Relational Logic and Axiomatic Context: Extending the Boundaries of Boolean and Fuzzy Paradigms

This paper introduces two novel mathematical frameworks: Relational Logic (RL) and Axiomatic Context (AC). By expanding the traditional truth value spaces from discrete sets {0, 1} or closed intervals [0, 1] to the extended real number line R¯ = R ∪ {−∞, +∞}, we redefine logic based on order comparison (“than”) rather than strict equality (“is”). We demonstrate that both classical Boolean logic and Zadeh’s Fuzzy Logic are special bounded cases within RL and AC. Furthermore, the proposed system exhibits paraconsistency, naturally absorbing contradictions as measurable dissonance signals without systemic collapse.