Saul Aaron Kripke (; November 13, 1940 – September 15, 2022) was an American
philosopher
A philosopher is a person who practices or investigates philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, meaning 'lover of wisdom'. The coining of the term has been attributed to the Greek th ...
and
logician in the
analytic tradition
Analytic philosophy is a branch and tradition of philosophy using analysis, popular in the Western world and particularly the Anglosphere, which began around the turn of the 20th century in the contemporary era in the United Kingdom, United Sta ...
. He was a Distinguished Professor of Philosophy at the
Graduate Center of the City University of New York
The Graduate School and University Center of the City University of New York (CUNY Graduate Center) is a public research institution and post-graduate university in New York City. Serving as the principal doctorate-granting institution of the ...
and
emeritus
''Emeritus'' (; female: ''emerita'') is an adjective used to designate a retired chair, professor, pastor, bishop, pope, director, president, prime minister, rabbi, emperor, or other person who has been "permitted to retain as an honorary title ...
professor at
Princeton University
Princeton University is a private university, private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial Colleges, fourth-oldest ins ...
. Since the 1960s, Kripke has been a central figure in a number of fields related to
mathematical logic
Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...
,
modal logic
Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend other ...
,
philosophy of language
In analytic philosophy, philosophy of language investigates the nature of language and the relations between language, language users, and the world. Investigations may include inquiry into the nature of meaning, intentionality, reference, ...
,
philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people's ...
,
metaphysics
Metaphysics is the branch of philosophy that studies the fundamental nature of reality, the first principles of being, identity and change, space and time, causality, necessity, and possibility. It includes questions about the nature of conscio ...
,
epistemology
Epistemology (; ), or the theory of knowledge, is the branch of philosophy concerned with knowledge. Epistemology is considered a major subfield of philosophy, along with other major subfields such as ethics, logic, and metaphysics.
Episte ...
, and
recursion theory. Much of his work remains unpublished or exists only as tape recordings and privately circulated manuscripts.
Kripke made influential and original contributions to
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
, especially
modal logic
Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend other ...
. His principal contribution is a
semantics
Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference, meaning, or truth. The term can be used to refer to subfields of several distinct disciplines, including philosophy
Philosophy (f ...
for modal logic involving
possible world
A possible world is a complete and consistent way the world is or could have been. Possible worlds are widely used as a formal device in logic, philosophy, and linguistics in order to provide a semantics for intensional logic, intensional and mod ...
s, now called
Kripke semantics
Kripke semantics (also known as relational semantics or frame semantics, and often confused with possible world semantics) is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke and André Jo ...
. He received the 2001
Schock Prize in Logic and Philosophy.
Kripke was also partly responsible for the revival of
metaphysics
Metaphysics is the branch of philosophy that studies the fundamental nature of reality, the first principles of being, identity and change, space and time, causality, necessity, and possibility. It includes questions about the nature of conscio ...
after the decline of
logical positivism
Logical positivism, later called logical empiricism, and both of which together are also known as neopositivism, is a movement in Western philosophy whose central thesis was the verification principle (also known as the verifiability criterion o ...
, claiming
necessity
Necessary or necessity may refer to:
* Need
** An action somebody may feel they must do
** An important task or essential thing to do at a particular time or by a particular moment
* Necessary and sufficient condition, in logic, something that is ...
is a metaphysical notion distinct from the
epistemic
Epistemology (; ), or the theory of knowledge, is the branch of philosophy concerned with knowledge. Epistemology is considered a major subfield of philosophy, along with other major subfields such as ethics, logic, and metaphysics.
Episte ...
notion of ''
a priori
("from the earlier") and ("from the later") are Latin phrases used in philosophy to distinguish types of knowledge, justification, or argument by their reliance on empirical evidence or experience. knowledge is independent from current ex ...
'', and that there are
necessary truths that are known ''
a posteriori
("from the earlier") and ("from the later") are Latin phrases used in philosophy to distinguish types of knowledge, justification, or argument by their reliance on empirical evidence or experience. knowledge is independent from current ex ...
