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Saul Aaron Kripke (; born November 13, 1940) is an American philosopher and logician. He is currently McCosh Professor of Philosophy, Emeritus, at Princeton University and teaches as a Distinguished Professor of Philosophy at the CUNY Graduate Center. Since the 1960s Kripke has been a central figure in a number of fields related to mathematical logic, philosophy of language, philosophy of mathematics, metaphysics, epistemology, and set theory. Much of his work remains unpublished or exists only as tape-recordings and privately circulated manuscripts. Kripke was the recipient of the 2001 Schock Prize in Logic and Philosophy. A recent poll conducted among philosophers ranked Kripke among the top ten most important philosophers of the past 200 years.^{[1]}
Kripke has made influential and original contributions to logic, especially modal logic. His work has profoundly influenced analytic philosophy, with his principal contribution being a semantics for modal logic, involving possible worlds as described in a system now called Kripke semantics.^{[2]} Another of his most important contributions is his argument that necessity is a 'metaphysical' notion, which should be separated from the epistemic notion of a priori, and that there are necessary truths which are a posteriori truths, such as "Water is H_{2}O." 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 S. Kripke.^{[3]} His father was the leader of Beth El Synagogue, the only Conservative congregation in Omaha, Nebraska, while his mother wrote educational Jewish books for children. Saul and his two sisters, Madeline and Netta, attended Dundee Grade School and Omaha Central High School. Kripke was labelled a prodigy, having taught himself Ancient Hebrew by the age of six, read the complete works of Shakespeare by nine, and mastered the works of Descartes and complex mathematical problems before finishing elementary school.^{[4]}^{[5]} He wrote his first completeness theorem in modal logic at the age of 17, and had it published a year later. After graduating from high school in 1958, Kripke attended Harvard University and graduated summa cum laude obtaining a bachelor's degree in mathematics. During his sophomore year at Harvard, Kripke taught a graduate-level logic course at nearby MIT. Upon graduation (1962) he received a Fulbright Fellowship, and in 1963 was appointed to the Society of Fellows.
After teaching briefly at Harvard, he moved to Rockefeller University in New York City in 1967, and then received a full-time position at Princeton University in 1977. 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 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, 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.
He is the second cousin once removed of the notable television writer, director, and producer Eric Kripke.
The Saul Kripke Center at the Graduate Center of the City University of New York is dedicated to preserving and promoting Kripke's work. The Saul Kripke Center is directed by Gary Ostertag. The SKC hold events related to Kripke's work and is currently working to create a digital archive of Kripke's previously unpublished recordings of lectures, lecture notes, and correspondence dating back to the 1950s.^{[6]} In his favorable review of Kripke's Philosophical Troubles, Mark Crimmins, a philosopher at Stanford 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."^{[7]}
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 absent prior to Kripke.
A Kripke frame or modal frame is a pair \langle W,R\rangle, 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 \langle W,R,\Vdash\rangle, where \langle W,R\rangle is a Kripke frame, and \Vdash is a relation between nodes of W and modal formulas, such that:
We read w\Vdash A as "w satisfies A", "A is satisfied in w", or "w forces A". The relation \Vdash 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:
We define Thm(C) to be the set of all formulas that are valid in C. Conversely, if X is a set of formulas, let Mod(X) be the class of all frames which validate every formula from X.
A modal logic (i.e., a set of formulas) L is sound with respect to a class of frames C, if L ⊆ Thm(C). L is complete with respect to C if L ⊇ Thm(C).
Semantics is useful for investigating a logic (i.e. a derivation system) only if the semantical entailment relation reflects its syntactical counterpart, the consequence 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 class C of Kripke frames, Thm(C) is a normal 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 logic L corresponds to a class of frames C, if C = Mod(L). 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 : \Box A\to A. T is valid in any reflexive frame \langle W,R\rangle: if w\Vdash \Box A, then w\Vdash A 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: u\Vdash p if and only if w R u. Then w\Vdash \Box p, thus w\Vdash p by T, which means w R w using the definition of \Vdash. 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 L_{1} ⊆ L_{2} are normal modal logics that correspond to the same class of frames, but L_{1} does not prove all theorems of L_{2}. Then L_{1} is Kripke incomplete. For example, the schema \Box(A\equiv\Box A)\to\Box A generates an incomplete logic, as it corresponds to the same class of frames as GL (viz. transitive and converse well-founded frames), but does not prove the GL-tautology \Box A\to\Box\Box A.
