Changes

Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0 (view source)

Revision as of 16:09, 5 July 2008

9,504 bytes removed , 16:09, 5 July 2008

a whiter shade of pale

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{| align="center" border="1" cellpadding="4" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:96%"

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{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"

|+ '''Table 3. Analogy of Real and Boolean Types'''

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|- style="background:ghostwhite"

! Real Domain '''R'''

! ←→

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{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"

|+ '''Table 3. Analogy of Real and Boolean Types'''

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|- style="background:~~paleturquoise~~"

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|- style="background:ghostwhite"

! Real Domain '''R'''

! ←→

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{| align="center" border="1" cellpadding="8" cellspacing="0" style="~~background:lightcyan;~~ text-align:center; width:96%"

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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:96%"

|+ '''Table 4. An Equivalence Based on the Propositions as Types Analogy

'''

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|- style="background:ghostwhite"

! Pattern

! Construction

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{| align="center" border="1" cellpadding="8" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:96%"

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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:96%"

|+ '''Table 5. A Bridge Over Troubled Waters'''

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|- style="background:~~paleturquoise~~"

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|- style="background:ghostwhite"

! Linear Space

! Liminal Space

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Propositional forms on one variable correspond to boolean functions ''f'' : '''B'''1 → '''B'''. In Table 6 these functions are listed in a variant form of [[truth table]], one which rotates the axes of the usual arrangement. Each function ''f''''i'' is indexed by the string of values that it takes on the points of the universe ''X'' • = [''x''] <math>\cong</math> '''B'''1. The binary index generated in this way is converted to its decimal equivalent, and these are used as conventional names for the ''f''''i'' , as shown in the first column of the Table. In their own right the 21 points of the universe ''X'' • are coordinated as a space of type '''B'''1, this in light of the universe ''X'' • being a functional domain where the coordinate projection ''x'' takes on its values in '''B'''.

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{| align="center" border="1" cellpadding="4" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:96%"

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{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"

|+ '''Table 6. Propositional Forms on One Variable'''

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|- style="background:~~paleturquoise~~"

+

|- style="background:ghostwhite"

! style="width:16%" | L1 Decimal

! style="width:16%" | L2 Binary

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! style="width:16%" | L5 English

! style="width:16%" | L6 Ordinary

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|- style="background:~~paleturquoise~~"

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|- style="background:ghostwhite"

|  

| align="right" | x :

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Propositional forms on two variables correspond to boolean functions ''f'' : '''B'''2 → '''B'''. In Table 7 each function ''f''''i'' is indexed by the values that it takes on the points of the universe ''X'' • = [''x'', ''y''] <math>\cong</math> '''B'''2. Converting the binary index thus generated to a decimal equivalent, we obtain the functional nicknames that are listed in the first column. The 22 points of the universe ''X'' • are coordinated as a space of type '''B'''2, as indicated under the heading of the Table, where the coordinate projections ''x'' and ''y'' run through the various combinations of their values in '''B'''.

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{| align="center" border="1" cellpadding="4" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:96%"

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{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"

|+ '''Table 7. Propositional Forms on Two Variables'''

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|- style="background:~~paleturquoise~~"

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|- style="background:ghostwhite"

! style="width:16%" | L1 Decimal

! style="width:16%" | L2 Binary

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! style="width:16%" | L5 English

! style="width:16%" | L6 Ordinary

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|- style="background:~~paleturquoise~~"

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|- style="background:ghostwhite"

|  

| align="right" | x :

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|  

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|- style="background:~~paleturquoise~~"

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|- style="background:ghostwhite"

|  

| align="right" | y :

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{| align="center" border="1" cellpadding="8" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:96%"

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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:96%"

|+ '''Table 8. Notation for the Differential Extension of Propositional Calculus'''

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|- style="background:~~paleturquoise~~"

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|- style="background:ghostwhite"

! Symbol

! Notation

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{| align="center" border="1" cellpadding="4" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:96%"

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{| align="center" border="1" cellpadding="4" cellspacing="0" style="text-align:left; width:96%"

|+ '''Table 9. Higher Order Differential Features'''

| width=50% |

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{| cellpadding="4"

| A

| =

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| width=50% |

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{| cellpadding="4"

| E0A

| =

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{| align="center" border="1" cellpadding="4" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:96%"

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{| align="center" border="1" cellpadding="4" cellspacing="0" style="text-align:left; width:96%"

|+ '''Table 10. A Realm of Intentional Features'''

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{| cellpadding="4"

| p0A

| =

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{| cellpadding="4~~" style="background:lightcyan~~"

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{| cellpadding="4"

| Q0A

| =

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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:96%"

|

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{| align="center" border="0" cellpadding="8" cellspacing="0" style="text-align:center; width:96%"

|   || From || (''A'') || and || (d''A'') || infer || (''A'') || next. ||  

|-

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{| align="center" border="1" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ text-align:center; width:96%"

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{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:96%"

|+ '''Table 11. A Pair of Commodious Trajectories'''

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|- style="background:ghostwhite"

! Time

! Trajectory 1

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|-

|

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{| align="center" border="0" cellpadding="6" cellspacing="0" style="text-align:center"

| 0

|-

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|

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{| align="center" border="0" cellpadding="6" cellspacing="0" style="text-align:center"

| ''A'' || d''A'' || (d2''A'')

|-

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|

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{| align="center" border="0" cellpadding="6" cellspacing="0" style="text-align:center"

| (''A'') || (d''A'') || d2''A''

|-

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To complete the construction of the extended universe of discourse E''X'' • = [''x''1, d''x''1] = [''A'', d''A''], one must add the set of differential propositions E''X''^ = {''g'' : E''X'' → '''B'''} <math>\cong</math> ('''B''' × '''D''' → '''B''') to the set of dispositions in E''X''. There are <math>2^{2^{2n}}</math> = 16 propositions in E''X''^, as detailed in Table 14.

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{| align="center" border="1" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:96%"

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{| align="center" border="1" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"

|+ '''Table 14. Differential Propositions'''

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|- style="background:~~paleturquoise~~"

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|- style="background:ghostwhite"

|  

| align="right" | A :

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|- style="background:~~paleturquoise~~"

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|- style="background:ghostwhite"

|  

| align="right" | dA :

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|-

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{| ~~style="background:lightcyan"~~

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{|

|

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{| ~~style="background:lightcyan"~~

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{|

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g1

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{|

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0 0 0 1

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{| ~~style="background:lightcyan"~~

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{|

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(A)(dA)

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{| ~~style="background:lightcyan"~~

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{|

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Neither A nor dA

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{|

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¬A &and; ¬dA

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{| ~~style="background:lightcyan"~~

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{|

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f1

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{|

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g3

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{|

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0 0 1 1

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{| ~~style="background:lightcyan"~~

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{|

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(A)

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{|

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Not A

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{|

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¬A

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{|

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{|

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g6

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{|

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0 1 1 0

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{| ~~style="background:lightcyan"~~

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{|

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(A, dA)

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{| ~~style="background:lightcyan"~~

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{|

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A not equal to dA

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{| ~~style="background:lightcyan"~~

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{|

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A ≠ dA

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{|

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{| ~~style="background:lightcyan"~~

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{|

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g5

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{|

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0 1 0 1

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{| ~~style="background:lightcyan"~~

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{|

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(dA)

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{|

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Not dA

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{|

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¬dA

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{|

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{|

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g7

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{|

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0 1 1 1

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{|

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(A dA)

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{|

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Not both A and dA

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{|

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¬A &or; ¬dA

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{| align="center" border="1" cellpadding="8" cellspacing="0" style="~~background:lightcyan;~~ text-align:center; width:96%"

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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:96%"

|+ '''Table 15. Tacit Extension of <math>[A]\!</math> to <math>[A, \operatorname{d}A]</math>'''

|

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{| align="center" border="0" cellpadding="8" cellspacing="0" style="text-align:center; width:96%"

|  

| <math>0\!</math>

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|

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{| align="center" border="0" cellpadding="8" cellspacing="0" style="text-align:center; width:96%"

| <math>r(q)\!</math>

| <math>=</math>

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Applied to the example of fourth gear curves, this scheme results in the data of Tables 17-a and 17-b, which exhibit one period for each orbit. The states in each orbit are listed as ordered pairs ‹''p''''i'', ''q''''j''›, where ''p''''i'' may be read as a temporal parameter that indicates the present time of the state, and where ''j'' is the decimal equivalent of the binary numeral ''s''. Informally and more casually, the Tables exhibit the states ''q''''s'' as subscripted with the numerators of their rational indices, taking for granted the constant denominators of 2''m'' = 24 = 16. Within this set-up, the temporal successions of states can be reckoned as given by a kind of ''parallel round-up rule''. That is, if ‹''d''''k'', ''d''''k''+1› is any pair of adjacent digits in the state index ''r'', then the value of ''d''''k'' in the next state is ''d''''k''′ = ''d''''k'' + ''d''''k''+1.

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{| 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="font-weight:bold; text-align:center; width:96%"

|+ '''Table 17-a. A Couple of Orbits in Fourth Gear: Orbit 1'''

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|- style="background:ghostwhite"

| Time

| State

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|  

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|- style="background:ghostwhite"

| ''p''''i''

| ''q''''j''

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|-

|

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{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center"

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{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center"

| ''p''0

|-

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|

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{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center"

| ''q''01

|-

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| colspan="5" |

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{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0. || 0 || 0 || 0 || 1

|-

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{| 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="font-weight:bold; text-align:center; width:96%"

|+ '''Table 17-b. A Couple of Orbits in Fourth Gear: Orbit 2'''

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|- style="background:~~paleturquoise~~"

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|- style="background:ghostwhite"

| Time

| State

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|  

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|- style="background:~~paleturquoise~~"

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|- style="background:ghostwhite"

| ''p''''i''

| ''q''''j''

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|-

|

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{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center"

| ''p''0

|-

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|

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+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center"

| ''q''25

|-

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| colspan="5" |

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+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 1. || 1 || 0 || 0 || 1

|-

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+

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

|+ '''Table 22. Disjunction ''f'' and Equality ''g'' '''

|

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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| ''u'' || ''v''

|}

|

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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| ''f'' || ''g''

|}

|-

|

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{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 0

|-

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+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 1

|-

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|+ '''Tables 23-i and 23-ii. Thematics of Disjunction and Equality (1)'''

|

−

{| align="center" border="1" cellpadding="0" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:90%"

+

{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"

|+ '''Table 23-i. Disjunction ''f'' '''

|

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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| ''u'' || ''v'' || ''f''

|}

|

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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| ''x'' || φ

|}

|-

|

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{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 0 || →

|-

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{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 1

|-

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|-

|

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{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 0 ||   

|-

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|}

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{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 1 || 0

|-

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{| align="center" border="1" cellpadding="0" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:90%"

+

{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"

|+ '''Table 23-ii. Equality ''g'' '''

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:~~paleturquoise~~; font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| ''u'' || ''v'' || ''g''

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:~~paleturquoise~~; font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| ''y'' || γ

|}

|-

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 0 || →

|-

Line 2,264: Line 2,264:

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 1 || 1

|-

Line 2,275: Line 2,275:

|-

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 0 ||   

|-

Line 2,285: Line 2,285:

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 0

|-

Line 2,304: Line 2,304:

|+ '''Tables 24-i and 24-ii. Thematics of Disjunction and Equality (2)'''

|

−

{| align="center" border="1" cellpadding="0" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:90%"

+

{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"

|+ '''Table 24-i. Disjunction ''f'' '''

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:~~paleturquoise~~; font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| ''u'' || ''v'' || ''f'' || ''x''

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:~~paleturquoise~~; font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| φ

|}

|-

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 0 || → || 0

|-

Line 2,326: Line 2,326:

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 1

|-

Line 2,337: Line 2,337:

|-

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 1 || 0 ||    || 0

|-

Line 2,347: Line 2,347:

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0

|-

Line 2,358: Line 2,358:

|}

|

−

{| align="center" border="1" cellpadding="0" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:90%"

+

{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"

|+ '''Table 24-ii. Equality ''g'' '''

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:~~paleturquoise~~; font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| ''u'' || ''v'' || ''g'' || ''y''

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:~~paleturquoise~~; font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| γ

|}

|-

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 0 ||    || 0

|-

Line 2,380: Line 2,380:

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0

|-

Line 2,391: Line 2,391:

|-

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 1 || 0 || → || 0

|-

Line 2,401: Line 2,401:

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 1

|-

Line 2,420: Line 2,420:

|+ '''Tables 25-i and 25-ii. Thematics of Disjunction and Equality (3)'''

|

−

{| align="center" border="1" cellpadding="0" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:90%"

+

{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"

|+ '''Table 25-i. Disjunction ''f'' '''

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:~~paleturquoise~~; font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| ''u'' || ''v'' || ''f'' || ''x''

