Editing Kripke semantics

{{Short description|Formal semantics for non-classical logic systems}}
{{more footnotes needed|date=April 2013}}
'''Kripke semantics''' (also known as '''relational semantics''' or '''frame semantics''', and often confused with [[Possible world|possible world semantics]])<ref>Possible world semantics is a broader term encompassing various approaches, including Kripke semantics. It generally refers to the idea of analyzing modal statements by considering alternative possible worlds where different propositions are true or false. While Kripke semantics is a specific type of possible world semantics, there are other ways to model possible worlds and their relationships. Kripke semantics is a specific form of possible world semantics that employs relational structures to represent the relationships between possible worlds and propositions in modal logic.{{Citation needed|date=January 2024}}</ref> is a formal [[semantics]] for [[non-classical logic]] systems created in the late 1950s and early 1960s by [[Saul Kripke]] and [[André Joyal]]. It was first conceived for [[modal logic]]s, and later adapted to [[intuitionistic logic]] and other non-classical systems. The development of Kripke semantics was a breakthrough in the theory of non-classical logics, because the [[model theory]] of such logics was almost non-existent before Kripke (algebraic semantics existed, but were considered 'syntax in disguise').

==Semantics of modal logic==
{{Main| Modal logic }}

The language of propositional  modal logic consists of a [[countable set|countably infinite set]] of [[propositional variable]]s, a set of truth-functional [[Logical connective|connectives]] (in this article <math>\to</math> and <math>\neg</math>), and the modal operator <math>\Box</math> ("necessarily"). The modal operator <math>\Diamond</math> ("possibly") is (classically) the [[duality (mathematics)#Duality in logic and set theory|dual]] of <math>\Box</math> and [[Classical modal logic|may be defined]] in terms of necessity like so: <math>\Diamond A := \neg\Box\neg A</math> ("possibly A" is defined as equivalent to "not necessarily not A").{{sfn|Shoham|Leyton-Brown|2008}}

===Basic definitions===
 
A '''Kripke frame''' or '''modal frame''' is a pair <math>\langle W,R\rangle</math>, where ''W'' is a (possibly 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]].{{sfn|Gasquet|Herzig|Said|Schwarzentruber|2013|pages=14–16}}

A '''Kripke model''' is a triple <math>\langle W,R,\Vdash\rangle</math>,<ref>Note that the ''notion'' of 'model' in the Kripke semantics of modal logic ''differs'' from the notion of 'model' in classical non-modal logics: In classical logics we say that some formula ''F'' ''has'' a 'model' if there exists some 'interpretation' of the variables of ''F'' which makes the formula ''F'' true; this specific interpretation is then ''a model of'' the ''formula F''. In the Kripke semantics of modal logic, by contrast, a 'model' is ''not'' a specific 'something' that makes a specific modal formula true; in Kripke semantics a 'model' must rather be understood as a larger ''universe of discourse'' within which ''any'' modal formulae can be meaningfully 'understood'. Thus: whereas the notion of 'has a model' in classical non-modal logic refers to some individual formula ''within'' that logic, the notion of 'has a model' in modal logic refers to the logic itself ''as a whole'' (i.e.: the entire system of its axioms and deduction rules).</ref> where
<math>\langle W,R\rangle</math> is a Kripke frame, and <math>\Vdash</math> is a relation between nodes of ''W'' and modal formulas, such that for all ''w''&nbsp;∈&nbsp;''W'' and modal formulas ''A'' and ''B'':

* <math>w\Vdash\neg A</math> if and only if <math>w\nVdash A</math>,
* <math>w\Vdash A\to B</math> if and only if <math>w\nVdash A</math> or <math>w\Vdash B</math>,
* <math>w\Vdash\Box A</math> if and only if <math>u\Vdash A</math> for all <math>u</math> such that <math>w\; R\; u</math>.
We read <math>w\Vdash A</math> as “''w'' satisfies
''A''”, “''A'' is satisfied in ''w''”, or
“''w'' forces ''A''”. The relation <math>\Vdash</math> is called the
''satisfaction relation'', ''evaluation'', or ''[[Forcing (mathematics)|forcing]] relation''.
The satisfaction relation is uniquely determined by its
value on propositional variables.