'', such as that
water
Water (chemical formula ) is an inorganic, transparent, tasteless, odorless, and nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living organisms (in which it acts as a ...
is H
2O. 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 introduces the concept of
name
A name is a term used for identification by an external observer. They can identify a class or category of things, or a single thing, either uniquely, or within a given context. The entity identified by a name is called its referent. A personal ...
s as
rigid designators, true in every possible world, as contrasted with
descriptions. It also contains Kripke's
causal theory of reference, disputing the
descriptivist theory found in
Gottlob Frege
Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic phil ...
's concept of
sense
A sense is a biological system used by an organism for sensation, the process of gathering information about the world through the detection of Stimulus (physiology), stimuli. (For example, in the human body, the brain which is part of the cen ...
and
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ...
's
theory of descriptions
The theory of descriptions is the philosopher Bertrand Russell's most significant contribution to the philosophy of language. It is also known as Russell's theory of descriptions (commonly abbreviated as RTD). In short, Russell argued that the ...
.
Kripke also gave an original reading of
Ludwig Wittgenstein
Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrian-British philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. He is considere ...
, known as "
Kripkenstein
''Wittgenstein on Rules and Private Language'' is a 1982 book by philosopher of language Saul Kripke in which he contends that the central argument of Ludwig Wittgenstein's ''Philosophical Investigations'' centers on a devastating rule-following p ...
", in his ''
Wittgenstein on Rules and Private Language
''Wittgenstein on Rules and Private Language'' is a 1982 book by philosopher of language Saul Kripke in which he contends that the central argument of Ludwig Wittgenstein's ''Philosophical Investigations'' centers on a devastating rule-following p ...
''. The book contains his rule-following argument, a paradox for
skepticism about
meaning.
Life and career
Saul Kripke was the oldest of three children born to
Dorothy K. Kripke and Rabbi
Myer S. Kripke. His father was the leader of Beth El Synagogue, the only Conservative congregation in
Omaha
Omaha ( ) is the largest city in the U.S. state of Nebraska and the county seat of Douglas County. Omaha is in the Midwestern United States on the Missouri River, about north of the mouth of the Platte River. The nation's 39th-largest city ...
,
Nebraska
Nebraska () is a state in the Midwestern region of the United States. It is bordered by South Dakota to the north; Iowa to the east and Missouri to the southeast, both across the Missouri River; Kansas to the south; Colorado to the southwe ...
; his mother wrote educational Jewish books for children. Saul and his two sisters,
Madeline
''Madeline'' is a media franchise that originated as a series of children's books written and illustrated by Ludwig Bemelmans, an Austrian-American author. The books have been adapted into numerous formats, spawning telefilms, television series a ...
and Netta, attended Dundee Grade School and
Omaha Central High School
Omaha Central High School, originally known as Omaha High School, is a fully accredited public high school located in downtown Omaha, Nebraska, United States. It is one of many public high schools located in Omaha. As of the 2015-16 academic year, ...
. Kripke was labeled a
prodigy
Prodigy, Prodigies or The Prodigy may refer to:
* Child prodigy, a child who produces meaningful output to the level of an adult expert performer
** Chess prodigy, a child who can beat experienced adult players at chess
Arts, entertainment, and ...
, teaching himself
Ancient Hebrew Ancient Hebrew (ISO 639-3 code ) is a blanket term for pre-modern varieties of the Hebrew language:
* Paleo-Hebrew (such as the Siloam inscription), a variant of the Phoenician alphabet
* Biblical Hebrew (including the use of Tiberian vocalization ...
by the age of six, reading
Shakespeare
William Shakespeare ( 26 April 1564 – 23 April 1616) was an English playwright, poet and actor. He is widely regarded as the greatest writer in the English language and the world's pre-eminent dramatist. He is often called England's nation ...
'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
Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend other ...
at 17, and had it published a year later. After graduating from high school in 1958, Kripke attended
Harvard University
Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of higher le ...
and graduated ''
summa cum laude
Latin honors are a system of Latin phrases used in some colleges and universities to indicate the level of distinction with which an academic degree has been earned. The system is primarily used in the United States. It is also used in some Sou ...
