Line 7,358: |
Line 7,358: |
| </pre> | | </pre> |
| | | |
− | ===Figure 70-b. Tangent Functor Ferris Wheel for F‹u, v› = ‹((u)(v)), ((u, v))›=== | + | ==Inquiry Driven Systems== |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |
| + | |+ Table 1. Sign Relation of Interpreter ''A'' |
| + | |- style="background:paleturquoise" |
| + | ! style="width:20%" | Object |
| + | ! style="width:20%" | Sign |
| + | ! style="width:20%" | Interpretant |
| + | |- |
| + | | ''A'' || "A" || "A" |
| + | |- |
| + | | ''A'' || "A" || "i" |
| + | |- |
| + | | ''A'' || "i" || "A" |
| + | |- |
| + | | ''A'' || "i" || "i" |
| + | |- |
| + | | ''B'' || "B" || "B" |
| + | |- |
| + | | ''B'' || "B" || "u" |
| + | |- |
| + | | ''B'' || "u" || "B" |
| + | |- |
| + | | ''B'' || "u" || "u" |
| + | |} |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%" |
| + | |+ Table 2. Sign Relation of Interpreter ''B'' |
| + | |- style="background:paleturquoise" |
| + | ! style="width:20%" | Object |
| + | ! style="width:20%" | Sign |
| + | ! style="width:20%" | Interpretant |
| + | |- |
| + | | ''A'' || "A" || "A" |
| + | |- |
| + | | ''A'' || "A" || "u" |
| + | |- |
| + | | ''A'' || "u" || "A" |
| + | |- |
| + | | ''A'' || "u" || "u" |
| + | |- |
| + | | ''B'' || "B" || "B" |
| + | |- |
| + | | ''B'' || "B" || "i" |
| + | |- |
| + | | ''B'' || "i" || "B" |
| + | |- |
| + | | ''B'' || "i" || "i" |
| + | |} |
| + | <br> |
| + | |
| + | <pre> |
| + | Table 3. Semiotic Partition of Interpreter A |
| + | "A" |
| + | "i" |
| + | "u" |
| + | "B" |
| + | </pre> |
| | | |
| <pre> | | <pre> |
− | o-----------------------o o-----------------------o o-----------------------o
| + | Table 4. Semiotic Partition of Interpreter B |
− | | dU | | dU | | dU | | + | "A" |
− | | o--o o--o | | o--o o--o | | o--o o--o | | + | "i" |
− | | /////\ /////\ | | /XXXX\ /XXXX\ | | /\\\\\ /\\\\\ | | + | "u" |
− | | ///////o//////\ | | /XXXXXXoXXXXXX\ | | /\\\\\\o\\\\\\\ | | + | "B" |
− | | //////// \//////\ | | /XXXXXX/ \XXXXXX\ | | /\\\\\\/ \\\\\\\\ | | + | </pre> |
− | | o/////// \//////o | | oXXXXXX/ \XXXXXXo | | o\\\\\\/ \\\\\\\o | | + | |
− | | |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| | | + | ==Logical Tables== |
− | | |/du//| |//dv/| | | |XXXXX| |XXXXX| | | |\du\\| |\\dv\| | | + | |
− | | |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| | | + | ===Higher Order Propositions=== |
− | | o//////\ ///////o | | oXXXXXX\ /XXXXXXo | | o\\\\\\\ /\\\\\\o | | + | |
− | | \//////\ //////// | | \XXXXXX\ /XXXXXX/ | | \\\\\\\\ /\\\\\\/ | | + | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
− | | \//////o/////// | | \XXXXXXoXXXXXX/ | | \\\\\\\o\\\\\\/ | | + | |+ '''Table 7. Higher Order Propositions (n = 1)''' |
− | | \///// \///// | | \XXXX/ \XXXX/ | | \\\\\/ \\\\\/ | | + | |- style="background:paleturquoise" |
− | | o--o o--o | | o--o o--o | | o--o o--o | | + | | \ ''x'' || 1 0 || ''F'' |
− | | | | | | | | + | |''m''||''m''||''m''||''m''||''m''||''m''||''m''||''m'' |
− | o-----------------------o o-----------------------o o-----------------------o
| + | |''m''||''m''||''m''||''m''||''m''||''m''||''m''||''m'' |
− | = du' @ (u)(v) o-----------------------o dv' @ (u)(v) =
| + | |- style="background:paleturquoise" |
− | = | dU' | =
| + | | ''F'' \ || || |
− | = | o--o o--o | =
| + | |00||01||02||03||04||05||06||07||08||09||10||11||12||13||14||15 |
− | = | /////\ /\\\\\ | =
| + | |- |
− | = | ///////o\\\\\\\ | =
| + | | ''F<sub>0</sub> || 0 0 || 0 ||0||1||0||1||0||1||0||1||0||1||0||1||0||1||0||1 |
− | = | ////////X\\\\\\\\ | =
| + | |- |
− | = | o///////XXX\\\\\\\o | =
| + | | ''F<sub>1</sub> || 0 1 || (x) ||0||0||1||1||0||0||1||1||0||0||1||1||0||0||1||1 |
− | = | |/////oXXXXXo\\\\\| | =
| + | |- |
− | = = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
| + | | ''F<sub>2</sub> || 1 0 || x ||0||0||0||0||1||1||1||1||0||0||0||0||1||1||1||1 |
− | | |/////oXXXXXo\\\\\| |
| + | |- |
− | | o//////\XXX/\\\\\\o |
| + | | ''F<sub>3</sub> || 1 1 || 1 ||0||0||0||0||0||0||0||0||1||1||1||1||1||1||1||1 |
− | | \//////\X/\\\\\\/ |
| + | |} |
− | | \//////o\\\\\\/ |
| + | <br> |
− | | \///// \\\\\/ |
| + | |
− | | o--o o--o |
| + | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
− | | |
| + | |+ '''Table 8. Interpretive Categories for Higher Order Propositions (n = 1)''' |
− | o-----------------------o
| + | |- style="background:paleturquoise" |
| + | |Measure||Happening||Exactness||Existence||Linearity||Uniformity||Information |
| + | |- |
| + | |''m''<sub>0</sub>||nothing happens|| || || || || |
| + | |- |
| + | |''m''<sub>1</sub>|| ||just false||nothing exists|| || || |
| + | |- |
| + | |''m''<sub>2</sub>|| ||just not x|| || || || |
| + | |- |
| + | |''m''<sub>3</sub>|| || ||nothing is x|| || || |
| + | |- |
| + | |''m''<sub>4</sub>|| ||just x|| || || || |
| + | |- |
| + | |''m''<sub>5</sub>|| || ||everything is x||F is linear|| || |
| + | |- |
| + | |''m''<sub>6</sub>|| || || || ||F is not uniform||F is informed |
| + | |- |
| + | |''m''<sub>7</sub>|| ||not just true|| || || || |
| + | |- |
| + | |''m''<sub>8</sub>|| ||just true|| || || || |
| + | |- |
| + | |''m''<sub>9</sub>|| || || || ||F is uniform||F is not informed |
| + | |- |
| + | |''m''<sub>10</sub>|| || ||something is not x||F is not linear|| || |
| + | |- |
| + | |''m''<sub>11</sub>|| ||not just x|| || || || |
| + | |- |
| + | |''m''<sub>12</sub>|| || ||something is x|| || || |
| + | |- |
| + | |''m''<sub>13</sub>|| ||not just not x|| || || || |
| + | |- |
| + | |''m''<sub>14</sub>|| ||not just false||something exists|| || || |
| + | |- |
| + | |''m''<sub>15</sub>||anything happens|| || || || || |
| + | |} |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |+ '''Table 9. Higher Order Propositions (n = 2)''' |
| + | |- style="background:paleturquoise" |
| + | | 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'' |
| + | |- style="background:paleturquoise" |
| + | | align=right | ''y'' : || 1010 || |
| + | |0||1||2||3||4||5||6||7||8||9||10||11||12 |
| + | |13||14||15||16||17||18||19||20||21||22||23 |
| + | |- |
| + | | ''f<sub>0</sub> || 0000 || ( ) |
| + | | 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 |
| + | | 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 |
| + | | 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 |
| + | |- |
| + | | ''f<sub>1</sub> || 0001 || (x)(y) |
| + | | || || 1 || 1 || 0 || 0 || 1 || 1 |
| + | | 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 |
| + | | 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 |
| + | |- |
| + | | ''f<sub>2</sub> || 0010 || (x) y |
| + | | || || || || 1 || 1 || 1 || 1 |
| + | | 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 |
| + | | 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 |
