| Line 2,579: |
Line 2,579: |
| | | | |
| | By way of initial orientation, Table 1 lists equivalent expressions for the sixteen functions in a number of different languages for zeroth order logic. | | By way of initial orientation, Table 1 lists equivalent expressions for the sixteen functions in a number of different languages for zeroth order logic. |
| | + | |
| | + | ===Variant 1=== |
| | | | |
| | {| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%" | | {| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%" |
| Line 2,717: |
Line 2,719: |
| | |}<br> | | |}<br> |
| | | | |
| − | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
| + | ===Variant 2=== |
| − | |+ '''Table 1. Propositional Forms on Two Variables'''
| |
| − | |- style="background:paleturquoise"
| |
| − | |
| |
| − | {| align="right" style="background:paleturquoise; text-align:right"
| |
| − | | u :
| |
| − | |-
| |
| − | | v :
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" style="background:paleturquoise; text-align:center"
| |
| − | | 1100
| |
| − | |-
| |
| − | | 1010
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" style="background:paleturquoise; text-align:center"
| |
| − | | f
| |
| − | |-
| |
| − | |
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" style="background:paleturquoise; text-align:center"
| |
| − | | θf
| |
| − | |-
| |
| − | |
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" style="background:paleturquoise; text-align:center"
| |
| − | | θf
| |
| − | |-
| |
| − | |
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
| |
| − | | f<sub>0</sub>
| |
| − | |-
| |
| − | | f<sub>1</sub>
| |
| − | |-
| |
| − | | f<sub>2</sub>
| |
| − | |-
| |
| − | | f<sub>3</sub>
| |
| − | |-
| |
| − | | f<sub>4</sub>
| |
| − | |-
| |
| − | | f<sub>5</sub>
| |
| − | |-
| |
| − | | f<sub>6</sub>
| |
| − | |-
| |
| − | | f<sub>7</sub>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
| |
| − | | 0000
| |
| − | |-
| |
| − | | 0001
| |
| − | |-
| |
| − | | 0010
| |
| − | |-
| |
| − | | 0011
| |
| − | |-
| |
| − | | 0100
| |
| − | |-
| |
| − | | 0101
| |
| − | |-
| |
| − | | 0110
| |
| − | |-
| |
| − | | 0111
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
| |
| − | | ()
| |
| − | |-
| |
| − | | (u)(v)
| |
| − | |-
| |
| − | | (u) v
| |
| − | |-
| |
| − | | (u)
| |
| − | |-
| |
| − | | u (v)
| |
| − | |-
| |
| − | | (v)
| |
| − | |-
| |
| − | | (u, v)
| |
| − | |-
| |
| − | | (u v)
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
| |
| − | | (( f , () ))
| |
| − | |-
| |
| − | | (( f , (u)(v) ))
| |
| − | |-
| |
| − | | (( f , (u) v ))
| |
| − | |-
| |
| − | | (( f , (u) ))
| |
| − | |-
| |
| − | | (( f , u (v) ))
| |
| − | |-
| |
| − | | (( f , (v) ))
| |
| − | |-
| |
| − | | (( f , (u, v) ))
| |
| − | |-
| |
| − | | (( f , (u v) ))
| |
| − | |}
| |
| − | |
| |
| − | {| align="left" cellpadding="2" style="background:lightcyan; text-align:left"
| |
| − | | f + 1
| |
| − | |-
| |
| − | | f + u + v + uv
| |
| − | |-
| |
| − | | f + v + uv + 1
| |
| − | |-
| |
| − | | f + u
| |
| − | |-
| |
| − | | f + u + uv + 1
| |
| − | |-
| |
| − | | f + v
| |
| − | |-
| |
| − | | f + u + v + 1
| |
| − | |-
| |
| − | | f + uv
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
| |
| − | | f<sub>8</sub>
| |
