Saul Aaron Kripke (/ˈkrɪpki/; November 13, 1940 – September 15, 2022) was an Americananalyticphilosopher andlogician. He was Distinguished Professor of Philosophy at theGraduate Center of the City University of New York andemeritus professor atPrinceton University. From the 1960s until his death, he was a central figure in a number of fields related tomathematical andmodal logic,philosophy of language andmathematics,metaphysics,epistemology, andrecursion theory.
Kripke made influential and original contributions tologic, especially modal logic. His principal contribution is asemantics for modal logic involvingpossible worlds, now calledKripke semantics.[6] He received the 2001Schock Prize in Logic and Philosophy.
Kripke was also partly responsible for the revival ofmetaphysics andessentialism after the decline oflogical positivism, claimingnecessity is a metaphysical notion distinct from theepistemic notion ofa priori, and that there arenecessary truths that are knowna posteriori, such as thatwater is H2O. A 1970 Princeton lecture series, published in book form in 1980 asNaming and Necessity, is considered one of the most important philosophical works of the 20th century. It introduced the concept ofnames asrigid designators, designating (picking out, denoting, referring to) the same object in every possible world, as contrasted withdescriptions. It also established Kripke'scausal theory of reference, disputing thedescriptivist theory found inGottlob Frege's concept ofsense andBertrand Russell'stheory 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.[7][8]
Kripke also gave an original reading ofLudwig Wittgenstein, known as "Kripkenstein", in hisWittgenstein on Rules and Private Language. The book contains his rule-following argument, a paradox forskepticism aboutmeaning. Much of his work remains unpublished or exists only as tape recordings and privately circulated manuscripts.
Saul Kripke was the oldest of three children born toDorothy K. Kripke andMyer S. Kripke.[9] His father was the leader of Beth El Synagogue, the only Conservative congregation inOmaha,Nebraska; his mother wroteJewish educational books for children. Saul and his two sisters,Madeline and Netta, attended Dundee Grade School andOmaha Central High School. Kripke was labeled aprodigy, teaching himselfAncient Hebrew by the age of six, readingShakespeare's complete works by nine, and mastering the works ofDescartes and complex mathematical problems before finishing elementary school.[10][11] He wrote his first completeness theorem inmodal logic at 17, and had it published a year later. After graduating from high school in 1958, Kripke attendedHarvard University and graduatedsumma cum laude in 1962 with abachelor's degree in mathematics. During his sophomore year at Harvard, he taught a graduate-level logic course at nearbyMIT.[12] Upon graduation he received aFulbright Fellowship, and in 1963 was appointed to theSociety 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."[13] His cousin isEric Kripke, known for writing TV shows likeThe Boys.[14]
After briefly teaching at Harvard, Kripke moved in 1968 toRockefeller University in New York City, where he taught until 1976. In 1978 he took a chaired professorship atPrinceton University.[15] In 1988 he received the university's Behrman Award for distinguished achievement in the humanities. In 2002 Kripke began teaching at theCUNY Graduate Center, and in 2003 he was appointed a distinguished professor of philosophy there.
Kripke has received honorary degrees from theUniversity of Nebraska Omaha (1977),Johns Hopkins University (1997),University of Haifa, Israel (1998), and theUniversity of Pennsylvania (2005). He was a member of theAmerican Philosophical Society and an elected Fellow of theAmerican Academy of Arts and Sciences, and in 1985 was a Corresponding Fellow of theBritish Academy.[16] He won theSchock Prize in Logic and Philosophy in 2001.[17]
Kripke married philosopherMargaret Gilbert in 1976. They divorced in 2000.[18]
Kripke died ofpancreatic cancer on September 15, 2022, in Plainsboro, New Jersey, at the age of 81.[19][20][21]
Kripke's contributions to philosophy include:
He has also contributed to recursion theory (seeadmissible ordinal andKripke–Platek set theory).
