## Functional Logic • Inquiry and Analogy • 11

### Inquiry and Analogy • Higher Order Propositional Expressions

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

A higher order proposition is a proposition about propositions.  If the original order of propositions is a class of indicator functions $f : X \to \mathbb{B}$ then the next higher order of propositions consists of maps of type $m : (X \to \mathbb{B}) \to \mathbb{B}.$

For example, consider the case where $X = \mathbb{B}.$  There are exactly four propositions one can make about the elements of $X.$  Each proposition has the concrete type $f: X \to \mathbb{B}$ and the abstract type $f : \mathbb{B} \to \mathbb{B}.$  From that beginning there are exactly sixteen higher order propositions one can make about the initial set of four propositions.  Each higher order proposition has the abstract type $m : (\mathbb{B} \to \mathbb{B}) \to \mathbb{B}.$

Table 11 lists the sixteen higher order propositions about propositions on one boolean variable, organized in the following fashion.

• Columns 1 and 2 taken together present a form of truth table for the four propositions $f : \mathbb{B} \to \mathbb{B}.$  Column 1 displays the names of the propositions $f_i,$ for $i$ = 1 to 4, while the entries in Column 2 show the value each proposition takes on the argument value 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 headed by a series of conventional names for the higher order propositions, also known as the measures $m_j,$ for $j$ = 0 to 15.  The entries in the body of the Table show the value each measure assigns to each proposition $f_i.$

$\text{Table 11. Higher Order Propositions}~ (n = 1)$

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