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:~~paleturquoise~~; font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| φ

|}

|-

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 0 || → || 0

|-

Line 2,442: Line 2,442:

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 1

|-

Line 2,453: Line 2,453:

|-

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 0 ||    || 1

|-

Line 2,463: Line 2,463:

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0

|-

Line 2,474: Line 2,474:

|}

|

−

{| align="center" border="1" cellpadding="0" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:90%"

+

{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"

|+ '''Table 25-ii. Equality ''g'' '''

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:~~paleturquoise~~; font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| ''u'' || ''v'' || ''g'' || ''y''

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:~~paleturquoise~~; font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| γ

|}

|-

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 0 ||    || 0

|-

Line 2,496: Line 2,496:

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0

|-

Line 2,507: Line 2,507:

|-

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 0 || → || 1

|-

Line 2,517: Line 2,517:

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 1

|-

Line 2,536: Line 2,536:

|+ '''Tables 26-i and 26-ii. Tacit Extension and Thematization'''

|

−

{| align="center" border="1" cellpadding="0" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:90%"

+

{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"

|+ '''Table 26-i. Disjunction ''f'' '''

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:~~paleturquoise~~; font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| ''u'' || ''v'' || ''x''

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:~~paleturquoise~~; font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| ε''f'' || θ''f''

|}

|-

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 0 || 0

|-

Line 2,558: Line 2,558:

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 1

|-

Line 2,569: Line 2,569:

|-

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 1 || 0 || 0

|-

Line 2,579: Line 2,579:

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 1 || 0

|-

Line 2,590: Line 2,590:

|}

|

−

{| align="center" border="1" cellpadding="0" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:90%"

+

{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"

|+ '''Table 26-ii. Equality ''g'' '''

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:~~paleturquoise~~; font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| ''u'' || ''v'' || ''y''

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:~~paleturquoise~~; font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| ε''g'' || θ''g''

|}

|-

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 0 || 0

|-

Line 2,612: Line 2,612:

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 1 || 0

|-

Line 2,623: Line 2,623:

|-

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 1 || 0 || 0

|-

Line 2,633: Line 2,633:

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 1

|-

Line 2,649: Line 2,649:

−

{| align="center" border="1" cellpadding="4" cellspacing="0" style="~~background:lightcyan;~~ 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%"

|+ Table 27. Thematization of Bivariate Propositions

−

|- style="background:~~paleturquoise~~"

+

|- style="background:ghostwhite"

|

−

{| align="right" style="background:~~paleturquoise~~; text-align:right"

+

{| align="right" style="background:ghostwhite; text-align:right"

| u :

|-

Line 2,659: Line 2,659:

|}

|

−

{| style="background:~~paleturquoise~~"

+

{| style="background:ghostwhite"

| 1100

|-

Line 2,665: Line 2,665:

|}

|

−

{| style="background:~~paleturquoise~~"

+

{| style="background:ghostwhite"

| f

|-

Line 2,671: Line 2,671:

|}

|

−

{| style="background:~~paleturquoise~~"

+

{| style="background:ghostwhite"

| θf

|-

Line 2,677: Line 2,677:

|}

|

−

{| style="background:~~paleturquoise~~"

+

{| style="background:ghostwhite"

| θf

|-

Line 2,684: Line 2,684:

|-

|

−

{| cellpadding="2~~" style="background:lightcyan~~"

+

{| cellpadding="2"

| f0

|-

Line 2,702: Line 2,702:

|}

|

−

{| cellpadding="2~~" style="background:lightcyan~~"

+

{| cellpadding="2"

| 0000

|-

Line 2,720: Line 2,720:

|}

|

−

{| cellpadding="2~~" style="background:lightcyan~~"

+

{| cellpadding="2"

| ()

|-

Line 2,738: Line 2,738:

|}

|

−

{| cellpadding="2~~" style="background:lightcyan~~"

+

{| cellpadding="2"

| (( f ,    ()    ))

|-

Line 2,756: Line 2,756:

|}

|

−

{| align="left" cellpadding="2" style="~~background:lightcyan;~~ text-align:left"

+

{| align="left" cellpadding="2" style="text-align:left"

|  f + 1

|-

Line 2,775: Line 2,775:

|-

|

−

{| cellpadding="2~~" style="background:lightcyan~~"

+

{| cellpadding="2"

| f8

|-

Line 2,793: Line 2,793:

|}

|

−

{| cellpadding="2~~" style="background:lightcyan~~"

+

{| cellpadding="2"

| 1000

|-

Line 2,811: Line 2,811:

|}

|

−

{| cellpadding="2~~" style="background:lightcyan~~"

+

{| cellpadding="2"

|   u  v  

|-

Line 2,829: Line 2,829:

|}

|

−

{| cellpadding="2~~" style="background:lightcyan~~"

+

{| cellpadding="2"

| (( f ,   u  v   ))

|-

Line 2,847: Line 2,847:

|}

|

−

{| align="left" cellpadding="2" style="~~background:lightcyan;~~ text-align:left"

+

{| align="left" cellpadding="2" style="text-align:left"

|  f + uv + 1

|-

Line 2,870: Line 2,870:

−

{| align="center" border="1" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:96%"

+

{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"

|+ Table 28. Propositions on Two Variables

|

−

{| align="center" border="0" cellpadding="4" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

−

|- style="background:~~paleturquoise~~"

+

|- style="background:ghostwhite"

| u || v ||  

|f00||f01||f02||f03

Line 2,896: Line 2,896:

−

{| align="center" border="1" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:96%"

+

{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"

|+ Table 29. Thematic Extensions of Bivariate Propositions

|

−

{| align="center" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

−

|- style="background:~~paleturquoise~~"

+

|- style="background:ghostwhite"

| u || v || f¢

| φ00 || φ01

Line 3,089: Line 3,089:

−

{| align="center" border="1" cellpadding="8" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:96%"

+

{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"

|

−

{| align="center" border="0" cellpadding="8" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:100%"

+

{| align="center" border="0" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"

| width="20%" |  

| width="20%" | ''x''

Line 3,415: Line 3,415:

−

{| align="center" border="1" cellpadding="12" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:96%"

+

{| align="center" border="1" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"

|

−

{| align="center" border="0" cellpadding="4" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:100%"

+

{| align="center" border="0" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"

| width="8%" | ''x''1

| width="4%" | =

Line 3,434: Line 3,434:

|-

|

−

{| align="center" border="0" cellpadding="4" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:100%"

+

{| align="center" border="0" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"

| width="8%" | d''x''1

| width="4%" | =

Line 3,498: Line 3,498:

