Kaxioms

Kaxioms is a fictional compact family of axioms for a modal epistemic logic designed to illustrate a minimal framework for formal knowledge in automated reasoning. The framework treats knowledge as a unary operator K applied to propositions, allowing one to express what is known about statements.

In the baseline system, the axioms include standard distribution: K(p → q) → (Kp → Kq); a truth axiom: Kp → p; and, for stronger systems, introspection principles such as KKp → Kp (positive introspection) and ¬Kp → K¬Kp (negative introspection). The rule of necessitation states that from p one may infer Kp, and Modus Ponens remains valid as in classical logic. Together with propositional connectives, these axioms yield a family of logics ranging from weaker to stronger forms, such as K, T, and S4 or S5 variants depending on which introspection principles are adopted.

Semantics use Kripke models with possible worlds and an accessibility relation R reflecting the agent’s epistemic reach. Kp is true at a world w if p holds in all worlds accessible from w. Frame conditions such as reflexivity, transitivity, or Euclideanness correspond to the properties of the resulting logic (e.g., T, 4, or 5). These choices shape how knowledge propagates through inference and across worlds.

Kaxioms has been explored in single-agent and multi-agent settings and serves as a base for extensions to dynamic epistemic logic and justification logics. It has potential applications in formal verification, AI planning under uncertainty, and the specification of knowledge-based systems. See also epistemic logic, modal logic, and justification logic.