| − | |-
| |
| − | | f<sub>9</sub>
| |
| − | |-
| |
| − | | f<sub>10</sub>
| |
| − | |-
| |
| − | | f<sub>11</sub>
| |
| − | |-
| |
| − | | f<sub>12</sub>
| |
| − | |-
| |
| − | | f<sub>13</sub>
| |
| − | |-
| |
| − | | f<sub>14</sub>
| |
| − | |-
| |
| − | | f<sub>15</sub>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
| |
| − | | 1000
| |
| − | |-
| |
| − | | 1001
| |
| − | |-
| |
| − | | 1010
| |
| − | |-
| |
| − | | 1011
| |
| − | |-
| |
| − | | 1100
| |
| − | |-
| |
| − | | 1101
| |
| − | |-
| |
| − | | 1110
| |
| − | |-
| |
| − | | 1111
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
| |
| − | | u v
| |
| − | |-
| |
| − | | ((u, v))
| |
| − | |-
| |
| − | | v
| |
| − | |-
| |
| − | | (u (v))
| |
| − | |-
| |
| − | | u
| |
| − | |-
| |
| − | | ((u) v)
| |
| − | |-
| |
| − | | ((u)(v))
| |
| − | |-
| |
| − | | (())
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
| |
| − | | (( f , u v ))
| |
| − | |-
| |
| − | | (( f , ((u, v)) ))
| |
| − | |-
| |
| − | | (( f , v ))
| |
| − | |-
| |
| − | | (( f , (u (v)) ))
| |
| − | |-
| |
| − | | (( f , u ))
| |
| − | |-
| |
| − | | (( f , ((u) v) ))
| |
| − | |-
| |
| − | | (( f , ((u)(v)) ))
| |
| − | |-
| |
| − | | (( f , (()) ))
| |
| − | |}
| |
| − | |
| |
| − | {| align="left" cellpadding="2" style="background:lightcyan; text-align:left"
| |
| − | | f + uv + 1
| |
| − | |-
| |
| − | | f + u + v
| |
| − | |-
| |
| − | | f + v + 1
| |
| − | |-
| |
| − | | f + u + uv
| |
| − | |-
| |
| − | | f + u + 1
| |
| − | |-
| |
| − | | f + v + uv
| |
| − | |-
| |
| − | | f + u + v + uv + 1
| |
| − | |-
| |
| − | | f
| |
| − | |}
| |
| − | |}
| |
| − | <br>
| |
| | | | |
| | {| align="center" border="1" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%" | | {| align="center" border="1" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%" |
| Line 3,175: |
Line 2,961: |
| | | 1 | | | 1 |
| | |}<br> | | |}<br> |
| | + | |
| | + | ===Variant 3=== |
| | + | |
| | + | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%" |
| | + | |+ '''Table 1. Propositional Forms on Two Variables''' |
| | + | |- style="background:paleturquoise" |
| | + | | |
| | + | {| align="right" style="background:paleturquoise; text-align:right" |
| | + | | u : |
| | + | |- |
| | + | | v : |
| | + | |} |
| | + | | |
| | + | {| align="center" style="background:paleturquoise; text-align:center" |
| | + | | 1100 |
| | + | |- |
| | + | | 1010 |
| | + | |} |
| | + | | |
| | + | {| align="center" style="background:paleturquoise; text-align:center" |
| | + | | f |
| | + | |- |
| | + | | |
| | + | |} |
| | + | | |
| | + | {| align="center" style="background:paleturquoise; text-align:center" |
| | + | | θf |
| | + | |- |
| | + | | |
| | + | |} |
| | + | | |
| | + | {| align="center" style="background:paleturquoise; text-align:center" |
| | + | | θf |
| | + | |- |
| | + | | |
| | + | |} |
| | + | |- |
| | + | | |
| | + | {| align="center" cellpadding="2" style="background:lightcyan; text-align:center" |
| | + | | f<sub>0</sub> |
| | + | |- |
| | + | | f<sub>1</sub> |
| | + | |- |
| | + | | f<sub>2</sub> |
| | + | |- |
| | + | | f<sub>3</sub> |
| | + | |- |
| | + | | f<sub>4</sub> |
| | + | |- |
| | + | | f<sub>5</sub> |
| | + | |- |
| | + | | f<sub>6</sub> |
| | + | |- |
| | + | | f<sub>7</sub> |
