To most people, relationships are secondary—interesting relationships happen to exist between individuals. As an Irishman, I find relationships much more important, fundamental in fact. We are defined by our relationships. Philosophically, this is considered inaccurate, as it suggests relationships can define things.
In classical metaphysics, things exist and relationships exist between things. Things exist first, then relate. Relations are external or accidental properties of things. There is no reason why relations must be symmetric or reflexive.
Turning this on its head, what if relationships are elemental and things arise from relationships? We have an elemental relationship between nodes A and B that are not things. We don’t care what the nodes of our relationships are, and we choose not to name them, but we can apply our framework to any system that can be described by logically constrained relationships as fundamental elements.
Let us assume there is no outside observer, so no perspective from which asymmetric relations can be established. Let us also assume that we can give each node an identity and that identity is not altered by its relationships. Even if A relates strongly to B and C, A’s self-strength stays at unity. A doesn’t lose “A-ness” by spreading its influence. Identity is preserved regardless of relational activity.
In this framework, we can say that the relationships will obey logical symmetry. We call this a logical constraint on what relationships can exist; other relationships are prohibited. If A relates to B, that relation must simultaneously constitute both A and B. A can only exist through its relations to B, which requires B to relate back to A.
Are there other logical constraints that we can impose on our relational ontology? We can assume that reality will not allow inconsistency, so that if A relates to B, and B relates to C, then the relationship A to C must be fundamental. Strong indirect paths force corresponding direct paths to be strong. If A to B or B to C weakens, then that weakens A to C.
We now have three key logical constraints on our relationships: symmetry, transitivity, and reflexivity. These can be expressed mathematically with respect to a matrix B containing our relationship strengths.
- Symmetry: B = B^T
If node A relates to node B with strength 0.7, then node B must relate back to node A with the same strength 0.7. - Transitivity: B² ≤ B
If A connects strongly to B, and B connects strongly to C, then A should connect to C at least as strongly. Here B² means matrix multiplication (paths of length 2), not squaring individual elements. - Reflexivity: diag(B) = I
Every node has maximum strength relationship with itself equal to unity.