Two of Kripke's earlier works, "A Completeness Theorem in Modal Logic" (1959)[22] and "Semantical Considerations on Modal Logic" (1963), the former written when he was a teenager, were onmodal logic. The most familiar logics in the modal family are constructed from a weak logic called K, named after Kripke. Kripke introduced the now-standardKripke 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 tointuitionistic 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.
AKripke frame ormodal frame is a pair, whereW is a non-empty set, andR is abinary relation onW. Elements ofW are callednodes orworlds, andR is known as theaccessibility relation. Depending on the properties of the accessibility relation (transitivity, reflexivity, etc.), the corresponding frame is described, by extension, as being transitive, reflexive, etc.
AKripke model is a triple, where is a Kripke frame, and is a relation between nodes ofW and modal formulas, such that:
We read as "w satisfiesA", "A is satisfied inw", or "w forcesA". The relation is called thesatisfaction relation,evaluation, orforcing relation. The satisfaction relation is uniquely determined by its value on propositional variables.
A formulaA isvalid in:
We define Thm(C) to be the set of all formulas that are valid inC. Conversely, ifX is a set of formulas, let Mod(X) be the class of all frames which validate every formula fromX.
A modal logic (i.e., a set of formulas)L issound with respect to a class of framesC, ifL ⊆ Thm(C).L iscomplete with respect toC ifL ⊇ Thm(C).
Semantics is useful for investigating a logic (i.e., a derivation system) only if the semanticalentailment relation reflects its syntactical counterpart, theconsequence relation (derivability). 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 classC of Kripke frames, Thm(C) is anormal modal logic (in particular, theorems of the minimal normal modal logic,K, are valid in every Kripke model). 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 logicLcorresponds to a class of framesC, ifC = Mod(L). In other words,C is the largest class of frames such thatL is sound wrtC. It follows thatL is Kripke complete if and only if it is complete of its corresponding class.
Consider the schemaT :.T is valid in anyreflexive frame: if, then sincew R w. On the other hand, a frame which validatesT has to be reflexive: fixw ∈ W, and define satisfaction of a propositional variablep as follows: if and only ifw R u. Then, thus byT, which meansw 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 ofL than to prove its completeness, thus correspondence serves as a guide to completeness proofs. Correspondence is also used to showincompleteness of modal logics: supposeL1 ⊆ L2 are normal modal logics that correspond to the same class of frames, butL1 does not prove all theorems ofL2. ThenL1 is Kripke incomplete. For example, the schema generates an incomplete logic, as it corresponds to the same class of frames asGL (viz. transitive and converse well-founded frames), but does not prove theGL-tautology.
For any normal modal logicL, a Kripke model (called thecanonical model) can be constructed, which validates precisely the theorems ofL, by an adaptation of the standard technique of usingmaximal consistent sets as models. Canonical Kripke models play a role similar to theLindenbaum–Tarski algebra construction in algebraic semantics.
A set of formulas isL-consistent if no contradiction can be derived from them using the axioms ofL, andmodus ponens. Amaximal L-consistent set (anL-MCS for short) is anL-consistent set which has no properL-consistent superset.
Thecanonical model ofL is a Kripke model, whereW is the set of allL-MCS, and the relationsR and are as follows:
The canonical model is a model ofL, as everyL-MCS contains all theorems ofL. ByZorn's lemma, eachL-consistent set is contained in anL-MCS, in particular every formula unprovable inL has a counterexample in the canonical model.
The main application of canonical models are completeness proofs. Properties of the canonical model ofK immediately imply completeness ofK with respect to the class of all Kripke frames. This argument doesnot work for arbitraryL, because there is no guarantee that the underlyingframe of the canonical model satisfies the frame conditions ofL.
We say that a formula or a setX of formulas iscanonical with respect to a propertyP of Kripke frames, if
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, andcompact.
