## Functional Logic • Inquiry and Analogy • 14

### Inquiry and Analogy • Umpire Operators

The $2^{16}$ measures of type $(\mathbb{B} \times \mathbb{B} \to \mathbb{B}) \to \mathbb{B}$ present a formidable array of propositions about propositions about 2-dimensional universes of discourse.  The early entries in their standard ordering define universes too amorphous to detain us for long on a first pass but as we turn toward the high end of the ordering we begin to recognize familiar structures worth examining from new angles.

Instrumental to our study we define a couple of higher order operators,

$\begin{matrix} \Upsilon : (\mathbb{B} \times \mathbb{B} \to \mathbb{B})^2 \to \mathbb{B} && \text{and} && \Upsilon_1 : (\mathbb{B} \times \mathbb{B} \to \mathbb{B}) \to \mathbb{B}, \end{matrix}$

referred to as the relative and absolute umpire operators, respectively.  If either operator is defined in terms of more primitive notions then the remaining operator can be defined in terms of the one first established.

Let $X = \langle u, v \rangle$ be a two-dimensional boolean space, $X \cong \mathbb{B} \times \mathbb{B},$ generated by two boolean variables or logical features $u$ and $v.$

Given an ordered pair of propositions $e, f : \langle u, v \rangle \to \mathbb{B}$ as arguments, the relative umpire operator reports the value $1$ if the first implies the second, otherwise it reports the value $0.$

$\begin{matrix} \Upsilon (e, f) = 1 && \text{if and only if} && e \Rightarrow f \end{matrix}$

Expressing it another way:

$\begin{matrix} \Upsilon (e, f) = 1 && \iff && \texttt{(} e \texttt{(} f \texttt{))} = 1 \end{matrix}$

In writing this, however, it is important to observe that the $1$ appearing on the left side and the $1$ appearing on the right side of the logical equivalence have different meanings.  Filling in the details, we have the following.

$\begin{matrix} \Upsilon (e, f) = 1 \in \mathbb{B} && \iff && \texttt{(} e \texttt{(} f \texttt{))} = 1 : \langle u, v \rangle \to \mathbb{B} \end{matrix}$

Writing types as subscripts and using the fact that $X = \langle u, v \rangle,$ it is possible to express this a little more succinctly as follows.

$\begin{matrix} \Upsilon (e, f) = 1_\mathbb{B} && \iff && \texttt{(} e \texttt{(} f \texttt{))} = 1_{X \to \mathbb{B}} \end{matrix}$

Finally, it is often convenient to write the first argument as a subscript.  Thus we have the following equation.

$\begin{matrix} \Upsilon_e (f) & = & \Upsilon (e, f). \end{matrix}$

The absolute umpire operator, also known as the umpire measure, is a higher order proposition $\Upsilon_1 : (\mathbb{B} \times \mathbb{B} \to \mathbb{B}) \to \mathbb{B}$ defined by the equation $\Upsilon_1 (f) = \Upsilon (1, f).$  In this case the subscript $1$ on the left and the argument $1$ on the right both refer to the constant proposition $1 : \mathbb{B} \times \mathbb{B} \to \mathbb{B}.$  In most settings where $\Upsilon_1$ is applied to arguments it is safe to omit the subscript $1$ since the number of arguments indicates which type of operator is meant.  Thus, we have the following identities and equivalents.

$\begin{matrix} \Upsilon f = \Upsilon_1 (f) = 1_\mathbb{B} & \iff & \texttt{(} 1 \texttt{(} f \texttt{))} = \mathbf{1} & \iff & f = 1_{\mathbb{B} \times \mathbb{B} \to \mathbb{B}} \end{matrix}$

The umpire measure $\Upsilon_1$ is defined at the level of boolean functions as mathematical objects but can also be understood in terms of the judgments it induces on the syntactic level.  In that interpretation $\Upsilon_1$ recognizes theorems of the propositional calculus over $[u, v],$ giving a score of $1$ to tautologies and a score of $0$ to everything else, regarding all contingent statements as no better than falsehoods.

One remark in passing for those who might prefer an alternative definition.  If we had originally taken $\Upsilon$ to mean the absolute measure then the relative measure could have been defined as $\Upsilon_e f = \Upsilon \texttt{(} e \texttt{(} f \texttt{))}.$

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