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Saul Kripke  

Full name  Saul Kripke 
Born  November 13, 1940 Bay Shore, New York 
Era  Contemporary philosophy 
Region  Western Philosophy 
School  Analytic 
Main interests  Mathematical Logic (particularly modal) Philosophy of language Metaphysics Set Theory Epistemology Philosophy of Mind History of Analytic Philosophy 
Notable ideas  Causal theory of reference Kripkenstein admissible ordinal rigid designator Kripke semantics 
Saul Aaron Kripke (born on November 13, 1940) is an American philosopher and logician. He is a professor emeritus at Princeton and teaches as a distinguished professor of philosophy at 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 Much of his work remains unpublished or exists only as taperecordings and privately circulated manuscripts (see "Unpublished Manuscripts and Online Lectures" below). Kripke was the recipient of the 2001 Schock Prize in Logic and Philosophy. A recent poll conducted by philosophers ranked Kripke among the top ten most important philosophers of the past 200 years.^{[1]}
Kripke has made important and original contributions to logic, especially modal logic, since he was a teenager. He has no advanced degree, only 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, after Quine had challenged the analytic/synthetic distinction.^{[2]} His most famous work is Naming and Necessity (1980).
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Saul Kripke is the eldest 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.^{[3]} 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.^{[4]} 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 nonhonorary degrees. During his sophomore year at Harvard, Kripke taught a graduatelevel 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 fulltime 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 contributions to philosophy include:
He has also contributed to settheory (see admissible ordinal and KripkePlatek 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 nowstandard Kripke semantics (also known as relational semantics or frame semantics) for modal logics. Kripke semantics is a formal semantics for nonclassical logic systems. It was first made for modal logics, and later adapted to intuitionistic logic and other nonclassical systems. The discovery of Kripke semantics was a breakthrough in the making of nonclassical 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 nonempty 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:
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 wrt 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 in general. There are Kripke incomplete normal modal logics, which is not a problem, 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 : . 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 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 generates an incomplete logic, as it corresponds to the same class of frames as GL (viz. transitive and converse wellfounded frames), but does not prove the GLtautology .
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 Lconsistent if no contradiction can be derived from them using the axioms of L, and Modus Ponens. A maximal Lconsistent set (an LMCS for short) is an Lconsistent set which has no proper Lconsistent superset.
The canonical model of L is a Kripke model , where W is the set of all LMCS, and the relations R and are as follows:
The canonical model is a model of L, as every LMCS contains all theorems of L. By Zorn's lemma, each Lconsistent set is contained in an LMCS, 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 cutfree 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 nonempty 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 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 worldrelative 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 worldrelative 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" (Fludernik). It has become especially useful in the analysis of hyperfiction.^{[5]}
Kripke semantics for the intuitionistic logic follows the same principles as the semantics of modal logic, but it uses a different definition of satisfaction.
An intuitionistic Kripke model is a triple , where is a partially ordered Kripke frame, and satisfies the following conditions:
Intuitionistic logic is sound and complete with respect to its Kripke semantics, and it has FMP.
Intuitionistic firstorder logic
Let L be a firstorder language. A Kripke model of L is a triple , where is an intuitionistic Kripke frame, M_{w} is a (classical) Lstructure 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 :
Here e(x→a) is the evaluation which gives x the value a, and otherwise agrees with e.
Kripke's three lectures constitute an attack on descriptivist theories of proper 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 reference 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 adumbrated 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: 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 fact is identical with some physical fact (See talk). 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 Cfibers firing — could not be necessary, given the possibility of pain that has nothing to do with Cfibers firing. Similar arguments have been proposed by David Chalmers.^{[citation needed]}
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 never been published and the transcript is officially available only in a reading copy in the university philosophy library, which cannot be copied or cited without Kripke's permission.
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.^{[6]} Smith identified six significant ideas to the New Theory which he claimed that 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, although 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. Several scholars have subsequently offered detailed responses arguing that no plagiarism occurred.^{[7]}.
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^{[8]}, 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. (Like 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 London is "joli" and that London is not beautiful. This cannot be explained by coreferring names having different semantic properties. According to Kripke, this shows that attributing additional semantic properties to names, will not explain what it is supposed to explain.
First published in 1982, Saul Kripke's Wittgenstein on Rules and Private Language contends that the central argument of the Philosophical Investigations centers on a devastating rulefollowing 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 rulefollowing paradox, but accepts it and offers a 'skeptical solution' to ameliorate the paradox's destructive effects. Whilst most commentators^{[citation needed]} accept that the Philosophical Investigations contains the rulefollowing 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,^{[citation needed]} 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 pseudoproblem that stems from a confused, selective reading of Wittgenstein. Kripke's position has recently been defended against these and other attacks by the Cambridge philosopher Martin Kusch (2006). Wittgenstein scholar David G. Stern considers the book to be "the most influential and widely discussed" work on Wittgenstein since the 1980s.^{[9]}
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 trick involves letting truth be a partially defined property over the set of grammatically wellformed 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, defining a truth predicate over just that segment: this 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 definitionstages; after a denumerable infinity of steps the language reaches a "fixed point" such that using Kripke's method to expand the truthpredicate 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 welldefined, 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 truthconditions; they are, in Kripke's terms, "ungrounded."
In late January 2006, Kripke attended a conference celebrating his 65th birthday and work at the Graduate Center of the City University of New York, and delivered a 70minute talk on "The First Person", discussing the meaning and reference of the pronoun "I".^{[3]} ^{[10]}
Kripke is an observant Jew.^{[11]} 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 world view and do not believe in materialism."^{[12]}



Saul Kripke (born 1940) is an American philosopher.