| | + | |} |
| | + | | |
| | + | {| align="center" cellpadding="2" style="background:lightcyan; text-align:center" |
| | + | | 0000 |
| | + | |- |
| | + | | 0001 |
| | + | |- |
| | + | | 0010 |
| | + | |- |
| | + | | 0011 |
| | + | |- |
| | + | | 0100 |
| | + | |- |
| | + | | 0101 |
| | + | |- |
| | + | | 0110 |
| | + | |- |
| | + | | 0111 |
| | + | |} |
| | + | | |
| | + | {| align="center" cellpadding="2" style="background:lightcyan; text-align:center" |
| | + | | () |
| | + | |- |
| | + | | (u)(v) |
| | + | |- |
| | + | | (u) v |
| | + | |- |
| | + | | (u) |
| | + | |- |
| | + | | u (v) |
| | + | |- |
| | + | | (v) |
| | + | |- |
| | + | | (u, v) |
| | + | |- |
| | + | | (u v) |
| | + | |} |
| | + | | |
| | + | {| align="center" cellpadding="2" style="background:lightcyan; text-align:center" |
| | + | | (( f , () )) |
| | + | |- |
| | + | | (( f , (u)(v) )) |
| | + | |- |
| | + | | (( f , (u) v )) |
| | + | |- |
| | + | | (( f , (u) )) |
| | + | |- |
| | + | | (( f , u (v) )) |
| | + | |- |
| | + | | (( f , (v) )) |
| | + | |- |
| | + | | (( f , (u, v) )) |
| | + | |- |
| | + | | (( f , (u v) )) |
| | + | |} |
| | + | | |
| | + | {| align="left" cellpadding="2" style="background:lightcyan; text-align:left" |
| | + | | f + 1 |
| | + | |- |
| | + | | f + u + v + uv |
| | + | |- |
| | + | | f + v + uv + 1 |
| | + | |- |
| | + | | f + u |
| | + | |- |
| | + | | f + u + uv + 1 |
| | + | |- |
| | + | | f + v |
| | + | |- |
| | + | | f + u + v + 1 |
| | + | |- |
| | + | | f + uv |
| | + | |} |
| | + | |- |
| | + | | |
| | + | {| align="center" cellpadding="2" style="background:lightcyan; text-align:center" |
| | + | | f<sub>8</sub> |
| | + | |- |
| | + | | f<sub>9</sub> |
| | + | |- |
| | + | | f<sub>10</sub> |
| | + | |- |
| | + | | f<sub>11</sub> |
| | + | |- |
| | + | | f<sub>12</sub> |
| | + | |- |
| | + | | f<sub>13</sub> |
| | + | |- |
| | + | | f<sub>14</sub> |
| | + | |- |
| | + | | f<sub>15</sub> |
| | + | |} |
| | + | | |
| | + | {| align="center" cellpadding="2" style="background:lightcyan; text-align:center" |
| | + | | 1000 |
| | + | |- |
| | + | | 1001 |
| | + | |- |
| | + | | 1010 |
| | + | |- |
| | + | | 1011 |
| | + | |- |
| | + | | 1100 |
| | + | |- |
| | + | | 1101 |
| | + | |- |
| | + | | 1110 |
| | + | |- |
| | + | | 1111 |
| | + | |} |
| | + | | |
| | + | {| align="center" cellpadding="2" style="background:lightcyan; text-align:center" |
| | + | | u v |
| | + | |- |
| | + | | ((u, v)) |
| | + | |- |
| | + | | v |
| | + | |- |
| | + | | (u (v)) |
| | + | |- |
| | + | | u |
| | + | |- |
| | + | | ((u) v) |
| | + | |- |
| | + | | ((u)(v)) |
| | + | |- |
| | + | | (()) |
| | + | |} |
| | + | | |
| | + | {| align="center" cellpadding="2" style="background:lightcyan; text-align:center" |
| | + | | (( f , u v )) |
| | + | |- |
| | + | | (( f , ((u, v)) )) |
| | + | |- |
| | + | | (( f , v )) |
| | + | |- |
| | + | | (( f , (u (v)) )) |
| | + | |- |
| | + | | (( f , u )) |
| | + | |- |
| | + | | (( f , ((u) v) )) |
| | + | |- |
| | + | | (( f , ((u)(v)) )) |
| | + | |- |
| | + | | (( f , (()) )) |
| | + | |} |
| | + | | |
| | + | {| align="left" cellpadding="2" style="background:lightcyan; text-align:left" |
| | + | | f + uv + 1 |
| | + | |- |
| | + | | f + u + v |
| | + | |- |