−

{| align="center" border="1" cellpadding="12" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:96%"

+

{| align="center" border="1" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"

|

−

{| align="center" border="0" cellpadding="4" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:100%"

+

{| align="center" border="0" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"

| width="8%" | ''x''1

| width="4%" | =

Line 3,517: Line 3,517:

|-

|

−

{| align="center" border="0" cellpadding="4" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:100%"

+

{| align="center" border="0" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"

| width="8%" | d''x''1

| width="4%" | =

Line 3,555: Line 3,555:

−

{| align="center" border="1" cellpadding="12" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:96%"

+

{| align="center" border="1" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"

|

−

{| align="center" border="0" cellpadding="4" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:100%"

+

{| align="center" border="0" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"

| width="8%" | d''x''1

| width="4%" | =

Line 3,595: Line 3,595:

−

{| align="center" border="1" cellpadding="12" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:96%"

+

{| align="center" border="1" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"

|

−

{| align="center" border="0" cellpadding="8" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:100%"

+

{| align="center" border="0" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"

| '''D'''

| =

Line 3,695: Line 3,695:

−

{| align="center" border="1" cellpadding="12" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:96%"

+

{| align="center" border="1" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"

|+ Table 36. Computation of <math>\epsilon</math>''J''

|

−

{| align="left" border="0" cellpadding="4" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"

| width="8%" | <math>\epsilon</math>''J''

| width="4%" | =

Line 3,716: Line 3,716:

|-

|

−

{| align="left" border="0" cellpadding="4" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"

| width="8%" | <math>\epsilon</math>''J''

| width="4%" | =

Line 3,794: Line 3,794:

−

{| align="center" border="1" cellpadding="8" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:96%"

+

{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"

|+ Table 38. Computation of E''J'' (Method 1)

|

−

{| align="left" border="0" cellpadding="0" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"

| width="8%" | E''J''

| width="4%" | =

Line 3,856: Line 3,856:

|-

|

−

{| align="left" border="0" cellpadding="0" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"

| width="8%" | E''J''

| width="23%" | = ''u'' ''v'' (d''u'')(d''v'')

Line 3,887: Line 3,887:

−

{| align="center" border="1" cellpadding="12" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:96%"

+

{| align="center" border="1" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"

|+ Table 39. Computation of E''J'' (Method 2)

|

−

{| align="left" border="0" cellpadding="0" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"

| width="8%" | E''J''

| colspan="2" | = ‹''u'' + d''u''› <math>\cdot</math> ‹''v'' + d''v''›

Line 3,974: Line 3,974:

−

{| align="center" border="1" cellpadding="4" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:96%"

+

{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"

|+ Table 41. Computation of D''J'' (Method 1)

|

−

{| align="left" border="0" cellpadding="0" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"

| width="8%" | D''J''

| width="4%" | =

Line 3,998: Line 3,998:

|-

|

−

{| align="left" border="0" cellpadding="0" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"

| width="8%" | D''J''

| width="3%" | =

Line 4,029: Line 4,029:

|-

|

−

{| align="left" border="0" cellpadding="0" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"

| width="8%" | D''J''

| width="3%" | =

Line 4,043: Line 4,043:

−

{| align="center" border="1" cellpadding="12" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:96%"

+

{| align="center" border="1" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"

|+ Table 42. Computation of D''J'' (Method 2)

|

−

{| align="left" border="0" cellpadding="0" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"

| width="8%" | D''J''

| width="4%" | =

Line 4,091: Line 4,091:

−

{| align="center" border="1" cellpadding="8" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:96%"

+

{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"

|+ Table 43. Computation of D''J'' (Method 3)

|

−

{| align="left" border="0" cellpadding="0" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"

| width="6%" | D''J''

| width="3%" | =

Line 4,104: Line 4,104:

|-

|

−

{| align="left" border="0" cellpadding="0" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"

| width="6%" | <math>\epsilon</math>''J''

| width="23%" | = ''u'' ''v'' (d''u'')(d''v'')

Line 4,119: Line 4,119:

|-

|

−

{| align="left" border="0" cellpadding="0" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"

| width="6%" | D''J''

| width="23%" | = 0 <math>\cdot</math> (d''u'')(d''v'')

Line 4,132: Line 4,132:

−

{| align="center" border="1" cellpadding="8" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:96%"

+

{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"

|

−

{| align="left" border="0" cellpadding="0" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"

| width="6%" | <math>\epsilon</math>''J''

| width="47%" | = {Dispositions from ''J'' to ''J'' }

Line 4,205: Line 4,205:

−

{| align="center" border="1" cellpadding="12" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:96%"

+

{| align="center" border="1" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"

|+ Table 45. Computation of d''J''

|

−

{| align="left" border="0" cellpadding="0" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"

| width="6%" | D''J''

| width="25%" | = ''u'' ''v'' ((d''u'')(d''v''))

Line 4,267: Line 4,267:

−

{| align="center" border="1" cellpadding="8" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:96%"

+

{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"

|+ Table 47. Computation of r''J''

|

−

{| align="left" border="0" cellpadding="0" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"

| width="6%" | r''J''

| width="5%" | =

Line 4,280: Line 4,280:

|-

|

−

{| align="left" border="0" cellpadding="0" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"

| width="6%" | D''J''

| width="25%" | = ''u'' ''v'' ((d''u'')(d''v''))

Line 4,295: Line 4,295:

|-

|

−

{| align="left" border="0" cellpadding="0" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"

| width="6%" | r''J''

| width="25%" | = ''u'' ''v''   d''u'' d''v''

Line 4,356: Line 4,356:

−

{| align="center" border="1" cellpadding="8" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:96%"

+

{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"

|+ Table 49. Computation Summary for ''J''

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| <math>\epsilon</math>''J''

| = || ''uv'' || <math>\cdot</math> || 1

Line 4,407: Line 4,407:

−

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

+

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

|+ Table 50. Computation of an Analytic Series in Terms of Coordinates

|

−

{| align="center" border="1" cellpadding="0" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:~~paleturquoise~~; font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| ''u''

| ''v''

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:~~paleturquoise~~; font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| d''u''

| d''v''

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:~~paleturquoise~~; font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| ''u''’

| ''v''’

Line 4,428: Line 4,428:

|-

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 0

|-

Line 4,438: Line 4,438:

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 0

|-

Line 4,448: Line 4,448:

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 0

|-

Line 4,459: Line 4,459:

|-

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 1

|-

Line 4,469: Line 4,469:

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 0

|-

Line 4,479: Line 4,479:

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 1

|-

Line 4,490: Line 4,490:

|-

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 1 || 0

|-

Line 4,500: Line 4,500:

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 0

|-

Line 4,510: Line 4,510:

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 1 || 0

|-

Line 4,521: Line 4,521:

|-

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 1 || 1

|-

Line 4,531: Line 4,531:

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 1 || 1

|-

Line 4,541: Line 4,541:

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 0

|-

Line 4,552: Line 4,552:

|}

|

−

{| align="center" border="1" cellpadding="0" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:~~paleturquoise~~; font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| <math>\epsilon</math>''J''

| E''J''

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:~~paleturquoise~~; font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| D''J''

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:~~paleturquoise~~; font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| d''J''

| d2''J''

Line 4,569: Line 4,569:

|-

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 0

|-

Line 4,579: Line 4,579:

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0

|-

Line 4,589: Line 4,589:

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 0

|-

Line 4,600: Line 4,600:

|-

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 0

|-

Line 4,610: Line 4,610:

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0

|-

Line 4,620: Line 4,620:

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 0

|-

Line 4,631: Line 4,631:

|-

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 0

|-

Line 4,641: Line 4,641:

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0

|-

Line 4,651: Line 4,651:

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 0

|-

Line 4,662: Line 4,662:

|-

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 1

|-

Line 4,672: Line 4,672:

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0

|-

Line 4,682: Line 4,682:

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 0

|-

Line 4,699: Line 4,699:

−

{| align="center" border="1" cellpadding="8" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:96%"

+

{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"

|

−

{| align="center" border="0" cellpadding="8" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

|   || ''u''’ || = || ''u'' + d''u'' || = || (''u'', d''u'') ||  

|-

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−

{| align="center" border="1" cellpadding="8" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:96%"

+

{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"

|

−

{| align="center" border="0" cellpadding="8" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| E''J''‹''u'', ''v'', d''u'', d''v''›

| =

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−

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

+

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

|+ Table 51. Computation of an Analytic Series in Symbolic Terms

|

−

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+

{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| ''u'' || ''v''

|}

|

−

{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:~~paleturquoise~~; font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| ''J''

|}

|

−

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+

{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| E''J''

|}

|

−

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+

{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| D''J''

|}

|

−

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+

{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| d''J''

|}

|

−

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+

{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| d2''J''

|}

|-

|

−

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+

{| align="center" border="0" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 0

|-

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|}

|

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+

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| 0

|-

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|}

|

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+

{| align="center" border="0" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

|  d''u''  d''v'' 

|-

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|}

|

−

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+

{| align="center" border="0" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

|   d''u''  d''v''  

|-

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|}

|

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+

{| align="center" border="0" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| ()

|-

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|}

|

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+

{| align="center" border="0" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| d''u'' d''v''

|-

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Table 54 provides basic notation and descriptive information for the objects and operators that are used used in this Example, giving the generic type (or broadest defined type) for each entity. Here, the operators '''W''' in {'''e''', '''E''', '''D''', '''d''', '''r'''} and their components W in {<math>\epsilon</math>, <math>\eta</math>, E, D, d, r} both have the same broad type '''W''', W : (''U'' • → ''X'' •) → (E''U'' • → E''X'' •), as would be expected of operators that map transformations ''J'' : ''U'' • → ''X'' • to extended transformations '''W'''''J'', W''J'' : E''U • → E''X'' •.

−

{| align="center" border="1" cellpadding="8" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:96%"

+

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:96%"

|+ '''Table 54. Cast of Characters: Expansive Subtypes of Objects and Operators'''

−

|- style="background:~~paleturquoise~~"

+

|- style="background:ghostwhite"

! Item

! Notation

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|-

| valign="top" |

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| W

|}

| valign="top" |

−

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| W :

|-

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|}

| valign="top" |

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| Operator

|}

| valign="top" |

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100"

|  

|-

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|-

|

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| <math>\epsilon</math>

|-

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| valign="top" |  

| colspan="2" |

−

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:60%"

| Tacit Extension Operator || <math>\epsilon</math>

|-

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|-

| valign="top" |

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| '''W'''

|}

| valign="top" |

−

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| '''W''' :

|-

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|}

| valign="top" |

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| Operator

|}

| valign="top" |

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100"

|  

|-

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|-

|

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| '''e'''

|-

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| valign="top" |  

| colspan="2" |

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:60%"

| Radius Operator || '''e''' = ‹<math>\epsilon</math>, <math>\eta</math>›

|-

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Table 55 supplies a more detailed outline of terminology for operators and their results. Here, I list the restrictive subtype (or narrowest defined subtype) that applies to each entity, and I indicate across the span of the Table the whole spectrum of alternative types that color the interpretation of each symbol. Accordingly, each of the component operator maps W''J'', since their ranges are 1-dimensional (of type '''B'''1 or '''D'''1), can be regarded either as propositions W''J'' : E''U'' → '''B''' or as logical transformations W''J'' : E''U'' • → ''X'' •. As a rule, the plan of the Table allows us to name each entry by detaching the adjective at the left of its row and prefixing it to the generic noun at the top of its column. In one case, however, it is customary to depart from this scheme. Because the phrase ''differential proposition'', applied to the result d''J'' : E''U'' → '''D''', does not distinguish it from the general run of differential propositions ''G'' : E''U'' → '''B''', it is usual to single out d''J'' as the ''tangent proposition'' of ''J''.

−

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+

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:96%"

|+ '''Table 55. Synopsis of Terminology: Restrictive and Alternative Subtypes'''

−

|- style="background:~~paleturquoise~~"

+

|- style="background:ghostwhite"

!  

