for every epistemically possible world. The knowledge generalization rule Another property we can derive is that if is valid (i.e. a tautology), then . This does not mean that if is true, then agent i knows . What it means is that if is true in every world that an agent considers to be a possible world, then the agent must know at every possible world. This principle is traditionally called N (Necessitation rule). This rule always preserves truth in relational semantics. The knowledge or truth axiom This axiom is also known as T. It says that if an agent knows facts, the facts must be true. This has often been taken as the major distinguishing feature between knowledge and belief. We can believe a statement to be true when it is false, but it would be impossible to know a false statement. This axiom can also be expressed in its contraposition as agents cannot know a false statement: This axiom is valid on any reflexive frame. The positive introspection axiom This property and the next state that an agent has introspection about its own knowledge, and are traditionally known as 4 and 5, respectively. The Positive Introspection Axiom, also known as the KK Axiom, says specifically that agents know that they know what they know. This axiom may seem less obvious than the ones listed previously, and Timothy Williamson has argued against its inclusion forcefully in his book, Knowledge and Its Limits. Equivalently, this modal axiom 4 says that agents do not know what they do not know that they know This axiom is valid on any transitive frame. The negative introspection axiom The Negative Introspection Axiom says that agents know that they do not know what they do not know. Or, equivalently, this modal axiom 5 says that agents know what they do not know that they do not know This axiom is valid on any Euclidean frame. Axiom systems Different modal logics can be derived from taking different subsets of these axioms, and these logics are normally named after the important axioms being employed. However, this is not always the case. KT45, the modal logic that results from the combining of K, T, 4, 5, and the Knowledge Generalization Rule, is primarily known as S5. This is why the properties of knowledge described above are often called the S5 Properties. However, it can be proven that modal