| | + | | f + v + 1 |
| | + | |- |
| | + | | f + u + uv |
| | + | |- |
| | + | | f + u + 1 |
| | + | |- |
| | + | | f + v + uv |
| | + | |- |
| | + | | f + u + v + uv + 1 |
| | + | |- |
| | + | | f |
| | + | |} |
| | + | |} |
| | + | <br> |
| | | | |
| | The next four Tables expand the expressions of <math>\operatorname{E}f</math> and <math>\operatorname{D}f</math> in two different ways, for each of the sixteen functions. Notice that the functions are given in a different order, here being collected into a set of seven natural classes. | | The next four Tables expand the expressions of <math>\operatorname{E}f</math> and <math>\operatorname{D}f</math> in two different ways, for each of the sixteen functions. Notice that the functions are given in a different order, here being collected into a set of seven natural classes. |
| Line 3,469: |
Line 3,475: |
| | o------o------------o------------o------------o------------o------------o | | o------o------------o------------o------------o------------o------------o |
| | </pre> | | </pre> |
| − |
| |
| − | ==Table 14. Differential Propositions==
| |
| − |
| |
| − | <pre>
| |
| − | Table 14. Differential Propositions
| |
| − | o-------o--------o---------o-----------o-------------------o----------o
| |
| − | | | A : 1 1 0 0 | | | |
| |
| − | | | dA : 1 0 1 0 | | | |
| |
| − | o-------o--------o---------o-----------o-------------------o----------o
| |
| − | | | | | | | |
| |
| − | | f_0 | g_0 | 0 0 0 0 | () | False | 0 |
| |
| − | | | | | | | |
| |
| − | o-------o--------o---------o-----------o-------------------o----------o
| |
| − | | | | | | | |
| |
| − | | | g_1 | 0 0 0 1 | (A)(dA) | Neither A nor dA | ~A & ~dA |
| |
| − | | | | | | | |
| |
| − | | | g_2 | 0 0 1 0 | (A) dA | Not A but dA | ~A & dA |
| |
| − | | | | | | | |
| |
| − | | | g_4 | 0 1 0 0 | A (dA) | A but not dA | A & ~dA |
| |
| − | | | | | | | |
| |
| − | | | g_8 | 1 0 0 0 | A dA | A and dA | A & dA |
| |
| − | | | | | | | |
| |
| − | o-------o--------o---------o-----------o-------------------o----------o
| |
| − | | | | | | | |
| |
| − | | f_1 | g_3 | 0 0 1 1 | (A) | Not A | ~A |
| |
| − | | | | | | | |
| |
| − | | f_2 | g_12 | 1 1 0 0 | A | A | A |
| |
| − | | | | | | | |
| |
| − | o-------o--------o---------o-----------o-------------------o----------o
| |
| − | | | | | | | |
| |
| − | | | g_6 | 0 1 1 0 | (A, dA) | A not equal to dA | A + dA |
| |
| − | | | | | | | |
| |
| − | | | g_9 | 1 0 0 1 | ((A, dA)) | A equal to dA | A = dA |
| |
| − | | | | | | | |
| |
| − | o-------o--------o---------o-----------o-------------------o----------o
| |
| − | | | | | | | |
| |
| − | | | g_5 | 0 1 0 1 | (dA) | Not dA | ~dA |
| |
| − | | | | | | | |
| |
| − | | | g_10 | 1 0 1 0 | dA | dA | dA |
| |
| − | | | | | | | |
| |
| − | o-------o--------o---------o-----------o-------------------o----------o
| |
| − | | | | | | | |
| |
| − | | | g_7 | 0 1 1 1 | (A dA) | Not both A and dA | ~A v ~dA |
| |
| − | | | | | | | |
| |
| − | | | g_11 | 1 0 1 1 | (A (dA)) | Not A without dA | A => dA |
| |
| − | | | | | | | |
| |
| − | | | g_13 | 1 1 0 1 | ((A) dA) | Not dA without A | A <= dA |
| |
| − | | | | | | | |
| |
| − | | | g_14 | 1 1 1 0 | ((A)(dA)) | A or dA | A v dA |
| |
| − | | | | | | | |
| |