The axioms T, 4, D, B, 5, H, G (and thus any combination of them) are canonical. GL and Grz are not canonical, because they are not compact. The axiom M by itself is not canonical (Goldblatt, 1991), but the combined logicS4.1 (in fact, evenK4.1) is canonical.
In general, it isundecidable whether a given axiom is canonical. We know a nice sufficient condition: H. Sahlqvist identified a broad class of formulas (now calledSahlqvist formulas) such that:
This is a powerful criterion: for example, all axioms listed above as canonical are (equivalent to) Sahlqvist formulas. A logic has thefinite model property (FMP) 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 (including all listed above) 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 setW equipped with binary relationsRi for eachi ∈ I. The definition of a satisfaction relation is modified as follows:
A simplified semantics, discovered by Tim Carlson, is often used for polymodalprovability logics.[23] ACarlson model is a structure with a single accessibility relationR, and subsetsDi ⊆ W for each modality. Satisfaction is defined as:
Carlson models are easier to visualize and to work with than usual polymodal Kripke models; there are, however, Kripke complete polymodal logics which are Carlson incomplete.
InSemantical Considerations on Modal Logic, published in 1963, Kripke responded to a difficulty with classicalquantification theory. The motivation for the world-relative approach was to represent the possibility that objects in one world may fail to exist in another. But if standard quantifier rules are used, every term must refer to something that exists in all the possible worlds. This seems incompatible with our ordinary practice of using terms to refer to things that exist contingently.
Kripke's response to this difficulty was to eliminate terms. He gave an example of a system that uses the world-relative interpretation and preserves the classical rules. But the costs are severe. First, his language is artificially impoverished, and second, the rules for the propositional modal logic must be weakened.
Kripke's possible worlds theory has been used by narratologists (beginning with Pavel and Dolezel) to understand "reader's manipulation of alternative plot developments, or the characters' planned or fantasized alternative action series." This application has become especially useful in the analysis ofhyperfiction.[24]
Kripke semantics forintuitionistic logic follows the same principles as the semantics of modal logic, but uses a different definition of satisfaction.
Anintuitionistic Kripke model is a triple, where is apartially ordered Kripke frame, and satisfies the following conditions:
Intuitionistic logic is sound and complete with respect to its Kripke semantics, and it has the Finite Model Property.
Intuitionistic first-order logic
LetL be afirst-order language. A Kripke model ofL is a triple, where is an intuitionistic Kripke frame,Mw is a(classical)L-structure for each nodew ∈ W, and the following compatibility conditions hold wheneveru ≤ v:
Given an evaluatione of variables by elements ofMw, we define the satisfaction relation:
Heree(x→a) is the evaluation which givesx the valuea, and otherwise agrees withe.
The three lectures that formNaming and Necessity constitute an attack on thedescriptivist theory of names. Kripke attributes variants of descriptivist theories toFrege,Russell,Wittgenstein, andJohn Searle, among others. According to descriptivist theories, proper names either are synonymous with descriptions, or have their reference determined by virtue of the name's being associated with a description or cluster of descriptions that an object uniquely satisfies. Kripke rejects both these kinds of descriptivism. He gives several examples purporting to renderdescriptivism implausible as a theory of how names get their references determined (e.g., surelyAristotle 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).
As an alternative, Kripke outlined acausal theory of reference, according to which a name refers to an object by virtue of a causal connection with the object as mediated through communities of speakers. He points out that proper names, in contrast to most descriptions, arerigid designators: that is, a proper name refers to the named object in everypossible world in which the object exists, while most descriptions designate different objects in different possible worlds. For example, "Richard Nixon" refers to the same person in every possible world in which Nixon exists, while "the person who won theUnited States presidential election of 1968" could refer toNixon, Humphrey, or others in different possible worlds.
Kripke also raised the prospect ofa posteriori necessities—facts that arenecessarily true, though they can be known only through empirical investigation. Examples include "Hesperus isPhosphorus", "Cicero isTully", "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 anda posteriori, and contingent and necessary do not map onto one another. Rather, analytic/synthetic is a semantic distinction,a priori/a posteriori is an epistemic distinction, and contingent/necessary is a metaphysical distinction.
