ShadmiDoron
September 11, 2006, 10:45 AM
The standard Set/Member relation is based on a xor connective.
For example: Any given x is a member xor not a member of set A and there is no intermediate state.
Fuzzy logic expands standard membership (0 xor 1) by using x, which defines the degree of a membership between 0 and 1 (0 or 1 are included too).
If 0<x<1, and [0,x] belongs to set A then (x,1] does not belong to set A ( [0,x] xor (x,1] )
So, in both cases a xor connective is used as the logical basis of the Set/Member relations.
0, 1, [0,x] or (x,1] (where 0<x<1) are all local mathematical objects because we can clearly define their locality (they are "in" xor "out" of some mathematical object).
An object that is not a set but can be a member of a set, is called an urelemnt (http://72.14.221.104/search?q=cache:JRO16vlJWNQJ:en.wikipedia.org/wiki/Urelement+urelement&hl=en&ct=clnk&cd=1 ).
A sub-object is a part of an object.
Since an urelement is not a set, it does include any sub-object as a part of it, or in other words, it is an atomic singleton.
x is a urelement.
If x is a member and not a member of A then x is a non-local mathematical object.
The best way to notate this is: _{_} , where __ is a urelement.
__ can be a local ( {__} xor __{ } ) or a non-local ( _{_} ) urelement.
If we wish to find the best way to notate a local urelement (a local atomic singleton) than we use . (a point).
An example: {.} xor .{ }
In the standard Set/Member relations each member is a sub-object xor not a sub-object of a set, and both the empty and any a non-empty set are defined by the ways that these sub-objects belongs xor don't belong to them.
Since . and __ are atomic singletons, they are not defined by each other (they are mutually independent exactly like two axioms) and we can expand the membership concept beyond the Set/Member dependency.
For more details please look at http://www.geocities.com/complementarytheory/TOUM.pdf .
Please reply youe comments.
Thank you,
Doron
For example: Any given x is a member xor not a member of set A and there is no intermediate state.
Fuzzy logic expands standard membership (0 xor 1) by using x, which defines the degree of a membership between 0 and 1 (0 or 1 are included too).
If 0<x<1, and [0,x] belongs to set A then (x,1] does not belong to set A ( [0,x] xor (x,1] )
So, in both cases a xor connective is used as the logical basis of the Set/Member relations.
0, 1, [0,x] or (x,1] (where 0<x<1) are all local mathematical objects because we can clearly define their locality (they are "in" xor "out" of some mathematical object).
An object that is not a set but can be a member of a set, is called an urelemnt (http://72.14.221.104/search?q=cache:JRO16vlJWNQJ:en.wikipedia.org/wiki/Urelement+urelement&hl=en&ct=clnk&cd=1 ).
A sub-object is a part of an object.
Since an urelement is not a set, it does include any sub-object as a part of it, or in other words, it is an atomic singleton.
x is a urelement.
If x is a member and not a member of A then x is a non-local mathematical object.
The best way to notate this is: _{_} , where __ is a urelement.
__ can be a local ( {__} xor __{ } ) or a non-local ( _{_} ) urelement.
If we wish to find the best way to notate a local urelement (a local atomic singleton) than we use . (a point).
An example: {.} xor .{ }
In the standard Set/Member relations each member is a sub-object xor not a sub-object of a set, and both the empty and any a non-empty set are defined by the ways that these sub-objects belongs xor don't belong to them.
Since . and __ are atomic singletons, they are not defined by each other (they are mutually independent exactly like two axioms) and we can expand the membership concept beyond the Set/Member dependency.
For more details please look at http://www.geocities.com/complementarytheory/TOUM.pdf .
Please reply youe comments.
Thank you,
Doron