For any normal modal logic L, a Kripke model (called the canonical 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 (an L-MCS for short) is an L-consistent set which has no proper L-consistent superset.
The canonical model of L is a Kripke model \langle W,R,\Vdash\rangle, where W is the set of all L-MCS, and the relations R and \Vdash 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
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 (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 logic S4.1 (in fact, even K4.1) 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 (now called Sahlqvist formulas) such that:
This is a powerful criterion: for example, all axioms listed above as canonical are (equivalent to) Sahlqvist formulas. A logic has the finite 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 \{\Box_i\mid\,i\in I\} as the set of its necessity operators consists of a non-empty set W equipped with binary relations R_{i} for each i ∈ I. The definition of a satisfaction relation is modified as follows:
A simplified semantics, discovered by Tim Carlson, is often used for polymodal provability logics. A Carlson model is a structure \langle W,R,\{D_i\}_{i\in I},\Vdash\rangle with a single accessibility relation R, and subsets D_{i} ⊆ 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.
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. If standard quantifier rules are used, however, 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. However, 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 of hyperfiction.^{[8]}
Kripke semantics for the intuitionistic logic follows the same principles as the semantics of modal logic, but uses a different definition of satisfaction.
An intuitionistic Kripke model is a triple \langle W,\le,\Vdash\rangle, where \langle W,\le\rangle is a partially ordered Kripke frame, and \Vdash 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
Let L be a first-order language. A Kripke model of L is a triple \langle W,\le,\{M_w\}_{w\in W}\rangle, where \langle W,\le\rangle is an intuitionistic Kripke frame, M_{w} is a (classical) L-structure for each node w ∈ W, and the following compatibility conditions hold whenever u ≤ v:
Given an evaluation e of variables by elements of M_{w}, we define the satisfaction relation w\Vdash A[e]:
Here e(x→a) is the evaluation which gives x the value a, and otherwise agrees with e.
The three lectures that form Naming and Necessity constitute an attack on descriptivist theory of names. Kripke attributes variants of descriptivist theories to Frege, Russell, Ludwig Wittgenstein and John 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 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, and yet it would seem wrong to deny that he was Aristotle).
As an alternative, Kripke outlined a causal 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, are rigid designators. That is, a proper name refers to the named object in every possible world in which the object exists, while most descriptions designate different objects in different possible worlds. For example, 'Nixon' refers to the same person in every possible world in which Nixon exists, while 'the person who won the United States presidential election of 1968' could refer to Nixon, Humphrey, or others in different possible worlds.
Kripke also raised the prospect of a posteriori necessities — facts that are necessarily true, though they can be known only through empirical investigation. Examples include "Hesperus is Phosphorus", "Cicero is Tully", "Water is H_{2}O" and other identity claims where two names refer to the same object.
Finally, Kripke gave an argument against identity materialism in the philosophy 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 an a posteriori necessary identity, but that such an identity — e.g., pain is C-fibers firing — could not be necessary, given the (clearly conceivable) possibility that pain be separate from the firing of C-fibers, or the firing of C-fibers be separate from pain (See: Zombies [Philosophy]). Similar arguments have been proposed by David Chalmers.^{[9]} In any event, the psychophysical identity theorist, according to Kripke, incurs a dialectical obligation to explain the apparent logical possibility of these circumstances, for in the opinion of such theorists they should be impossible.
Kripke delivered the John Locke lectures in philosophy at Oxford in 1973. Titled Reference and Existence, they are in many respects a continuation of Naming and Necessity, and deal with the subjects of fictional names and perceptual error. They have recently been published by Oxford University Press.