A formula ''A'' is '''valid''' in:
* a model <math>\langle W,R,\Vdash\rangle</math>, if <math>w\Vdash A</math> for all <math>w \in W</math>,
* a frame <math>\langle W,R\rangle</math>, if it is valid in <math>\langle W,R,\Vdash\rangle</math> for all possible choices of <math>\Vdash</math>,
* 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 '''[[Soundness|sound]]''' with
respect to a class of frames ''C'', if ''L''&nbsp;⊆&nbsp;Thm(''C''). ''L'' is
'''[[Completeness (logic)|complete]]''' wrt ''C'' if ''L''&nbsp;⊇&nbsp;Thm(''C'').

=== Correspondence and completeness ===
Semantics is useful for investigating a logic (i.e. a [[Formal system|derivation system]]) only if the [[Logical consequence#Semantic consequence|semantic consequence]] relation reflects its syntactical counterpart, the ''[[Logical consequence#Syntactic consequence|syntactic consequence]]'' relation (''derivability'').{{sfn|Giaquinto|2002}} It is vital to know which modal logics are sound and complete with respect to a class of Kripke frames, and to determine also which class that 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: while most of the modal systems studied are complete of classes of frames described by simple conditions, 
Kripke incomplete normal modal logics do exist. A natural example of such a system is [[Japaridze's polymodal logic]].

A normal modal logic ''L'' '''corresponds''' to a class of frames ''C'', if ''C''&nbsp;=&nbsp;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''' : <math>\Box A\to A</math>.
'''T''' is valid in any [[reflexive relation|reflexive]] frame <math>\langle W,R\rangle</math>: if
<math>w\Vdash \Box A</math>, then <math>w\Vdash A</math>
since ''w''&nbsp;''R''&nbsp;''w''. On the other hand, a frame which
validates '''T''' has to be reflexive: fix ''w''&nbsp;∈&nbsp;''W'', and
define satisfaction of a propositional variable ''p'' as follows:
<math>u\Vdash p</math> if and only if ''w''&nbsp;''R''&nbsp;''u''. Then
<math>w\Vdash \Box p</math>, thus <math>w\Vdash p</math>
by '''T''', which means ''w''&nbsp;''R''&nbsp;''w'' using the definition of
<math>\Vdash</math>. '''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''<sub>1</sub>&nbsp;⊆&nbsp;''L''<sub>2</sub> are normal modal logics that correspond to the same class of frames, but ''L''<sub>1</sub> does not prove all theorems of ''L''<sub>2</sub>. Then ''L''<sub>1</sub> is Kripke incomplete. For example, the schema <math>\Box(A\leftrightarrow\Box
A)\to\Box A</math> 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 <math>\Box
A\to\Box\Box A</math>.

==== Common modal axiom schemata ====
The following table lists common modal axioms together with their corresponding classes. The naming of the axioms often varies; Here, axiom '''K''' is named after [[Saul Kripke]]; axiom '''T''' is named after the [[Epistemic modal logic#The knowledge or truth axiom|truth axiom]] in [[epistemic logic]]; axiom '''D''' is named after [[deontic logic]]; axiom '''B''' is named after [[L. E. J. Brouwer]]; and axioms '''4''' and '''5''' are named based on [[C. I. Lewis]]'s numbering of [[Symbolic Logic|symbolic logic systems]].