| + | |- |
| + | | ''f<sub>3</sub> || 0011 || (x) |
| + | | || || || || || || || |
| + | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 |
| + | | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 |
| + | |- |
| + | | ''f<sub>4</sub> || 0100 || x (y) |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 |
| + | |- |
| + | | ''f<sub>5</sub> || 0101 || (y) |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | |- |
| + | | ''f<sub>6</sub> || 0110 || (x, y) |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | |- |
| + | | ''f<sub>7</sub> || 0111 || (x y) |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | |- |
| + | | ''f<sub>8</sub> || 1000 || x y |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | |- |
| + | | ''f<sub>9</sub> || 1001 || ((x, y)) |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | |- |
| + | | ''f<sub>10</sub> || 1010 || y |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | |- |
| + | | ''f<sub>11</sub> || 1011 || (x (y)) |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | |- |
| + | | ''f<sub>12</sub> || 1100 || x |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | |- |
| + | | ''f<sub>13</sub> || 1101 || ((x) y) |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | |- |
| + | | ''f<sub>14</sub> || 1110 || ((x)(y)) |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | |- |
| + | | ''f<sub>15</sub> || 1111 || (( )) |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | | || || || || || || || |
| + | |} |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |+ '''Table 10. Qualifiers of Implication Ordering: α<sub>''i'' </sub>''f'' = Υ(''f''<sub>''i''</sub> ⇒ ''f'')''' |
| + | |- style="background:paleturquoise" |
| + | | align=right | ''x'' : || 1100 || ''f'' |
| + | |α||α||α||α||α||α||α||α |
| + | |α||α||α||α||α||α||α||α |
| + | |- style="background:paleturquoise" |
| + | | align=right | ''y'' : || 1010 || |
| + | |15||14||13||12||11||10||9||8||7||6||5||4||3||2||1||0 |
| + | |- |
| + | | ''f<sub>0</sub> || 0000 || ( ) |
| + | | || || || || || || || |
| + | | || || || || || || || 1 |
| + | |- |
| + | | ''f<sub>1</sub> || 0001 || (x)(y) |
| + | | || || || || || || || |
| + | | || || || || || || 1 || 1 |
| + | |- |
| + | | ''f<sub>2</sub> || 0010 || (x) y |
| + | | || || || || || || || |
| + | | || || || || || 1 || || 1 |
| + | |- |
| + | | ''f<sub>3</sub> || 0011 || (x) |
| + | | || || || || || || || |
| + | | || || || || 1 || 1 || 1 || 1 |
| + | |- |
| + | | ''f<sub>4</sub> || 0100 || x (y) |
| + | | || || || || || || || |
| + | | || || || 1 || || || || 1 |
| + | |- |
| + | | ''f<sub>5</sub> || 0101 || (y) |
| + | | || || || || || || || |
| + | | || || 1 || 1 || || || 1 || 1 |
| + | |- |
| + | | ''f<sub>6</sub> || 0110 || (x, y) |
| + | | || || || || || || || |
| + | | || 1 || || 1 || || 1 || || 1 |
| + | |- |
| + | | ''f<sub>7</sub> || 0111 || (x y) |
| + | | || || || || || || || |
| + | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 |
| + | |- |
| + | | ''f<sub>8</sub> || 1000 || x y |
| + | | || || || || || || || 1 |
| + | | || || || || || || || 1 |
| + | |- |
| + | | ''f<sub>9</sub> || 1001 || ((x, y)) |
| + | | || || || || || || 1 || 1 |
| + | | || || || || || || 1 || 1 |
| + | |- |
| + | | ''f<sub>10</sub> || 1010 || y |
| + | | || || || || || 1 || || 1 |
| + | | || || || || || 1 || || 1 |
| + | |- |
| + | | ''f<sub>11</sub> || 1011 || (x (y)) |
| + | | || || || || 1 || 1 || 1 || 1 |
| + | | || || || || 1 || 1 || 1 || 1 |
| + | |- |
| + | | ''f<sub>12</sub> || 1100 || x |
| + | | || || || 1 || || || || 1 |
| + | | || || || 1 || || || || 1 |
| + | |- |
| + | | ''f<sub>13</sub> || 1101 || ((x) y) |
| + | | || || 1 || 1 || || || 1 || 1 |
| + | | || || 1 || 1 || || || 1 || 1 |
| + | |- |
| + | | ''f<sub>14</sub> || 1110 || ((x)(y)) |
| + | | || 1 || || 1 || || 1 || || 1 |
| + | | || 1 || || 1 || || 1 || || 1 |
| + | |- |
| + | | ''f<sub>15</sub> || 1111 || (( )) |
| + | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 |
| + | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 |
| + | |} |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |+ '''Table 11. Qualifiers of Implication Ordering: β<sub>''i'' </sub>''f'' = Υ(''f'' ⇒ ''f''<sub>''i''</sub>)''' |
| + | |- style="background:paleturquoise" |
| + | | align=right | ''x'' : || 1100 || ''f'' |
| + | |β||β||β||β||β||β||β||β |
| + | |β||β||β||β||β||β||β||β |
| + | |- style="background:paleturquoise" |
| + | | align=right | ''y'' : || 1010 || |
| + | |0||1||2||3||4||5||6||7||8||9||10||11||12||13||14||15 |
| + | |- |
| + | | ''f<sub>0</sub> || 0000 || ( ) |
| + | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 |
| + | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 |
| + | |- |
| + | | ''f<sub>1</sub> || 0001 || (x)(y) |
| + | | || 1 || || 1 || || 1 || || 1 |
| + | | || 1 || || 1 || || 1 || || 1 |
| + | |- |
| + | | ''f<sub>2</sub> || 0010 || (x) y |
| + | | || || 1 || 1 || || || 1 || 1 |
| + | | || || 1 || 1 || || || 1 || 1 |
| + | |- |
| + | | ''f<sub>3</sub> || 0011 || (x) |
| + | | || || || 1 || || || || 1 |
| + | | || || || 1 || || || || 1 |
| + | |- |
| + | | ''f<sub>4</sub> || 0100 || x (y) |
| + | | || || || || 1 || 1 || 1 || 1 |
| + | | || || || || 1 || 1 || 1 || 1 |
| + | |- |
| + | | ''f<sub>5</sub> || 0101 || (y) |
| + | | || || || || || 1 || || 1 |
| + | | || || || || || 1 || || 1 |
| + | |- |
| + | | ''f<sub>6</sub> || 0110 || (x, y) |
| + | | || || || || || || 1 || 1 |
| + | | || || || || || || 1 || 1 |
| + | |- |
| + | | ''f<sub>7</sub> || 0111 || (x y) |
| + | | || || || || || || || 1 |
| + | | || || || || || || || 1 |
| + | |- |
| + | | ''f<sub>8</sub> || 1000 || x y |
| + | | || || || || || || || |
| + | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 |
| + | |- |
| + | | ''f<sub>9</sub> || 1001 || ((x, y)) |
| + | | || || || || || || || |
| + | | || 1 || || 1 || || 1 || || 1 |
| + | |- |
| + | | ''f<sub>10</sub> || 1010 || y |
| + | | || || || || || || || |
| + | | || || 1 || 1 || || || 1 || 1 |
| + | |- |
| + | | ''f<sub>11</sub> || 1011 || (x (y)) |
| + | | || || || || || || || |
| + | | || || || 1 || || || || 1 |
| + | |- |
| + | | ''f<sub>12</sub> || 1100 || x |
| + | | || || || || || || || |
| + | | || || || || 1 || 1 || 1 || 1 |
| + | |- |
| + | | ''f<sub>13</sub> || 1101 || ((x) y) |
| + | | || || || || || || || |
| + | | || || || || || 1 || || 1 |
| + | |- |
| + | | ''f<sub>14</sub> || 1110 || ((x)(y)) |
| + | | || || || || || || || |
| + | | || || || || || || 1 || 1 |
| + | |- |
| + | | ''f<sub>15</sub> || 1111 || (( )) |
| + | | || || || || || || || |
| + | | || || || || || || || 1 |
| + | |} |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |+ '''Table 13. Syllogistic Premisses as Higher Order Indicator Functions''' |