In a 1995 paper, philosopher Quentin Smith argued that key concepts in Kripke's new theory of reference had originated from the work of Ruth Barcan Marcus more than a decade earlier.^{[10]} Smith identified six significant ideas to the New Theory that he claimed Marcus had developed: (1) The idea that proper names are direct references, which don't consist of contained definitions. (2) While one can single out a single thing by a description, this description is not equivalent with 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 the actual name of the concept was coined by Kripke.(6) The idea of a posteriori identity. Smith proceeded to argue that Kripke failed to understand Marcus' theory at the time, yet 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.^{[11]}^{[12]}
Kripke's main propositions in Naming and Necessity concerning proper names are that the meaning of a name simply is the object it refers to and that a name's referent is determined by a causal link between some sort of "baptism" and the utterance of the name. Nevertheless he acknowledges the possibility that propositions containing names may have some additional semantic properties,^{[13]} properties that could explain why two names referring to the same person may give different truth values in propositions about beliefs. For example, Lois Lane believes that Superman can fly, although she does not believe that Clark Kent can fly. This can be accounted for if the names "Superman" and "Clark Kent", though referring to the same person, have distinct semantic properties.
In the article "A Puzzle about Belief" Kripke seems to oppose even this possibility. His argument can be reconstructed in the following way: The idea that two names referring to the same object may have different semantic properties is supposed to explain that coreferring names behave differently in propositions about beliefs (as in Lois Lane's case). But the same phenomenon occurs even with coreferring names that obviously have the same semantic properties:
Kripke invites us to imagine a French, monolingual boy, Pierre, who believes the following: "Londres est joli." ("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 in which he lives, so he comes to believe that London is not beautiful. If Kripke's account is correct, Pierre now believes both that "Londres" is "joli" and that "London" is not beautiful. This cannot be explained by coreferring names having different semantic properties. According to Kripke, this demonstrates that attributing additional semantic properties to names does not explain what it is intended to.
First published in 1982, Kripke's Wittgenstein on Rules and Private Language contends that the central argument of Wittgenstein's Philosophical Investigations centers on a devastating rule-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." (p. 60) Kripke argues 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.
Whilst most commentators accept that the Philosophical Investigations contains the rule-following paradox as Kripke presents it, few have concurred with Kripke when he attributes a skeptical solution to Wittgenstein. It should be noted that Kripke himself expresses doubts in Wittgenstein on Rules and Private Language as to whether Wittgenstein would endorse his interpretation of the Philosophical 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" (p. 5).
The portmanteau "Kripkenstein" has been coined as a jesting nickname for Kripke's reading of the Philosophical Investigations. The real significance of "Kripkenstein" was to put forward a clear statement of a new kind of skepticism, dubbed "meaning skepticism", which is the idea that for an isolated individual there is no fact in virtue of which he/she means 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 as Gordon Baker and Peter Hacker, who argue that his meaning skepticism is a pseudo-problem that stems from a confused, selective reading of Wittgenstein. Kripke's position has, however recently been defended against these and other attacks by the Cambridge philosopher Martin Kusch (2006), and Wittgenstein scholar David G. Stern considers the book to be "the most influential and widely discussed" work on Wittgenstein since the 1980s.^{[14]}
In his 1975 article "Outline of a Theory of Truth", Kripke showed that a language can consistently contain its own truth predicate, which was deemed impossible by Alfred Tarski, a pioneer in the area of 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 which 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."
Nevertheless, it has been shown by Gödel 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. It has thus been claimed ^{[15]} that Kripke's suggestion does lead to contradiction: while its truth predicate is only partial, it does give truth value (true/false) to propositions such as the one built in Tarski's proof, and is therefore inconsistent. While there is still a debate on whether Tarski's proof can be implemented to every variation of such a partial truth system, none have been shown to be consistent by acceptable proving methods used in mathematical logic.
Kripke's proposal is also 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 this will create a new version of the liar paradox , called the strengthened 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.^{[16]}
Kripke is an observant Jew.^{[17]} Discussing how his religious views influenced his philosophical views (in an interview with Andreas Saugstad) he stated: "I don't have the prejudices many have today, I don't believe in a naturalist world view. I don't base my thinking on prejudices or a worldview and do not believe in materialism."^{[18]}
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