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→‎Functional Quantifiers: \langle \rangle

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==Functional Quantifiers==

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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:

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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"

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

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

|}

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The auxiliary notations:

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

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

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

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

define two series of measures:

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

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

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

|- style="background:ghostwhite"

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

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

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

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

|- style="background:ghostwhite"

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

Jon Awbrey

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