{| class="wikitable"
! Name !! Axiom !! Frame condition
|-
! K 
| <math>\Box (A\to B)\to(\Box A\to \Box B)</math> 
| holds true for any frames
|-
! T 
| <math>\Box A\to A</math> 
| [[reflexive relation|reflexive]]: <math>w\,R\,w</math>
|-
! Q
| <math>\Box\Box A\to\Box A</math>
| [[dense relation|dense]]: <math> w\,R\,u\Rightarrow \exists v\,(w\,R\,v \land v\,R\,u)</math>
|-
! 4 
| <math>\Box A\to\Box\Box A</math>
| [[transitive relation|transitive]]: <math>w\,R\,v \wedge v\,R\,u \Rightarrow w\,R\,u</math>
|-
! D 
| <math>\Box A\to\Diamond A</math>  or <math>\Diamond\top</math> or <math>\neg\Box\bot</math>
| [[serial relation|serial]]: <math>\forall w\,\exists v\,(w\,R\,v)</math>
|-
! B 
| <math>A\to\Box\Diamond A</math> or <math>\Diamond\Box A\to A</math>
| [[symmetric relation|symmetric]] :  <math>w\,R\,v \Rightarrow v\,R\,w</math>
|-
! 5 
| <math>\Diamond A\to\Box\Diamond A</math> 
| [[Euclidean relation|Euclidean]]: <math>w\,R\,u\land w\,R\,v\Rightarrow u\,R\,v</math>
|-
! GL
| <math>\Box(\Box A\to A)\to\Box A</math> 
| ''R'' transitive, ''R''<sup>−1</sup> [[well-founded]]
|-
! Grz<ref>After [[Andrzej Grzegorczyk]].</ref>
| <math>\Box(\Box(A\to\Box A)\to A)\to A</math> 
| ''R'' reflexive and transitive, ''R''<sup>−1</sup>−Id well-founded
|-
! H
| <math>\Box(\Box A\to B)\lor\Box(\Box B\to A)</math>
| <math>w\,R\,u\land w\,R\,v\Rightarrow u\,R\,v\lor v\,R\,u</math><ref>{{Cite book |last=Boolos |first=George |title=The Logic of Provability |publisher=Cambridge University Press |year=1993 |isbn=0-521-43342-8 |pages=148,149}}</ref>
|-
! M
| <math>\Box\Diamond A\to\Diamond\Box A</math>
| (a complicated [[second-order logic|second-order]] property)
|-
! G
| <math>\Diamond\Box A\to\Box\Diamond A</math>
| convergent: <math>w\,R\,u\land w\,R\,v\Rightarrow\exists x\,(u\,R\,x\land v\,R\,x)</math>
|-
! -
| <math> A\to\Box A</math> 
| discrete: <math>w\,R\,v\Rightarrow w=v</math>
|-
! -
| <math>\Diamond A\to\Box A</math>
| [[partial function]]:  <math> w\,R\,u\land w\,R\,v\Rightarrow u=v</math>
|-
! -
| <math>\Diamond A\leftrightarrow\Box A</math>
| function: <math> \forall w\,\exists!u\, w\,R\,u</math> (<math> \exists!</math> is the [[uniqueness quantification]])
|-
!-
| <math>\Box A</math> or <math>\Box \bot</math>
| empty: <math> \forall w\,\forall u\, \neg ( w\, R\,u)</math>
|-
|}
Axiom '''K''' can also be [[Rewriting|rewritten]] as <math>\Box [(A\to B)\land A]\to \Box B</math>, which logically establishes [[modus ponens]] as a [[rule of inference]] in every possible world.

Note that for axiom '''D''', <math>\Diamond A</math> implicitly implies <math>\Diamond\top</math>, which means that for every possible world in the model, there is always at least one possible world accessible from it (which could be itself). This implicit implication <math>\Diamond A \rightarrow \Diamond\top</math> is similar to the implicit implication by [[Quantifier (logic)#Range of quantification|existential quantifier on the range of quantification]].

==== Common modal systems ====
{{:Normal modal logic}}

===Canonical models===

For any normal modal logic, ''L'', a Kripke model (called the '''canonical model''') can be constructed that refutes precisely the non-theorems of
''L'', by an adaptation of the standard technique of using [[maximal consistent set]]s 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 it using the theorems of ''L'', and Modus Ponens. A ''maximal L-consistent set'' (an ''L''-''MCS''
for short) is an ''L''-consistent set that has no proper ''L''-consistent superset.

The '''canonical model''' of ''L'' is a Kripke model
<math>\langle W,R,\Vdash\rangle</math>, where ''W'' is the set of all ''L''-''MCS'',
and the relations ''R'' and <math>\Vdash</math> are as follows:
: <math>X\;R\;Y</math> if and only if for every formula <math>A</math>, if <math>\Box A\in X</math> then <math>A\in Y</math>,
: <math>X\Vdash A</math> if and only if <math>A\in X</math>.
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 that satisfies ''P'',
* for any normal modal logic ''L'' that 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
[[compactness theorem|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 [[decision problem|undecidable]] whether a given axiom is
canonical. We know a nice sufficient condition: [[Henrik Sahlqvist]] identified a broad class of formulas (now called
[[Sahlqvist formula]]s) such that
* a Sahlqvist formula is canonical,
* the class of frames corresponding to a Sahlqvist formula is [[first-order logic|first-order]] definable,
* there is an algorithm that 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.

===Finite model property===

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 (logic)|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 [[#Model constructions|filtration]] or
[[#Model constructions|unravelling]]. As another possibility,
completeness proofs based on [[cut-elimination|cut-free]]
[[sequent calculus|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 with respect to a class of
[[modal algebra]]s, 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.

