Functional Logic • Inquiry and Analogy • 13

Inquiry and Analogy • Higher Order Propositional Expressions

Higher Order Propositions and Logical Operators (n = 2)

By way of reviewing notation and preparing to extend it to higher order universes of discourse, let’s first consider the universe of discourse $X^\bullet = [\mathcal{X}] = [x_1, x_2] = [u, v],$ based on two logical features or boolean variables $u$ and $v.$

The universe of discourse $X^\bullet$ consists of two parts, a set of points and a set of propositions.

The points of $X^\bullet$ form the space:

$\begin{matrix} X & = & \langle \mathcal{X} \rangle & = & \langle u, v \rangle & = & \{ (u, v) \} & \cong & \mathbb{B}^2. \end{matrix}$

Each point in $X$ may be indicated by means of a singular proposition, that is, a proposition which describes it uniquely. This form of representation leads to the following enumeration of points.

$\begin{matrix} X & = & \{ ~ \texttt{(} u \texttt{)(} v \texttt{)} ~,~ \texttt{(} u \texttt{)} ~ v ~,~ u ~ \texttt{(} v \texttt{)} ~,~ u ~ v ~ \} & \cong & \mathbb{B}^2. \end{matrix}$

Each point in $X$ may also be described by means of its coordinates, that is, by the ordered pair of values in $\mathbb{B}$ which the coordinate propositions $u$ and $v$ take on that point. This form of representation leads to the following enumeration of points.

$\begin{matrix} X & = & \{\ (0, 0),\ (0, 1),\ (1, 0),\ (1, 1)\ \} & \cong & \mathbb{B}^2. \end{matrix}$

The propositions of $X^\bullet$ form the space:

$\begin{matrix} X^\uparrow & = & (X \to \mathbb{B}) & = & \{ f : X \to \mathbb{B} \} & \cong & (\mathbb{B}^2 \to \mathbb{B}). \end{matrix}$

As always, it is frequently convenient to omit a few of the finer markings of distinctions among isomorphic structures, so long as one is aware of their presence and knows when it is crucial to call upon them again.

The next higher order universe of discourse built on $X^\bullet$ is $X^{\bullet 2} = [X^\bullet] = [[u, v]],$ which may be developed in the following way. The propositions of $X^\bullet$ become the points of $X^{\bullet 2},$ and the mappings of the type $m : (X \to \mathbb{B}) \to \mathbb{B}$ become the propositions of $X^{\bullet 2}.$ In addition, it is convenient to equip the discussion with a selected set of higher order operators on propositions, all of which have the form $w : (\mathbb{B}^2 \to \mathbb{B})^k \to \mathbb{B}.$

To save a few words in the remainder of this discussion, I will use the terms measure and qualifier to refer to all types of higher order propositions and operators. To describe the present setting in picturesque terms, the propositions of $[u, v]$ may be regarded as a gallery of sixteen venn diagrams, while the measures $m : (X \to \mathbb{B}) \to \mathbb{B}$ are analogous to a body of judges or a panel of critical viewers, each of whom evaluates each of the pictures as a whole and reports the ones that find favor or not. In this way, each judge $m_j$ partitions the gallery of pictures into two aesthetic portions, the pictures $m_j^{-1}(1)$ that $m_j$ likes and the pictures $m_j^{-1}(0)$ that $m_j$ dislikes.

There are $2^{16} = 65536$ measures of the form $m : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}.$ Table 13 shows the first 24 of their number in the style of higher order truth table I used before. The column headed $m_j$ shows the value of the measure $m_j$ on each of the propositions $f_i : \mathbb{B}^2 \to \mathbb{B}$ for $i$ = 0 to 15. The arrangement of measures in the order indicated will be referred to as their standard ordering. In this scheme of things, the index $j$ of the measure $m_j$ is the decimal equivalent of the bit string in the corresponding column of the Table, reading the binary digits in order from bottom to top.

$\text{Table 13. Higher Order Propositions}~ (n = 2)$

Resources

Logic Syllabus
Boolean Function
Boolean-Valued Function
Logical Conjunction
Minimal Negation Operator
Introduction to Inquiry Driven Systems
Functional Logic • Part 1 • Part 2 • Part 3
Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: Conceptual Graphs • Cybernetics • Laws of Form • Ontolog Forum
cc: FB | Peirce Matters • Structural Modeling • Systems Science

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