5 May
2022
5 May
'22
9:30 p.m.
Cf: Functional Logic • Inquiry and Analogy • 11
https://inquiryintoinquiry.com/2022/05/05/functional-logic-inquiry-and-anal…
Inquiry and Analogy • Higher Order Propositional Expressions
https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#HOPE
Higher Order Propositions and Logical Operators (n = 1)
https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#HOPE_1
All,
A “higher order proposition” is, 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.
For example, consider the case where X = B. There are exactly four
propositions one can make about the elements of X. Each proposition
has the concrete type f : X → B and the abstract type f : B → B. Then
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 : (B → B) → B.
Table 11 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 propositions
f : B → B, turned on its side from the way one is most likely
accustomed to see truth tables, with the row leaders in Column 1
displaying the names of the functions f_i, for i = 1 to 4, while
the entries in Column 2 give the values of each function for the
argument values 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 collection 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 each m_j assigns to each f_i.
Table 11. Higher Order Propositions (n = 1)
https://inquiryintoinquiry.files.wordpress.com/2022/05/higher-order-proposi…
Regards,
Jon
