Kripke has made important and original contributions to logic, especially modal logic, since he was a teenager. Unusually for a professional philosopher, his only degree is an undergraduate degree from Harvard. His work has profoundly influenced analytic philosophy and his principal contribution is a metaphysical description of modality, involving possible worlds as described in a system now called Kripke semantics. Another of his most important contributions is his insistence that there are necessary a posteriori truths, such as "Water is H2O." He has also contributed an original reading of Wittgenstein, referred to as "Kripkenstein." His most famous work is Naming and Necessity (1980).

Saul Kripke is the oldest of three children born to Dorothy K. Kripke and Rabbi Myer Kripke. His father was the leader of Beth El Synagogue, the only Conservative congregation in Omaha, Nebraska. His mother wrote Jewish educational children's books. Saul and his two sisters, Madeline and Netta, attended Dundee Grade School in Omaha and Omaha Central High School. Saul was an extraordinary child prodigy. He had taught himself Ancient Hebrew by the age of six. By the age of nine, he had read the complete works of Shakespeare, studied Descartes and (working entirely on his own) had mastered complex problems in geometry, algebra and calculus. He wrote his first completeness theorem in modal logic at the age of 17 (and it was published when he was 18). After graduating from high school in 1958, Kripke attended Harvard University and graduated summa cum laude obtaining a bachelor's degree in mathematics. He has no other non-honorary degrees. During his sophomore year at Harvard, Kripke taught a graduate-level logic course at nearby MIT. Upon graduation (1962) he received a Fulbright Fellowship. In 1963 he was appointed to the Society of Fellows. For some years he taught at Harvard, moved to Rockefeller University in New York City in 1967, then to Princeton University full-time in 1977. In 1988 he received Princeton's Behrman Award for distinguished achievement in the humanities. In 2002 Kripke started teaching at the CUNY Graduate Center in midtown Manhattan, and was appointed a distinguished professor of philosophy there in 2003. He was married to philosopher Margaret Gilbert.

He has received honorary degrees from the University of Nebraska, Omaha (1977), Johns Hopkins University (1997), University of Haifa, Israel (1998), and the University of Pennsylvania (2005). He is a member of the American Philosophical Society. Kripke is also an elected Fellow of the American Academy of Arts and Sciences and a Corresponding Fellow of the British Academy. He won the Schock Prize in Logic and Philosophy in 2001.

Kripke's contributions to philosophy include:

He has also contributed to set-theory (see admissible ordinal and Kripke-Platek set theory)

Two of Kripke's earlier works ("A Completeness Theorem in Modal Logic" and "Semantical Considerations on Modal Logic"), the former written while he was still a teenager, were on the subject of modal logic. The most familiar logics in the modal family are constructed from a weak logic called K, named after Kripke for his contributions to modal logic. Kripke introduced the now-standard Kripke semantics (also known as relational semantics or frame semantics) for modal logics. Kripke semantics is a formal semantics for non-classical logic systems. It was first made for modal logics, and later adapted to intuitionistic logic and other non-classical systems. The discovery of Kripke semantics was a breakthrough in the making of non-classical logics, because the model theory of such logics was nonexistent before Kripke.

A Kripke frame or modal frame is a pair , where W is a non-empty set, and R is a binary relation on W. Elements of W are called nodes or worlds, and R is known as the accessibility relation. Depending on the properties of the accessibility relation (transitivity, reflexivity, etc.), the corresponding frame is described, by extension, as being transitive, reflexive, etc.

A Kripke model is a triple , where is a Kripke frame, and is a relation between nodes of W and modal formulas, such that:

We read as “w satisfies A”, “A is satisfied in w”, or “w forces A”. The relation is called the satisfaction relation, evaluation, or forcing relation. The satisfaction relation is uniquely determined by its value on propositional variables.

A formula A is valid in: