A Dictionary of Philosophy, Third Edition
. A relation holding from a to b is symmetric if it must hold from b to a (sibling of). It is asymmetric if it cannot (parent of), and non-symmetric if it may or may not (brother of). ‘Non-symmetric’ occasionally includes ‘asymmetric’.
It is anti-symmetric if its holding from a to b and from b to a implies that a and b are the same object. Not greater than among numbers is anti-symmetric, but not older than among people is not, but only non-symmetric: different people can have the same age, but different numbers cannot have the same size, so if a is not greater than b nor b than a, then a and b are the same number. Sometimes ‘weak anti-symmetric’ is used for this, with ‘anti-symmetric’ replacing ‘asymmetric’.
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