ones such as , , , and so on. Thus, the language of basic propositional logic can be defined recursively as follows. If is an atomic formula, then is a formula of . If is a formula of , then is too. If and are formulas of , then is too. If is a formula of , then is too. If is a formula of , then is too. Modal operators can be added to other kinds of logic by introducing rules analogous to #4 and #5 above. Modal predicate logic is one widely used variant which includes formulas such as . In systems of modal logic where and are duals, can be taken as an abbreviation for , thus eliminating the need for a separate syntactic rule to introduce it. However, separate syntactic rules are necessary in systems where the two operators are not interdefinable. Common notational variants include symbols such as and in systems of modal logic used to represent knowledge and and in those used to represent belief. These notations are particularly common in systems which use multiple modal operators simultaneously. For instance, a combined epistemic-deontic logic could use the formula read as "I know P is permitted". Systems of modal logic can include infinitely many modal operators distinguished by indices, i.e. , , , and so on. Semantics Relational semantics Basic notions The standard semantics for modal logic is called the relational semantics. In this approach, the truth of a formula is determined relative to a point which is often called a possible world. For a formula that contains a modal operator, its truth value can depend on what is true at other accessible worlds. Thus, the relational semantics interprets formulas of modal logic using models defined as follows. A relational model is a tuple where: is a set of possible worlds is a binary relation on is a valuation function which assigns a truth value to each pair of an atomic formula and a world, (i.e. where is the set of atomic formulae) The set is often called the universe. The binary relation is called an accessibility relation, and it controls which worlds can "see" each other for the sake of determining what is true. For example, means that the world is accessible from world . That is to say, the state of affairs known as is a live possibility for . Finally, the function