({ Clicking on a green or brown colored
Method displays or hides the
Application of that
Method, and also displays or hides another Method or the
Reason in that
Paragraph. })
(^ KnowledgeBase ^)
[[ Abstract relations ]]
[( Things and features )]
({ We tell things apart by their distinguishing features. In this knowledge
base, +ft// +a +b will express the relation that a
thing +a is a feature of a thing +b. })
[- Being the same -]
({ Things are the same if they have the same features. Thus the statement
that a thing +a is the same as a thing +b is an
abbreviation of the statement that for every thing +c,
+c is a feature of +a if and only if +c
is a feature of +b. })
| +Df |
+sm// +a +b |
[c = +ft// +c +a +ft// +c +b |
({ A thing is the same as itself. })
| +1 |
+Tb |
= +ft// +c +a +ft// +c +a |
| +2 |
+Gn/ +1 |
[c = +ft// +c +a +ft// +c +a |
| +Sm1 | +Pr | +Ab/ +2 |
+sm// +a +a |
| +1 |
+Ex |
) - [c = +ft// +c +a +ft// +c +b = +ft// +c +a +ft// +c +b |
| +Sm2 | +Pr | +Ab/ +1 |
) - +sm// +a +b = +ft// +c +a +ft// +c +b |
({ The relation of being the same is symmetrical. })
| +1 |
+Tb/ +Sm2 |
) - +sm// +a +b = +ft// +c +b +ft// +c +a |
| +2 |
+Gn/ +1 |
[c ) - +sm// +a +b = +ft// +c +b +ft// +c +a |
| +3 |
+Fc/ +2 |
) - +sm// +a +b [c = +ft// +c +b +ft// +c +a |
| +4 |
+Ab/ +3 |
) - +sm// +a +b +sm// +b +a |
| +1 |
+Ex |
) - [b ) - +sm// +a +b +sm// +b +a ) - +sm// +a +c +sm// +c +a |
| +2 |
+Gn/ +4 |
[b ) - +sm// +a +b +sm// +b +a |
| +5 | +Rp/ +4 | +Tb// +1 +2 |
) - +sm// +a +c +sm// +c +a |
| +1 |
+Ex |
) - [a ) - +sm// +a +c +sm// +c +a ) - +sm// +b +c +sm// +c +b |
| +2 |
+Gn/ +5 |
[a ) - +sm// +a +c +sm// +c +a |
| +6 | +Rp/ +5 | +Tb// +1 +2 |
) - +sm// +b +c +sm// +c +b |
| +1 |
+Ex |
) - [c ) - +sm// +b +c +sm// +c +b ) - +sm// +b +a +sm// +a +b |
| +2 |
+Gn/ +6 |
[c ) - +sm// +b +c +sm// +c +b |
| +7 | +Rp/ +6 | +Tb// +1 +2 |
) - +sm// +b +a +sm// +a +b |
| +Sm3 | +Pr | +Tb// +4 +7 |
= +sm// +a +b +sm// +b +a |
({ The relation of being the same is transmitted. })
| +1 |
+Ex |
) - [c = +ft// +c +b +ft// +c +d = +ft// +c +b +ft// +c +d |
| +2 |
+Ab/ +1 |
) - +sm// +b +d = +ft// +c +b +ft// +c +d |
| +3 |
+Tb// +Sm2 +2 |
) - ( +sm// +a +b +sm// +b +d = +ft// +c +a +ft// +c +d |
| +4 |
+Gn/ +3 |
[c ) - ( +sm// +a +b +sm// +b +d = +ft// +c +a +ft// +c +d |
| +5 |
+Fc/ +4 |
) - ( +sm// +a +b +sm// +b +d [c = +ft// +c +a +ft// +c +d |
| +Sm4 | +Pr | +Ab/ +5 |
) - ( +sm// +a +b +sm// +b +d +sm// +a +d |
| +1 |
+Ex |
) - [a = +sm// +a +b +sm// +b +a = +sm// +d +b +sm// +b +d |
| +2 |
+Gn/ +Sm3 |
[a = +sm// +a +b +sm// +b +a |
| +1 | +Rp/ +Sm3 | +Tb// +1 +2 |
= +sm// +d +b +sm// +b +d |
| +Sm5 | +Pr | +Tb// +Sm4 +1 |
) - ( +sm// +a +b +sm// +d +b +sm// +a +d |
| +Sm6 |
+Tb// +Sm3 +Sm4 |
) - ( +sm// +b +a +sm// +b +d +sm// +a +d |
| +1 |
+Ex |
) - [a ) - ( +sm// +a +b +sm// +b +d +sm// +a +d ) - ( +sm// +c +b +sm// +b +d +sm// +c +d |
| +2 |
+Gn/ +Sm4 |
[a ) - ( +sm// +a +b +sm// +b +d +sm// +a +d |
| +1 | +Rp/ +Sm4 | +Tb// +1 +2 |
) - ( +sm// +c +b +sm// +b +d +sm// +c +d |
| +1a |
+Ex |
) - [d ) - ( +sm// +c +b +sm// +b +d +sm// +c +d ) - ( +sm// +c +b +sm// +b +a +sm// +c +a |
| +2 |
+Gn/ +1 |
[d ) - ( +sm// +c +b +sm// +b +d +sm// +c +d |
| +2 | +Rp/ +1 | +Tb// +1a +2 |
) - ( +sm// +c +b +sm// +b +a +sm// +c +a |
| +1 |
+Ex |
) - [c ) - ( +sm// +c +b +sm// +b +a +sm// +c +a ) - ( +sm// +d +b +sm// +b +a +sm// +d +a |
| +2a |
+Gn/ +2 |
[c ) - ( +sm// +c +b +sm// +b +a +sm// +c +a |
| +3 | +Rp/ +2 | +Tb// +1 +2a |
) - ( +sm// +d +b +sm// +b +a +sm// +d +a |
| +1 |
+Ex |
) - [b ) - ( +sm// +d +b +sm// +b +a +sm// +d +a ) - ( +sm// +d +c +sm// +c +a +sm// +d +a |
| +2 |
+Gn/ +3 |
[b ) - ( +sm// +d +b +sm// +b +a +sm// +d +a |
| +4 | +Rp/ +3 | +Tb// +1 +2 |
) - ( +sm// +d +c +sm// +c +a +sm// +d +a |
| +1 |
+Ex |
) - [d ) - ( +sm// +d +c +sm// +c +a +sm// +d +a ) - ( +sm// +b +c +sm// +c +a +sm// +b +a |
| +2 |
+Gn/ +4 |
[d ) - ( +sm// +d +c +sm// +c +a +sm// +d +a |
| +5 | +Rp/ +4 | +Tb// +1 +2 |
) - ( +sm// +b +c +sm// +c +a +sm// +b +a |
| +5a |
+Tb/ +Sm3 |
) - +sm// +a +b +sm// +b +a |
| +1 |
+Ex |
) - [b ) - +sm// +a +b +sm// +b +a ) - +sm// +a +c +sm// +c +a |
| +2 |
+Gn/ +5a |
[b ) - +sm// +a +b +sm// +b +a |
| +6 | +Rp/ +5a | +Tb// +1 +2 |
) - +sm// +a +c +sm// +c +a |
| +1 |
+Ex |
) - [a ) - +sm// +a +c +sm// +c +a ) - +sm// +b +c +sm// +c +b |
| +2 |
+Gn/ +6 |
[a ) - +sm// +a +c +sm// +c +a |
| +7 | +Rp/ +6 | +Tb// +1 +2 |
) - +sm// +b +c +sm// +c +b |
| +8 |
+Tb// +2 +7 |
) - ( +sm// +b +c +sm// +b +a +sm// +c +a |