| − | o-------o--------o---------o-----------o-------------------o----------o
| |
| − | | | | | | | |
| |
| − | | f_3 | g_15 | 1 1 1 1 | (()) | True | 1 |
| |
| − | | | | | | | |
| |
| − | o-------o--------o---------o-----------o-------------------o----------o
| |
| − | </pre>
| |
| − |
| |
| − | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
| |
| − | |+ '''Table 14. Differential Propositions'''
| |
| − | |- style="background:paleturquoise"
| |
| − | |
| |
| − | | align="right" | A :
| |
| − | | 1 1 0 0
| |
| − | |
| |
| − | |
| |
| − | |
| |
| − | |- style="background:paleturquoise"
| |
| − | |
| |
| − | | align="right" | dA :
| |
| − | | 1 0 1 0
| |
| − | |
| |
| − | |
| |
| − | |
| |
| − | |-
| |
| − | | f<sub>0</sub>
| |
| − | | g<sub>0</sub>
| |
| − | | 0 0 0 0
| |
| − | | ( )
| |
| − | | False
| |
| − | | 0
| |
| − | |-
| |
| − | |
| |
| − | | g<sub>1</sub>
| |
| − | | 0 0 0 1
| |
| − | | (A)(dA)
| |
| − | | Neither A nor dA
| |
| − | | ¬A ∧ ¬dA
| |
| − | |-
| |
| − | |
| |
| − | | g<sub>2</sub>
| |
| − | | 0 0 1 0
| |
| − | | (A) dA
| |
| − | | Not A but dA
| |
| − | | ¬A ∧ dA
| |
| − | |-
| |
| − | |
| |
| − | | g<sub>4</sub>
| |
| − | | 0 1 0 0
| |
| − | | A (dA)
| |
| − | | A but not dA
| |
| − | | A ∧ ¬dA
| |
| − | |-
| |
| − | |
| |
| − | | g<sub>8</sub>
| |
| − | | 1 0 0 0
| |
| − | | A dA
| |
| − | | A and dA
| |
| − | | A ∧ dA
| |
| − | |-
| |
| − | | f<sub>1</sub>
| |
| − | | g<sub>3</sub>
| |
| − | | 0 0 1 1
| |
| − | | (A)
| |
| − | | Not A
| |
| − | | ¬A
| |
| − | |-
| |
| − | | f<sub>2</sub>
| |
| − | | g<sub>12</sub>
| |
| − | | 1 1 0 0
| |
| − | | A
| |
| − | | A
| |
| − | | A
| |
| − | |-
| |
| − | |
| |
| − | | g<sub>6</sub>
| |
| − | | 0 1 1 0
| |
| − | | (A, dA)
| |
| − | | A not equal to dA
| |
| − | | A ≠ dA
| |
| − | |-
| |
| − | |
| |
| − | | g<sub>9</sub>
| |
| − | | 1 0 0 1
| |
| − | | ((A, dA))
| |
| − | | A equal to dA
| |
| − | | A = dA
| |
| − | |-
| |
| − | |
| |
| − | | g<sub>5</sub>
| |
| − | | 0 1 0 1
| |
| − | | (dA)
| |
| − | | Not dA
| |
| − | | ¬dA
| |
| − | |-
| |
| − | |
| |
| − | | g<sub>10</sub>
| |
| − | | 1 0 1 0
| |
| − | | dA
| |
| − | | dA
| |
| − | | dA
| |
| − | |-
| |
| − | |
| |
| − | | g<sub>7</sub>
| |
| − | | 0 1 1 1
| |
| − | | (A dA)
| |
| − | | Not both A and dA
| |
| − | | ¬A ∨ ¬dA
| |
| − | |-
| |
| − | |
| |
| − | | g<sub>11</sub>
| |
| − | | 1 0 1 1
| |
| − | | (A (dA))
| |
| − | | Not A without dA
| |
| − | | A → dA
| |
| − | |-
| |
| − | |
| |
| − | | g<sub>13</sub>
| |
| − | | 1 1 0 1
| |
| − | | ((A) dA)
| |
| − | | Not dA without A
| |
| − | | A ← dA
| |
| − | |-
| |
| − | |
| |
| − | | g<sub>14</sub>
| |
| − | | 1 1 1 0
| |
| − | | ((A)(dA))
| |
| − | | A or dA
| |
| − | | A ∨ dA
| |
| − | |-
| |
| − | | f<sub>3</sub>
| |
| − | | g<sub>15</sub>
| |
| − | | 1 1 1 1
| |
| − | | (( ))
| |
| − | | True
| |
| − | | 1
| |
| − | |}
| |
| − | <br>
| |
| − |
| |
| − | {| align="center" border="1" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
| |
| − | |+ '''Table 14. Differential Propositions'''
| |
| − | |- style="background:paleturquoise"
| |
| − | |
| |
| − | | align="right" | <math>x\!</math> :
| |
| − | | 1 1 0 0
| |
| − | |
| |
| − | |
| |
| − | |
| |
| − | |- style="background:paleturquoise"
| |
| − | |
| |
| − | | align="right" | dA :
| |