! Operator

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|-

|

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| Tacit

|-

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|}

|

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| <math>\epsilon</math> :

|-

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|}

|

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| <math>\epsilon</math>''J'' :

|-

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|}

|

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| <math>\epsilon</math>''J'' :

|-

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|-

|

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+

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| Trope

|-

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|}

|

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| <math>\eta</math> :

|-

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|

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| <math>\eta</math>''J'' :

|-

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|}

|

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| <math>\eta</math>''J'' :

|-

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|-

|

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| Enlargement

|-

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|}

|

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+

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| E :

|-

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|}

|

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| E''J'' :

|-

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|}

|

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| E''J'' :

|-

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|-

|

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| Difference

|-

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|}

|

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| D :

|-

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|}

|

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| D''J'' :

|-

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|}

|

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| D''J'' :

|-

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|-

|

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| Differential

|-

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|}

|

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+

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| d :

|-

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|}

|

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| d''J'' :

|-

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|

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| d''J'' :

|-

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|-

|

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| Remainder

|-

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|}

|

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| r :

|-

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|}

|

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| r''J'' :

|-

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|}

|

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| r''J'' :

|-

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|-

|

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| Radius

|-

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|}

|

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| '''e''' = ‹<math>\epsilon</math>, <math>\eta</math>› :

|-

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|}

|

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

|  

|-

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|}

|

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| '''e'''''J'' :

|-

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|-

|

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| Secant

|-

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|}

|

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| '''E''' = ‹<math>\epsilon</math>, E› :

|-

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|}

|

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

|  

|-

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|}

|

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| '''E'''''J'' :

|-

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|-

|

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| Chord

|-

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|}

|

−

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| '''D''' = ‹<math>\epsilon</math>, D› :

|-

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|}

|

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|  

|-

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|}

|

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| '''D'''''J'' :

|-

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|-

|

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+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| Tangent

|-

Line 5,339: Line 5,339:

|}

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| '''T''' = ‹<math>\epsilon</math>, d› :

|-

Line 5,347: Line 5,347:

|}

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| d''J'' :

|-

Line 5,355: Line 5,355:

|}

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| '''T'''''J'' :

|-

Line 5,438: Line 5,438:

−

{| align="center" border="1" cellpadding="8" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:96%"

+

{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"

|

−

{| align="center" border="0" cellpadding="8" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| align="left" | ''F''

| =

Line 5,482: Line 5,482:

But that's it, and no further. Neglect of these distinctions in range and target universes of higher dimensions is bound to cause a hopeless confusion. To guard against these adverse prospects, Tables 58 and 59 lay the groundwork for discussing a typical map ''F'' : ['''B'''2] → ['''B'''2], and begin to pave the way, to some extent, for discussing any transformation of the form ''F'' : ['''B'''''n''] → ['''B'''''k''].

−

{| align="center" border="1" cellpadding="8" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:96%"

+

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:96%"

|+ '''Table 58. Cast of Characters: Expansive Subtypes of Objects and Operators'''

−

|- style="background:~~paleturquoise~~"

+

|- style="background:ghostwhite"

! Item

! Notation

Line 5,497: Line 5,497:

| valign="top" | ''X'' •

| valign="top" |

−

{| align="left" border="0" cellpadding="0" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="0" cellspacing="0" style="text-align:left; width:100%"

| = [''x'', ''y'']

|-

Line 5,512: Line 5,512:

| valign="top" | E''X'' •

| valign="top" |

−

{| align="left" border="0" cellpadding="0" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="0" cellspacing="0" style="text-align:left; width:100%"

| = [''x'', ''y'', d''x'', d''y'']

|-

Line 5,526: Line 5,526:

|-

| valign="top" |

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

|  

|-

Line 5,534: Line 5,534:

|}

| valign="top" |

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| ''f'', ''g'' : ''U'' → '''B'''

|-

Line 5,542: Line 5,542:

|}

| valign="top" |

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| Proposition

|}

| valign="top" |

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100"

| '''B'''''n'' → '''B'''

|-

Line 5,555: Line 5,555:

|-

| valign="top" |

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| W

|}

| valign="top" |

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| W :

|-

Line 5,577: Line 5,577:

|}

| valign="top" |

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| Operator

|}

| valign="top" |

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100"

|  

|-

Line 5,600: Line 5,600:

|-

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| <math>\epsilon</math>

|-

Line 5,613: Line 5,613:

| valign="top" |  

| colspan="2" |

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:60%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:60%"

| Tacit Extension Operator || <math>\epsilon</math>

|-

Line 5,626: Line 5,626:

|-

| valign="top" |

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| '''W'''

|}

| valign="top" |

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| '''W''' :

|-

Line 5,648: Line 5,648:

|}

| valign="top" |

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| Operator

|}

| valign="top" |

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100"

|  

|-

Line 5,671: Line 5,671:

|-

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| '''e'''

|-

Line 5,682: Line 5,682:

| valign="top" |  

| colspan="2" |

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:60%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:60%"

| Radius Operator || '''e''' = ‹<math>\epsilon</math>, <math>\eta</math>›

|-

Line 5,693: Line 5,693:

|}

−

{| align="center" border="1" cellpadding="4" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:96%"

+

{| align="center" border="1" cellpadding="4" cellspacing="0" style="text-align:left; width:96%"

|+ '''Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes'''

−

|- style="background:~~paleturquoise~~"

+

|- style="background:ghostwhite"

|  

| align="center" | '''Operator or Operand'''

Line 5,703: Line 5,703:

| Operand

| valign="top" |

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| ''F'' = ‹''F''1, ''F''2›

|-

Line 5,709: Line 5,709:

|}

| valign="top" |

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| ''F''''i'' : 〈''u'', ''v''〉 → '''B'''

|-

Line 5,715: Line 5,715:

|}

| valign="top" |

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100"

| ''F'' : [''u'', ''v''] → [''x'', ''y'']

|-

Line 5,722: Line 5,722:

|-

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| Tacit

|-

Line 5,728: Line 5,728:

|}

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| <math>\epsilon</math> :

|-

Line 5,736: Line 5,736:

|}

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| <math>\epsilon</math>''F''''i'' :

|-

Line 5,744: Line 5,744:

|}

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| <math>\epsilon</math>''F'' :

|-

Line 5,753: Line 5,753:

|-

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| Trope

|-

Line 5,759: Line 5,759:

|}

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| <math>\eta</math> :

|-

Line 5,767: Line 5,767:

|}

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| <math>\eta</math>''F''''i'' :

|-

Line 5,775: Line 5,775:

|}

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| <math>\eta</math>''F'' :

|-

Line 5,784: Line 5,784:

|-

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| Enlargement

|-

Line 5,790: Line 5,790:

|}

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| E :

|-

Line 5,798: Line 5,798:

|}

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| E''F''''i'' :

|-

Line 5,806: Line 5,806:

|}

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| E''F'' :

|-

Line 5,815: Line 5,815:

|-

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| Difference

|-

Line 5,821: Line 5,821:

|}

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| D :

|-

Line 5,829: Line 5,829:

|}

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| D''F''''i'' :

|-

Line 5,837: Line 5,837:

|}

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| D''F'' :

|-

Line 5,846: Line 5,846:

|-

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| Differential

|-

Line 5,852: Line 5,852:

|}

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| d :

|-

Line 5,860: Line 5,860:

|}

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| d''F''''i'' :

