Functional Logic • Inquiry and Analogy • 11

Inquiry and AnalogyHigher 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)
Higher Order Propositions (n = 1)


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This entry was posted in Abduction, Analogy, Argument, Aristotle, C.S. Peirce, Constraint, Deduction, Determination, Diagrammatic Reasoning, Diagrams, Differential Logic, Functional Logic, Hypothesis, Indication, Induction, Inference, Information, Inquiry, Logic, Logic of Science, Mathematics, Pragmatic Semiotic Information, Probable Reasoning, Propositional Calculus, Propositions, Reasoning, Retroduction, Semiotics, Sign Relations, Syllogism, Triadic Relations, Visualization and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , . Bookmark the permalink.

2 Responses to Functional Logic • Inquiry and Analogy • 11

  1. Pingback: Survey of Abduction, Deduction, Induction, Analogy, Inquiry • 2 | Inquiry Into Inquiry

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