| + | | A |
| + | | align=left | Universal Affirmative |
| + | | align=left | All |
| + | | x || is || y |
| + | | align=left | Indicator of " x (y)" = 0 |
| + | |- |
| + | | E |
| + | | align=left | Universal Negative |
| + | | align=left | All |
| + | | x || is || (y) |
| + | | align=left | Indicator of " x y " = 0 |
| + | |- |
| + | | I |
| + | | align=left | Particular Affirmative |
| + | | align=left | Some |
| + | | x || is || y |
| + | | align=left | Indicator of " x y " = 1 |
| + | |- |
| + | | O |
| + | | align=left | Particular Negative |
| + | | align=left | Some |
| + | | x || is || (y) |
| + | | align=left | Indicator of " x (y)" = 1 |
| + | |} |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |+ '''Table 14. Relation of Quantifiers to Higher Order Propositions''' |
| + | |- style="background:paleturquoise" |
| + | |Mnemonic||Category||Classical Form||Alternate Form||Symmetric Form||Operator |
| + | |- |
| + | | E<br>Exclusive |
| + | | Universal<br>Negative |
| + | | align=left | All x is (y) |
| + | | align=left | |
| + | | align=left | No x is y |
| + | | (''L''<sub>11</sub>) |
| + | |- |
| + | | A<br>Absolute |
| + | | Universal<br>Affirmative |
| + | | align=left | All x is y |
| + | | align=left | |
| + | | align=left | No x is (y) |
| + | | (''L''<sub>10</sub>) |
| + | |- |
| + | | |
| + | | |
| + | | align=left | All y is x |
| + | | align=left | No y is (x) |
| + | | align=left | No (x) is y |
| + | | (''L''<sub>01</sub>) |
| + | |- |
| + | | |
| + | | |
| + | | align=left | All (y) is x |
| + | | align=left | No (y) is (x) |
| + | | align=left | No (x) is (y) |
| + | | (''L''<sub>00</sub>) |
| + | |- |
| + | | |
| + | | |
| + | | align=left | Some (x) is (y) |
| + | | align=left | |
| + | | align=left | Some (x) is (y) |
| + | | ''L''<sub>00</sub> |
| + | |- |
| + | | |
| + | | |
| + | | align=left | Some (x) is y |
| + | | align=left | |
| + | | align=left | Some (x) is y |
| + | | ''L''<sub>01</sub> |
| + | |- |
| + | | O<br>Obtrusive |
| + | | Particular<br>Negative |
| + | | align=left | Some x is (y) |
| + | | align=left | |
| + | | align=left | Some x is (y) |
| + | | ''L''<sub>10</sub> |
| + | |- |
| + | | I<br>Indefinite |
| + | | Particular<br>Affirmative |
| + | | align=left | Some x is y |
| + | | align=left | |
| + | | align=left | Some x is y |
| + | | ''L''<sub>11</sub> |
| + | |} |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
| + | |+ '''Table 15. Simple Qualifiers of Propositions (n = 2)''' |
| + | |- style="background:paleturquoise" |
| + | | align=right | ''x'' : || 1100 || ''f'' |
| + | | (''L''<sub>11</sub>) |
| + | | (''L''<sub>10</sub>) |
| + | | (''L''<sub>01</sub>) |
| + | | (''L''<sub>00</sub>) |
| + | | ''L''<sub>00</sub> |
| + | | ''L''<sub>01</sub> |
| + | | ''L''<sub>10</sub> |
| + | | ''L''<sub>11</sub> |
| + | |- style="background:paleturquoise" |
| + | | 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) |
| + | | align=left | some (x) <br> is (y) |
| + | | align=left | some (x) <br> is y |
| + | | align=left | some x <br> is (y) |
| + | | align=left | some x <br> is y |
| + | |- |
| + | | ''f<sub>0</sub> || 0000 || ( ) |
| + | | 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0 |
| + | |- |
| + | | ''f<sub>1</sub> || 0001 || (x)(y) |
| + | | 1 || 1 || 1 || 0 || 1 || 0 || 0 || 0 |
| + | |- |
| + | | ''f<sub>2</sub> || 0010 || (x) y |
| + | | 1 || 1 || 0 || 1 || 0 || 1 || 0 || 0 |
| + | |- |
| + | | ''f<sub>3</sub> || 0011 || (x) |
| + | | 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0 |
| + | |- |
| + | | ''f<sub>4</sub> || 0100 || x (y) |
| + | | 1 || 0 || 1 || 1 || 0 || 0 || 1 || 0 |
| + | |- |
| + | | ''f<sub>5</sub> || 0101 || (y) |
| + | | 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 |
| + | |- |
| + | | ''f<sub>6</sub> || 0110 || (x, y) |
| + | | 1 || 0 || 0 || 1 || 0 || 1 || 1 || 0 |
| + | |- |
| + | | ''f<sub>7</sub> || 0111 || (x y) |
| + | | 1 || 0 || 0 || 0 || 1 || 1 || 1 || 0 |
| + | |- |
| + | | ''f<sub>8</sub> || 1000 || x y |
| + | | 0 || 1 || 1 || 1 || 0 || 0 || 0 || 1 |
| + | |- |
| + | | ''f<sub>9</sub> || 1001 || ((x, y)) |
| + | | 0 || 1 || 1 || 0 || 1 || 0 || 0 || 1 |
| + | |- |
| + | | ''f<sub>10</sub> || 1010 || y |
| + | | 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 |
| + | |- |
| + | | ''f<sub>11</sub> || 1011 || (x (y)) |
| + | | 0 || 1 || 0 || 0 || 1 || 1 || 0 || 1 |
| + | |- |
| + | | ''f<sub>12</sub> || 1100 || x |
| + | | 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 |
| + | |- |
| + | | ''f<sub>13</sub> || 1101 || ((x) y) |
| + | | 0 || 0 || 1 || 0 || 1 || 0 || 1 || 1 |
| + | |- |
| + | | ''f<sub>14</sub> || 1110 || ((x)(y)) |
| + | | 0 || 0 || 0 || 1 || 0 || 1 || 1 || 1 |
| + | |- |
| + | | ''f<sub>15</sub> || 1111 || (( )) |
| + | | 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 |
| + | |} |
| + | <br> |
| + | |
| + | Table 7. Higher Order Propositions (n = 1) |
| + | o------o-----o-----o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o---o |
| + | | \ x | 1 0 | F |m |m |m |m |m |m |m |m |m |m |m |m |m |m |m |m | |
| + | | F \ | | |00|01|02|03|04|05|06|07|08|09|10|11|12|13|14|15 | |
| + | o------o-----o-----o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o---o |
| + | | | | | | |
| + | | F_0 | 0 0 | 0 | 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 | |
| + | | | | | | |
| + | | F_1 | 0 1 | (x) | 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 | |
| + | | | | | | |
| + | | F_2 | 1 0 | x | 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 | |
| + | | | | | | |
| + | | F_3 | 1 1 | 1 | 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 | |
| + | | | | | | |
| + | o------o-----o-----o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o---o |
| + | <br> |
| + | |
| + | Table 8. Interpretive Categories for Higher Order Propositions (n = 1) |
| + | o-------o----------o------------o------------o----------o----------o-----------o |
| + | |Measure| Happening| Exactness | Existence | Linearity|Uniformity|Information| |
| + | o-------o----------o------------o------------o----------o----------o-----------o |
| + | | m_0 | nothing | | | | | | |
| + | | | happens | | | | | | |
| + | o-------o----------o------------o------------o----------o----------o-----------o |
| + | | m_1 | | | nothing | | | | |
| + | | | | just false | exists | | | | |
| + | o-------o----------o------------o------------o----------o----------o-----------o |
| + | | m_2 | | | | | | | |
| + | | | | just not x | | | | | |
| + | o-------o----------o------------o------------o----------o----------o-----------o |
| + | | m_3 | | | nothing | | | | |
| + | | | | | is x | | | | |