|-

Line 5,868: Line 5,868:

|}

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| d''F'' :

|-

Line 5,877: Line 5,877:

|-

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| Remainder

|-

Line 5,883: Line 5,883:

|}

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| r :

|-

Line 5,891: Line 5,891:

|}

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| r''F''''i'' :

|-

Line 5,899: Line 5,899:

|}

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| r''F'' :

|-

Line 5,908: Line 5,908:

|-

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| Radius

|-

Line 5,914: Line 5,914:

|}

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| '''e''' = ‹<math>\epsilon</math>, <math>\eta</math>› :

|-

Line 5,922: Line 5,922:

|}

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

|  

|-

Line 5,930: Line 5,930:

|}

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| '''e'''''F'' :

|-

Line 5,939: Line 5,939:

|-

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| Secant

|-

Line 5,945: Line 5,945:

|}

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| '''E''' = ‹<math>\epsilon</math>, E› :

|-

Line 5,953: Line 5,953:

|}

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

|  

|-

Line 5,961: Line 5,961:

|}

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| '''E'''''F'' :

|-

Line 5,970: Line 5,970:

|-

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| Chord

|-

Line 5,976: Line 5,976:

|}

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| '''D''' = ‹<math>\epsilon</math>, D› :

|-

Line 5,984: Line 5,984:

|}

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

|  

|-

Line 5,992: Line 5,992:

|}

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| '''D'''''F'' :

|-

Line 6,001: Line 6,001:

|-

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| Tangent

|-

Line 6,007: Line 6,007:

|}

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| '''T''' = ‹<math>\epsilon</math>, d› :

|-

Line 6,015: Line 6,015:

|}

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| d''F''''i'' :

|-

Line 6,023: Line 6,023:

|}

|

−

{| align="left" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"

| '''T'''''F'' :

|-

Line 6,037: Line 6,037:

−

{| align="center" border="1" cellpadding="8" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:96%"

+

{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"

|

−

{| align="center" border="0" cellpadding="8" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

|  

| ''x''

Line 6,062: Line 6,062:

−

{| align="center" border="1" cellpadding="8" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:96%"

+

{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"

|

−

{| align="center" border="0" cellpadding="8" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

|  

| ‹''x'', ''y''›

Line 6,079: Line 6,079:

−

{| align="center" border="1" cellpadding="4" cellspacing="0" style="~~background:lightcyan;~~ 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%"

|+ '''Table 60. Propositional Transformation'''

−

|- style="background:~~paleturquoise~~"

+

|- style="background:ghostwhite"

| width="25%" | ''u''

| width="25%" | ''v''

Line 6,088: Line 6,088:

|-

| width="25%" |

−

{| align="center" border="0" cellpadding="4" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0

|-

Line 6,098: Line 6,098:

|}

| width="25%" |

−

{| align="center" border="0" cellpadding="4" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0

|-

Line 6,108: Line 6,108:

|}

| width="25%" |

−

{| align="center" border="0" cellpadding="4" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0

|-

Line 6,118: Line 6,118:

|}

| width="25%" |

−

{| align="center" border="0" cellpadding="4" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 1

|-

Line 6,155: Line 6,155:

Table 64 shows how the action of the transformation ''F'' on cells or points is computed in terms of coordinates.

−

{| align="center" border="1" cellpadding="8" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:96%"

+

{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"

|+ '''Table 64. Transformation of Positions'''

−

|- style="background:~~paleturquoise~~"

+

|- style="background:ghostwhite"

| ''u''  ''v''

| ''x''

Line 6,168: Line 6,168:

|-

| width="12%" |

−

{| align="center" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0  0

|-

Line 6,178: Line 6,178:

|}

| width="12%" |

−

{| align="center" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0

|-

Line 6,188: Line 6,188:

|}

| width="12%" |

−

{| align="center" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 1

|-

Line 6,198: Line 6,198:

|}

| width="12%" |

−

{| align="center" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0

|-

Line 6,208: Line 6,208:

|}

| width="12%" |

−

{| align="center" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0

|-

Line 6,218: Line 6,218:

|}

| width="12%" |

−

{| align="center" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 1

|-

Line 6,228: Line 6,228:

|}

| width="12%" |

−

{| align="center" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0

|-

Line 6,238: Line 6,238:

|}

| width="12%" |

−

{| align="center" border="0" cellpadding="2" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| ↑

|-

Line 6,262: Line 6,262:

−

{| align="center" border="1" cellpadding="4" cellspacing="0" style="~~background:lightcyan;~~ 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%"

|+ '''Table 65. Induced Transformation on Propositions'''

−

|- style="background:~~paleturquoise~~"

+

|- style="background:ghostwhite"

| ''X'' •

| colspan="3" |

−

{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:~~paleturquoise~~; font-weight:bold; text-align:center; width:80%"

+

{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:80%"

| ←

| ''F'' = ‹''f'' , ''g''›

Line 6,273: Line 6,273:

|}

| ''U'' •

−

|- style="background:~~paleturquoise~~"

+

|- style="background:ghostwhite"

| rowspan="2" | ''f''''i''‹''x'', ''y''›

|

−

{| align="right" style="background:~~paleturquoise~~; text-align:right"

+

{| align="right" style="background:ghostwhite; text-align:right"

| ''u'' =

|-

Line 6,282: Line 6,282:

|}

|

−

{| align="center" style="background:~~paleturquoise~~; text-align:center"

+

{| align="center" style="background:ghostwhite; text-align:center"

| 1 1 0 0

|-

Line 6,288: Line 6,288:

|}

|

−

{| align="left" style="background:~~paleturquoise~~; text-align:left"

+

{| align="left" style="background:ghostwhite; text-align:left"

| = ''u''

|-

Line 6,294: Line 6,294:

|}

| rowspan="2" | ''f''''j''‹''u'', ''v''›

−

|- style="background:~~paleturquoise~~"

+

|- style="background:ghostwhite"

|

−

{| align="right" style="background:~~paleturquoise~~; text-align:right"

+

{| align="right" style="background:ghostwhite; text-align:right"

| ''x'' =

|-

Line 6,302: Line 6,302:

|}

|

−

{| align="center" style="background:~~paleturquoise~~; text-align:center"

+

{| align="center" style="background:ghostwhite; text-align:center"

| 1 1 1 0

|-

Line 6,308: Line 6,308:

|}

|

−

{| align="left" style="background:~~paleturquoise~~; text-align:left"

+

{| align="left" style="background:ghostwhite; text-align:left"

| = ''f''‹''u'', ''v''›

|-

Line 6,315: Line 6,315:

|-

|

−

{| cellpadding="2~~" style="background:lightcyan~~"