===Multimodal logics===
{{See also|Multimodal logic}}

Kripke semantics has a straightforward generalization to logics with
more than one modality. A Kripke frame for a language with
<math>\{\Box_i\mid\,i\in I\}</math> as the set of its necessity operators
consists of a non-empty set ''W'' equipped with binary relations
''R<sub>i</sub>'' for each ''i''&nbsp;∈&nbsp;''I''. The definition of a
satisfaction relation is modified as follows:

: <math>w\Vdash\Box_i A</math> if and only if <math>\forall u\,(w\;R_i\;u\Rightarrow u\Vdash A).</math>

A simplified semantics, discovered by Tim Carlson, is often used for
polymodal [[provability logic]]s. A '''Carlson model''' is a structure
<math>\langle W,R,\{D_i\}_{i\in I},\Vdash\rangle</math>
with a single accessibility relation ''R'', and subsets
''D<sub>i</sub>''&nbsp;⊆&nbsp;''W'' for each modality. Satisfaction is
defined as

: <math>w\Vdash\Box_i A</math> if and only if <math>\forall u\in D_i\,(w\;R\;u\Rightarrow u\Vdash A).</math>

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.

==Semantics of intuitionistic logic==

Kripke semantics for [[intuitionistic logic]] follows the same principles as the semantics of modal logic, but it uses a different definition of satisfaction.{{sfn|Troelstra|van Dalen|1988|loc=5. Kripke semantics|pages=75-87}}

An '''intuitionistic Kripke model''' is a triple
<math>\langle W,\le,\Vdash\rangle</math>, where <math>\langle W,\le\rangle</math> is a [[preorder]]ed Kripke frame, and <math>\Vdash</math> satisfies the following conditions:{{sfn|Simpson|1994|page=20|loc=2.2 The semantics of intuitionistic logic}}
* if ''p'' is a propositional variable, <math>w\le u</math>, and <math>w\Vdash p</math>, then <math>u\Vdash p</math> (''persistency'' condition (cf. [[monotonicity]])),
* <math>w\Vdash A\land B</math> if and only if <math>w\Vdash A</math> and <math>w\Vdash B</math>,
* <math>w\Vdash A\lor B</math> if and only if <math>w\Vdash A</math> or <math>w\Vdash B</math>,
* <math>w\Vdash A\to B</math> if and only if for all <math>u\ge w</math>, <math>u\Vdash A</math> implies <math>u\Vdash B</math>,
* not <math>w\Vdash\bot</math>.

The negation of ''A'', ¬''A'', could be defined as an abbreviation for ''A'' → ⊥.  If for all ''u'' such that ''w'' ≤ ''u'', not ''u'' <big>⊩</big> ''A'', then ''w'' <big>⊩</big> ''A'' → ⊥ is [[vacuous truth|vacuously true]], so ''w'' <big>⊩</big> ¬''A''.

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 logic|first-order]] language. A Kripke
model of ''L'' is a triple
<math>\langle W,\le,\{M_w\}_{w\in W}\rangle</math>, where
<math>\langle W,\le\rangle</math> is an intuitionistic Kripke frame, ''M<sub>w</sub>'' is a
(classical) ''L''-structure for each node ''w''&nbsp;∈&nbsp;''W'', and
the following compatibility conditions hold whenever ''u''&nbsp;≤&nbsp;''v'':
* the domain of ''M<sub>u</sub>'' is included in the domain of ''M<sub>v</sub>'',
* realizations of function symbols in ''M<sub>u</sub>'' and ''M<sub>v</sub>'' agree on elements of ''M<sub>u</sub>'',
* for each ''n''-ary predicate ''P'' and elements ''a''<sub>1</sub>,...,''a<sub>n</sub>''&nbsp;∈&nbsp;''M<sub>u</sub>'': if ''P''(''a''<sub>1</sub>,...,''a<sub>n</sub>'') holds in ''M<sub>u</sub>'', then it holds in ''M<sub>v</sub>''.
Given an evaluation ''e'' of variables by elements of ''M<sub>w</sub>'', we
define the satisfaction relation <math>w\Vdash A[e]</math>:
* <math>w\Vdash P(t_1,\dots,t_n)[e]</math> if and only if <math>P(t_1[e],\dots,t_n[e])</math> holds in ''M<sub>w</sub>'',
* <math>w\Vdash(A\land B)[e]</math> if and only if <math>w\Vdash A[e]</math> and <math>w\Vdash B[e]</math>,
* <math>w\Vdash(A\lor B)[e]</math> if and only if <math>w\Vdash A[e]</math> or <math>w\Vdash B[e]</math>,
* <math>w\Vdash(A\to B)[e]</math> if and only if for all <math>u\ge w</math>, <math>u\Vdash A[e]</math> implies <math>u\Vdash B[e]</math>,
* not <math>w\Vdash\bot[e]</math>,
* <math>w\Vdash(\exists x\,A)[e]</math> if and only if there exists an <math>a\in M_w</math> such that <math>w\Vdash A[e(x\to a)]</math>,
* <math>w\Vdash(\forall x\,A)[e]</math> if and only if for every <math>u\ge w</math> and every <math>a\in M_u</math> , <math>u\Vdash A[e(x\to a)]</math>.
Here ''e''(''x''→''a'') is the evaluation which gives ''x'' the
value ''a'', and otherwise agrees with ''e''.{{refn|See a slightly different formalization in {{harvtxt|Moschovakis|2022}}}}