'' in 1962 with a
bachelor's degree
A bachelor's degree (from Middle Latin ''baccalaureus'') or baccalaureate (from Modern Latin ''baccalaureatus'') is an undergraduate academic degree awarded by colleges and universities upon completion of a course of study lasting three to six ...
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
The Fulbright Program, including the Fulbright–Hays Program, is one of several United States Cultural Exchange Programs with the goal of improving intercultural relations, cultural diplomacy, and intercultural competence between the people of ...
, and in 1963 was appointed to the
Society of Fellows
The Society of Fellows is a group of scholars selected at the beginnings of their careers by Harvard University for their potential to advance academic wisdom, upon whom are bestowed distinctive opportunities to foster their individual and intell ...
. 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."
After briefly teaching at Harvard, Kripke moved in 1968 to
Rockefeller University
The Rockefeller University is a private biomedical research and graduate-only university in New York City, New York. It focuses primarily on the biological and medical sciences and provides doctoral and postdoctoral education. It is classif ...
in New York City, where he taught until 1976. In 1978 he took a chaired professorship at
Princeton University
Princeton University is a private university, private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial Colleges, fourth-oldest ins ...
. 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
A university () is an institution of higher (or tertiary) education and research which awards academic degrees in several academic disciplines. Universities typically offer both undergraduate and postgraduate programs. In the United States, the ...
, Omaha (1977),
Johns Hopkins University
Johns Hopkins University (Johns Hopkins, Hopkins, or JHU) is a private university, private research university in Baltimore, Maryland. Founded in 1876, Johns Hopkins is the oldest research university in the United States and in the western hem ...
(1997),
University of Haifa
The University of Haifa ( he, אוניברסיטת חיפה Arabic: جامعة حيفا) is a university located on Mount Carmel in Haifa, Israel. Founded in 1963, the University of Haifa received full academic accreditation in 1972, becoming Is ...
, Israel (1998), and the
University of Pennsylvania
The University of Pennsylvania (also known as Penn or UPenn) is a private research university in Philadelphia. It is the fourth-oldest institution of higher education in the United States and is ranked among the highest-regarded universitie ...
(2005). He was a member of the
American Philosophical Society
The American Philosophical Society (APS), founded in 1743 in Philadelphia, is a scholarly organization that promotes knowledge in the sciences and humanities through research, professional meetings, publications, library resources, and communit ...
and an elected Fellow of the
American Academy of Arts and Sciences
The American Academy of Arts and Sciences (abbreviation: AAA&S) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, and ...
, and in 1985 was a Corresponding Fellow of the
British Academy
The British Academy is the United Kingdom's national academy for the humanities and the social sciences.
It was established in 1902 and received its royal charter in the same year. It is now a fellowship of more than 1,000 leading scholars span ...
. He won the
Schock Prize in Logic and Philosophy in 2001.
Kripke was married to philosopher
Margaret Gilbert
Margaret Gilbert (born 1942) is a British philosopher best known for her founding contributions to the analytic philosophy of social phenomena. She has also made substantial contributions to other philosophical fields including political philosop ...
. He is the second cousin once removed of television writer, director, and producer
Eric Kripke
Eric Kripke (born 1974) is an American writer and television producer. He came to prominence as the creator of the fantasy drama series ''Supernatural'' (2005–2020) which aired on The CW. He served as the showrunner during the first five seasons ...
.
Kripke died of
pancreatic cancer
Pancreatic cancer arises when cell (biology), cells in the pancreas, a glandular organ behind the stomach, begin to multiply out of control and form a Neoplasm, mass. These cancerous cells have the malignant, ability to invade other parts of t ...
on September 15, 2022, in Plainsboro, New Jersey, at the age of 81.
Work

Kripke's contributions to philosophy include:
#
Kripke semantics
Kripke semantics (also known as relational semantics or frame semantics, and often confused with possible world semantics) is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke and André Jo ...
for
modal and related logics, published in several essays beginning in his teens.
# His 1970 Princeton lectures ''
Naming and Necessity'' (published in 1972 and 1980), which significantly restructured
philosophy of language
In analytic philosophy, philosophy of language investigates the nature of language and the relations between language, language users, and the world. Investigations may include inquiry into the nature of meaning, intentionality, reference, ...
.
# His interpretation of
Wittgenstein
Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrians, Austrian-British people, British philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy o ...
.
# His theory of
truth
Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to represent reality or otherwise correspond to it, such as beliefs ...
.
He has also contributed to recursion theory (see
admissible ordinal and
Kripke–Platek set theory).
Modal logic
Two of Kripke's earlier works, "A Completeness Theorem in Modal Logic" (1959) and "Semantical Considerations on Modal Logic" (1963), the former written when he was a teenager, were on
modal logic
Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend other ...
. 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
Kripke semantics (also known as relational semantics or frame semantics, and often confused with possible world semantics) is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke and André Jo ...
(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 before Kripke.
A Kripke frame or modal frame is a pair
, where ''W'' is a non-empty set, and ''R'' is a
binary relation
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of ele ...
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:
*
if and only if
,
*
if and only if
or
,
*
if and only if
implies
.
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(''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 :
. 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 well-founded frames), but does not prove the GL-
tautology .
Canonical models
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 set In mathematical logic, a theory is complete if it is consistent and for every closed formula in the theory's language, either that formula or its negation is provable. That is, for every sentence \varphi, the theory T contains the sentence or it ...
s as models. Canonical Kripke models play a role similar to the
Lindenbaum–Tarski algebra
In mathematical logic, the Lindenbaum–Tarski algebra (or Lindenbaum algebra) of a logical theory ''T'' consists of the equivalence classes of sentences of the theory (i.e., the quotient, under the equivalence relation ~ defined such that ''p'' ...
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
In propositional logic, ''modus ponens'' (; MP), also known as ''modus ponendo ponens'' (Latin for "method of putting by placing") or implication elimination or affirming the antecedent, is a deductive argument form and rule of inference. ...
. 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
, where ''W'' is the set of all ''L''-''MCS'', and the relations ''R'' and
are as follows:
:
if and only if for every formula
, if
then
,
:
if and only if
.
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
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
.
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 formula In modal logic, Sahlqvist formulas are a certain kind of modal formula with remarkable properties. The Sahlqvist correspondence theorem states that every Henrik Sahlqvist, Sahlqvist formula is Kripke semantics#Canonical models, canonical, and corres ...
s) such that:
* a Sahlqvist formula is canonical,
* the class of frames corresponding to a Sahlqvist formula is
first-order
In mathematics and other formal sciences, first-order or first order most often means either:
* "linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of high ...
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 (equivalent to) Sahlqvist formulas. A logic has the
finite model property In mathematical logic, a logic L has the finite model property (fmp for short) if any non-theorem of L is falsified by some ''finite'' model of L. Another way of putting this is to say that L has the fmp if for every formula ''A'' of L, ''A'' is an ...
(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 set ''W'' equipped with binary relations ''R
i'' for each ''i'' ∈ ''I''. The definition of a satisfaction relation is modified as follows:
:
if and only if
Carlson models
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:
:
if and only if
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
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
. 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 of
hyperfiction
Hypertext fiction is a genre of electronic literature, characterized by the use of hypertext links that provide a new context for non-linearity in literature and reader interaction. The reader typically chooses links to move from one node of text t ...
.
Intuitionistic logic
Kripke semantics for
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
, where
is a
partially ordered Kripke frame, and
satisfies the following conditions:
* if ''p'' is a propositional variable,
, and
, then
(''persistency'' condition),
*
if and only if
and
,
*
if and only if
or
,
*
if and only if for all
,
implies
,
* not
.
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
In mathematics and other formal sciences, first-order or first order most often means either:
* "linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of high ...
language. A Kripke model of ''L'' is a triple
, where
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'':
* the domain of ''M
u'' is included in the domain of ''M
v'',
* realizations of function symbols in ''M
u'' and ''M
v'' agree on elements of ''M
u'',
* for each ''n''-ary predicate ''P'' and elements ''a''
1,...,''a
n'' ∈ ''M
u'': if ''P''(''a''
1,...,''a
n'') holds in ''M
u'', then it holds in ''M
v''.
Given an evaluation ''e'' of variables by elements of ''M
w'', we define the satisfaction relation