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Directory:Jon Awbrey/Papers/Functional Logic : Quantification Theory (view source)
Revision as of 16:26, 18 November 2008
, 16:26, 18 November 2008no edit summary
|+ '''Table 3. Higher Order Propositions (''n'' = 2)'''
|+ '''Table 3. Higher Order Propositions (''n'' = 2)'''
|- style="background:paleturquoise"
|- style="background:paleturquoise"
| align=right | ''x'' : || 1100 || ''f''
| align="right" | ''x'' : || 1100 || ''f''
|''m''||''m''||''m''||''m''||''m''||''m''||''m''||''m''
|''m''||''m''||''m''||''m''||''m''||''m''||''m''||''m''
|''m''||''m''||''m''||''m''||''m''||''m''||''m''||''m''
|''m''||''m''||''m''||''m''||''m''||''m''||''m''||''m''
|''m''||''m''||''m''||''m''||''m''||''m''||''m''||''m''
|''m''||''m''||''m''||''m''||''m''||''m''||''m''||''m''
|- style="background:paleturquoise"
|- style="background:paleturquoise"
| align=right | ''y'' : || 1010 ||
| align="right" | ''y'' : || 1010 ||
|0||1||2||3||4||5||6||7||8||9||10||11||12
|0||1||2||3||4||5||6||7||8||9||10||11||12
|13||14||15||16||17||18||19||20||21||22||23
|13||14||15||16||17||18||19||20||21||22||23
The auxiliary notations:
The auxiliary notations:
: α<sub>''i''</sub> ''f'' = Υ(''f''<sub>''i''</sub>, ''f''),
: <math>\alpha_i f = \Upsilon (f_i, f),\!</math>
: β<sub>''i''</sub> ''f'' = Υ(''f'', ''f''<sub>''i''</sub>),
: <math>\beta_i f = \Upsilon (f, f_i),\!</math>
define two series of measures:
define two series of measures:
: α<sub>''i''</sub>, β<sub>''i''</sub> : ('''B'''<sup>2</sup> → '''B''') → '''B''',
: <math>\alpha_i, \beta_i : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B},</math>
incidentally providing compact names for the column headings of the next two Tables.
incidentally providing compact names for the column headings of the next two Tables.
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ '''Table 4. Qualifiers of Implication Ordering: α<sub>''i'' </sub>''f'' = Υ(''f''<sub>''i''</sub> ⇒ ''f'')'''
|+ '''Table 4. Qualifiers of Implication Ordering: <math>\alpha_i f = \Upsilon (f_i \Rightarrow f)</math>'''
|- style="background:paleturquoise"
|- style="background:paleturquoise"
| align=right | ''x'' : || 1100 || ''f''
| align="right" | ''x'' : || 1100 || ''f''
|α||α||α||α||α||α||α||α
|α||α||α||α||α||α||α||α
|α||α||α||α||α||α||α||α
|α||α||α||α||α||α||α||α
|- style="background:paleturquoise"
|- style="background:paleturquoise"
| align=right | ''y'' : || 1010 ||
| align="right" | ''y'' : || 1010 ||
|15||14||13||12||11||10||9||8||7||6||5||4||3||2||1||0
|15||14||13||12||11||10||9||8||7||6||5||4||3||2||1||0
|-
|-
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
|+ '''Table 5. Qualifiers of Implication Ordering: β<sub>''i'' </sub>''f'' = Υ(''f'' ⇒ ''f''<sub>''i''</sub>)'''
|+ '''Table 5. Qualifiers of Implication Ordering: <math>\beta_i f = \Upsilon (f \Rightarrow f_i)</math>'''
|- style="background:paleturquoise"
|- style="background:paleturquoise"
| align=right | ''x'' : || 1100 || ''f''
| align="right" | ''x'' : || 1100 || ''f''
|β||β||β||β||β||β||β||β
|β||β||β||β||β||β||β||β
|β||β||β||β||β||β||β||β
|β||β||β||β||β||β||β||β
|- style="background:paleturquoise"
|- style="background:paleturquoise"
| align=right | ''y'' : || 1010 ||
| align="right" | ''y'' : || 1010 ||
|0||1||2||3||4||5||6||7||8||9||10||11||12||13||14||15
|0||1||2||3||4||5||6||7||8||9||10||11||12||13||14||15
|-
|-
{| cellpadding=2
{| cellpadding=2
| width=36 |
| width="36" |
| α<sub>00</sub> ''f'' = 1
| α<sub>00</sub> ''f'' = 1
| iff || ''f''<sub>00</sub> ⇒ ''f'',
| iff || ''f''<sub>00</sub> ⇒ ''f'',
{|
{|
| width=36 | || ''L''<sub>00</sub> ''f''
| width="36" | || ''L''<sub>00</sub> ''f''
|-
|-
| || = || ''L''<sub>"(x)(y)"</sub> ''f''
| || = || ''L''<sub>"(x)(y)"</sub> ''f''
{|
{|
| width=36 | || ''L''<sub>01</sub> ''f''
| width="36" | || ''L''<sub>01</sub> ''f''
|-
|-
| || = || ''L''<sub>"(x) y "</sub> ''f''
| || = || ''L''<sub>"(x) y "</sub> ''f''
{|
{|
| width=36 | || ''L''<sub>10</sub> ''f''
| width="36" | || ''L''<sub>10</sub> ''f''
|-
|-
| || = || ''L''<sub>" x (y)"</sub> ''f''
| || = || ''L''<sub>" x (y)"</sub> ''f''
{|
{|
| width=36 | || ''L''<sub>11</sub> ''f''
| width="36" | || ''L''<sub>11</sub> ''f''
|-
|-
| || = || ''L''<sub>" x y"</sub> ''f''
| || = || ''L''<sub>" x y"</sub> ''f''
|+ '''Table 9. Simple Qualifiers of Propositions (''n'' = 2)'''
|+ '''Table 9. Simple Qualifiers of Propositions (''n'' = 2)'''
|- style="background:paleturquoise"
|- style="background:paleturquoise"
| align=right | ''x'' : || 1100 || ''f''
| align="right" | ''x'' : || 1100 || ''f''
| (''L''<sub>11</sub>)
| (''L''<sub>11</sub>)
| (''L''<sub>10</sub>)
| (''L''<sub>10</sub>)
| ''L''<sub>11</sub>
| ''L''<sub>11</sub>
|- style="background:paleturquoise"
|- style="background:paleturquoise"
| align=right | ''y'' : || 1010 ||
| align="right" | ''y'' : || 1010 ||
| align=left | no x <br> is y
| align=left | no x <br> is y
| align=left | no x <br> is (y)
| align=left | no x <br> is (y)