===Kripke–Joyal semantics===

As part of the independent development of [[sheaf theory]], it was realised around 1965 that Kripke semantics was intimately related to the treatment of [[existential quantification]] in  [[topos theory]].{{sfn|Goldblatt|2006b}} That is, the 'local' aspect of existence for sections of a sheaf was a kind of logic of the 'possible'. Though this development was the work of a number of people, the name '''Kripke–Joyal semantics''' is often used in this connection.

==Model constructions==

As in classical [[model theory]], there are methods for
constructing a new Kripke model from other models.

The natural [[homomorphism]]s in Kripke semantics are called
'''p-morphisms''' (which is short for ''pseudo-epimorphism'', but the
latter term is rarely used). A p-morphism of Kripke frames
<math>\langle W,R\rangle</math> and <math>\langle W',R'\rangle</math> is a mapping
<math>f\colon W\to W'</math> such that
* ''f'' preserves the accessibility relation, i.e., ''u&nbsp;R&nbsp;v'' implies ''f''(''u'')&nbsp;''R’''&nbsp;''f''(''v''),
* whenever ''f''(''u'')&nbsp;''R’''&nbsp;''v''’, there is a ''v''&nbsp;∈&nbsp;''W'' such that ''u&nbsp;R&nbsp;v'' and ''f''(''v'')&nbsp;=&nbsp;''v''’.
A p-morphism of Kripke models <math>\langle W,R,\Vdash\rangle</math> and
<math>\langle W',R',\Vdash'\rangle</math> is a p-morphism of their
underlying frames <math>f\colon W\to W'</math>, which
satisfies
: <math>w\Vdash p</math> if and only if <math>f(w)\Vdash'p</math>, for any propositional variable ''p''.

P-morphisms are a special kind of [[bisimulation]]s. In general, a
'''bisimulation''' between frames <math>\langle W,R\rangle</math> and
<math>\langle W',R'\rangle</math> is a relation
''B&nbsp;⊆&nbsp;W&nbsp;×&nbsp;W’'', which satisfies
the following “zig-zag” property:
* if ''u&nbsp;B&nbsp;u’'' and ''u&nbsp;R&nbsp;v'', there exists ''v’''&nbsp;∈&nbsp;''W’'' such that ''v&nbsp;B&nbsp;v’'' and ''u’&nbsp;R’&nbsp;v’'',
* if ''u&nbsp;B&nbsp;u’'' and ''u’&nbsp;R’&nbsp;v’'', there exists ''v''&nbsp;∈&nbsp;''W'' such that ''v&nbsp;B&nbsp;v’'' and ''u&nbsp;R&nbsp;v''.
A bisimulation of models is additionally required to preserve forcing
of [[atomic formula]]s:
: if ''w&nbsp;B&nbsp;w’'', then <math>w\Vdash p</math> if and only if <math>w'\Vdash'p</math>, for any propositional variable ''p''.
The key property which follows from this definition is that
bisimulations (hence also p-morphisms) of models preserve the
satisfaction of ''all'' formulas, not only propositional variables.

We can transform a Kripke model into a [[tree (graph theory)|tree]] using
'''unravelling'''. Given a model <math>\langle W,R,\Vdash\rangle</math> and a fixed
node ''w''<sub>0</sub>&nbsp;∈&nbsp;''W'', we define a model
<math>\langle W',R',\Vdash'\rangle</math>, where ''W’'' is the
set of all finite sequences
<math>s=\langle w_0,w_1,\dots,w_n\rangle</math> such
that ''w<sub>i</sub>&nbsp;R&nbsp;w<sub>i+1</sub>'' for all
''i''&nbsp;<&nbsp;''n'', and <math>s\Vdash p</math> if and only if
<math>w_n\Vdash p</math> for a propositional variable
''p''. The definition of the accessibility relation ''R’''
varies; in the simplest case we put
:<math>\langle w_0,w_1,\dots,w_n\rangle\;R'\;\langle w_0,w_1,\dots,w_n,w_{n+1}\rangle</math>,
but many applications need the reflexive and/or transitive closure of
this relation, or similar modifications.

