Advanced Symbolic Logic

Properties of Relational Predicates
'Rxy' = 'x bears R to y'

Symmetry: (∀x)(∀y)(Rxy Ryx)
Asymmetry: (∀x)(∀y)((Rxy -Ryx)
Nonsymmetry: (x)(y)(Rxy Ryx) (x)(y)(Rxy -Ryx)

Transitivity: (∀x)(∀y)(∀z)((Rxy Ryz) Rxz)
Intransitivity: (∀x)(∀y)(∀z)((Rxy Ryz) -Rxz)
Nontransitivity: (x)(y)(z)((Rxy Ryz) Rxz)) (x)(y)(z)((Rxy Ryz) -Rxz))

Total Reflexivity: (∀x)Rxx
Reflexivity: (∀x)(∀y)((Rxy v Ryx) Rxx)
Irreflexivity: (∀x)-Rxx
Nonreflexivity: (x)(y)((Rxy v Ryx) -Rxx) (x)Rxx

Every dyadic relation has at least one property from each set above. For instance, the relation 'is greater than' is asymmetrical, transitive, and irreflexive.

'is identical to' = symmetrical, transitive, totally reflexive
'is next to' = symmetrical, intransitive, irreflexive
'is the successor to' = asymmetrical, intransitive, irreflexive
'is the author of' = asymmetrical, intransitive, irreflexive
'is the negation of' = asymmetrical, intransitive, irreflexive
'has same color eyes as' = symmetrical, transitive, totally reflexive
'admires' = nonsymmetrical, nontransitive, nonreflexive
'contradicts' = symmetical, transitive, nonreflexive
'to the left of' = asymmetrical, transitive, irreflexive
'is younger than' = asymmetrical, transitive, irreflexive
'is divisible by' = asymmetrical, intransitive, reflexive
'is less than by 1' = asymmetrical, intransitive, irreflexive
'kills' = nonsymetrical, nontransitive, nonreflexive
‘is as tall as’ = symmetrical, transitive, totally reflexive

H. Hamner Hill
Department of Political Science, Philosophy, and Religion
Southeast Missouri State University