Saul Kripke


Saul Aaron Kripke was an American analytic philosopher and logician. He was Distinguished Professor of Philosophy at the Graduate Center of the City University of New York and emeritus professor at Princeton University. From the 1960s until his death, he was a central figure in a number of fields related to mathematical and modal logic, philosophy of language and mathematics, metaphysics, epistemology, and recursion theory.
Kripke made influential and original contributions to logic, especially modal logic. His principal contribution is a semantics for modal logic involving possible worlds, now called Kripke semantics. He received the 2001 Schock Prize in Logic and Philosophy.
Kripke was also partly responsible for the revival of metaphysics and essentialism after the decline of logical positivism, claiming necessity is a metaphysical notion distinct from the epistemic notion of a priori, and that there are necessary truths that are known a posteriori, such as that water is H2O. 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. It introduced the concept of names as rigid designators, designating the same object in every possible world, as contrasted with descriptions. It also established Kripke's causal theory of reference, disputing the descriptivist theory found in Gottlob Frege's concept of sense and Bertrand Russell's theory of descriptions. Kripke is often seen in opposition to the other great late-20th-century philosopher to eschew logical positivism: W. V. O. Quine. Quine rejected essentialism and modal logic.
Kripke also gave an original reading of Ludwig Wittgenstein, known as "Kripkenstein", in his Wittgenstein on Rules and Private Language. The book contains his rule-following argument, a paradox for skepticism about meaning. Much of his work remains unpublished or exists only as tape recordings and privately circulated manuscripts.

Life and career

Saul Kripke was the oldest of three children born to Dorothy K. Kripke and Myer S. Kripke. His father was the leader of Beth El Synagogue, the only Conservative congregation in Omaha, Nebraska; his mother wrote Jewish educational books for children. Saul and his two sisters, Madeline and Netta, attended Dundee Grade School and Omaha Central High School. Kripke was labeled a prodigy, teaching himself Ancient Hebrew by the age of six, reading Shakespeare's complete works by nine, and mastering the works of Descartes and complex mathematical problems before finishing elementary school. He wrote his first completeness theorem in modal logic at 17, and had it published a year later. After graduating from high school in 1958, Kripke attended Harvard University and graduated summa cum laude in 1962 with a bachelor's degree in mathematics. During his sophomore year at Harvard, he taught a graduate-level logic course at nearby MIT. Upon graduation he received a Fulbright Fellowship, and in 1963 was appointed to the Society of Fellows. Kripke later said, "I wish I could have skipped college. I got to know some interesting people but I can't say I learned anything. I probably would have learned it all anyway just reading on my own." His cousin is Eric Kripke, known for writing TV shows like The Boys.
After briefly teaching at Harvard, Kripke moved in 1968 to Rockefeller University in New York City, where he taught until 1976. In 1978 he took a chaired professorship at Princeton University. In 1988 he received the university's Behrman Award for distinguished achievement in the humanities. In 2002 Kripke began teaching at the CUNY Graduate Center, and in 2003 he was appointed a distinguished professor of philosophy there.
Kripke has received honorary degrees from the University of Nebraska Omaha, Johns Hopkins University, University of Haifa, Israel, and the University of Pennsylvania. He was a member of the American Philosophical Society and an elected Fellow of the American Academy of Arts and Sciences, and in 1985 was a Corresponding Fellow of the British Academy. He won the Schock Prize in Logic and Philosophy in 2001.
Kripke married philosopher Margaret Gilbert in 1976. They divorced in 2000.
Kripke died of pancreatic cancer on September 15, 2022, in Plainsboro, New Jersey, at the age of 81.

Work

Kripke's contributions to philosophy include:
  1. Kripke semantics for modal and related logics, published in several essays beginning in his teens.
  2. His 1970 Princeton lectures Naming and Necessity, which significantly restructured philosophy of language.
  3. His interpretation of Wittgenstein.
  4. His theory of truth.
He has also contributed to recursion theory.