'''Filtration''' is a useful construction which one can use to prove [[Kripke semantics#Finite model property|FMP]] for many logics. Let ''X'' be a set of
formulas closed under taking subformulas. An ''X''-filtration of a
model <math>\langle W,R,\Vdash\rangle</math> is a mapping ''f'' from ''W'' to a model
<math>\langle W',R',\Vdash'\rangle</math> such that
* ''f'' is a [[surjection]],
* ''f'' preserves the accessibility relation, and (in both directions) satisfaction of variables ''p''&nbsp;∈&nbsp;''X'',
* if ''f''(''u'')&nbsp;''R’''&nbsp;''f''(''v'') and <math>u\Vdash\Box A</math>, where <math>\Box A\in X</math>, then <math>v\Vdash A</math>.
It follows that ''f'' preserves satisfaction of all formulas from
''X''. In typical applications, we take ''f'' as the projection
onto the [[quotient set|quotient]] of ''W'' over the relation
: ''u&nbsp;≡<sub>X</sub>&nbsp;v'' if and only if for all ''A''&nbsp;∈&nbsp;''X'', <math>u\Vdash A</math> if and only if <math>v\Vdash A</math>.
As in the case of unravelling, the definition of the accessibility
relation on the quotient varies.

==General frame semantics==
The main defect of Kripke semantics is the existence of Kripke incomplete logics, and logics which are complete but not compact. It can be remedied by equipping Kripke frames with extra structure which restricts the set of possible valuations, using ideas from algebraic semantics. This gives rise to the [[general frame]] semantics.

==Computer science applications==
{{main|Kripke structure|state transition system|model checking}}
Blackburn et al. (2001) point out that because a relational structure is simply a set together with a collection of relations on that set, it is unsurprising that relational structures are to be found just about everywhere. As an example from [[theoretical computer science]], they give [[labeled transition system]]s, which model [[computer program|program execution]]. Blackburn et al. thus claim because of this connection that modal languages are ideally suited in providing "internal, local perspective on relational structures." (p. xii)

==History and terminology==
{{Unreferenced section|date=October 2009}}
Similar work that predated Kripke's revolutionary semantic breakthroughs:{{sfn|Stokhof|2008|loc=See the last two paragraphs in Section 3 '''Quasi-historical Interlude: the Road from Vienna to Los Angeles'''.}}
* [[Rudolf Carnap]] seems to have been the first to have the idea that one can give a '''possible world semantics''' for the modalities of necessity and possibility by means of giving the valuation function a parameter that ranges over Leibnizian possible worlds.  Bayart develops this idea further, but neither gave recursive definitions of satisfaction in the style introduced by Tarski;
* J.C.C. McKinsey and [[Alfred Tarski]] developed an approach to modeling modal logics that is still influential in modern research, namely the algebraic approach, in which Boolean algebras with operators are used as models. [[Bjarni Jónsson]] and Tarski established the representability of Boolean algebras with operators in terms of frames. If the two ideas had been put together, the result would have been precisely frame models, which is to say Kripke models, years before Kripke. But no one (not even Tarski) saw the connection at the time.
*[[Arthur Prior]], building on unpublished work of [[C. A. Meredith]], developed a translation of sentential modal logic into classical predicate logic that, if he had combined it with the usual model theory for the latter, would have produced a model theory equivalent to Kripke models for the former. But his approach was resolutely syntactic and anti-model-theoretic.
* [[Stig Kanger]] gave a rather more complex approach to the interpretation of modal logic, but one that contains many of the key ideas of Kripke's approach.  He first noted the relationship between conditions on accessibility relations and [[C.I. Lewis|Lewis]]-style axioms for modal logic.  Kanger failed, however, to give a completeness proof for his system;
* [[Jaakko Hintikka]] gave a semantics in his papers introducing epistemic logic that is a simple variation of Kripke's semantics, equivalent to the characterisation of valuations by means of maximal consistent sets.  He doesn't give inference rules for epistemic logic, and so cannot give a completeness proof;
* [[Richard Montague]] had many of the key ideas contained in Kripke's work, but he did not regard them as significant, because he had no completeness proof, and so did not publish until after Kripke's papers had created a sensation in the logic community;
* [[Evert Willem Beth]] presented a semantics of intuitionistic logic based on trees, which closely resembles Kripke semantics, except for using a more cumbersome definition of satisfaction.