Finally, Kripke gave an argument againstidentity materialism in thephilosophy of mind, the view that every mental particular is identical with some physical particular. Kripke argued that the only way to defend this identity is as ana posteriori necessary identity, but that such an identity—e.g., that pain isC-fibers firing—could not be necessary, given the (clearly conceivable) possibility that pain could be separate from the firing of C-fibers, or the firing of C-fibers be separate from pain. (Similar arguments have since been made byDavid Chalmers.[25]) In any event, the psychophysical identity theorist, according to Kripke, incurs a dialectical obligation to explain the apparent logical possibility of these circumstances, since according to such theorists they should be impossible.
Kripke delivered theJohn Locke Lectures in philosophy atOxford in 1973. TitledReference and Existence, they were in many respects a continuation ofNaming and Necessity, and deal with the subjects of fictional names and perceptual error. In 2013 Oxford University Press published the lectures as a book, also titledReference and Existence.
In a 1995 paper, philosopherQuentin Smith argued that key concepts in Kripke's new theory of reference originated in the work ofRuth Barcan Marcus more than a decade earlier.[26] 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 thenecessity of identity; (5) the concept of arigid designator, though Kripke coined that term; and (6)a posteriori identity. Smith argued that Kripke failed to understand Marcus's theory at the time but later adopted many of its key conceptual themes in his New Theory of Reference.
Other scholars have subsequently offered detailed responses arguing that no plagiarism occurred.[27][28]
InNaming and Necessity, Kripke argues fordirect reference theory (that the meaning of a name is simply the object it refers to). Nevertheless, he acknowledges the possibility that propositions containing names may have some additional semantic properties,[29] properties that could explain why two names referring to the same person may give differenttruth values in propositions about beliefs. For example, Lois Lane believes that Superman can fly, although she does not believe that Clark Kent can fly. According to themediated reference theory of names, this is explained by the fact that the names "Superman" and "Clark Kent", though referring to the same person, have distinct semantic properties.
But in his article "A Puzzle about Belief" (1988) Kripke seems to oppose even this possibility. His argument can be reconstructed as follows: The idea that two names referring to the same object may have different semantic properties is supposed to explain the fact that the intersubstitution ofcoreferring names in propositions about beliefs can alter truth value (as in Lois Lane's case). 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"). Pierre moves to London without realizing that London = Londres. He then learns English the same way a child would learn the language, that is, not by translating words from French to English. Pierre learns the name "London" from the unattractive part of the city where he lives, and so comes to believe that London is not beautiful. Pierre will now assent to the sentences "Londres est jolie" and "London is not beautiful". With only translation and disquotation, the puzzle can be generated: Pierre both believes that London is pretty and doesn't believe that London is pretty. This paradox arises without making use of intersubstitution of coreferring names. Kripke shows later in the article how this puzzle can be generatedwithin a single language, using only disquotation. The upshot of this, according to Kripke, is that intersubstitution of coreferring names cannot be blamed for the difficulty created by belief contexts. If this is right, contra proponents of mediated reference theory, the inconsistency of belief contexts involving coreferring names cannot be taken as evidence against his direct reference theory of names.
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First published in 1982, Kripke'sWittgenstein on Rules and Private Language contends that the central argument ofWittgenstein'sPhilosophical Investigations centers on a devastatingrule-following paradox that undermines the possibility of our ever following rules in our use of language. Kripke writes that this paradox is "the most radical and original skeptical problem that philosophy has seen to date", and that Wittgenstein does not reject the argument that leads to the rule-following paradox, but accepts it and offers a "skeptical solution" to ameliorate the paradox's destructive effects.
