Sign Relation - Dyadic Aspects of Sign Relations

Dyadic Aspects of Sign Relations

For an arbitrary triadic relation L ⊆ O × S × I, whether it is a sign relation or not, there are six dyadic relations that can be obtained by projecting L on one of the planes of the OSI-space O × S × I. The six dyadic projections of a triadic relation L are defined and notated as follows:

LOS	=	projOS(L)	=	{ (o, s) ∈ O × S : (o, s, i) ∈ L for some i ∈ I }
LSO	=	projSO(L)	=	{ (s, o) ∈ S × O : (o, s, i) ∈ L for some i ∈ I }
LIS	=	projIS(L)	=	{ (i, s) ∈ I × S : (o, s, i) ∈ L for some o ∈ O }
LSI	=	projSI(L)	=	{ (s, i) ∈ S × I : (o, s, i) ∈ L for some o ∈ O }
LOI	=	projOI(L)	=	{ (o, i) ∈ O × I : (o, s, i) ∈ L for some s ∈ S }
LIO	=	projIO(L)	=	{ (i, o) ∈ I × O : (o, s, i) ∈ L for some s ∈ S }

By way of unpacking the set-theoretic notation, here is what the first definition says in ordinary language. The dyadic relation that results from the projection of L on the OS-plane O × S is written briefly as L_OS or written more fully as proj_OS(L), and it is defined as the set of all ordered pairs (o, s) in the cartesian product O × S for which there exists an ordered triple (o, s, i) in L for some interpretant i in the interpretant domain I.

In the case where L is a sign relation, which it becomes by satisfying one of the definitions of a sign relation, some of the dyadic aspects of L can be recognized as formalizing aspects of sign meaning that have received their share of attention from students of signs over the centuries, and thus they can be associated with traditional concepts and terminology. Of course, traditions may vary as to the precise formation and usage of such concepts and terms. Other aspects of meaning have not received their fair share of attention, and thus remain anonymous on the contemporary scene of sign studies.