Functional Logic • Inquiry and Analogy • 15

Inquiry and Analogy • Measure for Measure

Let us define two families of measures,

$\alpha_i, \beta_i : (\mathbb{B} \times \mathbb{B} \to \mathbb{B}) \to \mathbb{B} ~\text{for}~ i = 0 ~\text{to}~ 15,$

by means of the following equations:

$\begin{matrix} \alpha_i f & = & \Upsilon (f_i, f) & = & \Upsilon (f_i \Rightarrow f), \\[6pt] \beta_i f & = & \Upsilon (f, f_i) & = & \Upsilon (f \Rightarrow f_i). \end{matrix}$

Table 14 shows the value of each $\alpha_i$ on each of the 16 boolean functions $f: \mathbb{B} \times \mathbb{B} \to \mathbb{B}.$ In terms of the implication ordering on the 16 functions, $\alpha_i f = 1$ says that $f$ is above or identical to $f_i$ in the implication lattice, that is, $f \ge f_i$ in the implication ordering.

$\text{Table 14. Qualifiers of the Implication Ordering}~ \alpha_i f = \Upsilon (f_i, f)$

Table 15 shows the value of each $\beta_i$ on each of the 16 boolean functions $f: \mathbb{B} \times \mathbb{B} \to \mathbb{B}.$ In terms of the implication ordering on the 16 functions, $\beta_i f = 1$ says that $f$ is below or identical to $f_i$ in the implication lattice, that is, $f \le f_i$ in the implication ordering.

$\text{Table 15. Qualifiers of the Implication Ordering}~ \beta_i f = \Upsilon (f, f_i)$

Applied to a given proposition $f,$ the qualifiers $\alpha_i$ and $\beta_i$ tell whether $f$ is above $f_i$ or below $f_i,$ respectively, in the implication ordering. By way of example, let us trace the effects of several such measures, namely, those which occupy the limiting positions in the Tables.

$\begin{array}{*{8}{r}} \alpha_{0} f = 1 & \mathrm{iff} & f_{0} \Rightarrow f & \mathrm{iff} & 0 \Rightarrow f, & \mathrm{hence} & \alpha_{0} f = 1 & \mathrm{for~all} ~ f. \\[4pt] \alpha_{15} f = 1 & \mathrm{iff} & f_{15} \Rightarrow f & \mathrm{iff} & 1 \Rightarrow f, & \mathrm{hence} & \alpha_{15} f = 1 & \mathrm{iff} ~ f = 1. \\[4pt] \beta_{0} f = 1 & \mathrm{iff} & f \Rightarrow f_{0} & \mathrm{iff} & f \Rightarrow 0, & \mathrm{hence} & \beta_{0} f = 1 & \mathrm{iff} ~ f = 0. \\[4pt] \beta_{15} f = 1 & \mathrm{iff} & f \Rightarrow f_{15} & \mathrm{iff} & f \Rightarrow 1, & \mathrm{hence} & \beta_{15} f = 1 & \mathrm{for~all} ~ f. \end{array}$

Expressed in terms of the propositional forms they value positively, $\alpha_{0} = \beta_{15}$ is a totally indiscriminate measure, accepting all propositions $f : \mathbb{B} \times \mathbb{B} \to \mathbb{B},$ whereas $\alpha_{15}$ and $\beta_{0}$ are measures valuing the constant propositions $1 : \mathbb{B} \times \mathbb{B} \to \mathbb{B}$ and $0 : \mathbb{B} \times \mathbb{B} \to \mathbb{B},$ respectively, above all others.

Finally, in conformity with the use of fiber notation to indicate sets of models, it is natural to use notations like the following to denote sets of propositions satisfying the umpires in question.

$\begin{matrix} [| \alpha_i |] & = & \alpha_i^{-1}(1), \\[6pt] [| \beta_i |] & = & \beta_i^{-1}(1), \\[6pt] [| \Upsilon_p |] & = & \Upsilon_p^{-1}(1). \end{matrix}$

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

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