## Functional Logic • Inquiry and Analogy • 21

### Inquiry and Analogy • Generalized Umpire Operators

To get a better handle on the space of higher order propositions and continue developing our functional approach to quantification theory, we’ll need a number of specialized tools.  To begin, we define a higher order operator $\Upsilon,$ called the umpire operator, which takes 1, 2, or 3 propositions as arguments and returns a single truth value as the result.  Operators with optional numbers of arguments are called multigrade operators, typically defined as unions over function types.  Expressing $\Upsilon$ in that form gives the following formula.

$\Upsilon : \bigcup_{\ell = 1, 2, 3} ((\mathbb{B}^k \to \mathbb{B})^\ell \to \mathbb{B}).$

In contexts of application, that is, where a multigrade operator is actually being applied to arguments, the number of arguments in the argument list tells which of the optional types is “operative”.  In the case of $\Upsilon,$ the first and last arguments appear as indices, the one in the middle serving as the main argument while the other two arguments serve to modify the sense of the operation in question.  Thus, we have the following forms.

$\begin{matrix} \Upsilon_p^r q = \Upsilon (p, q, r) \\[10pt] \Upsilon_p^r : (\mathbb{B}^k \to \mathbb{B}) \to \mathbb{B} \end{matrix}$

The operation $\Upsilon_p^r q$ evaluates the proposition $q$ on each model of the proposition $p$ and combines the results according to the method indicated by the connective parameter $r.$  In principle, the index $r$ may specify any logical connective on as many as $2^k$ arguments but in practice we usually have a much simpler form of combination in mind, typically either products or sums.  By convention, each of the accessory indices $p, r$ is assigned a default value understood to be in force when the corresponding argument place is left blank, specifically, the constant proposition $1 : \mathbb{B}^k \to \mathbb{B}$ for the lower index $p$ and the continued conjunction or continued product operation $\textstyle\prod$ for the upper index $r.$  Taking the upper default value gives license to the following readings.

$\begin{matrix} \Upsilon_p (q) = \Upsilon (p, q) = \Upsilon (p, q, \textstyle\prod). \\[10pt] \Upsilon_p = \Upsilon (p, \underline{~~}, \textstyle\prod) : (\mathbb{B}^k \to \mathbb{B}) \to \mathbb{B}. \end{matrix}$

This means $\Upsilon_p (q) = 1$ if and only if $q$ holds for all models of $p.$  In propositional terms, this is tantamount to the assertion that $p \Rightarrow q,$ or that $\texttt{(} p \texttt{(} q \texttt{))} = 1.$

Throwing in the lower default value permits the following abbreviations.

$\begin{matrix} \Upsilon q = \Upsilon (q) = \Upsilon_1 (q) = \Upsilon (1, q, \textstyle\prod). \\[10pt] \Upsilon = \Upsilon (1, \underline{~~}, \textstyle\prod)) : (\mathbb{B}^k\ \to \mathbb{B}) \to \mathbb{B}. \end{matrix}$

This means $\Upsilon q = 1$ if and only if $q$ holds for the whole universe of discourse in question, that is, if and only $q$ is the constantly true proposition $1 : \mathbb{B}^k \to \mathbb{B}.$  The ambiguities of this usage are not a problem so long as we distinguish the context of definition from the context of application and restrict all shorthand notations to the latter.

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