Saul Aaron Kripke (/ˈkrɪpki/; November 13, 1940 – September 15, 2022) was an American analytic philosopher and logician.

A 1970 Princeton lecture series, published in book form in 1980 as Naming and Necessity, is considered one of the most important philosophical works of the 20th century.

[9] His father was the leader of Beth El Synagogue, the only Conservative congregation in Omaha, Nebraska; his mother wrote Jewish educational books for children.

[14] After briefly teaching at Harvard, Kripke moved in 1968 to Rockefeller University in New York City, where he taught until 1976.

In 2002 Kripke began teaching at the CUNY Graduate Center, and in 2003 he was appointed a distinguished professor of philosophy there.

A modal logic (i.e., a set of formulas) L is sound with respect to a class of frames C, if L ⊆ Thm(C).

since w R w. On the other hand, a frame which validates T has to be reflexive: fix w ∈ W, and define satisfaction of a propositional variable p as follows:

Properties of the canonical model of K immediately imply completeness of K with respect to the class of all Kripke frames.

It follows from the preceding discussion that any logic axiomatized by a canonical set of formulas is Kripke complete, and compact.

Refinements and extensions of the canonical model construction often work, using tools such as filtration or unravelling.

As another possibility, completeness proofs based on cut-free sequent calculi usually produce finite models directly.

As an example, Robert Bull proved using this method that every normal extension of S4.3 has FMP, and is Kripke complete.

The definition of a satisfaction relation is modified as follows: A simplified semantics, discovered by Tim Carlson, is often used for polymodal provability logics.

In Semantical Considerations on Modal Logic, published in 1963, Kripke responded to a difficulty with classical quantification theory.

The motivation for the world-relative approach was to represent the possibility that objects in one world may fail to exist in another.

satisfies the following conditions: Intuitionistic logic is sound and complete with respect to its Kripke semantics, and it has the Finite Model Property.

Kripke attributes variants of descriptivist theories to Frege, Russell, Wittgenstein, and John Searle, among others.

He gives several examples purporting to render descriptivism implausible as a theory of how names get their references determined (e.g., surely Aristotle could have died at age two and so not satisfied any of the descriptions we associate with his name, but it would seem wrong to deny that he was still Aristotle).

Examples include "Hesperus is Phosphorus", "Cicero is Tully", "Water is H2O", and other identity claims where two names refer to the same object.

According to Kripke, the Kantian distinctions between analytic and synthetic, a priori and a posteriori, and contingent and necessary do not map onto one another.

In 2013 Oxford University Press published the lectures as a book, also titled Reference and Existence.

In a 1995 paper, philosopher Quentin Smith argued that key concepts in Kripke's new theory of reference originated in the work of Ruth Barcan Marcus more than a decade earlier.

[25] Smith identified six significant ideas in the New Theory that he claimed Marcus had developed: (1) that proper names are direct references that do not consist of contained definitions; (2) that while one can single out a single thing by a description, this description is not equivalent to a proper name of this thing; (3) the modal argument that proper names are directly referential, and not disguised descriptions; (4) a formal modal logic proof of the necessity of identity; (5) the concept of a rigid designator, though Kripke coined that term; and (6) a posteriori identity.

But the same phenomenon occurs even without the intersubstitution of coreferring names: Kripke invites us to imagine a French, monolingual boy, Pierre, who believes the proposition expressed by "Londres est jolie" ("London is beautiful").

The upshot of this, according to Kripke, is that intersubstitution of coreferring names cannot be blamed for the difficulty created by belief contexts.

Most commentators accept that Philosophical Investigations contains the rule-following paradox as Kripke presents it, but few have agreed with his attributing a skeptical solution to Wittgenstein.

Kripke's book generated a large secondary literature, divided between those who find his skeptical problem interesting and perceptive, and others, such as Gordon Baker, Peter Hacker, and Colin McGinn, who argue that his meaning skepticism is a pseudo-problem that stems from a confused, selective reading of Wittgenstein.

The approach involves letting truth be a partially defined property over the set of grammatically well-formed sentences in the language.

Such a fixed point can then be taken as the basic form of a natural language containing its own truth predicate.

Gödel's first incompleteness theorem demonstrates that self-reference cannot be avoided naively, since propositions about seemingly unrelated objects (such as integers) can have an informal self-referential meaning, and this idea – manifested by the diagonal lemma – is the basis for Tarski's theorem that truth cannot be consistently defined.