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The values of the sixteen <math>\beta_i</math> on each of the sixteen boolean functions <math>f : \mathbb{B}^2 \to \mathbb{B}</math> are shown in Table 5. Expressed in terms of the implication ordering on the sixteen functions, <math>\beta_i f = 1</math> says that <math>f</math> is ''below or identical to'' <math>f_i</math> in the implication lattice, that is, <math>\le f_i</math> in the implication ordering.
The values of the sixteen <math>\beta_i</math> on each of the sixteen boolean functions <math>f : \mathbb{B}^2 \to \mathbb{B}</math> are shown in Table 5. Expressed in terms of the implication ordering on the sixteen functions, <math>\beta_i f = 1</math> says that <math>f</math> is ''below or identical to'' <math>f_i</math> in the implication lattice, that is, <math>\le f_i</math> in the implication ordering.
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\end{array}</math>
\end{array}</math>
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Thus, <math>\alpha_{0} = \beta_{15}</math> is a totally indiscriminate measure, one that accepts all propositions <math>f : \mathbb{B}^2 \to \mathbb{B},</math> whereas <math>\alpha_{15}</math> and <math>\beta_{0}</math> are measures that value the constant propositions <math>1 : \mathbb{B}^2 \to \mathbb{B}</math> and <math>0 : \mathbb{B}^2 \to \mathbb{B},</math> respectively, above all others.
Thus, <math>\alpha_{0} = \beta_{15}</math> is a totally indiscriminate measure, one that accepts all propositions <math>f : \mathbb{B}^2 \to \mathbb{B},</math> whereas <math>\alpha_{15}</math> and <math>\beta_{0}</math> are measures that value the constant propositions <math>1 : \mathbb{B}^2 \to \mathbb{B}</math> and <math>0 : \mathbb{B}^2 \to \mathbb{B},</math> respectively, above all others.
| <math>(\Upsilon_p)^{-1}(1).</math>
| <math>(\Upsilon_p)^{-1}(1).</math>
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==Functional Conception of Quantification Theory==
Up till now quantification theory has been based on the assumption of individual variables ranging over universal collections of perfectly determinate elements. Merely to write down quantified formulas like <math>\forall_{x \in X} f(x)</math> and <math>\exists_{x \in X} f(x)</math> involves a subscription to such notions, as shown by the membership relations invoked in their indices. Reflected on pragmatic and constructive principles, these ideas begin to appear as problematic hypotheses whose warrants are not beyond question, projects of exhaustive determination that overreach the powers of finite information and control to manage. Therefore, it is worth considering how we might shift the medium of quantification theory closer to familiar ground, toward the predicates themselves that represent our continuing acquaintance with phenomena.