| + | o-------o----------o------------o------------o----------o----------o-----------o |
| + | | m_4 | | | | | | | |
| + | | | | just x | | | | | |
| + | o-------o----------o------------o------------o----------o----------o-----------o |
| + | | m_5 | | | everything | F is | | | |
| + | | | | | is x | linear | | | |
| + | o-------o----------o------------o------------o----------o----------o-----------o |
| + | | m_6 | | | | | F is not | F is | |
| + | | | | | | | uniform | informed | |
| + | o-------o----------o------------o------------o----------o----------o-----------o |
| + | | m_7 | | not | | | | | |
| + | | | | just true | | | | | |
| + | o-------o----------o------------o------------o----------o----------o-----------o |
| + | | m_8 | | | | | | | |
| + | | | | just true | | | | | |
| + | o-------o----------o------------o------------o----------o----------o-----------o |
| + | | m_9 | | | | | F is | F is not | |
| + | | | | | | | uniform | informed | |
| + | o-------o----------o------------o------------o----------o----------o-----------o |
| + | | m_10 | | | something | F is not | | | |
| + | | | | | is not x | linear | | | |
| + | o-------o----------o------------o------------o----------o----------o-----------o |
| + | | m_11 | | not | | | | | |
| + | | | | just x | | | | | |
| + | o-------o----------o------------o------------o----------o----------o-----------o |
| + | | m_12 | | | something | | | | |
| + | | | | | is x | | | | |
| + | o-------o----------o------------o------------o----------o----------o-----------o |
| + | | m_13 | | not | | | | | |
| + | | | | just not x | | | | | |
| + | o-------o----------o------------o------------o----------o----------o-----------o |
| + | | m_14 | | not | something | | | | |
| + | | | | just false | exists | | | | |
| + | o-------o----------o------------o------------o----------o----------o-----------o |
| + | | m_15 | anything | | | | | | |
| + | | | happens | | | | | | |
| + | o-------o----------o------------o------------o----------o----------o-----------o |
| + | <br> |
| + | |
| + | Table 9. Higher Order Propositions (n = 2) |
| + | o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o |
| + | | | x | 1100 | f |m|m|m|m|m|m|m|m|m|m|m|m|m|m|m|m|.| |
| + | | | y | 1010 | |0|0|0|0|0|0|0|0|0|0|1|1|1|1|1|1|.| |
| + | | f \ | | |0|1|2|3|4|5|6|7|8|9|0|1|2|3|4|5|.| |
| + | o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o |
| + | | | | | | |
| + | | f_0 | 0000 | () |0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 | |
| + | | | | | | |
| + | | f_1 | 0001 | (x)(y) | 1 1 0 0 1 1 0 0 1 1 0 0 1 1 | |
| + | | | | | | |
| + | | f_2 | 0010 | (x) y | 1 1 1 1 0 0 0 0 1 1 1 1 | |
| + | | | | | | |
| + | | f_3 | 0011 | (x) | 1 1 1 1 1 1 1 1 | |
| + | | | | | | |
| + | | f_4 | 0100 | x (y) | | |
| + | | | | | | |
| + | | f_5 | 0101 | (y) | | |
| + | | | | | | |
| + | | f_6 | 0110 | (x, y) | | |
| + | | | | | | |
| + | | f_7 | 0111 | (x y) | | |
| + | | | | | | |
| + | o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o |
| + | | | | | | |
| + | | f_8 | 1000 | x y | | |
| + | | | | | | |
| + | | f_9 | 1001 | ((x, y)) | | |
| + | | | | | | |
| + | | f_10 | 1010 | y | | |
| + | | | | | | |
| + | | f_11 | 1011 | (x (y)) | | |
| + | | | | | | |
| + | | f_12 | 1100 | x | | |
| + | | | | | | |
| + | | f_13 | 1101 | ((x) y) | | |
| + | | | | | | |
| + | | f_14 | 1110 | ((x)(y)) | | |
| + | | | | | | |
| + | | f_15 | 1111 | (()) | | |
| + | | | | | | |
| + | o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o |
| + | <br> |
| + | |
| + | Table 10. Qualifiers of Implication Ordering: !a!_i f = !Y!(f_i => f) |
| + | o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o |
| + | | | x | 1100 | f |a |a |a |a |a |a |a |a |a |a |a |a |a |a |a |a | |
| + | | | y | 1010 | |1 |1 |1 |1 |1 |1 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 | |
| + | | f \ | | |5 |4 |3 |2 |1 |0 |9 |8 |7 |6 |5 |4 |3 |2 |1 |0 | |
| + | o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o |
| + | | | | | | |
| + | | f_0 | 0000 | () | 1 | |
| + | | | | | | |
| + | | f_1 | 0001 | (x)(y) | 1 1 | |
| + | | | | | | |
| + | | f_2 | 0010 | (x) y | 1 1 | |
| + | | | | | | |
| + | | f_3 | 0011 | (x) | 1 1 1 1 | |
| + | | | | | | |
| + | | f_4 | 0100 | x (y) | 1 1 | |
| + | | | | | | |
| + | | f_5 | 0101 | (y) | 1 1 1 1 | |
| + | | | | | | |
| + | | f_6 | 0110 | (x, y) | 1 1 1 1 | |
| + | | | | | | |
| + | | f_7 | 0111 | (x y) | 1 1 1 1 1 1 1 1 | |
| + | | | | | | |
| + | | f_8 | 1000 | x y | 1 1 | |
| + | | | | | | |
| + | | f_9 | 1001 | ((x, y)) | 1 1 1 1 | |
| + | | | | | | |
| + | | f_10 | 1010 | y | 1 1 1 1 | |
| + | | | | | | |
| + | | f_11 | 1011 | (x (y)) | 1 1 1 1 1 1 1 1 | |
| + | | | | | | |
| + | | f_12 | 1100 | x | 1 1 1 1 | |
| + | | | | | | |
| + | | f_13 | 1101 | ((x) y) | 1 1 1 1 1 1 1 1 | |
| + | | | | | | |
| + | | f_14 | 1110 | ((x)(y)) | 1 1 1 1 1 1 1 1 | |
| + | | | | | | |
| + | | f_15 | 1111 | (()) |1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 | |
| + | | | | | | |
| + | o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o |
| + | <br> |
| + | |
| + | Table 11. Qualifiers of Implication Ordering: !b!_i f = !Y!(f => f_i) |
| + | o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o |
| + | | | x | 1100 | f |b |b |b |b |b |b |b |b |b |b |b |b |b |b |b |b | |
| + | | | y | 1010 | |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |1 |1 |1 |1 |1 |1 | |
| + | | f \ | | |0 |1 |2 |3 |4 |5 |6 |7 |8 |9 |0 |1 |2 |3 |4 |5 | |
| + | o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o |
| + | | | | | | |
| + | | f_0 | 0000 | () |1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 | |
| + | | | | | | |
| + | | f_1 | 0001 | (x)(y) | 1 1 1 1 1 1 1 1 | |
| + | | | | | | |
| + | | f_2 | 0010 | (x) y | 1 1 1 1 1 1 1 1 | |
| + | | | | | | |
| + | | f_3 | 0011 | (x) | 1 1 1 1 | |
| + | | | | | | |
| + | | f_4 | 0100 | x (y) | 1 1 1 1 1 1 1 1 | |
| + | | | | | | |
| + | | f_5 | 0101 | (y) | 1 1 1 1 | |
| + | | | | | | |
| + | | f_6 | 0110 | (x, y) | 1 1 1 1 | |
| + | | | | | | |
| + | | f_7 | 0111 | (x y) | 1 1 | |
| + | | | | | | |
| + | | f_8 | 1000 | x y | 1 1 1 1 1 1 1 1 | |
| + | | | | | | |
| + | | f_9 | 1001 | ((x, y)) | 1 1 1 1 | |
| + | | | | | | |
| + | | f_10 | 1010 | y | 1 1 1 1 | |
| + | | | | | | |
| + | | f_11 | 1011 | (x (y)) | 1 1 | |
| + | | | | | | |
| + | | f_12 | 1100 | x | 1 1 1 1 | |
| + | | | | | | |
| + | | f_13 | 1101 | ((x) y) | 1 1 | |
| + | | | | | | |
| + | | f_14 | 1110 | ((x)(y)) | 1 1 | |
| + | | | | | | |
| + | | f_15 | 1111 | (()) | 1 | |
| + | | | | | | |
| + | o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o |
| + | <br> |
| + | |
| + | Table 13. Syllogistic Premisses as Higher Order Indicator Functions |
