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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 "∀_{''x''∈''X''} ''Fx''" and "∃_{''x''∈''X''} ''Fx''" 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===

By way of equipping this inquiry with a bit of concrete material, I begin with a consideration of "higher order propositional expressions" (HOPE's), in particular, those that stem from the propositions on 1 and 2 variables.

By way of equipping this inquiry with a bit of concrete material, I begin with a consideration of ''higher order propositional expressions'' (HOPE's), in particular, those that stem from the propositions on 1 and 2 variables.

====Higher Order Propositions and Logical Operators (''n'' <nowiki>=</nowiki> 1)====

====Higher Order Propositions and Logical Operators (''n'' = 1)====

A "higher order proposition" is, very roughly speaking, a proposition about propositions.  If the original order of propositions is a class of indicator functions {''F'' : ''X'' &rarr; '''B'''}, then the next higher order of propositions consists of maps of the type ''m'' : (''X'' &rarr; '''B''') &rarr; '''B''', where, as usual, '''B''' = {0,&nbsp;1}.

A ''higher order proposition'' is, very roughly speaking, a proposition about propositions.  If the original order of propositions is a class of indicator functions {''F'' : ''X'' &rarr; '''B'''}, then the next higher order of propositions consists of maps of the type ''m'' : (''X'' &rarr; '''B''') &rarr; '''B''', where, as usual, '''B''' = {0,&nbsp;1}.

For example, consider the case where ''X'' = '''B'''¹ = '''B'''.  Then there are exactly four propositions of the form ''F'' : '''B''' &rarr; '''B''', and exactly sixteen higher order propositions, all of the type ''m'' : ('''B''' &rarr; '''B''') &rarr; '''B'''.  Table 7 lists the sixteen higher order propositions about propositions on one boolean variable, organized in the following fashion:

For example, consider the case where ''X'' = '''B'''¹ = '''B'''.  Then there are exactly four propositions of the form ''F'' : '''B''' &rarr; '''B''', and exactly sixteen higher order propositions, all of the type ''m'' : ('''B''' &rarr; '''B''') &rarr; '''B'''.  Table 7 lists the sixteen higher order propositions about propositions on one boolean variable, organized in the following fashion: