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==Toward a Functional Conception of Quantificational Logic==

==Toward a Functional Conception of Quantificational Logic==

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} Fx</math> and <math>\exists_{x \in X} Fx</math> involves a subscription to such notions, as shown by the membership relations invoked in their indices.  Reflected on pragmatic and constructive principles, however, 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 scene of quantification theory closer to familiar ground, toward the predicates themselves that represent our continuing acquaintance with phenomena.

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, however, 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 scene of quantification theory closer to familiar ground, toward the predicates themselves that represent our continuing acquaintance with phenomena.

===Higher Order Propositional Expressions===

===Higher Order Propositional Expressions===

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I am going to put off explaining Table 8, that presents a sample of what I call "Interpretive Categories for Higher Order Propositions", until after we get beyond the 1-dimensional case, since these lower dimensional cases tend to be a bit "condensed" or "degenerate" in their structures, and a lot of what is going on here will almost automatically become clearer as soon as we get even two logical variables into the mix.

I am going to put off explaining Table&nbsp;8, that presents a sample of what I call ''interpretive categories'' for higher order propositions, until after we get beyond the 1-dimensional case, since these lower dimensional cases tend to be a bit ''condensed'' or ''degenerate'' in their structures, and a lot of what is going on here will almost automatically become clearer as soon as we get even two logical variables into the mix.

{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"

{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"

|+ '''Table 8.  Interpretive Categories for Higher Order Propositions (n = 1)'''

|+ '''Table 8.  Interpretive Categories for Higher Order Propositions (''n'' = 1)'''

|- style="background:paleturquoise"

|- style="background:paleturquoise"

|Measure||Happening||Exactness||Existence||Linearity||Uniformity||Information

|Measure||Happening||Exactness||Existence||Linearity||Uniformity||Information

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====Higher Order Propositions and Logical Operators (n <nowiki>=</nowiki> 2)====

====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 us first consider the universe of discourse ''X''° = [''X''] = [''x''₁, ''x''₂] = [''x'', ''y''], based on two logical features or boolean variables ''x'' and ''y''.

By way of reviewing notation and preparing to extend it to higher order universes of discourse, let us first consider the universe of discourse ''X''° = [''X''] = [''x''₁, ''x''₂] = [''x'', ''y''], based on two logical features or boolean variables ''x'' and ''y''.