| + | o---o------------------------o-----------------o---------------------------o |
| + | | | | | | |
| + | | A | Universal Affirmative | All x is y | Indicator of " x (y)" = 0 | |
| + | | | | | | |
| + | | E | Universal Negative | All x is (y) | Indicator of " x y " = 0 | |
| + | | | | | | |
| + | | I | Particular Affirmative | Some x is y | Indicator of " x y " = 1 | |
| + | | | | | | |
| + | | O | Particular Negative | Some x is (y) | Indicator of " x (y)" = 1 | |
| + | | | | | | |
| + | o---o------------------------o-----------------o---------------------------o |
| + | <br> |
| + | |
| + | Table 14. Relation of Quantifiers to Higher Order Propositions |
| + | o------------o------------o-----------o-----------o-----------o-----------o |
| + | | Mnemonic | Category | Classical | Alternate | Symmetric | Operator | |
| + | | | | Form | Form | Form | | |
| + | o============o============o===========o===========o===========o===========o |
| + | | E | Universal | All x | | No x | (L_11) | |
| + | | Exclusive | Negative | is (y) | | is y | | |
| + | o------------o------------o-----------o-----------o-----------o-----------o |
| + | | A | Universal | All x | | No x | (L_10) | |
| + | | Absolute | Affrmtve | is y | | is (y) | | |
| + | o------------o------------o-----------o-----------o-----------o-----------o |
| + | | | | All y | No y | No (x) | (L_01) | |
| + | | | | is x | is (x) | is y | | |
| + | o------------o------------o-----------o-----------o-----------o-----------o |
| + | | | | All (y) | No (y) | No (x) | (L_00) | |
| + | | | | is x | is (x) | is (y) | | |
| + | o------------o------------o-----------o-----------o-----------o-----------o |
| + | | | | Some (x) | | Some (x) | L_00 | |
| + | | | | is (y) | | is (y) | | |
| + | o------------o------------o-----------o-----------o-----------o-----------o |
| + | | | | Some (x) | | Some (x) | L_01 | |
| + | | | | is y | | is y | | |
| + | o------------o------------o-----------o-----------o-----------o-----------o |
| + | | O | Particular | Some x | | Some x | L_10 | |
| + | | Obtrusive | Negative | is (y) | | is (y) | | |
| + | o------------o------------o-----------o-----------o-----------o-----------o |
| + | | I | Particular | Some x | | Some x | L_11 | |
| + | | Indefinite | Affrmtve | is y | | is y | | |
| + | o------------o------------o-----------o-----------o-----------o-----------o |
| + | <br> |
| + | |
| + | Table 15. Simple Qualifiers of Propositions (n = 2) |
| + | o------o------o----------o-----o-----o-----o-----o-----o-----o-----o-----o |
| + | | | x | 1100 | f |(L11)|(L10)|(L01)|(L00)| L00 | L01 | L10 | L11 | |
| + | | | y | 1010 | |no x|no x|no ~x|no ~x|sm ~x|sm ~x|sm x|sm x| |
| + | | f \ | | |is y|is ~y|is y|is ~y|is ~y|is y|is ~y|is y| |
| + | o------o------o----------o-----o-----o-----o-----o-----o-----o-----o-----o |
| + | | | | | | |
| + | | f_0 | 0000 | () | 1 1 1 1 0 0 0 0 | |
| + | | | | | | |
| + | | f_1 | 0001 | (x)(y) | 1 1 1 0 1 0 0 0 | |
| + | | | | | | |
| + | | f_2 | 0010 | (x) y | 1 1 0 1 0 1 0 0 | |
| + | | | | | | |
| + | | f_3 | 0011 | (x) | 1 1 0 0 1 1 0 0 | |
| + | | | | | | |
| + | | f_4 | 0100 | x (y) | 1 0 1 1 0 0 1 0 | |
| + | | | | | | |
| + | | f_5 | 0101 | (y) | 1 0 1 0 1 0 1 0 | |
| + | | | | | | |
| + | | f_6 | 0110 | (x, y) | 1 0 0 1 0 1 1 0 | |
| + | | | | | | |
| + | | f_7 | 0111 | (x y) | 1 0 0 0 1 1 1 0 | |
| + | | | | | | |
| + | | f_8 | 1000 | x y | 0 1 1 1 0 0 0 1 | |
| + | | | | | | |
| + | | f_9 | 1001 | ((x, y)) | 0 1 1 0 1 0 0 1 | |
| + | | | | | | |
| + | | f_10 | 1010 | y | 0 1 0 1 0 1 0 1 | |
| + | | | | | | |
| + | | f_11 | 1011 | (x (y)) | 0 1 0 0 1 1 0 1 | |
| + | | | | | | |
| + | | f_12 | 1100 | x | 0 0 1 1 0 0 1 1 | |
| + | | | | | | |
| + | | f_13 | 1101 | ((x) y) | 0 0 1 0 1 0 1 1 | |
| + | | | | | | |
| + | | f_14 | 1110 | ((x)(y)) | 0 0 0 1 0 1 1 1 | |
| + | | | | | | |
| + | | f_15 | 1111 | (()) | 0 0 0 0 1 1 1 1 | |
| + | | | | | | |
| + | o------o------o----------o-----o-----o-----o-----o-----o-----o-----o-----o |
| + | <br> |
| + | |
| + | ===[[Zeroth Order Logic]]=== |
| | | |
− | o-----------------------o o-----------------------o o-----------------------o
| + | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
− | | dU | | dU | | dU | | + | |+ '''Table 1. Propositional Forms on Two Variables''' |
− | | o--o o--o | | o--o o--o | | o--o o--o | | + | |- style="background:paleturquoise" |
− | | / \ /////\ | | /\\\\\ /XXXX\ | | /\\\\\ /\\\\\ | | + | ! style="width:15%" | L<sub>1</sub> |
− | | / o//////\ | | /\\\\\\oXXXXXX\ | | /\\\\\\o\\\\\\\ | | + | ! style="width:15%" | L<sub>2</sub> |
− | | / //\//////\ | | /\\\\\\//\XXXXXX\ | | /\\\\\\/ \\\\\\\\ | | + | ! style="width:15%" | L<sub>3</sub> |
− | | o ////\//////o | | o\\\\\\////\XXXXXXo | | o\\\\\\/ \\\\\\\o | | + | ! style="width:15%" | L<sub>4</sub> |
− | | | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| | | + | ! style="width:15%" | L<sub>5</sub> |
− | | | du |/////|//dv/| | | |\\\\\|/////|XXXXX| | | |\du\\| |\\dv\| | | + | ! style="width:15%" | L<sub>6</sub> |
− | | | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| | | + | |- style="background:paleturquoise" |
− | | o \//////////o | | o\\\\\\\////XXXXXXo | | o\\\\\\\ /\\\\\\o | | + | | |
− | | \ \///////// | | \\\\\\\\//XXXXXX/ | | \\\\\\\\ /\\\\\\/ | | + | | align="right" | x : |
− | | \ o/////// | | \\\\\\\oXXXXXX/ | | \\\\\\\o\\\\\\/ | | + | | 1 1 0 0 |
− | | \ / \///// | | \\\\\/ \XXXX/ | | \\\\\/ \\\\\/ | | + | | |
− | | o--o o--o | | o--o o--o | | o--o o--o | | + | | |
− | | | | | | | | + | | |
− | o-----------------------o o-----------------------o o-----------------------o
| + | |- style="background:paleturquoise" |
− | = du' @ (u) v o-----------------------o dv' @ (u) v =
| + | | |
− | = | dU' | =
| + | | align="right" | y : |
− | = | o--o o--o | =
| + | | 1 0 1 0 |
− | = | /////\ /\\\\\ | =
| + | | |
− | = | ///////o\\\\\\\ | =
| + | | |
− | = | ////////X\\\\\\\\ | =
| + | | |
− | = | o///////XXX\\\\\\\o | =
| + | |- |
− | = | |/////oXXXXXo\\\\\| | =
| + | | f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || ( ) || false || 0 |
− | = = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
| + | |- |
− | | |/////oXXXXXo\\\\\| |
| + | | f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || ¬x ∧ ¬y |
− | | o//////\XXX/\\\\\\o |
| + | |- |
− | | \//////\X/\\\\\\/ |
| + | | f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || ¬x ∧ y |
− | | \//////o\\\\\\/ |
| + | |- |
− | | \///// \\\\\/ |
| + | | f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || ¬x |
− | | o--o o--o |
| + | |- |
− | | |
| + | | f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x ∧ ¬y |
− | o-----------------------o
| + | |- |
| + | | f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || ¬y |
| + | |- |
| + | | f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x ≠ y |
| + | |- |
| + | | f<sub>7</sub> || f<sub>0111</sub> || 0 1 1 1 || (x y) || not both x and y || ¬x ∨ ¬y |
| + | |- |
| + | | f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x y || x and y || x ∧ y |
| + | |- |
| + | | f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y |
| + | |- |
| + | | f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y |
| + | |- |
| + | | f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x → y |
| + | |- |
| + | | f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x |
| + | |- |
| + | | f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x ← y |
| + | |- |
| + | | f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y || x ∨ y |
| + | |- |
| + | | f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || (( )) || true || 1 |
| + | |} |
| + | <br> |
| | | |
− | o-----------------------o o-----------------------o o-----------------------o
| + | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:90%" |
− | | dU | | dU | | dU | | + | |+ '''Table 1. Propositional Forms on Two Variables''' |
− | | o--o o--o | | o--o o--o | | o--o o--o | | + | |- style="background:aliceblue" |
− | | /////\ / \ | | /XXXX\ /\\\\\ | | /\\\\\ /\\\\\ | | + | ! style="width:15%" | L<sub>1</sub> |
− | | ///////o \ | | /XXXXXXo\\\\\\\ | | /\\\\\\o\\\\\\\ | | + | ! style="width:15%" | L<sub>2</sub> |
− | | /////////\ \ | | /XXXXXX//\\\\\\\\ | | /\\\\\\/ \\\\\\\\ | | + | ! style="width:15%" | L<sub>3</sub> |
− | | o//////////\ o | | oXXXXXX////\\\\\\\o | | o\\\\\\/ \\\\\\\o | | + | ! style="width:15%" | L<sub>4</sub> |
− | | |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| | | + | ! style="width:15%" | L<sub>5</sub> |
− | | |/du//|/////| dv | | | |XXXXX|/////|\\\\\| | | |\du\\| |\\dv\| | | + | ! style="width:15%" | L<sub>6</sub> |
− | | |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| | | + | |- style="background:aliceblue" |
− | | o//////\//// o | | oXXXXXX\////\\\\\\o | | o\\\\\\\ /\\\\\\o | | + | | |
− | | \//////\// / | | \XXXXXX\//\\\\\\/ | | \\\\\\\\ /\\\\\\/ | | + | | align="right" | x : |
− | | \//////o / | | \XXXXXXo\\\\\\/ | | \\\\\\\o\\\\\\/ | | + | | 1 1 0 0 |
− | | \///// \ / | | \XXXX/ \\\\\/ | | \\\\\/ \\\\\/ | | + | | |
− | | o--o o--o | | o--o o--o | | o--o o--o | | + | | |
− | | | | | | | | + | | |
− | o-----------------------o o-----------------------o o-----------------------o
| + | |- style="background:aliceblue" |
− | = du' @ u (v) o-----------------------o dv' @ u (v) =
| + | | |
− | = | dU' | =
| + | | align="right" | y : |
− | = | o--o o--o | =
| + | | 1 0 1 0 |
− | = | /////\ /\\\\\ | =
| + | | |
− | = | ///////o\\\\\\\ | =
| + | | |
− | = | ////////X\\\\\\\\ | =
| + | | |
− | = | o///////XXX\\\\\\\o | =
| + | |- |
− | = | |/////oXXXXXo\\\\\| | =
| + | | f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || ( ) || false || 0 |
− | = = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
| + | |- |
− | | |/////oXXXXXo\\\\\| |
| + | | f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || ¬x ∧ ¬y |
− | | o//////\XXX/\\\\\\o |
| + | |- |
− | | \//////\X/\\\\\\/ |
| + | | f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || ¬x ∧ y |
− | | \//////o\\\\\\/ |
| + | |- |
− | | \///// \\\\\/ |
| + | | f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || ¬x |
− | | o--o o--o |
| + | |- |
− | | |
| + | | f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x ∧ ¬y |
− | o-----------------------o
| + | |- |
| + | | f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || ¬y |
| + | |- |
| + | | f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x ≠ y |
| + | |- |
| + | | f<sub>7</sub> || f<sub>0111</sub> || 0 1 1 1 || (x y) || not both x and y || ¬x ∨ ¬y |
| + | |- |
| + | | f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x y || x and y || x ∧ y |
| + | |- |
| + | | f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y |
| + | |- |
| + | | f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y |
| + | |- |
| + | | f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x → y |
| + | |- |
| + | | f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x |
| + | |- |
| + | | f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x ← y |
| + | |- |
| + | | f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y || x ∨ y |
| + | |- |
| + | | f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || (( )) || true || 1 |
| + | |} |
| + | <br> |
| | | |
− | o-----------------------o o-----------------------o o-----------------------o
| + | ===Template Draft=== |
− | | dU | | dU | | dU |
| |
− | | o--o o--o | | o--o o--o | | o--o o--o |
| |
− | | / \ / \ | | /\\\\\ /\\\\\ | | /\\\\\ /\\\\\ |
| |
− | | / o \ | | /\\\\\\o\\\\\\\ | | /\\\\\\o\\\\\\\ |
| |
− | | / / \ \ | | /\\\\\\/ \\\\\\\\ | | /\\\\\\/ \\\\\\\\ |
| |
− | | o / \ o | | o\\\\\\/ \\\\\\\o | | o\\\\\\/ \\\\\\\o |
| |
− | | | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |
| |
− | | | du | | dv | | | |\\\\\| |\\\\\| | | |\du\\| |\\dv\| |
| |
− | | | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |
| |
− | | o \ / o | | o\\\\\\\ /\\\\\\o | | o\\\\\\\ /\\\\\\o |
| |
− | | \ \ / / | | \\\\\\\\ /\\\\\\/ | | \\\\\\\\ /\\\\\\/ |
| |
− | | \ o / | | \\\\\\\o\\\\\\/ | | \\\\\\\o\\\\\\/ |
| |
− | | \ / \ / | | \\\\\/ \\\\\/ | | \\\\\/ \\\\\/ |
| |
− | | o--o o--o | | o--o o--o | | o--o o--o |
| |
− | | | | | | |
| |
− | o-----------------------o o-----------------------o o-----------------------o
| |
− | = du' @ u v o-----------------------o dv' @ u v =
| |
− | = | dU' | =
| |
− | = | o--o o--o | =
| |
− | = | /////\ /\\\\\ | =
| |
− | = | ///////o\\\\\\\ | =
| |
− | = | ////////X\\\\\\\\ | =
| |
− | = | o///////XXX\\\\\\\o | =
| |
− | = | |/////oXXXXXo\\\\\| | =
| |
− | = = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
| |
− | | |/////oXXXXXo\\\\\| |
| |
− | | o//////\XXX/\\\\\\o |
| |
− | | \//////\X/\\\\\\/ |
| |
− | | \//////o\\\\\\/ |
| |
− | | \///// \\\\\/ |
| |
− | | o--o o--o |
| |
− | | |
| |
− | o-----------------------o
| |
| | | |
− | o-----------------------o o-----------------------o o-----------------------o
| + | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:98%" |
− | | U | |\U\\\\\\\\\\\\\\\\\\\\\| |\U\\\\\\\\\\\\\\\\\\\\\| | + | |+ '''Propositional Forms on Two Variables''' |
− | | o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\| | + | |- style="background:aliceblue" |
− | | /////\ /////\ | |\\\\\/////\\/////\\\\\\| |\\\\\/ \\/ \\\\\\| | + | ! style="width:14%" | L<sub>1</sub> |
− | | ///////o//////\ | |\\\\///////o//////\\\\\| |\\\\/ o \\\\\| | + | ! style="width:14%" | L<sub>2</sub> |
− | | /////////\//////\ | |\\\////////X\//////\\\\| |\\\/ /\\ \\\\| | + | ! style="width:14%" | L<sub>3</sub> |
− | | o//////////\//////o | |\\o///////XXX\//////o\\| |\\o /\\\\ o\\| | + | ! style="width:14%" | L<sub>4</sub> |
− | | |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\| | + | ! style="width:14%" | L<sub>5</sub> |
− | | |//u//|/////|//v//| | |\\|//u//|XXXXX|//v//|\\| |\\| u |\\\\\| v |\\| | + | ! style="width:14%" | L<sub>6</sub> |
− | | |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\| | + | ! style="width:14%" | Name |
− | | o//////\//////////o | |\\o//////\XXX///////o\\| |\\o \\\\/ o\\| | + | |- style="background:aliceblue" |
− | | \//////\///////// | |\\\\//////\X////////\\\| |\\\\ \\/ /\\\| | + | | |
− | | \//////o/////// | |\\\\\//////o///////\\\\| |\\\\\ o /\\\\| | + | | align="right" | x : |
− | | \///// \///// | |\\\\\\/////\\/////\\\\\| |\\\\\\ /\\ /\\\\\| | + | | 1 1 0 0 |
− | | o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\| | + | | |
− | | | |\\\\\\\\\\\\\\\\\\\\\\\| |\\\\\\\\\\\\\\\\\\\\\\\| | + | | |
− | o-----------------------o o-----------------------o o-----------------------o
| + | | |
− | = u' o-----------------------o v' =
| + | | |
− | = | U' | =
| + | |- style="background:aliceblue" |
− | = | o--o o--o | =
| + | | |
− | = | /////\ /\\\\\ | =
| + | | align="right" | y : |
− | = | ///////o\\\\\\\ | =
| + | | 1 0 1 0 |
− | = | ////////X\\\\\\\\ | =
| + | | |
− | = | o///////XXX\\\\\\\o | =
| + | | |
− | = | |/////oXXXXXo\\\\\| | =
| + | | |
− | = = = = = = = = = = =|/u'//|XXXXX|\\v'\|= = = = = = = = = = =
| + | | |
− | | |/////oXXXXXo\\\\\| |
| + | |- |
− | | o//////\XXX/\\\\\\o |
| + | | f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || ( ) || false || 0 || Falsity |
− | | \//////\X/\\\\\\/ |
| + | |- |
− | | \//////o\\\\\\/ |
| + | | f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || ¬x ∧ ¬y || [[NNOR]] |
− | | \///// \\\\\/ |
| + | |- |
− | | o--o o--o |
| + | | f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || ¬x ∧ y || Insuccede |
− | | |
| + | |- |
− | o-----------------------o
| + | | f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || ¬x || Not One |
| + | |- |
| + | | f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x ∧ ¬y || Imprecede |
| + | |- |
| + | | f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || ¬y || Not Two |
| + | |- |
| + | | f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x ≠ y || Inequality |
| + | |- |
| + | | f<sub>7</sub> || f<sub>0111</sub> || 0 1 1 1 || (x y) || not both x and y || ¬x ∨ ¬y || NAND |
| + | |- |
| + | | f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x y || x and y || x ∧ y || [[Conjunction]] |
| + | |- |
| + | | f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y || Equality |
| + | |- |
| + | | f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y || Two |
| + | |- |
| + | | f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x → y || [[Logical implcation|Implication]] |
| + | |- |
| + | | f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x || One |
| + | |- |
| + | | f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x ← y || [[Logical involution|Involution]] |
| + | |- |
| + | | f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y || x ∨ y || [[Disjunction]] |
| + | |- |
| + | | f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || (( )) || true || 1 || Tautology |
| + | |} |
| + | <br> |
| | | |
− | Figure 70-b. Tangent Functor Ferris Wheel for F<u, v> = <((u)(v)), ((u, v))>
| + | ===[[Truth Tables]]=== |
− | </pre> | + | |
| + | ====[[Logical negation]]==== |
| + | |
| + | '''Logical negation''' is an [[logical operation|operation]] on one [[logical value]], typically the value of a [[proposition]], that produces a value of ''true'' when its operand is false and a value of ''false'' when its operand is true. |
| + | |
| + | The [[truth table]] of '''NOT p''' (also written as '''~p''' or '''¬p''') is as follows: |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:40%" |
| + | |+ '''Logical Negation''' |
| + | |- style="background:aliceblue" |
| + | ! style="width:20%" | p |
| + | ! style="width:20%" | ¬p |
| + | |- |
| + | | F || T |
| + | |- |
| + | | T || F |
| + | |} |
| + | <br> |
| + | |
| + | The logical negation of a proposition '''p''' is notated in different ways in various contexts of discussion and fields of application. Among these variants are the following: |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; width:40%" |
| + | |+ '''Variant Notations''' |
| + | |- style="background:aliceblue" |
| + | ! style="text-align:center" | Notation |
| + | ! Vocalization |
| + | |- |
| + | | style="text-align:center" | <math>\bar{p}</math> |
| + | | bar ''p'' |
| + | |- |
| + | | style="text-align:center" | <math>p'\!</math> |
| + | | ''p'' prime,<p> ''p'' complement |
| + | |- |
| + | | style="text-align:center" | <math>!p\!</math> |
| + | | bang ''p'' |
| + | |} |
| + | <br> |
| + | |
| + | No matter how it is notated or symbolized, the logical negation ¬''p'' is read as "it is not the case that ''p''", or usually more simply as "not ''p''". |
| + | |
| + | * Within a system of [[classical logic]], double negation, that is, the negation of the negation of a proposition ''p'', is [[logically equivalent]] to the initial proposition ''p''. Expressed in symbolic terms, ¬(¬''p'') ⇔ ''p''. |
| + | |
| + | * Within a system of [[intuitionistic logic]], however, ¬¬''p'' is a weaker statement than ''p''. On the other hand, the logical equivalence ¬¬¬''p'' ⇔ ¬''p'' remains valid. |
| + | |
| + | Logical negation can be defined in terms of other logical operations. For example, ~''p'' can be defined as ''p'' → ''F'', where → is [[material implication]] and ''F'' is absolute falsehood. Conversely, one can define ''F'' as ''p'' & ~''p'' for any proposition ''p'', where & is [[logical conjunction]]. The idea here is that any [[contradiction]] is false. While these ideas work in both classical and intuitionistic logic, they don't work in [[Brazilian logic]], where contradictions are not necessarily false. But in classical logic, we get a further identity: ''p'' → ''q'' can be defined as ~''p'' ∨ ''q'', where ∨ is [[logical disjunction]]. |
| + | |
| + | Algebraically, logical negation corresponds to the ''complement'' in a [[Boolean algebra]] (for classical logic) or a [[Heyting algebra]] (for intuitionistic logic). |
| + | |
| + | ====[[Logical conjunction]]==== |
| + | |
| + | '''Logical conjunction''' is an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' if and only if both of its operands are true. |
| + | |