Modal logic

Two of Kripke's earlier works, "A Completeness Theorem in Modal Logic" and "Semantical Considerations on Modal Logic", the former written when he was a teenager, were on modal logic. The most familiar logics in the modal family are constructed from a weak logic called K, named after Kripke. Kripke introduced the now-standard Kripke 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 absent 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, the corresponding frame is described, by extension, as being transitive, reflexive, etc.
A Kripke model is a triple, where is a Kripke frame, and Turnstile | 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:
  • a model, if for all w ∈ W,
  • a frame, if it is valid in for all possible choices of,
  • a class C of frames or models, if it is valid in every member of C.
We define Thm to be the set of all formulas that are valid in C. Conversely, if X is a set of formulas, let Mod be the class of all frames which validate every formula from X.
A modal logic L is sound with respect to a class of frames C, if L ⊆ Thm. L is complete with respect to C if L ⊇ Thm.
Semantics is useful for investigating a logic only if the semantical entailment relation reflects its syntactical counterpart, the consequence relation. It is vital to know which modal logics are sound and complete with respect to a class of Kripke frames, and for them, to determine which class it is.
For any class C of Kripke frames, Thm is a normal modal logic. However, the converse does not hold generally. There are Kripke incomplete normal modal logics, which is unproblematic, because most of the modal systems studied are complete of classes of frames described by simple conditions.
A normal modal logic L corresponds to a class of frames C, if C = Mod. In other words, C is the largest class of frames such that L is sound wrt C. It follows that L is Kripke complete if and only if it is complete of its corresponding class.
Consider the schema T :. T is valid in any reflexive frame : if, then 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: if and only if w ''R u''. Then, thus by T, which means w ''R w'' using the definition of. T corresponds to the class of reflexive Kripke frames.
It is often much easier to characterize the corresponding class of L than to prove its completeness, thus correspondence serves as a guide to completeness proofs. Correspondence is also used to show incompleteness of modal logics: suppose L1 ⊆ L2 are normal modal logics that correspond to the same class of frames, but L1 does not prove all theorems of L2. Then L1 is Kripke incomplete. For example, the schema generates an incomplete logic, as it corresponds to the same class of frames as GL, but does not prove the GL-tautology.

Canonical models

For any normal modal logic L, a Kripke model can be constructed, which validates precisely the theorems of L, by an adaptation of the standard technique of using maximal consistent sets as models. Canonical Kripke models play a role similar to the Lindenbaum–Tarski algebra construction in algebraic semantics.
A set of formulas is L-''consistent if no contradiction can be derived from them using the axioms of L'', and modus ponens. A maximal L-consistent set is an L-consistent set which has no proper L-consistent superset.
The canonical model of L is a Kripke model, where W is the set of all L-''MCS, and the relations R'' and are as follows:
The canonical model is a model of L, as every L-''MCS contains all theorems of L''. By Zorn's lemma, each L-consistent set is contained in an L-''MCS, in particular every formula unprovable in L'' has a counterexample in the canonical model.
The main application of canonical models are completeness proofs. Properties of the canonical model of K immediately imply completeness of K with respect to the class of all Kripke frames. This argument does not work for arbitrary L, because there is no guarantee that the underlying frame of the canonical model satisfies the frame conditions of L.
We say that a formula or a set X of formulas is canonical with respect to a property P of Kripke frames, if
  • X is valid in every frame which satisfies P,
  • for any normal modal logic L which contains X, the underlying frame of the canonical model of L satisfies P.
A union of canonical sets of formulas is itself canonical. It follows from the preceding discussion that any logic axiomatized by a canonical set of formulas is Kripke complete, and compact.
The axioms T, 4, D, B, 5, H, G are canonical. GL and Grz are not canonical, because they are not compact. The axiom M by itself is not canonical, but the combined logic S4.1 is canonical.
In general, it is undecidable whether a given axiom is canonical. We know a nice sufficient condition: H. Sahlqvist identified a broad class of formulas such that:
  • a Sahlqvist formula is canonical,
  • the class of frames corresponding to a Sahlqvist formula is first-order definable,
  • there is an algorithm which computes the corresponding frame condition to a given Sahlqvist formula.
This is a powerful criterion: for example, all axioms listed above as canonical are Sahlqvist formulas. A logic has the finite model property if it is complete with respect to a class of finite frames. An application of this notion is the decidability question: it follows from Post's theorem that a recursively axiomatized modal logic L which has FMP is decidable, provided it is decidable whether a given finite frame is a model of L. In particular, every finitely axiomatizable logic with FMP is decidable.
There are various methods for establishing FMP for a given logic. 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.
Most of the modal systems used in practice have FMP.
In some cases, we can use FMP to prove Kripke completeness of a logic: every normal modal logic is complete wrt a class of modal algebras, and a finite modal algebra can be transformed into a Kripke frame. As an example, Robert Bull proved using this method that every normal extension of S4.3 has FMP, and is Kripke complete.
Kripke semantics has a straightforward generalization to logics with more than one modality. A Kripke frame for a language with
as the set of its necessity operators consists of a non-empty set W equipped with binary relations Ri for each i ∈ I. The definition of a satisfaction relation is modified as follows: