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How do possible worlds help explain the truth conditions for knowledge claims?

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Marble bust of Aristotle with a beard and draped robe, showing detailed facial features and wavy hair, set against a dark background. — Truth Detail of a Roman copy (2nd century bce) of a Greek alabaster portrait bust of Aristotle (c. 325 bce); in the collection of the Museo Nazionale Romano, Rome.

In the context of epistemic logic, possible worlds serve as a framework to define the truth conditions for knowledge claims. This approach relies on the concept of an accessibility relation, which creates a dichotomy between scenarios that are compatible with what an agent knows and those that are not.

The scenarios compatible with an agent's knowledge are referred to as their "epistemic alternatives." Within this semantic structure, the truth conditions for a knowledge claim are defined as follows: a person $P$ knows that a proposition $A$ is true if and only if $A$ is true in all of $P$ ’s epistemic alternatives.1

By utilizing this model, logicians can analyze how knowledge functions in practice. If an agent knows that $A$ , they are logically entitled to disregard any possible world in which $A$ is false. This framework also helps explain why certain laws of first-order logic, such as the substitutivity of identity, may fail; an agent might know that a statement is true in the actual world but fail to know it is true in all of their epistemic alternatives.1