| + | The [[truth table]] of '''p AND q''' (also written as '''p ∧ q''', '''p & q''', or '''p<math>\cdot</math>q''') is as follows: |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%" |
| + | |+ '''Logical Conjunction''' |
| + | |- style="background:aliceblue" |
| + | ! style="width:15%" | p |
| + | ! style="width:15%" | q |
| + | ! style="width:15%" | p ∧ q |
| + | |- |
| + | | F || F || F |
| + | |- |
| + | | F || T || F |
| + | |- |
| + | | T || F || F |
| + | |- |
| + | | T || T || T |
| + | |} |
| + | <br> |
| + | |
| + | ====[[Logical disjunction]]==== |
| + | |
| + | '''Logical disjunction''' is an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' if and only if both of its operands are false. |
| + | |
| + | The [[truth table]] of '''p OR q''' (also written as '''p ∨ q''') is as follows: |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%" |
| + | |+ '''Logical Disjunction''' |
| + | |- style="background:aliceblue" |
| + | ! style="width:15%" | p |
| + | ! style="width:15%" | q |
| + | ! style="width:15%" | p ∨ q |
| + | |- |
| + | | F || F || F |
| + | |- |
| + | | F || T || T |
| + | |- |
| + | | T || F || T |
| + | |- |
| + | | T || T || T |
| + | |} |
| + | <br> |
| + | |
| + | ====[[Logical equality]]==== |
| + | |
| + | '''Logical equality''' is an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' if and only if both operands are false or both operands are true. |
| + | |
| + | The [[truth table]] of '''p EQ q''' (also written as '''p = q''', '''p ↔ q''', or '''p ≡ q''') is as follows: |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%" |
| + | |+ '''Logical Equality''' |
| + | |- style="background:aliceblue" |
| + | ! style="width:15%" | p |
| + | ! style="width:15%" | q |
| + | ! style="width:15%" | p = q |
| + | |- |
| + | | F || F || T |
| + | |- |
| + | | F || T || F |
| + | |- |
| + | | T || F || F |
| + | |- |
| + | | T || T || T |
| + | |} |
| + | <br> |
| + | |
| + | ====[[Exclusive disjunction]]==== |
| + | |
| + | '''Exclusive disjunction''' is an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' just in case exactly one of its operands is true. |
| + | |
| + | The [[truth table]] of '''p XOR q''' (also written as '''p + q''', '''p ⊕ q''', or '''p ≠ q''') is as follows: |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%" |
| + | |+ '''Exclusive Disjunction''' |
| + | |- style="background:aliceblue" |
| + | ! style="width:15%" | p |
| + | ! style="width:15%" | q |
| + | ! style="width:15%" | p XOR q |
| + | |- |
| + | | F || F || F |
| + | |- |
| + | | F || T || T |
| + | |- |
| + | | T || F || T |
| + | |- |
| + | | T || T || F |
| + | |} |
| + | <br> |
| + | |
| + | The following equivalents can then be deduced: |
| + | |
| + | : <math>\begin{matrix} |
| + | p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\ |
| + | \\ |
| + | & = & (p \lor q) & \land & (\lnot p \lor \lnot q) \\ |
| + | \\ |
| + | & = & (p \lor q) & \land & \lnot (p \land q) |
| + | \end{matrix}</math> |
| + | |
| + | '''Generalized''' or '''n-ary''' XOR is true when the number of 1-bits is odd. |
| + | |
| + | ====[[Logical implication]]==== |
| + | |
| + | The '''material conditional''' and '''logical implication''' are both associated with an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' if and only if the first operand is true and the second operand is false. |
| + | |
| + | The [[truth table]] associated with the material conditional '''if p then q''' (symbolized as '''p → q''') and the logical implication '''p implies q''' (symbolized as '''p ⇒ q''') is as follows: |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%" |
| + | |+ '''Logical Implication''' |
| + | |- style="background:aliceblue" |
| + | ! style="width:15%" | p |
| + | ! style="width:15%" | q |
| + | ! style="width:15%" | p ⇒ q |
| + | |- |
| + | | F || F || T |
| + | |- |
| + | | F || T || T |
| + | |- |
| + | | T || F || F |
| + | |- |
| + | | T || T || T |
| + | |} |
| + | <br> |
| + | |
| + | ====[[Logical NAND]]==== |
| + | |
| + | The '''NAND operation''' is a [[logical operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' if and only if both of its operands are true. In other words, it produces a value of ''true'' if and only if at least one of its operands is false. |
| + | |
| + | The [[truth table]] of '''p NAND q''' (also written as '''p | q''' or '''p ↑ q''') is as follows: |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%" |
| + | |+ '''Logical NAND''' |
| + | |- style="background:aliceblue" |
| + | ! style="width:15%" | p |
| + | ! style="width:15%" | q |
| + | ! style="width:15%" | p ↑ q |
| + | |- |
| + | | F || F || T |
| + | |- |
| + | | F || T || T |
| + | |- |
| + | | T || F || T |
| + | |- |
| + | | T || T || F |
| + | |} |
| + | <br> |
| + | |
| + | ====[[Logical NNOR]]==== |
| + | |
| + | The '''NNOR operation''' is a [[logical operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' if and only if both of its operands are false. In other words, it produces a value of ''false'' if and only if at least one of its operands is true. |
| + | |
| + | The [[truth table]] of '''p NNOR q''' (also written as '''p ⊥ q''' or '''p ↓ q''') is as follows: |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%" |
| + | |+ '''Logical NOR''' |
| + | |- style="background:aliceblue" |
| + | ! style="width:15%" | p |
| + | ! style="width:15%" | q |
| + | ! style="width:15%" | p ↓ q |
| + | |- |
| + | | F || F || T |
| + | |- |
| + | | F || T || F |
| + | |- |
| + | | T || F || F |
| + | |- |
| + | | T || T || F |
| + | |} |
| + | <br> |
| + | |
| + | ===Exclusive Disjunction=== |
| + | |
| + | A + B = (A ∧ !B) ∨ (!A ∧ B) |
| + | = {(A ∧ !B) ∨ !A} ∧ {(A ∧ !B) ∨ B} |
| + | = {(A ∨ !A) ∧ (!B ∨ !A)} ∧ {(A ∨ B) ∧ (!B ∨ B)} |
| + | = (!A ∨ !B) ∧ (A ∨ B) |
| + | = !(A ∧ B) ∧ (A ∨ B) |
| + | |
| + | |
| + | p + q = (p ∧ !q) ∨ (!p ∧ B) |
| + | |
| + | = {(p ∧ !q) ∨ !p} ∧ {(p ∧ !q) ∨ q} |
| + | |
| + | = {(p ∨ !q) ∧ (!q ∨ !p)} ∧ {(p ∨ q) ∧ (!q ∨ q)} |
| + | |
| + | = (!p ∨ !q) ∧ (p ∨ q) |
| + | |
| + | = !(p ∧ q) ∧ (p ∨ q) |
| + | |
| + | |
| + | p + q = (p ∧ ~q) ∨ (~p ∧ q) |
| + | |
| + | = ((p ∧ ~q) ∨ ~p) ∧ ((p ∧ ~q) ∨ q) |
| + | |
| + | = ((p ∨ ~q) ∧ (~q ∨ ~p)) ∧ ((p ∨ q) ∧ (~q ∨ q)) |
| + | |
| + | = (~p ∨ ~q) ∧ (p ∨ q) |
| + | |
| + | = ~(p ∧ q) ∧ (p ∨ q) |
| + | |
| + | : <math>\begin{matrix} |
| + | p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\ |
| + | & = & ((p \land \lnot q) \lor \lnot p) & \and & ((p \land \lnot q) \lor q) \\ |
| + | & = & ((p \lor \lnot q) \land (\lnot q \lor \lnot p)) & \land & ((p \lor q) \land (\lnot q \lor q)) \\ |
| + | & = & (\lnot p \lor \lnot q) & \land & (p \lor q) \\ |
| + | & = & \lnot (p \land q) & \land & (p \lor q) |
| + | \end{matrix}</math> |
| | | |
| ==Logical Tables== | | ==Logical Tables== |