Saul Aaron Kripke (; November 13, 1940 – September 15, 2022) was an American
philosopher and
logician
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 ...
in the
analytic tradition. He was a Distinguished Professor of Philosophy at the
Graduate Center of the City University of New York and
emeritus professor at
Princeton University
Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the ...
. Since the 1960s, Kripke has been a central figure in a number of fields related to
mathematical logic
Mathematical logic is the study of 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 formal ...
,
modal logic,
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 peop ...
,
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.
Epis ...
, and
recursion theory
Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has sinc ...
. Much of his work remains unpublished or exists only as tape recordings and privately circulated manuscripts.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 premise ...
, especially
modal logic. 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, linguistics and comp ...
for modal logic involving
possible worlds, now called
Kripke semantics. He received the 2001
Schock Prize
The Rolf Schock Prizes were established and endowed by bequest of philosopher and artist Rolf Schock (1933–1986). The prizes were first awarded in Stockholm
Stockholm () is the capital and largest city of Sweden as well as the largest ...
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, claiming
necessity is a metaphysical notion distinct from the
epistemic notion of ''
a priori'', and that there are
necessary truths that are known ''
a posteriori'', such as that
water
Water (chemical formula ) is an Inorganic compound, inorganic, transparent, tasteless, odorless, and Color of water, nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living ...
is H
2O. A 1970 Princeton lecture series, published in book form in 1980 as ''
Naming and Necessity
''Naming and Necessity'' is a 1980 book with the transcript of three lectures, given by the philosopher Saul Kripke, at Princeton University in 1970, in which he dealt with the debates of proper names in the philosophy of language. The transcript ...
'', is considered one of the most important philosophical works of the 20th century. It introduces the concept of
names as
rigid designator
In modal logic and the philosophy of language, a term is said to be a rigid designator or absolute substantial term when it designates (picks out, denotes, refers to) the same thing in ''all possible worlds'' in which that thing exists. A designat ...
s, true in every possible world, as contrasted with
description
Description is the pattern of narrative development that aims to make vivid a place, object, character, or group. Description is one of four rhetorical modes (also known as ''modes of discourse''), along with exposition, argumentation, and narra ...
s. It also contains Kripke's
causal theory of reference A causal theory of reference or historical chain theory of reference is a theory of how terms acquire specific referents based on evidence. Such theories have been used to describe many referring terms, particularly logical terms, proper names, and ...
, 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 ph ...
'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 stimuli. (For example, in the human body, the brain which is part of the central nervous system re ...
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 con ...
, known as "
Kripkenstein", in his ''
Wittgenstein on Rules and Private Language''. The book contains his rule-following argument, a paradox for
skepticism
Skepticism, also spelled scepticism, is a questioning attitude or doubt toward knowledge claims that are seen as mere belief or dogma. For example, if a person is skeptical about claims made by their government about an ongoing war then the p ...
about
meaning.
Life and career
Saul Kripke was the oldest of three children born to
Dorothy K. Kripke and Rabbi
Myer S. Kripke
Myer Samuel Kripke (January 21, 1914 – April 11, 2014) was an American rabbi, scholar, and philanthropist. He was based in Omaha, Nebraska.
Early life
Kripke was born on January 21, 1914 in Toledo, Ohio, to parents Jacob "J. Michael" Kripke ...
. His father was the leader of Beth El Synagogue, the only Conservative congregation in
Omaha,
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 sout ...
; 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
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, teaching himself
Ancient Hebrew 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 natio ...
'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 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 high ...
and graduated ''
summa cum laude'' 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
The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the m ...
. Upon graduation he received a
Fulbright Fellowship, and in 1963 was appointed to the
Society of Fellows. Kripke later said, "I wish I could have skipped college. I got to know some interesting people but I can't say I learned anything. I probably would have learned it all anyway just reading on my own."
After briefly teaching at Harvard, Kripke moved in 1968 to
Rockefeller University in New York City, where he taught until 1976. In 1978 he took a chaired professorship at
Princeton University
Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the ...
. 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
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 C ...
, 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, ...
, Omaha (1977),
Johns Hopkins University
Johns Hopkins University (Johns Hopkins, Hopkins, or JHU) is a private research university in Baltimore, Maryland. Founded in 1876, Johns Hopkins is the oldest research university in the United States and in the western hemisphere. It consi ...
(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 ...
, 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, a ...
, 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 spa ...
. He won the
Schock Prize
The Rolf Schock Prizes were established and endowed by bequest of philosopher and artist Rolf Schock (1933–1986). The prizes were first awarded in Stockholm
Stockholm () is the capital and largest city of Sweden as well as the largest ...
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.
Kripke died of
pancreatic cancer on September 15, 2022, in Plainsboro, New Jersey, at the age of 81.
Work
Kripke's contributions to philosophy include:
#
Kripke semantics for
modal and related logics, published in several essays beginning in his teens.
# His 1970 Princeton lectures ''
Naming and Necessity
''Naming and Necessity'' is a 1980 book with the transcript of three lectures, given by the philosopher Saul Kripke, at Princeton University in 1970, in which he dealt with the debates of proper names in the philosophy of language. The transcript ...
'' (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 belie ...
.
He has also contributed to recursion theory (see
admissible ordinal In set theory, an ordinal number ''α'' is an admissible ordinal if L''α'' is an admissible set (that is, a transitive model of Kripke–Platek set theory); in other words, ''α'' is admissible when ''α'' is a limit ordinal and L''α'' ⊧ Σ0- ...
and
Kripke–Platek set theory
The Kripke–Platek set theory (KP), pronounced , is an axiomatic set theory developed by Saul Kripke and Richard Platek.
The theory can be thought of as roughly the predicative part of ZFC and is considerably weaker than it.
Axioms
In its fo ...
).
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. 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 (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
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems ...
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 on ''W''. Elements of ''W'' are called ''nodes'' or ''worlds'', and ''R'' is known as the
accessibility relation
An accessibility relation is a relation which plays a key role in assigning truth values to sentences in the relational semantics for modal logic. In relational semantics, a modal formula's truth value at a '' possible world'' w can depend on ...
. 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
Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically ''follows from'' one or more statements. A valid logical argument is one ...
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 logic, a normal modal logic is a set ''L'' of modal formulas such that ''L'' contains:
* All propositional tautologies;
* All instances of the Kripke schema: \Box(A\to B)\to(\Box A\to\Box B)
and it is closed under:
* Detachment rule (''modus po ...
(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 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
, 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.
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:
* a Sahlqvist formula is canonical,
* the class of frames corresponding to a Sahlqvist formula is
first-order 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 (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 logic Provability logic is a modal logic, in which the box (or "necessity") operator is interpreted as 'it is provable that'. The point is to capture the notion of a proof predicate of a reasonably rich formal theory, such as Peano arithmetic.
Examples
...
s. 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. 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.
Intuitionistic logic
Kripke semantics for
intuitionistic logic
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems ...
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
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...
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 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