| +Sm7 | +Pr | +Tb// +5 +8 |
) - +sm// +b +c = +sm// +b +a +sm// +c +a |
| +1 |
+Ex |
) - [b = +sm// +a +b +sm// +b +a = +sm// +a +c +sm// +c +a |
| +2 |
+Gn/ +Sm3 |
[b = +sm// +a +b +sm// +b +a |
| +1 | +Rp/ +Sm3 | +Tb// +1 +2 |
= +sm// +a +c +sm// +c +a |
| +Sm8 | +Pr | +Tb/// +Sm3 +Sm7 +1 |
) - +sm// +b +c = +sm// +a +b +sm// +a +c |
[- Being a feature -]
({ A thing will be called a feature if it is a feature of something. })
| +Df |
+ft/ +a |
]b +ft// +a +b |
({ The Adverb can be changed as in the
following example. })
| +1 |
+Ex |
) - [b - +ft// +a +b - +ft// +a +c |
| +2 |
+Gn/ +1 |
[c ) - [b - +ft// +a +b - +ft// +a +c |
| +3 |
+Fc/ +2 |
) - [b - +ft// +a +b [c - +ft// +a +c |
| +4 |
+Ex |
) - [c - +ft// +a +c - +ft// +a +b |
| +5 |
+Gn/ +4 |
[b ) - [c - +ft// +a +c - +ft// +a +b |
| +6 |
+Fc/ +5 |
) - [c - +ft// +a +c [b - +ft// +a +b |
| +7 |
+Tb// +3 +6 |
= - [b - +ft// +a +b - [c - +ft// +a +c |
| +8 |
+Sm/ +7 |
= ]b +ft// +a +b - [c - +ft// +a +c |
| +9 |
+Sm/ +8 |
= ]b +ft// +a +b ]c +ft// +a +c |
| +Ft1 | +Pr | +Ab/ +9 |
= +ft/ +a ]c +ft// +a +c |
[- No features -]
({ The thing with no features will be represented by +0. It is assumed
to be a feature. })
| +Df |
+Nf |
( [a - +ft// +a +0 +ft/ +0 |
| +1 |
+Ex |
) - [a - +ft// +a +0 - +ft// +c +0 |
| +2 |
+Tb/ +1 |
) - ( [a - +ft// +a +0 +ft/ +0 - +ft// +c +0 |
| +Nf1 | +Pr | +Ab/ +2 |
) - +Nf - +ft// +c +0 |
| +1 |
+Ex |
) - [b ) - +sm// +a +b = +ft// +c +a +ft// +c +b ) - +sm// +a +0 = +ft// +c +a +ft// +c +0 |
| +2 |
+Gn/ +Sm2 |
[b ) - +sm// +a +b = +ft// +c +a +ft// +c +b |
| +1 | +Rp/ +Sm2 | +Tb// +1 +2 |
) - +sm// +a +0 = +ft// +c +a +ft// +c +0 |
| +Nf2 | +Pr | +Tb// +1 +Nf1 |
) - ( +Nf +sm// +a +0 - +ft// +c +a |
| +1 |
+Gn/ +Nf2 |
[c ) - ( +Nf +sm// +a +0 - +ft// +c +a |
| +2 |
+Fc/ +1 |
) - ( +Nf +sm// +a +0 [c - +ft// +c +a |
| +3 |
+Tb/ +2 |
) - +Nf ) - +sm// +a +0 [c - +ft// +c +a |
| +4 |
+Ex |
) - [c - +ft// +c +a - +ft// +c +a |
| +5 |
+Tb// +Nf1 +4 |
) - ( +Nf [c - +ft// +c +a = +ft// +c +a +ft// +c +0 |
| +6 |
+Gn/ +5 |
[c ) - ( +Nf [c - +ft// +c +a = +ft// +c +a +ft// +c +0 |
| +7 |
+Fc/ +6 |
) - ( +Nf [c - +ft// +c +a [c = +ft// +c +a +ft// +c +0 |
| +8 |
+Tb/ +7 |
) - +Nf ) - [c - +ft// +c +a [c = +ft// +c +a +ft// +c +0 |
| +9 |
+Ab/ +8 |
) - +Nf ) - [c - +ft// +c +a +sm// +a +0 |
| +Nf3 | +Pr | +Tb// +3 +9 |
) - +Nf = +sm// +a +0 [c - +ft// +c +a |
| +1 |
+Tb |
) - ( [a - +ft// +a +0 +ft/ +0 +ft/ +0 |
| +Nf4 | +Pr | +Ab/ +1 |
) - +Nf +ft/ +0 |
[- Specified features -]
({ If +a and +b are features, then the thing whose
features are +a and +b will be represented
by +th// +a +b. So a thing +c is a feature of
+th// +a +b if and only if either +c is the same as
+a or +c is the same as +b. })
| +Df |
+Th |
[a [b ) - ( +ft/ +a +ft/ +b [c = +ft// +c +th// +a +b ) +sm// +c +a +sm// +c +b |
({ If a thing is the same as a feature, then that thing is also a feature. })
| +1 |
+Ex |
) - [a [b ) - ( +ft/ +a +ft/ +b [c = +ft// +c +th// +a +b ) +sm// +c +a +sm// +c +b
[b ) - ( +ft/ +a +ft/ +b [c = +ft// +c +th// +a +b ) +sm// +c +a +sm// +c +b |
| +2 |
+Ab/ +1 |
) - +Th [b ) - ( +ft/ +a +ft/ +b [c = +ft// +c +th// +a +b ) +sm// +c +a +sm// +c +b |
| +3 |
+Ex |
) - [b ) - ( +ft/ +a +ft/ +b [c = +ft// +c +th// +a +b ) +sm// +c +a +sm// +c +b
) - ( +ft/ +a +ft/ +a [c = +ft// +c +th// +a +a ) +sm// +c +a +sm// +c +a |
| +4 |
+Tb// +2 +3 |
) - +Th ) - +ft/ +a [c = +ft// +c +th// +a +a ) +sm// +c +a +sm// +c +a |
| +5 |
+Ex |
) - [c = +ft// +c +th// +a +a ) +sm// +c +a +sm// +c +a
= +ft// +b +th// +a +a ) +sm// +b +a +sm// +b +a |
| +6 |
+Tb// +4 +5 |
) - +Th ) - ( +ft/ +a +sm// +b +a +ft// +b +th// +a +a |
| +7 |
+Ex |
) - [c - +ft// +b +c - +ft// +b +th// +a +a |
| +8 |
+Tb// +6 +7 |
) - +Th ) - ( +ft/ +a +sm// +b +a - [c - +ft// +b +c |
| +9 |
+Sm/ +8 |
) - +Th ) - ( +ft/ +a +sm// +b +a ]c +ft// +b +c |
| +1 |
+Ex |
) - [a = +ft/ +a ]c +ft// +a +c = +ft/ +b ]c +ft// +b +c |
| +2 |
+Gn/ +Ft1 |
[a = +ft/ +a ]c +ft// +a +c |
| +10 | +Rp/ +Ft1 | +Tb// +1 +2 |
= +ft/ +b ]c +ft// +b +c |
| +Ft2 | +Pr | +Tb// +9 +10 |
) - +Th ) - ( +ft/ +a +sm// +b +a +ft/ +b |
| +1 |
+Ex |
) - [a [b ) - ( +ft/ +a +ft/ +b [c = +ft// +c +th// +a +b ) +sm// +c +a +sm// +c +b
[b ) - ( +ft/ +a +ft/ +b [c = +ft// +c +th// +a +b ) +sm// +c +a +sm// +c +b |
| +2 |
+Ab/ +1 |
) - +Th
[b ) - ( +ft/ +a +ft/ +b [c = +ft// +c +th// +a +b ) +sm// +c +a +sm// +c +b |
| +3 |
+Ex |
) - [b ) - ( +ft/ +a +ft/ +b [c = +ft// +c +th// +a +b ) +sm// +c +a +sm// +c +b
) - ( +ft/ +a +ft/ +b [c = +ft// +c +th// +a +b ) +sm// +c +a +sm// +c +b |
| +4 |
+Ex |
) - [c = +ft// +c +th// +a +b ) +sm// +c +a +sm// +c +b
= +ft// +c +th// +a +b ) +sm// +c +a +sm// +c +b |
| +Th1 | +Pr | +Tb/// +2 +3 +4 |
) - +Th ) - ( +ft/ +a +ft/ +b
= +ft// +c +th// +a +b ) +sm// +c +a +sm// +c +b |
| +1 |
+Ex |
) - [c ) - +Th ) - ( +ft/ +a +ft/ +b = +ft// +c +th// +a +b ) +sm// +c +a +sm// +c +b ) - +Th ) - ( +ft/ +a +ft/ +b = +ft// +a +th// +a +b ) +sm// +a +a +sm// +a +b |
| +2 |
+Gn/ +Th1 |
[c ) - +Th ) - ( +ft/ +a +ft/ +b = +ft// +c +th// +a +b ) +sm// +c +a +sm// +c +b |
| +1 | +Rp/ +Th1 | +Tb// +1 +2 |
) - +Th ) - ( +ft/ +a +ft/ +b
= +ft// +a +th// +a +b ) +sm// +a +a +sm// +a +b |
| +Th2 | +Pr | +Tb// +Sm1 +1 |
) - +Th ) - ( +ft/ +a +ft/ +b +ft// +a +th// +a +b |
| +1 |
+Ex |
) - [c - +ft// +c +th// +a +b - +ft// +a +th// +a +b |
| +2 |
+Tb// +Th2 +1 |
) - +Th ) - ( +ft/ +a +ft/ +b - [c - +ft// +c +th// +a +b |
| +Th3 | +Pr | +Sm/ +2 |
) - +Th ) - ( +ft/ +a +ft/ +b ]c +ft// +c +th// +a +b |
| +1 |
+Ex |
) - [c ) - +Th ) - ( +ft/ +a +ft/ +b = +ft// +c +th// +a +b ) +sm// +c +a +sm// +c +b ) - +Th ) - ( +ft/ +a +ft/ +b = +ft// +b +th// +a +b ) +sm// +b +a +sm// +b +b |
| +2 |
+Gn/ +Th1 |
[c ) - +Th ) - ( +ft/ +a +ft/ +b = +ft// +c +th// +a +b ) +sm// +c +a +sm// +c +b |
| +1 | +Rp/ +Th1 | +Tb// +1 +2 |
) - +Th ) - ( +ft/ +a +ft/ +b
= +ft// +b +th// +a +b ) +sm// +b +a +sm// +b +b |
| +1 |
+Ex |
) - [a +sm// +a +a +sm// +b +b |
| +2 |
+Gn/ +Sm1 |
[a +sm// +a +a |
| +2 | +Rp/ +Sm1 | +Tb// +1 +2 |
+sm// +b +b |
| +Th4 | +Pr | +Tb// +1 +2 |
) - +Th ) - ( +ft/ +a +ft/ +b +ft// +b +th// +a +b |
({ +th// +a +b is symmetrical. })
| +1 |
+Ex |
) - [a ) - +Th ) - ( +ft/ +a +ft/ +b = +ft// +c +th// +a +b ) +sm// +c +a +sm// +c +b ) - +Th ) - ( +ft/ +d +ft/ +b = +ft// +c +th// +d +b ) +sm// +c +d +sm// +c +b |
| +2 |
+Gn/ +Th1 |
[a ) - +Th ) - ( +ft/ +a +ft/ +b = +ft// +c +th// +a +b ) +sm// +c +a +sm// +c +b |
| +1 | +Rp/ +Th1 | +Tb// +1 +2 |
) - +Th ) - ( +ft/ +d +ft/ +b = +ft// +c +th// +d +b ) +sm// +c +d +sm// +c +b |
| +1a |
+Ex |
) - [b ) - +Th ) - ( +ft/ +d +ft/ +b = +ft// +c +th// +d +b ) +sm// +c +d +sm// +c +b ) - +Th ) - ( +ft/ +d +ft/ +a = +ft// +c +th// +d +a ) +sm// +c +d +sm// +c +a |
| +2 |
+Gn/ +1 |
