Changes

==Functional Quantifiers==

==Functional Quantifiers==

The ''umpire measure'' of type <math>\Upsilon : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}</math> takes a proposition of type <math>\mathbb{B}^2 \to \mathbb{B}</math> as argument, giving the constant proposition <math>1 : \mathbb{B}^2 \to \mathbb{B}</math> a value of 1 and everything else a value of 0.  Expressed in symbolic form:

The ''umpire measure'' of type <math>\Upsilon : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}</math> takes a single proposition of type <math>\mathbb{B}^2 \to \mathbb{B}</math> as argument, giving the constant proposition <math>1 : \mathbb{B}^2 \to \mathbb{B}</math> a value of 1 and everything else a value of 0.  Expressed in symbolic form:

{| align="center" cellpadding="8"

{| align="center" cellpadding="8"

| <math>\Upsilon p = 1 \quad \Leftrightarrow \quad p = 1.</math>

| <math>\Upsilon \langle p \rangle = 1 \quad \Leftrightarrow \quad p = 1.</math>

|}

|}

The auxiliary notations:

The auxiliary notations:

: <math>\alpha_i f = \Upsilon (f_i, f) = \Upsilon (f_i \Rightarrow f)</math>

: <math>\alpha_i f = \Upsilon \langle f_i, f \rangle = \Upsilon \langle f_i \Rightarrow f \rangle</math>

: <math>\beta_i f = \Upsilon (f, f_i) = \Upsilon (f \Rightarrow f_i)</math>

: <math>\beta_i f = \Upsilon \langle f, f_i \rangle = \Upsilon \langle f \Rightarrow f_i \rangle</math>

define two series of measures:

define two series of measures:

{| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"

{| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"

|+ '''Table 1.  Qualifiers of Implication Ordering:&nbsp; <math>\alpha_i f = \Upsilon (f_i \Rightarrow f)</math>'''

|+ '''Table 1.  Qualifiers of Implication Ordering:&nbsp; <math>\alpha_i f = \Upsilon \langle f_i \Rightarrow f \rangle</math>'''

|- style="background:ghostwhite"

|- style="background:ghostwhite"

| align="right" | <math>p:</math><br><math>q:</math>

| align="right" | <math>p:</math><br><math>q:</math>

{| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"

{| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"

|+ '''Table 2.  Qualifiers of Implication Ordering:&nbsp; <math>\beta_i f = \Upsilon (f \Rightarrow f_i)</math>'''

|+ '''Table 2.  Qualifiers of Implication Ordering:&nbsp; <math>\beta_i f = \Upsilon \langle f \Rightarrow f_i \rangle</math>'''

|- style="background:ghostwhite"

|- style="background:ghostwhite"

| align="right" | <math>p:</math><br><math>q:</math>

| align="right" | <math>p:</math><br><math>q:</math>