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Homogeneous relation

A dyadic relation xRy is homogeneous when permuting referent x and relatum y in R, the resulting formula, i.e. yRx, is plausible but neither mandatory nor impossible.

Homogeneous polyadic predicate was defined by R. Carnap. When "from a sentence of n arguments, another sentence always arises as a result of any permutation of n arguments.The majority of the terms of relational theory refer to homogeneous binary predicates."^[1].

The homogeneity is not a new property of xRy apart from reflexive/irreflexive, symmetrical/anti-symmetrical and transitive/intransitive. It is a label[1] that applies when any of mentioned properties not always hold but only sometimes e.g. 'x functionally determines y' (x→y) is known to be reflexive, non-symmetrical and transitive; adding "x→y is homogeneous", means that y→x can hold for some couple of the set <x,y>€U² where x≠y.

Let explain a case of homogeneity applying to the three main properties of a relation e.g. 'x loves y' (x♥y); (x♥y), as we know, is non-reflexive, non-symmetrical, non-transitive, then saying "x♥y is homogeneous" gives three times more information that the first example.

Finally, "xPy ('x father of y') is an heterogeneous relation" summarizes that xPy is irreflexive, anti-symmetrical and intransitive.

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