| − | | 1 0 1 0
| |
| − | |
| |
| − | |
| |
| − | |
| |
| − | |-
| |
| − | | f<sub>0</sub>
| |
| − | | g<sub>0</sub>
| |
| − | | 0 0 0 0
| |
| − | | ( )
| |
| − | | False
| |
| − | | 0
| |
| − | |-
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | <br>
| |
| − | <br>
| |
| − | <br>
| |
| − |
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | g<sub>1</sub><br>
| |
| − | g<sub>2</sub><br>
| |
| − | g<sub>4</sub><br>
| |
| − | g<sub>8</sub>
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | 0 0 0 1<br>
| |
| − | 0 0 1 0<br>
| |
| − | 0 1 0 0<br>
| |
| − | 1 0 0 0
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | (A)(dA)<br>
| |
| − | (A) dA <br>
| |
| − | A (dA)<br>
| |
| − | A dA
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | Neither A nor dA<br>
| |
| − | Not A but dA<br>
| |
| − | A but not dA<br>
| |
| − | A and dA
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | ¬A ∧ ¬dA<br>
| |
| − | ¬A ∧ dA<br>
| |
| − | A ∧ ¬dA<br>
| |
| − | A ∧ dA
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | f<sub>1</sub><br>
| |
| − | f<sub>2</sub>
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | g<sub>3</sub><br>
| |
| − | g<sub>12</sub>
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | 0 0 1 1<br>
| |
| − | 1 1 0 0
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | (A)<br>
| |
| − | A
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | Not A<br>
| |
| − | A
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | ¬A<br>
| |
| − | A
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | <br>
| |
| − |
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | g<sub>6</sub><br>
| |
| − | g<sub>9</sub>
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | 0 1 1 0<br>
| |
| − | 1 0 0 1
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | (A, dA)<br>
| |
| − | ((A, dA))
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | A not equal to dA<br>
| |
| − | A equal to dA
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | A ≠ dA<br>
| |
| − | A = dA
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | <br>
| |
| − |
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | g<sub>5</sub><br>
| |
| − | g<sub>10</sub>
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | 0 1 0 1<br>
| |
| − | 1 0 1 0
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | (dA)<br>
| |
| − | dA
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | Not dA<br>
| |
| − | dA
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | ¬dA<br>
| |
| − | dA
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | <br>
| |
| − | <br>
| |
| − | <br>
| |
| − |
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | g<sub>7</sub><br>
| |
| − | g<sub>11</sub><br>
| |
| − | g<sub>13</sub><br>
| |
| − | g<sub>14</sub>
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | 0 1 1 1<br>
| |
| − | 1 0 1 1<br>
| |
| − | 1 1 0 1<br>
| |
| − | 1 1 1 0
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | (A dA)<br>
| |
| − | (A (dA))<br>
| |
| − | ((A) dA)<br>
| |
| − | ((A)(dA))
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | Not both A and dA<br>
| |
| − | Not A without dA<br>
| |
| − | Not dA without A<br>
| |
| − | A or dA
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | ¬A ∨ ¬dA<br>
| |
| − | A → dA<br>
