logical Operators (''n'' = 1)====

A ''higher order proposition'' is, very roughly speaking, a proposition about propositions.  If the original order of propositions is a class of indicator functions {''F'' : ''X'' → '''B'''}, then the next higher order of propositions consists of maps of the type ''m'' : (''X'' → '''B''') → '''B''', where, as usual, '''B''' = {0, 1}.

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>

For example, consider the case where ''X'' = '''B'''¹ = '''B'''. Then there are exactly four propositions of the form ''F'' : '''B''' &rarr; '''B''', and exactly sixteen higher order propositions, all of the type ''m'' : ('''B''' &rarr; '''B''') &rarr; '''B'''. Table 7 lists the sixteen higher order propositions about propositions on one boolean variable, organized in the following fashion:

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>

Columns 1 and 2 form a truth table for the four ''F'' : '''B''' &rarr; '''B''', perhaps turned on its side from the way one is accustomed to see truth tables, with the row leaders in Column 1 displaying the names of the functions ''F''_''i'', ''i'' = 1 to 4, while the entries in Column 2 give the values of each function for the argument values that are listed in the column head.  Column 3 displays one of the usual expressions for the proposition in question.  The last sixteen columns are topped by a set of conventional names for the higher order propositions, also known as the "measures" ''m''_''j'', for ''j'' = 0 to 15, where the entries in the body of the Table record the values that each ''m''_''j'' assigns to each ''F''_''i''.

Table&nbsp;10 lists the sixteen higher order propositions about propositions on one boolean variable, organized in the following fashion:  Columns&nbsp;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&nbsp;1 displaying the names of the functions <math>F_i,\!</math> for <math>i = 1\ \operatorname{to}\ 4,\!</math>  while the entries in Column&nbsp;2 give the values of each function for the argument values that are listed in the corresponding column head.  Column&nbsp;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>

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