==See also==
* [[Alexandrov topology]]
* [[Normal modal logic]]
* [[Two-dimensionalism]]
* [[Induction puzzles#Muddy Children Puzzle|Muddy Children Puzzle]]

== Notes ==
<references/>

==References==
*{{cite book |first1=P. |last1=Blackburn |author2-link=Maarten de Rijke |first2=M. |last2=de Rijke |first3=Yde |last3=Venema |title=Modal Logic |url=https://books.google.com/books?id=-n7uBgAAQBAJ |date=2002 |publisher=Cambridge University Press |isbn=978-1-316-10195-7}}
*{{cite book |last1=Bull |first1=Robert A. |first2=K. |last2=Segerberg |chapter=Basic Modal Logic |editor-first=D.M. |editor-last=Gabbay |editor2-first=F. |editor2-last=Guenthner |series=Handbook of Philosophical Logic |volume=2 |title=Extensions of Classical Logic |chapter-url=https://books.google.com/books?id=IoTtCAAAQBAJ&pg=PA1 |date=2012 |publisher=Springer |isbn=978-94-009-6259-0 |pages=1–88 |orig-year=1984}}
*{{cite book |first1=A. |last1=Chagrov |first2=M. |last2=Zakharyaschev |title=Modal Logic |url=https://books.google.com/books?id=dhgi5NF4RtcC |year=1997 |publisher=Clarendon Press |isbn=978-0-19-853779-3}}
*{{cite book |first1=M.J. |last1=Cresswell |author1-link=Max Cresswell|first2=G.E. |last2=Hughes |author2-link=George Edward Hughes|title=A New Introduction to Modal Logic |url=https://books.google.com/books?id=TkwrBgAAQBAJ |year=2012 |orig-year=1996 |publisher=Routledge |isbn=978-1-134-80028-5}}
*{{cite book |last=Van Dalen |first=Dirk |author-link=Dirk van Dalen|orig-year=1986 |chapter=Intuitionistic Logic |editor-first=Dov M. |editor-last=Gabbay |editor2-first=Franz |editor2-last=Guenthner |series=Handbook of Philosophical Logic |title=Alternatives to Classical Logic |chapter-url=https://books.google.com/books?id=NwDwCAAAQBAJ&pg=PA225 |year=2013 |publisher=Springer |isbn=978-94-009-5203-4 |pages=225–339 |volume=3}}
*{{cite book |author-link=Michael Dummett |first=Michael A. E. |last=Dummett |title=Elements of Intuitionism |url=https://books.google.com/books?id=JVFzknbGBVAC |year=2000 |publisher=Clarendon Press |isbn=978-0-19-850524-2 |edition=2nd}}
*{{cite book |first=Melvin |last=Fitting |authorlink = Melvin Fitting|title=Intuitionistic Logic, Model Theory and Forcing |url=https://archive.org/details/intuitionisticlo0000fitt |url-access=registration |year=1969 |publisher=North-Holland |isbn=978-0-444-53418-7}}
*{{cite book|last1=Gasquet|first1=Olivier|last2=Herzig|first2=Andreas|last3=Said|first3=Bilal|last4=Schwarzentruber|first4=François|title=Kripke's Worlds: An Introduction to Modal Logics via Tableaux|date=2013|publisher=Springer |isbn=978-3764385033|pages=XV, 198|url=https://books.google.com/books?id=qFS-BAAAQBAJ|access-date=24 December 2014}}
*{{cite book|last=Giaquinto|first=Marcus|title=The Search for Certainty : A Philosophical Account of Foundations of Mathematics: A Philosophical Account of Foundations of Mathematics|date=2002|publisher=Oxford University Press|isbn=019875244X|page=256|url=https://books.google.com/books?id=gyNbrpYXzxoC|access-date=24 December 2014}}
*{{cite book |author-link=Robert Goldblatt |first=Robert |last=Goldblatt |chapter=Mathematical Modal Logic: a View of its Evolution |chapter-url=http://www.msor.vuw.ac.nz/~rob/papers/modalhist.pdf |format=PDF |editor-first=Dov M. |editor-last=Gabbay |editor2-first=John |editor2-last=Woods |title=Logic and the Modalities in the Twentieth Century |url=https://books.google.com/books?id=gpwZYXsomEsC&pg=PP1 |date=2006a|publisher=Elsevier |isbn=978-0-08-046303-2 |pages=1–98 |series=Handbook of the History of Logic |volume=7}}
*{{cite book |author-link=Robert Goldblatt |first=Robert |last=Goldblatt |chapter=A Kripke-Joyal Semantics for Noncommutative Logic in Quantales |chapter-url=http://homepages.ecs.vuw.ac.nz/~rob/papers/aiml11printed.pdf |editor-first=G.|editor-last=Governatori|editor2-first=I. |editor2-last=Hodkinson|editor3-first=Y. |editor3-last=Venema |title=Advances in Modal Logic |publisher=College Publications |location=London |year=2006b|isbn=1904987206 |pages=209–225 |volume=6}}
*{{cite book |author-link=Saunders Mac Lane |author2-link=Ieke Moerdijk |first1=Saunders |last1=Mac Lane |first2=Ieke |last2=Moerdijk |title=Sheaves in Geometry and Logic: A First Introduction to Topos Theory |url=https://books.google.com/books?id=LZWLBAAAQBAJ |year=2012 |orig-year=1991 |publisher=Springer |isbn=978-1-4612-0927-0 }}
*{{cite book |last1=Shoham |first1=Yoav |first2=Kevin |last2=Leyton-Brown |title=Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations |date=2008 |publisher=Cambridge University Press |isbn=978-0521899437 |page=397|url=https://books.google.com/books?id=bMR_qScakukC}}
*{{cite thesis|last=Simpson|first=Alex K.|date=1994|title=The Proof Theory and Semantics of Intuitionistic Modal Logic|url=https://era.ed.ac.uk/handle/1842/407|hdl=1842/407|publisher=[[University of Edinburgh|Edinburgh Research Archive (ERA)]]}}
*{{cite book|last=Stokhof|first=Martin|year=2008|editor1-last=Zamuner|editor1-first=Edoardo|editor2-last=Levy|editor2-first=David K.|title=Wittgenstein's Enduring Arguments|publisher=Routledge|location=London|chapter=The architecture of meaning: Wittgenstein’s ''Tractatus'' and formal semantics|pages=211–244|isbn=9781134107070|chapter-url=https://books.google.com/books?id=MUp8AgAAQBAJ&pg=PA211}}
* {{cite book | last1 = Troelstra | first1 = A. S. | author-link1 = Anne Sjerp Troelstra | last2 = van Dalen | first2 = D. | author-link2 = Dirk van Dalen | title = Constructivism in Mathematics: An Introduction, Volume 1 | url = https://archive.org/details/StudiesInLogicAndTheFoundationsOfMathematics121AnneS.TroelstraDirkVanDalenConstr | series = Studies in Logic and the Foundations of Mathematics | language = en | volume = 121 | year = 1988 | publisher = North-Holland | location = Amsterdam | isbn = 9780444702661 }}