| |
| − | A ← dA<br>
| |
| − | A ∨ dA
| |
| − | |}
| |
| − | |-
| |
| − | | f<sub>3</sub>
| |
| − | | g<sub>15</sub>
| |
| − | | 1 1 1 1
| |
| − | | (( ))
| |
| − | | True
| |
| − | | 1
| |
| − | |}<br>
| |
| − |
| |
| − | ==Table 27. Thematization of Bivariate Propositions==
| |
| − |
| |
| − | <pre>
| |
| − | Table 27. Thematization of Bivariate Propositions
| |
| − | o---------o---------o----------o--------------------o--------------------o
| |
| − | | u : 1 1 0 0 | f | theta (f) | theta (f) |
| |
| − | | v : 1 0 1 0 | | | |
| |
| − | o---------o---------o----------o--------------------o--------------------o
| |
| − | | | | | | |
| |
| − | | f_0 | 0 0 0 0 | () | (( f , () )) | f + 1 |
| |
| − | | | | | | |
| |
| − | | f_1 | 0 0 0 1 | (u)(v) | (( f , (u)(v) )) | f + u + v + uv |
| |
| − | | | | | | |
| |
| − | | f_2 | 0 0 1 0 | (u) v | (( f , (u) v )) | f + v + uv + 1 |
| |
| − | | | | | | |
| |
| − | | f_3 | 0 0 1 1 | (u) | (( f , (u) )) | f + u |
| |
| − | | | | | | |
| |
| − | | f_4 | 0 1 0 0 | u (v) | (( f , u (v) )) | f + u + uv + 1 |
| |
| − | | | | | | |
| |
| − | | f_5 | 0 1 0 1 | (v) | (( f , (v) )) | f + v |
| |
| − | | | | | | |
| |
| − | | f_6 | 0 1 1 0 | (u, v) | (( f , (u, v) )) | f + u + v + 1 |
| |
| − | | | | | | |
| |
| − | | f_7 | 0 1 1 1 | (u v) | (( f , (u v) )) | f + uv |
| |
| − | | | | | | |
| |
| − | o---------o---------o----------o--------------------o--------------------o
| |
| − | | | | | | |
| |
| − | | f_8 | 1 0 0 0 | u v | (( f , u v )) | f + uv + 1 |
| |
| − | | | | | | |
| |
| − | | f_9 | 1 0 0 1 | ((u, v)) | (( f , ((u, v)) )) | f + u + v |
| |
| − | | | | | | |
| |
| − | | f_10 | 1 0 1 0 | v | (( f , v )) | f + v + 1 |
| |
| − | | | | | | |
| |
| − | | f_11 | 1 0 1 1 | (u (v)) | (( f , (u (v)) )) | f + u + uv |
| |
| − | | | | | | |
| |
| − | | f_12 | 1 1 0 0 | u | (( f , u )) | f + u + 1 |
| |
| − | | | | | | |
| |
| − | | f_13 | 1 1 0 1 | ((u) v) | (( f , ((u) v) )) | f + v + uv |
| |
| − | | | | | | |
| |
| − | | f_14 | 1 1 1 0 | ((u)(v)) | (( f , ((u)(v)) )) | f + u + v + uv + 1 |
| |
| − | | | | | | |
| |
| − | | f_15 | 1 1 1 1 | (()) | (( f , (()) )) | f |
| |
| − | | | | | | |
| |
| − | o---------o---------o----------o--------------------o--------------------o
| |
| − | </pre>
| |
| − |
| |
| − | <br><font face="courier new">
| |
| − | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
| |
| − | |+ Table 27. Thematization of Bivariate Propositions
| |
| − | |- style="background:paleturquoise"
| |
| − | |
| |
| − | {| align="right" style="background:paleturquoise; text-align:right"
| |
| − | | u :
| |
| − | |-
| |
| − | | v :
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:paleturquoise"
| |
| − | | 1100
| |
| − | |-
| |
| − | | 1010
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:paleturquoise"
| |
| − | | f
| |
| − | |-
| |
| − | |
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:paleturquoise"
| |
| − | | θf
| |
| − | |-
| |
| − | |
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:paleturquoise"
| |
| − | | θf
| |
| − | |-
| |
| − | |
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
| |
| − | | f<sub>0</sub>
| |
| − | |-
| |
| − | | f<sub>1</sub>
| |
| − | |-
| |
| − | | f<sub>2</sub>
| |
| − | |-
| |
| − | | f<sub>3</sub>
| |
