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| 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. |
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− | {| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:96%" | + | <br> |
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| + | {| align="center" border="1" cellpadding="0" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" |
| |+ '''Table 1. Propositional Forms on Two Variables''' | | |+ '''Table 1. Propositional Forms on Two Variables''' |
− | |- style="background:ghostwhite; height:36px" | + | |- style="background:#e6e6ff; height:36px" |
| | <math>\mathcal{L}_1</math> | | | <math>\mathcal{L}_1</math> |
| | <math>\mathcal{L}_2</math> | | | <math>\mathcal{L}_2</math> |
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| | <math>\mathcal{L}_5</math> | | | <math>\mathcal{L}_5</math> |
| | <math>\mathcal{L}_6</math> | | | <math>\mathcal{L}_6</math> |
− | |- style="background:ghostwhite; height:48px" | + | |- style="background:#e6e6ff; height:48px" |
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− | {| align="right" style="background:ghostwhite; text-align:right" | + | {| align="right" style="background:#e6e6ff; text-align:right" |
| |- | | |- |
| | <math>x\!</math> : | | | <math>x\!</math> : |
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| |} | | |} |
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− | {| align="center" style="background:ghostwhite" | + | {| align="center" style="background:#e6e6ff" |
| |- | | |- |
| | 1 1 0 0 | | | 1 1 0 0 |
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| |} | | |} |
| |} | | |} |
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| <br> | | <br> |
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| Table 2 exhibits the same information in a different order, grouping the sixteen functions into seven natural classes. | | Table 2 exhibits the same information in a different order, grouping the sixteen functions into seven natural classes. |
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− | {| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:96%" | + | <br> |
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| + | {| align="center" border="1" cellpadding="0" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" |
| |+ '''Table 2. Propositional Forms on Two Variables''' | | |+ '''Table 2. Propositional Forms on Two Variables''' |
− | |- style="background:ghostwhite; height:36px" | + | |- style="background:#e6e6ff; height:36px" |
| | <math>\mathcal{L}_1</math> | | | <math>\mathcal{L}_1</math> |
| | <math>\mathcal{L}_2</math> | | | <math>\mathcal{L}_2</math> |
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| | <math>\mathcal{L}_5</math> | | | <math>\mathcal{L}_5</math> |
| | <math>\mathcal{L}_6</math> | | | <math>\mathcal{L}_6</math> |
− | |- style="background:ghostwhite; height:48px" | + | |- style="background:#e6e6ff; height:48px" |
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− | {| align="right" style="background:ghostwhite; text-align:right" | + | {| align="right" style="background:#e6e6ff; text-align:right" |
| |- | | |- |
| | <math>x\!</math> : | | | <math>x\!</math> : |
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| |} | | |} |
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− | {| align="center" style="background:ghostwhite" | + | {| align="center" style="background:#e6e6ff" |
| |- | | |- |
| | 1 1 0 0 | | | 1 1 0 0 |
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| | <p><math>1\!</math></p> | | | <p><math>1\!</math></p> |
| |} | | |} |
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| <br> | | <br> |
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| 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. |
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− | {| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%" | + | <br> |
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| + | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" |
| |+ '''Table 3. <math>\operatorname{E}f</math> Expanded Over Differential Features <math>\{ \operatorname{d}x, \operatorname{d}y \}</math>''' | | |+ '''Table 3. <math>\operatorname{E}f</math> Expanded Over Differential Features <math>\{ \operatorname{d}x, \operatorname{d}y \}</math>''' |
− | |- style="background:ghostwhite; height:48px" | + | |- style="background:#e6e6ff; height:48px" |
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| | <math>f\!</math> | | | <math>f\!</math> |
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| | <math>((~))\!</math> | | | <math>((~))\!</math> |
| | <math>((~))\!</math> | | | <math>((~))\!</math> |
− | |- style="background:ghostwhite; height:36px" | + | |- style="background:#e6e6ff; height:36px" |
| | colspan="2" | Fixed Point Total : | | | colspan="2" | Fixed Point Total : |
| | <math>4\!</math> | | | <math>4\!</math> |
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| | <math>16\!</math> | | | <math>16\!</math> |
| |} | | |} |
| + | |
| <br> | | <br> |
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− | {| 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="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" |
| |+ '''Table 4. <math>\operatorname{D}f</math> Expanded Over Differential Features <math>\{ \operatorname{d}x, \operatorname{d}y \}</math>''' | | |+ '''Table 4. <math>\operatorname{D}f</math> Expanded Over Differential Features <math>\{ \operatorname{d}x, \operatorname{d}y \}</math>''' |
− | |- style="background:ghostwhite; height:36px" | + | |- style="background:#e6e6ff; height:36px" |
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| | <math>f\!</math> | | | <math>f\!</math> |
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| | <math>(~)\!</math> | | | <math>(~)\!</math> |
| |} | | |} |
| + | |
| <br> | | <br> |
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− | {| 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="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" |
| |+ '''Table 5. <math>\operatorname{E}f</math> Expanded Over Ordinary Features <math>\{ x, y \}\!</math>''' | | |+ '''Table 5. <math>\operatorname{E}f</math> Expanded Over Ordinary Features <math>\{ x, y \}\!</math>''' |
− | |- style="background:ghostwhite; height:36px" | + | |- style="background:#e6e6ff; height:36px" |
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| | <math>f\!</math> | | | <math>f\!</math> |
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| | <math>((~))\!</math> | | | <math>((~))\!</math> |
| |} | | |} |
| + | |
| <br> | | <br> |
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− | {| 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="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" |
| |+ '''Table 6. <math>\operatorname{D}f</math> Expanded Over Ordinary Features <math>\{ x, y \}\!</math>''' | | |+ '''Table 6. <math>\operatorname{D}f</math> Expanded Over Ordinary Features <math>\{ x, y \}\!</math>''' |
− | |- style="background:ghostwhite; height:36px" | + | |- style="background:#e6e6ff; height:36px" |
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| | <math>f\!</math> | | | <math>f\!</math> |
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| | <math>(~)\!</math> | | | <math>(~)\!</math> |
| |} | | |} |
| + | |
| <br> | | <br> |
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