==External links==
{{Commons category|Kripke models}}
*{{cite web|first=John P.|last=Burgess|title=Kripke Models|url=http://www.princeton.edu/~jburgess/Kripke1.doc|url-status=dead|archive-url=https://web.archive.org/web/20041020014707/http://www.princeton.edu/~jburgess/Kripke1.doc|archive-date=2004-10-20}}
*{{cite book |first1=V. |last1=Detlovs |last2=Podnieks |first2=K. |chapter=4.4 Constructive Propositional Logic &mdash; Kripke Semantics |chapter-url=http://www.ltn.lv/~podnieks/mlog/ml4a.htm#s44 |title=Introduction to Mathematical Logic |publisher=University of Latvia |url=https://dspace.lu.lv/dspace/handle/7/34986}} N.B: Constructive = intuitionistic.
*{{cite SEP|last=Garson|first=James|date=23 January 2023|title=Modal Logic|url-id=logic-modal}}
* {{springer|title=Kripke models|id=p/k055850}}
*{{cite SEP|last=Moschovakis|first=Joan|date=16 December 2022|title=Intuitionistic Logic|url-id=logic-intuitionistic}}

{{DEFAULTSORT:Kripke Semantics}}
[[Category:Modal logic]]
[[Category:Model theory]]
[[Category:Mathematical logic]]
[[Category:Philosophical logic]]
[[Category:Sheaf theory]]
[[Category:Non-classical logic]]

[[de:Kripke-Semantik]]