is known as a valuation function. It determines which atomic formulas are true at which worlds. Then we recursively define the truth of a formula at a world in a model : iff iff iff and iff for every element of , if then iff for some element of , it holds that and According to this semantics, a formula is necessary with respect to a world if it holds at every world that is accessible from . It is possible if it holds at some world that is accessible from . Possibility thereby depends upon the accessibility relation , which allows us to express the relative nature of possibility. For example, we might say that given our laws of physics it is not possible for humans to travel faster than the speed of light, but that given other circumstances it could have been possible to do so. Using the accessibility relation we can translate this scenario as follows: At all of the worlds accessible to our own world, it is not the case that humans can travel faster than the speed of light, but at one of these accessible worlds there is another world accessible from those worlds but not accessible from our own at which humans can travel faster than the speed of light. Frames and completeness The choice of accessibility relation alone can sometimes be sufficient to guarantee the truth or falsity of a formula. For instance, consider a model whose accessibility relation is reflexive. Because the relation is reflexive, we will have that for any regardless of which valuation function is used. For this reason, modal logicians sometimes talk about frames, which are the portion of a relational model excluding the valuation function. A relational frame is a pair where is a set of possible worlds, is a binary relation on . The different systems of modal logic are defined using frame conditions. A frame is called: reflexive if w R w, for every w in G symmetric if w R u implies u R w, for all w and u in G transitive if w R u and u R q together imply w R q, for all w, u, q in G. serial if, for every w in G there is some u in G such that w R u. Euclidean if, for every u, t, and w, w R u and w R