In semantics for logics using possible worlds—especially conditional logics—a host of related semantic conditions concerning how to formalize the notion of similarity between possible worlds. Three distinct constraints are offered when similarity is construed as relative to a formula $\phi$ (captured by an accessibility relation $R_{\phi }$ for each $\phi$):

1. 1 for any possible world there exists at most one possible world $w^{\prime }$ that is maximally similar to $w$ with respect to any $\phi$

2. 2 for any possible world there exists precisely one possible world that is maximally similar to $w$ with respect to all consistent $\phi$

3. 3 for any possible world there exists precisely one possible world that is maximally similar to $w$ with respect to all formulae $\phi$

Clearly, [3] entails [2], [2] entails [1], and all three entail the limit assumption, that is, that there exists a set of worlds (possibly empty or a singleton) whose members are maximally similar to $w$. The third form of the uniqueness assumption suggests a need of impossible worlds, i.e., states at which contradictions may be true, leading philosopher Robert Stalnaker (1940– ) to include an absurd world—a possible world at which every sentence is true—among each set of otherwise consistent possible worlds.