[b ) - +Th ) - ( +ft/ +d +ft/ +b = +ft// +c +th// +d +b ) +sm// +c +d +sm// +c +b |
| +2 | +Rp/ +1 | +Tb// +1a +2 |
) - +Th ) - ( +ft/ +d +ft/ +a = +ft// +c +th// +d +a ) +sm// +c +d +sm// +c +a |
| +1 |
+Ex |
) - [d ) - +Th ) - ( +ft/ +d +ft/ +a = +ft// +c +th// +d +a ) +sm// +c +d +sm// +c +a ) - +Th ) - ( +ft/ +b +ft/ +a = +ft// +c +th// +b +a ) +sm// +c +b +sm// +c +a |
| +2a |
+Gn/ +2 |
[d ) - +Th ) - ( +ft/ +d +ft/ +a = +ft// +c +th// +d +a ) +sm// +c +d +sm// +c +a |
| +3 | +Rp/ +2 | +Tb// +1 +2a |
) - +Th ) - ( +ft/ +b +ft/ +a = +ft// +c +th// +b +a ) +sm// +c +b +sm// +c +a |
| +4 |
+Tb// +Th1 +3 |
) - ( +Th ( +ft/ +a +ft/ +b = +ft// +c +th// +a +b +ft// +c +th// +b +a |
| +5 |
+Gn/ +4 |
[c ) - ( +Th ( +ft/ +a +ft/ +b = +ft// +c +th// +a +b +ft// +c +th// +b +a |
| +6 |
+Fc/ +5 |
) - ( +Th ( +ft/ +a +ft/ +b [c = +ft// +c +th// +a +b +ft// +c +th// +b +a |
| +Th5 | +Pr | +Ab/ +6 |
) - ( +Th ( +ft/ +a +ft/ +b +sm// +th// +a +b +th// +b +a |
({ The relation of being the same is transmitted for +th//. })
| +1 |
+Ex |
) - [a ) - ( +sm// +a +b +sm// +b +d +sm// +a +d ) - ( +sm// +c +b +sm// +b +d +sm// +c +d |
| +2 |
+Gn/ +Sm4 |
[a ) - ( +sm// +a +b +sm// +b +d +sm// +a +d |
| +1 | +Rp/ +Sm4 | +Tb// +1 +2 |
) - ( +sm// +c +b +sm// +b +d +sm// +c +d |
| +1a |
+Ex |
) - [b ) - ( +sm// +c +b +sm// +b +d +sm// +c +d ) - ( +sm// +c +a +sm// +a +d +sm// +c +d |
| +2 |
+Gn/ +1 |
[b ) - ( +sm// +c +b +sm// +b +d +sm// +c +d |
| +2 | +Rp/ +1 | +Tb// +1a +2 |
) - ( +sm// +c +a +sm// +a +d +sm// +c +d |
| +1a |
+Ex |
) - [d ) - ( +sm// +c +b +sm// +b +d +sm// +c +d ) - ( +sm// +c +b +sm// +b +a +sm// +c +a |
| +2 |
+Gn/ +1 |
[d ) - ( +sm// +c +b +sm// +b +d +sm// +c +d |
| +3 | +Rp/ +1 | +Tb// +1a +2 |
) - ( +sm// +c +b +sm// +b +a +sm// +c +a |
| +1 |
+Ex |
) - [b ) - ( +sm// +c +b +sm// +b +a +sm// +c +a ) - ( +sm// +c +d +sm// +d +a +sm// +c +a |
| +2 |
+Gn/ +3 |
[b ) - ( +sm// +c +b +sm// +b +a +sm// +c +a |
| +4 | +Rp/ +3 | +Tb// +1 +2 |
) - ( +sm// +c +d +sm// +d +a +sm// +c +a |
| +4a |
+Tb/ +Sm3 |
) - +sm// +a +b +sm// +b +a |
| +1 |
+Ex |
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({ The thing whose only feature is +a will be represented by
+th/ +a. })
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