Most commentators accept thatPhilosophical Investigations contains the rule-following paradox as Kripke presents it, but few have agreed with his attributing a skeptical solution to Wittgenstein. Kripke himself expresses doubts inWittgenstein on Rules and Private Language as to whether Wittgenstein would endorse his interpretation ofPhilosophical Investigations. He says that the work should not be read as an attempt to give an accurate statement of Wittgenstein's views, but rather as an account of Wittgenstein's argument "as it struck Kripke, as it presented a problem for him".
The portmanteau "Kripkenstein" has been coined for Kripke's interpretation ofPhilosophical Investigations. Kripkenstein's main significance was a clear statement of a new kind of skepticism, dubbed "meaning skepticism": the idea that for isolated individuals there is no fact in virtue of which they mean one thing rather than another by the use of a word. Kripke's "skeptical solution" to meaning skepticism is to ground meaning in the behavior of a community.
Kripke's book generated a large secondary literature, divided between those who find his skeptical problem interesting and perceptive, and others, such asGordon Baker,Peter Hacker, andColin McGinn, who argue that his meaning skepticism is a pseudo-problem that stems from a confused, selective reading of Wittgenstein. Kripke's position has been defended against these and other attacks by the Cambridge philosopherMartin Kusch, and Wittgenstein scholar David G. Stern considers Kripke's book "the most influential and widely discussed" work on Wittgenstein since the 1980s.[30]
In his 1975 article "Outline of a Theory of Truth", Kripke showed that a language can consistently contain its owntruth predicate, something deemed impossible byAlfred Tarski, a pioneer in formal theories of truth. The approach involves letting truth be a partially defined property over the set of grammatically well-formed sentences in the language. Kripke showed how to do this recursively by starting from the set of expressions in a language that do not contain the truth predicate, and defining a truth predicate over just that segment: this action adds new sentences to the language, and truth is in turn defined for all of them. Unlike Tarski's approach, however, Kripke's lets "truth" be the union of all of these definition-stages; after a denumerable infinity of steps the language reaches a "fixed point" such that using Kripke's method to expand the truth-predicate does not change the language any further. Such a fixed point can then be taken as the basic form of a natural language containing its own truth predicate. But this predicate is undefined for any sentences that do not, so to speak, "bottom out" in simpler sentences not containing a truth predicate. That is, " 'Snow is white' is true" is well-defined, as is " ' "Snow is white" is true' is true," and so forth, but neither "This sentence is true" nor "This sentence is not true" receive truth-conditions; they are, in Kripke's terms, "ungrounded."
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 thediagonal lemma – is the basis forTarski's theorem that truth cannot be consistently defined. But Kripke's truth predicate does not give a truth value (true/false) to propositions such as the one built in Tarski's proof, since it is provable byinduction that it is undefined at stage for every finite.
Kripke's proposal is problematic in the sense that while the language contains a "truth" predicate of itself (at least a partial one), some of its sentences – such as the liar sentence ("this sentence is false") – have an undefined truth value, but the language does not contain its own "undefined" predicate. In fact it cannot, as that would create a new version of theliar paradox, thestrengthened liar paradox ("this sentence is false or undefined"). Thus while the liar sentence is undefined in the language, the language cannot express that it is undefined.[31]
The mission of the Saul Kripke Center at theGraduate Center of the City University of New York is to preserve and to promote Kripke's work. Its director is Yale Weiss. The Saul Kripke Center holds events related to Kripke's work and is creating a digital archive of previously unpublished recordings of his lectures, lecture notes, and correspondence dating to the 1950s.[32] In his review of Kripke'sPhilosophical Troubles, philosopher Mark Crimmins wrote, "That four of the most admired and discussed essays in 1970s philosophy are here is enough to make this first volume of Saul Kripke's collected articles a must-have... The reader's delight will grow as hints are dropped that there is a great deal more to come in this series being prepared by Kripke and an ace team of philosopher-editors at the Saul Kripke Center at The Graduate Center of the City University of New York."[33]
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