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(^ 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. })

+Sm1 +Pr +sm// +a +a
+Sm2 +Pr ) - +sm// +a +b = +ft// +c +a +ft// +c +b

({ The relation of being the same is symmetrical. })

+Sm3 +Pr = +sm// +a +b +sm// +b +a

({ The relation of being the same is transmitted. })

+Sm4 +Pr ) - ( +sm// +a +b +sm// +b +d +sm// +a +d
+Sm5 +Pr ) - ( +sm// +a +b +sm// +d +b +sm// +a +d
+Sm6 +Tb// +Sm3 +Sm4 ) - ( +sm// +b +a +sm// +b +d +sm// +a +d
+Sm7 +Pr ) - +sm// +b +c = +sm// +b +a +sm// +c +a
+Sm8 +Pr ) - +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. })

+Ft1 +Pr = +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
+Nf1 +Pr ) - +Nf - +ft// +c +0
+Nf2 +Pr ) - ( +Nf +sm// +a +0 - +ft// +c +a
+Nf3 +Pr ) - +Nf = +sm// +a +0 [c - +ft// +c +a
+Nf4 +Pr ) - +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. })

+Ft2 +Pr ) - +Th ) - ( +ft/ +a +sm// +b +a +ft/ +b
+Th1 +Pr ) - +Th ) - ( +ft/ +a +ft/ +b = +ft// +c +th// +a +b ) +sm// +c +a +sm// +c +b
+Th2 +Pr ) - +Th ) - ( +ft/ +a +ft/ +b +ft// +a +th// +a +b
+Th3 +Pr ) - +Th ) - ( +ft/ +a +ft/ +b ]c +ft// +c +th// +a +b
+Th4 +Pr ) - +Th ) - ( +ft/ +a +ft/ +b +ft// +b +th// +a +b

({ +th// +a +b is symmetrical. })

+Th5 +Pr ) - ( +Th ( +ft/ +a +ft/ +b +sm// +th// +a +b +th// +b +a

({ The relation of being the same is transmitted for +th//. })

+Th6 +Pr ) - ( +Th ( +ft/ +a ( +ft/ +b +sm// +a +d +sm// +th// +a +b +th// +d +b
+Th7 +Pr ) - ( +Th ( +ft/ +a ( +ft/ +b +sm// +a +d +sm// +th// +b +a +th// +b +d
+Th8 +Pr ) - ( +Th ( +ft/ +a ( +ft/ +b ( +ft/ +d +ft/ +f ) - +sm// +th// +a +b +th// +d +f ) ( +sm// +a +d +sm// +b +f ( +sm// +a +f +sm// +b +d
+Th9 +Pr ) - ( +Th ( +ft/ +b ( +ft/ +d +ft/ +f ) - +sm// +th// +d +b +th// +d +f +sm// +b +f

({ The thing whose only feature is +a will be represented by +th/ +a. })

+Df +th/ +a +th// +a +a
+Th10 +Pr ) - +Th ) - +ft/ +a = +ft// +c +th/ +a +sm// +c +a
+Th11 +Pr ) - +Th ) - +ft/ +a +ft// +a +th/ +a
+Th12 +Pr ) - ( +Th ( +ft/ +a +sm// +a +d +sm// +th/ +a +th/ +d
+Th13 +Pr ) - ( +Th ( +ft/ +a ( +ft/ +b +sm// +th/ +a +th/ +b +sm// +a +b
+Th14 +Pr ) - ( +Th ( +ft/ +a ( +ft/ +b +ft/ +c ) - +sm// +th// +a +b +th/ +c +sm// +a +b
+Th15 +Pr ) - ( +Nf +Th ) - +ft/ +a - +sm// +th/ +a +0
+Th16 +Pr ) - ( +Nf +Th ) - +ft/ +a - +sm// +0 +th/ +a
+Th17 +Pr ) - +Th ) - +ft/ +a ) - +sm// +b +th/ +a +ft// +a +b

(_ Amalgamation _)

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