Changes

==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} 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.

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===

====Higher Order Propositions and Logical Operators (''n'' = 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 <math>F : X \to \mathbb{B},</math> then the next higher order of propositions consists of maps of the type <math>m : (X \to \mathbb{B}) \to \mathbb{B}.</math>

A ''higher order proposition'' is, very roughly speaking, a proposition about propositions.  If the original order of propositions is a class of indicator functions <math>f : X \to \mathbb{B},</math> then the next higher order of propositions consists of maps of the type <math>m : (X \to \mathbb{B}) \to \mathbb{B}.</math>

For example, consider the case where <math>X = \mathbb{B}.</math>  Then there are exactly four propositions <math>F : \mathbb{B} \to \mathbb{B},</math> and exactly sixteen higher order propositions that are based on this set, all bearing the type <math>m : (\mathbb{B} \to \mathbb{B}) \to \mathbb{B}.</math>

For example, consider the case where <math>X = \mathbb{B}.</math>  Then there are exactly four propositions <math>f : \mathbb{B} \to \mathbb{B},</math> and exactly sixteen higher order propositions that are based on this set, all bearing the type <math>m : (\mathbb{B} \to \mathbb{B}) \to \mathbb{B}.</math>

Table&nbsp;1 lists the sixteen higher order propositions about propositions on one boolean variable, organized in the following fashion:  Columns&nbsp;1 and 2 form a truth table for the four <math>F : \mathbb{B} \to \mathbb{B},</math> turned on its side from the way that one is most likely accustomed to see truth tables, with the row leaders in Column&nbsp;1 displaying the names of the functions <math>F_i,\!</math> for <math>i\!</math> = 1 to 4, while the entries in Column&nbsp;2 give the values of each function for the argument values that are listed in the corresponding column head.  Column&nbsp;3 displays one of the more usual expressions for the proposition in question.  The last sixteen columns are topped by a collection of conventional names for the higher order propositions, also known as the ''measures'' <math>m_j,\!</math> for <math>j\!</math> = 0 to 15, where the entries in the body of the Table record the values that each <math>m_j\!</math> assigns to each <math>F_i.\!</math>

Table&nbsp;1 lists the sixteen higher order propositions about propositions on one boolean variable, organized in the following fashion:  Columns&nbsp;1 and 2 form a truth table for the four <math>f : \mathbb{B} \to \mathbb{B},</math> turned on its side from the way that one is most likely accustomed to see truth tables, with the row leaders in Column&nbsp;1 displaying the names of the functions <math>f_i,\!</math> for <math>i\!</math> = 1 to 4, while the entries in Column&nbsp;2 give the values of each function for the argument values that are listed in the corresponding column head.  Column&nbsp;3 displays one of the more usual expressions for the proposition in question.  The last sixteen columns are topped by a collection of conventional names for the higher order propositions, also known as the ''measures'' <math>m_j,\!</math> for <math>j\!</math> = 0 to 15, where the entries in the body of the Table record the values that each <math>m_j\!</math> assigns to each <math>f_i.\!</math>

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

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

| align="right" | <math>x</math>:

| align="right" | <math>x</math>:

| 1 0

| 1 0

| <math>F</math>

| <math>f</math>

| <math>m_0</math>

| <math>m_0</math>

| <math>m_1</math>

| <math>m_1</math>

| <math>m_{15}</math>

| <math>m_{15}</math>

|-

|-

| <math>F_0</math>

| <math>f_0</math>

| 0 0

| 0 0

| <math>0\!</math>

| <math>0\!</math>

| 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1

| 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1

|-

|-

| <math>F_1</math>

| <math>f_1</math>

| 0 1

| 0 1

| <math>(x)\!</math>

| <math>(x)\!</math>

| 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1

| 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1

|-

|-

| <math>F_2</math>

| <math>f_2</math>

| 1 0

| 1 0

| <math>x\!</math>

| <math>x\!</math>

| 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1

| 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1

|-

|-

| <math>F_3</math>

| <math>f_3</math>

| 1 1

| 1 1

| <math>1\!</math>

| <math>1\!</math>

|+ '''Table 2.  Interpretive Categories for Higher Order Propositions (''n'' = 1)'''

|+ '''Table 2.  Interpretive Categories for Higher Order Propositions (''n'' = 1)'''

|- style="background:ghostwhite"

|- style="background:ghostwhite"

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

| Measure

| Happening

| Exactness

| Existence

| Linearity

| Uniformity

| Information

|-

|-

|''m''₀||nothing happens||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;

| <math>m_0\!</math>

| nothing happens

| &nbsp;

| &nbsp;

| &nbsp;

| &nbsp;

| &nbsp;

|-

|-

|''m''₁||&nbsp;||just false||nothing exists||&nbsp;||&nbsp;||&nbsp;

| <math>m_1\!</math>

| &nbsp;

| just false

| nothing exists

| &nbsp;

| &nbsp;

| &nbsp;

|-

|-

|''m''₂||&nbsp;||just not x||&nbsp;||&nbsp;||&nbsp;||&nbsp;

| <math>m_2\!</math>

| &nbsp;

| just not ''x''

| &nbsp;

| &nbsp;

| &nbsp;

| &nbsp;

|-

|-

|''m''₃||&nbsp;||&nbsp;||nothing is x||&nbsp;||&nbsp;||&nbsp;

| <math>m_3\!</math>

| &nbsp;

| &nbsp;

| nothing is ''x''

| &nbsp;

| &nbsp;

| &nbsp;

|-

|-

|''m''₄||&nbsp;||just x||&nbsp;||&nbsp;||&nbsp;||&nbsp;

| <math>m_4\!</math>

| &nbsp;

| just ''x''

| &nbsp;

| &nbsp;

| &nbsp;

| &nbsp;

|-

|-

|''m''₅||&nbsp;||&nbsp;||everything is x||F is linear||&nbsp;||&nbsp;

| <math>m_5\!</math>

| &nbsp;

| &nbsp;

| everything is ''x''

| ''f'' is linear

| &nbsp;

| &nbsp;

|-

|-

|''m''₆||&nbsp;||&nbsp;||&nbsp;||&nbsp;||F is not uniform||F is informed

| <math>m_6\!</math>

| &nbsp;

| &nbsp;

| &nbsp;

| &nbsp;

| ''f'' is not uniform

| ''f'' is informed

|-

|-

|''m''₇||&nbsp;||not just true||&nbsp;||&nbsp;||&nbsp;||&nbsp;

| <math>m_7\!</math>

| &nbsp;

| not just true

| &nbsp;

| &nbsp;

| &nbsp;

| &nbsp;

|-

|-

|''m''₈||&nbsp;||just true||&nbsp;||&nbsp;||&nbsp;||&nbsp;

| <math>m_8\!</math>

| &nbsp;

| just true

| &nbsp;

| &nbsp;

| &nbsp;

| &nbsp;

|-

|-

|''m''₉||&nbsp;||&nbsp;||&nbsp;||&nbsp;||F is uniform||F is not informed

| <math>m_9\!</math>

| &nbsp;

| &nbsp;

| &nbsp;

| &nbsp;

| ''f'' is uniform

| ''f'' is not informed

|-

|-

|''m''₁₀||&nbsp;||&nbsp;||something is not x||F is not linear||&nbsp;||&nbsp;

| <math>m_{10}\!</math>

| &nbsp;

| &nbsp;

| something is not ''x''

| ''f'' is not linear

| &nbsp;

| &nbsp;

|-

|-

|''m''₁₁||&nbsp;||not just x||&nbsp;||&nbsp;||&nbsp;||&nbsp;

| <math>m_{11}\!</math>

| &nbsp;

| not just ''x''

| &nbsp;

| &nbsp;

| &nbsp;

| &nbsp;

|-

|-

|''m''₁₂||&nbsp;||&nbsp;||something is x||&nbsp;||&nbsp;||&nbsp;

| <math>m_{12}\!</math>

| &nbsp;

| &nbsp;

| something is ''x''

| &nbsp;

| &nbsp;

| &nbsp;

|-

|-

|''m''₁₃||&nbsp;||not just not x||&nbsp;||&nbsp;||&nbsp;||&nbsp;

| <math>m_{13}\!</math>

| &nbsp;

| not just not ''x''

| &nbsp;

| &nbsp;

| &nbsp;

| &nbsp;

|-

|-

|''m''₁₄||&nbsp;||not just false||something exists||&nbsp;||&nbsp;||&nbsp;

| <math>m_{14}\!</math>

| &nbsp;

| not just false

| something exists

| &nbsp;

| &nbsp;

| &nbsp;

|-

|-

|''m''₁₅||anything happens||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;

| <math>m_{15}\!</math>

| anything happens

| &nbsp;

| &nbsp;

| &nbsp;

| &nbsp;

| &nbsp;

|}<br>

|}<br>