we have both that and also that for all Topological approaches subsume relational ones, allowing non-normal modal logics. The extra structure they provide also allows a transparent way of modeling certain concepts such as the evidence or justification one has for one's beliefs. Topological semantics is widely used in recent work in formal epistemology and has antecedents in earlier work such as David Lewis and Angelika Kratzer's logics for counterfactuals. Axiomatic systems The first formalizations of modal logic were axiomatic. Numerous variations with very different properties have been proposed since C. I. Lewis began working in the area in 1912. Hughes and Cresswell (1996), for example, describe 42 normal and 25 non-normal modal logics. Zeman (1973) describes some systems Hughes and Cresswell omit. Modern treatments of modal logic begin by augmenting the propositional calculus with two unary operations, one denoting "necessity" and the other "possibility". The notation of C. I. Lewis, much employed since, denotes "necessarily p" by a prefixed "box" (□p) whose scope is established by parentheses. Likewise, a prefixed "diamond" (◇p) denotes "possibly p". Similar to the quantifiers in first-order logic, "necessarily p" (□p) does not assume the range of quantification (the set of accessible possible worlds in Kripke semantics) to be non-empty, whereas "possibly p" (◇p) often implicitly assumes (viz. the set of accessible possible worlds is non-empty). Regardless of notation, each of these operators is definable in terms of the other in classical modal logic: □p (necessarily p) is equivalent to ("not possible that not-p") ◇p (possibly p) is equivalent to ("not necessarily not-p") Hence □ and ◇ form a dual pair of operators. In many modal logics, the necessity and possibility operators satisfy the following analogues of de Morgan's laws from Boolean algebra: "It is not necessary that X" is logically equivalent to "It is possible that not X". "It is not possible that X" is logically equivalent to "It is necessary that not X". Precisely what axioms and rules must be added to the propositional calculus to create a usable system of modal logic is a matter of philosophical opinion, often driven by the theorems one wishes to prove; or, in computer science, it is a matter of what sort of computational or deductive system one wishes to model. Many modal logics, known collectively as normal modal logics, include the following rule and axiom: N, Necessitation Rule: If p is a theorem/tautology (of any