| − | |-
| |
| − | | f<sub>4</sub>
| |
| − | |-
| |
| − | | f<sub>5</sub>
| |
| − | |-
| |
| − | | f<sub>6</sub>
| |
| − | |-
| |
| − | | f<sub>7</sub>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
| |
| − | | 0000
| |
| − | |-
| |
| − | | 0001
| |
| − | |-
| |
| − | | 0010
| |
| − | |-
| |
| − | | 0011
| |
| − | |-
| |
| − | | 0100
| |
| − | |-
| |
| − | | 0101
| |
| − | |-
| |
| − | | 0110
| |
| − | |-
| |
| − | | 0111
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
| |
| − | | ()
| |
| − | |-
| |
| − | | (u)(v)
| |
| − | |-
| |
| − | | (u) v
| |
| − | |-
| |
| − | | (u)
| |
| − | |-
| |
| − | | u (v)
| |
| − | |-
| |
| − | | (v)
| |
| − | |-
| |
| − | | (u, v)
| |
| − | |-
| |
| − | | (u v)
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
| |
| − | | (( f , () ))
| |
| − | |-
| |
| − | | (( f , (u)(v) ))
| |
| − | |-
| |
| − | | (( f , (u) v ))
| |
| − | |-
| |
| − | | (( f , (u) ))
| |
| − | |-
| |
| − | | (( f , u (v) ))
| |
| − | |-
| |
| − | | (( f , (v) ))
| |
| − | |-
| |
| − | | (( f , (u, v) ))
| |
| − | |-
| |
| − | | (( f , (u v) ))
| |
| − | |}
| |
| − | |
| |
| − | {| align="left" cellpadding="2" style="background:lightcyan; text-align:left"
| |
| − | | f + 1
| |
| − | |-
| |
| − | | f + u + v + uv
| |
| − | |-
| |
| − | | f + v + uv + 1
| |
| − | |-
| |
| − | | f + u
| |
| − | |-
| |
| − | | f + u + uv + 1
| |
| − | |-
| |
| − | | f + v
| |
| − | |-
| |
| − | | f + u + v + 1
| |
| − | |-
| |
| − | | f + uv
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
| |
| − | | f<sub>8</sub>
| |
| − | |-
| |
| − | | f<sub>9</sub>
| |
| − | |-
| |
| − | | f<sub>10</sub>
| |
| − | |-
| |
| − | | f<sub>11</sub>
| |
| − | |-
| |
| − | | f<sub>12</sub>
| |
| − | |-
| |
| − | | f<sub>13</sub>
| |
| − | |-
| |
| − | | f<sub>14</sub>
| |
| − | |-
| |
| − | | f<sub>15</sub>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
| |
| − | | 1000
| |
| − | |-
| |
| − | | 1001
| |
| − | |-
| |
| − | | 1010
| |
| − | |-
| |
| − | | 1011
| |
| − | |-
| |
| − | | 1100
| |
| − | |-
| |
| − | | 1101
| |
| − | |-
| |
| − | | 1110
| |
| − | |-
| |
| − | | 1111
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
| |
| − | | u v
| |
| − | |-
| |
| − | | ((u, v))
| |
| − | |-
| |
| − | | v
| |
| − | |-
| |
| − | | (u (v))
| |
| − | |-
| |
| − | | u
| |
| − | |-
| |
| − | | ((u) v)
| |
| − | |-
| |
| − | | ((u)(v))
| |
| − | |-
| |
| − | | (())
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" cellpadding="2" style="background:lightcyan; text-align:center"
| |
| − | | (( f , u v ))
| |
| − | |-
| |
| − | | (( f , ((u, v)) ))
| |
| − | |-
| |
| − | | (( f , v ))
| |
| − | |-
| |
| − | | (( f , (u (v)) ))
| |
| − | |-
| |
| − | | (( f , u ))
| |
| − | |-
| |
| − | | (( f , ((u) v) ))
| |
| − | |-
| |
| − | | (( f , ((u)(v)) ))
| |
| − | |-
| |
| − | | (( f , (()) ))
| |
| − | |}
| |
| − | |
| |
| − | {| align="left" cellpadding="2" style="background:lightcyan; text-align:left"
| |
| − | | f + uv + 1
| |
| − | |-
| |
| − | | f + u + v
| |
| − | |-
| |
| − | | f + v + 1
| |
| − | |-
| |
| − | | f + u + uv
| |
| − | |-
| |
| − | | f + u + 1
| |
| − | |-
| |
| − | | f + v + uv
| |
| − | |-
| |
| − | | f + u + v + uv + 1
| |
| − | |-
| |
| − | | f
| |
| − | |}
| |
| − | |}
| |
| − | </font><br>
| |
