Kripke, Saul (1940–)

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Saul Kripke is an American logician and philosopher born in New York in 1940. After earning a BA from Harvard University in 1962, he held positions at Harvard, Rockefeller, Princeton, New York Universities, and elsewhere.

Modal Logic

Saul Kripke has worked in many branches of logic (higher recursion theory, set theory, models of arithmetic, and relevance logic), but the work best known to philosophers, and much cited in the literature of linguistic semantics, computer science, and other disciplines, is his development of Kripke models for modal and related logics. At the level of sentential logic such a model consists of a set X (of "states of the world," often misleadingly called "worlds"), a binary relation R (of "relative possibility") thereon, plus an assignment to each atomic formula p of the set of those x in X at which p is true. The assignment extends to all formulas, taking "Necessarily A " to be true at x if A is true at every y with xRy.

Kripke was the first to publish proofs of completeness theorems to the effect that truth at all x in all models with R reflexive (and transitive) (and symmetric) coincides with provability in the modal logic T (respectively S4 ) (respectively S5 ), and he obtained similar results for other modal logics. Announced in "Semantic Considerations on Modal Logic" (1963), and presented in detail in a subsequent series of technical papers, Kripke's work covers modal and intuitionistic sentential and predicate logic, and includes besides completeness theorems results on decidability and undecidability.

Semantic Paradoxes

Also well known is Kripke's work on semantic paradoxes in "Outline of a Theory of Truth" (1975). A truth-predicate in a language L permitting quotation or equivalent means of self-reference would be a predicate T such that the following biconditional holds with any sentence of L in the blanks:
"T ('__________')" is true if and only if "__________" is true.
The liar paradox shows there cannot be a truth-predicate in L if L has no truth-value gaps. Given a partial interpretation I of a predicate U (under which U is declared true of some items, declared false of others, or not declared either of the rest), any treatment of truth-value gaps, such as Stephen Cole Kleene's three-valued or Bas van Fraassen's supervaluational approach, will dictate which sentences containing U are to be declared true, declared false, or not declared either. If U is being thought of as "is true," this amounts to dictating a new partial interpretation I* of U. For a fixed point, or partial interpretation having I = I*, the biconditional displayed earlier holds.

Kripke's work, besides more purely philosophical contributions, shows how to obtain a minimal fixed point (contained in any other, and explicating an intuitive notion of groundedness ), a maximal intrinsic fixed point (not declaring true anything declared false by any other fixed point), and many others, for any reasonable treatments of gaps.

Wittgenstein and Skepticism

Turning from logic to philosophy of language and its applications to analytic metaphysics, Kripke has written two much-discussed books that are almost entirely independent of each other. In Wittgenstein on Rules and Private Language (1982) he advances as noteworthy, though not as sound, an argument inspired by his reading of Ludwig Wittgenstein's Philosophical Investigations (1953/1993) that is not unqualifiedly attributed to Wittgenstein. On Kripke's reading the target of the argument is any theory (such as that of the Tractatus Logico-Philosophicus ) that conceives of meaning as given by conditions for truth, conceived as correspondence with facts. Kripke compares Wittgenstein as he reads him to David Hume (more specifically, to a version of Hume that takes seriously his protestations that he is only a mitigated, not an extreme skeptic). So read, Wittgenstein's attack on correspondence theories of meaning consists, like Hume's attack on rationalist theories of inference, of two phases.

First there is a "skeptical paradox." Consider an ascription of meaning, say that according to which by "plus" I mean plus, so that 125 is the right answer to the question "what is 68 plus 57?" as I mean it. To what fact does this correspond? Not the record of how I have worked sums in the past. (Perhaps I have never worked this one before, and many rules are compatible with all the ones I have worked so far.) Not my ability to state general rules for doing sums, since this only raises the question what fact corresponds to my meaning what I do by the words in these rules. Not my behavioral dispositions (nor anything in the structure or functioning of my brain causally underlying them) since what answer I am disposed to give is one question, and what answer would be the right one for me to give is another question; and I am disposed to give wrong answers fairly often even for medium-sized numbers, and to give no answer at all for really big ones. Further considerations rule out also introspectable feelings accompanying calculation. No candidates seem to remain, so it seems that there is no fact to which an ascription of meaning corresponds. The conclusion is that if meaning consists in conditions for truth and truth of correspondence with facts, then ascriptions of meaning like "What I mean by 'plus' is plus " are neither true nor meaningful, and no one ever means anything by anything.

Second, there is a "skeptical solution," defying short summary. This solution identifies the meaningfulness of a sentence with the possession not of truth-conditions but of a potential for use within a speech community. The aspects of useof usage and utilitythat are emphasized are on the one hand the conditions under which assertion of a sentence is warranted, and on the other hand the applications warranted when a sentence is accepted.

One objection, anticipated by Kripke, is that Wittgenstein does accept talk of "truth" and "facts" in a deflated sense, in which sense to say, "It is true or a fact that by 'plus' I mean plus," amounts to no more than saying, "By 'plus' I mean plus," which on Kripke's reading Wittgenstein never denies. So a straightforward statement of Wittgenstein's view as the thesis that there are no "facts" corresponding to meaning ascriptions will not do. But as Kripke notes, one of the tasks of a reading of Wittgenstein is precisely to explain why he does not state his view in straightforward philosophical theses. Other objections to Kripke's interpretation, which has Wittgenstein opposing one theory of meaning to another, have been advanced by those who interpret Wittgenstein as a "therapist" who aims to treat philosophical questions not by developing philosophical theories (of meaning or of anything else) to answer these questions, but by developing methods to cure one of wanting to ask such questions. But such a reading may be less utterly irreconcilable with the reading of Wittgenstein as skeptic than its proponents generally recognize, since after all historical skepticism was itself a form of psychotherapy, aiming to achieve philosophic ataraxia by cultivating indifference to unanswerable questions.

