Entails is the basic ordering predicate on relations. It expresses subsumption: one recognition includes another.
Formal Signature
Entails : (Rel, Rel) → Truth
Definition
Entails(r, s) holds when r is contained in s --- that is, every distinction recognized by r is also recognized by s. In set-theoretic terms, r ⊆ s. This gives the relation algebra its partial-order structure.
Entails is reflexive (every relation entails itself), transitive (if r entails s and s entails t, then r entails t), and antisymmetric (if r entails s and s entails r, then r = s).
Derivational context
Entails arises early in Movement I: Logical Origination as a requirement of consistent distinguishing. Once distinctions have been drawn, they must be comparable: if one distinction refines another, this relationship must be tracked. Entails formalizes this comparison — one recognition includes another. The transitivity and reflexivity of Entails are not assumptions but consequences of what it means for distinctions to be coherent. In mathematical terms, Entails gives the partial-order structure, but relationally it is the basic condition of consistent inclusion.
Relations to Other Terms
- Excludes --- Excludes(r, s) is equivalent to Entails(r, Negate(s))
- Implies --- the residuated implication defined via Entails and Together
- Distinguish --- produces the relations that Entails orders