Many ternary properties are interesting, and many of them are learned in school before properties of binary relations. I define any ternary relation as a relation that is expressed as a three-place predicate. For example, consider the three place predicate, '_____added to_____yields_____'. This particular 3-adic relation has a number of interesting properties, including:
the commutative property: (All x)(All y)(All z)(Rxyz -> Ryxz)
the associative property: (All w)(All x)(All y)(All z)(Rs(w,x)yz -> Rws(x,y)z), where the function s takes the sum of its inputs.
et cetera.
Note that these two properties are also properties of the relation '_____multiplied by_____yields_____'.
These two ternary relations differ, however, on this property (the additive identity property):
(All x)(Rx0x)
For all quantities, x added to 0 yields x, but it is not the case that for all quantities x, x multiplied by 0 yields x.
Notice that the 3-adic relation '_____and_____sit to either side of_____on the sofa' also shares the commutative property.