Reference and Metaphysics

Kripke's most famous work is Naming and Necessity (1980), which consists of a transcription (with addenda and a preface written a decade later) of lectures given at Princeton in 1970. Only a rough, brief treatment will be possible here, leaving entirely to one side the influential ancillary papers "Speaker's Reference and Semantic Reference" and "A Puzzle about Belief," related work of Keith Donnellan (on proper names) and Hilary Putnam (on natural kind terms), and Kripke's provocative discussion of several side topics (among them the contingent a priori and the identity theory in philosophy of mind).

Kripke maintains the following doctrines about naming, illustrating them with examples, many of which have become famous. The reference of a proper name (e.g. "Phosphorus," "Feynman," "Newton") is not determined by some associated definite description (or cluster of descriptions, which is to say, description of the form "the object of which most of the following is true "). The description a speaker associates with a name may be incorrect. (The speaker may describe Isaac Newton as "the man who was hit on the head by an apple and thereby struck with the idea of a force of gravity.") Even if correct, it may fail to be uniquely identifying. (The speaker may be able to describe Richard Feynman [19181988] only as "a famous physicist," which does not distinguish him from Murray Gell-Man [1929].) Even if correct and uniquely identifying, it may be so only contingently, so that in speaking of certain counterfactual situations the description may denote something else or nothing at all. (Phosphorus, though it is the brightest object regularly visible in the eastern sky before sunrise, might have only been second brightest, in which case "the brightest " would have denoted something else; while if it had been tied for brightest, "the brightest " would denote nothing.) By contrast, names designate rigidly, continuing to designate the same thing even when discussing counterfactual hypotheses. (If I say, "If there had been a brighter object, Phosphorus would have been only second brightest," I am still speaking of Phosphorus.)

A better picture than the description theory of how a name comes to denote its bearer would be this: The first user of the proper name or "initial baptist" may fix its reference by some description (possibly involving demonstratives and requiring supplementation by ostension, for example, "that bright object over there by the eastern horizon"). The second user may use the name with the intention of referring to whatever the first user was referring to, while perhaps ignorant of the original de-scription. And so on in a historical chain. (Some commentators say causal chain, but it is important to note that there need not be any causal connection between initial baptist and thing named, which may be a mathematical object.) Kripke also offers an analogous picture of how a natural kind term comes to denote the kind of things it does.

Kripke also maintains the following doctrines about necessity, partly as corollaries to the above doctrines about naming. A true identity linking proper names (e.g., "Hesperus is Phosphorus") is necessary (as a consequence of rigidity, since even in a counterfactual situation each name will continue to denote the bearer it actually denotes, and therefore the two will continue to denote the same object, if they actually do so). But such an identity is not a priori (the identity of the heavenly body spotted at dawn and called "Phosphorus" with the one seen at dusk and dubbed "Hesperus" being an empirical astronomical discovery). Therefore, the metaphysical notion of necessity, "what could not have been otherwise," must be distinguished from epistemological notions like "what can be known a priori to be so."

There are other examples of metaphysical necessities, many involving natural kind terms: the facts of identity of heat with random molecular motion, of water with H2O, of gold with the element of atomic number 79, and more; that a given object (e.g., a table) is composed of the material it is composed of (wood rather than ice); that a given person or organism has the ancestry he, she, or it does (e.g., that Elizabeth II is the daughter of George VI, and if he had had no daughter, she would never have been born). This is so even though in none of these examples does one have a priori knowledge. (There would be no internal logical contradiction in a tabloid press article claiming Elizabeth II to be the daughter of Harry Truman.) Historically, from Immanuel Kant to Gottlob Frege to Rudolf Carnap and beyond, necessity had tended to dwindle to aprioricity, which in turn had tended to dwindle to analyticity; Kripke's sharp reversal of this trend is perhaps his most important single contribution to philosophy.

See also Liar Paradox, The; Modal Logic; Philosophy of Language; Philosophy of Mind; Putnam, Hilary.


Wittgenstein, Ludwig. Philosophical Investigations (1953). Translated by G. E. M. Anscombe. Malden, MA: Blackwell, 1993.

works by kripke

"Semantic Considerations on Modal Logic." Acta Philosophica Fennica 16 (1963): 8394.

"Outline of a Theory of Truth." Journal of Philosophy 72 (1975): 690716.

"Speaker's Reference and Semantic Reference." Midwest Studies in Philosophy 2 (1977): 255276.

"A Puzzle about Belief." In Meaning and Use: Papers Presented at the Second Jerusalem Philosophical Encounter, April 1976, edited by Avishai Margalit. Dordrecht, Netherlands: Reidel, 1979.

Naming and Necessity. Cambridge, MA: Harvard University Press, 1980.

Wittgenstein on Rules and Private Language: An Elementary Exposition. Cambridge, MA: Harvard University Press, 1982.

John P. Burgess (1996, 2005)

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