+

{| cellpadding="2"

| ''f''0

|-

Line 6,333: Line 6,333:

|}

|

−

{| cellpadding="2~~" style="background:lightcyan~~"

+

{| cellpadding="2"

| ()

|-

Line 6,351: Line 6,351:

|}

|

−

{| cellpadding="2~~" style="background:lightcyan~~"

+

{| cellpadding="2"

| 0 0 0 0

|-

Line 6,369: Line 6,369:

|}

|

−

{| cellpadding="2~~" style="background:lightcyan~~"

+

{| cellpadding="2"

| ()

|-

Line 6,387: Line 6,387:

|}

|

−

{| cellpadding="2~~" style="background:lightcyan~~"

+

{| cellpadding="2"

| ''f''0

|-

Line 6,406: Line 6,406:

|-

|

−

{| cellpadding="2~~" style="background:lightcyan~~"

+

{| cellpadding="2"

| ''f''8

|-

Line 6,424: Line 6,424:

|}

|

−

{| cellpadding="2~~" style="background:lightcyan~~"

+

{| cellpadding="2"

|   ''x''  ''y''  

|-

Line 6,442: Line 6,442:

|}

|

−

{| cellpadding="2~~" style="background:lightcyan~~"

+

{| cellpadding="2"

| 1 0 0 0

|-

Line 6,460: Line 6,460:

|}

|

−

{| cellpadding="2~~" style="background:lightcyan~~"

+

{| cellpadding="2"

|   ''u''  ''v''  

|-

Line 6,478: Line 6,478:

|}

|

−

{| cellpadding="2~~" style="background:lightcyan~~"

+

{| cellpadding="2"

| ''f''8

|-

Line 6,507: Line 6,507:

−

{| align="center" border="1" cellpadding="12" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:96%"

+

{| align="center" border="1" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"

|

−

{| align="left" border="0" cellpadding="12" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"

| width="8%" | E''G''''i''

| width="4%" | =

Line 6,520: Line 6,520:

−

{| align="center" border="1" cellpadding="12" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:96%"

+

{| align="center" border="1" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"

|

−

{| align="left" border="0" cellpadding="12" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"

| width="8%" | D''G''''i''

| width="4%" | =

Line 6,545: Line 6,545:

−

{| align="center" border="1" cellpadding="12" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:96%"

+

{| align="center" border="1" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"

|

−

{| align="left" border="0" cellpadding="12" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"

| width="8%" | E''f''

| width="4%" | =

Line 6,560: Line 6,560:

−

{| align="center" border="1" cellpadding="12" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:96%"

+

{| align="center" border="1" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"

|

−

{| align="left" border="0" cellpadding="12" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:left; width:100%"

+

{| align="left" border="0" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"

| width="8%" | D''f''

| width="4%" | =

Line 6,581: Line 6,581:

−

{| align="center" border="1" cellpadding="8" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:96%"

+

{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"

|+ '''Table 66-i. Computation Summary for ''f''‹''u'', ''v''› = ((''u'')(''v''))'''

|

−

{| align="left" border="0" cellpadding="1" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="left" border="0" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| <math>\epsilon</math>''f''

| = || ''uv'' || <math>\cdot</math> || 1

Line 6,619: Line 6,619:

−

{| align="center" border="1" cellpadding="8" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:96%"

+

{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"

|+ '''Table 66-ii. Computation Summary for g‹''u'', ''v''› = ((''u'', ''v''))'''

|

−

{| align="left" border="0" cellpadding="1" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="left" border="0" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| <math>\epsilon</math>''g''

| = || ''uv'' || <math>\cdot</math> || 1

Line 6,658: Line 6,658:

Table 67 shows how to compute the analytic series for ''F'' = ‹''f'', ''g''› = ‹((''u'')(''v'')), ((''u'', ''v''))› in terms of coordinates, and Table 68 recaps these results in symbolic terms, agreeing with earlier derivations.

−

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

+

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

|+ '''Table 67. Computation of an Analytic Series in Terms of Coordinates'''

|

−

{| align="center" border="1" cellpadding="0" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:~~paleturquoise~~; font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| ''u''

| ''v''

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:~~paleturquoise~~; font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| d''u''

| d''v''

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:~~paleturquoise~~; font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| ''u''’

| ''v''’

Line 6,679: Line 6,679:

|-

| valign="top" |

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 0

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 0

|-

Line 6,693: Line 6,693:

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 0

|-

Line 6,704: Line 6,704:

|-

| valign="top" |

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 1

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 0

|-

Line 6,718: Line 6,718:

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 1

|-

Line 6,729: Line 6,729:

|-

| valign="top" |

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 1 || 0

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 0

|-

Line 6,743: Line 6,743:

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 1 || 0

|-

Line 6,754: Line 6,754:

|-

| valign="top" |

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 1 || 1

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 1 || 1

|-

Line 6,768: Line 6,768:

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 0

|-

Line 6,779: Line 6,779:

|}

|

−

{| align="center" border="1" cellpadding="0" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:~~paleturquoise~~; font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| <math>\epsilon</math>''f''

| <math>\epsilon</math>''g''

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:~~paleturquoise~~; font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| E''f''

| E''g''

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:~~paleturquoise~~; font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| D''f''

| D''g''

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:~~paleturquoise~~; font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| d''f''

| d''g''

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:~~paleturquoise~~; font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"

| d2''f''

| d2''g''

Line 6,807: Line 6,807:

|-

| valign="top" |

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 1

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 1

|-

Line 6,821: Line 6,821:

|}

|

−

{| align="center" border="0" cellpadding="6" cellspacing="0" style="~~background:lightcyan;~~ font-weight:bold; text-align:center; width:100%"

+

{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

| 0 || 0

|-

Line 6,831: Line 6,831:

|}

|

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|+ '''Table 68. Computation of an Analytic Series in Symbolic Terms'''

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|- style="background:ghostwhite"

| ''u''  ''v''

| ''f''  ''g''

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|

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| 0  0

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| 0  1

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| ((d''u'')(d''v''))

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| (d''u'', d''v'')

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| (d''u'', d''v'')

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| (d''u'', d''v'')

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| d''u'' d''v''

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| ( )

|-

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|  

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[[Category:Computer Science]]

[[Category:Cybernetics]]

+

[[Category:Differential Logic]]

[[Category:Discrete Systems]]

[[Category:Dynamical Systems]]

Jon Awbrey

12,089

edits