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Kripke semantics (also known as relational semantics or frame semantics, and often confused with
possible world semantics A possible world is a complete and consistent way the world is or could have been. They are widely used as a formal device in logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize ...
) is a formal
semantics Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference Reference is a relationship between objects in which one object designates, or acts as a means by which to connect to or link to, another ...
for
non-classical logic Non-classical logics (and sometimes alternative logics) are formal system A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of t ...
systems created in the late 1950s and early 1960s by
Saul Kripke Saul Aaron Kripke (; born November 13, 1940) is an American philosopher American(s) may refer to: * American, something of, from, or related to the United States of America, commonly known as the United States ** Americans, citizens and nationa ...

and
André Joyal André Joyal (; born 1943) is a professor of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are ...
. It was first conceived for
modal logic Modal logic is a collection of formal system A formal system is an used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus ...
intuitionistic logic Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic Mathematical logic, also called formal logic, is a subfield of mathematics exploring the formal applications of logic to mathematics. I ...
and other non-classical systems. The development of Kripke semantics was a breakthrough in the theory of non-classical logics, because the
model theory In 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 ...
of such logics was almost non-existent before Kripke (algebraic semantics existed, but were considered 'syntax in disguise').

# Semantics of modal logic

The language of propositional modal logic consists of a countably infinite set of
propositional variable In 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 p ...
s, a set of truth-functional
connectives In Mathematical logic, logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant used to connect two or more formulas. For instance in the syntax (logic), syntax of proposi ...
(in this article $\to$ and $\neg$), and the modal operator $\Box$ ("necessarily"). The modal operator $\Diamond$ ("possibly") is (classically) the
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality ** . . . see more cases in :Duality theories * Dual ...
of $\Box$ and may be defined in terms of necessity like so: $\Diamond A := \neg\Box\neg A$ ("possibly A" is defined as equivalent to "not necessarily not A").

## Basic definitions

A Kripke frame or modal frame is a pair $\langle W,R\rangle$, where ''W'' is a (possibly empty) set, and ''R'' is a
binary relation Binary may refer to: Science and technology Mathematics * Binary number In mathematics and digital electronics Digital electronics is a field of electronics The field of electronics is a branch of physics and electrical engineeri ...
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 Kripke semantics (also known as relational semantics or frame semantics, and often confused with possible world s ...
. A Kripke model is a triple $\langle W,R,\Vdash\rangle$, where $\langle W,R\rangle$ is a Kripke frame, and $\Vdash$ is a relation between nodes of ''W'' and modal formulas, such that for all ''w'' ∈ ''W'' and modal formulas ''A'' and ''B'': * $w\Vdash\neg A$ if and only if $w\nVdash A$, * $w\Vdash A\to B$ if and only if $w\nVdash A$ or $w\Vdash B$, * $w\Vdash\Box A$ if and only if $u\Vdash A$ for all $u$ such that $w\; R\; u$. We read $w\Vdash A$ as “''w'' satisfies ''A''”, “''A'' is satisfied in ''w''”, or “''w'' forces ''A''”. The relation $\Vdash$ 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 $\langle W,R,\Vdash\rangle$, if $w\Vdash A$ for all ''w'' ∈ ''W'', * a frame $\langle W,R\rangle$, if it is valid in $\langle W,R,\Vdash\rangle$ for all possible choices of $\Vdash$, * 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 In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...

with respect to a class of frames ''C'', if ''L'' ⊆ Thm(''C''). ''L'' is
complete Complete may refer to: Logic * Completeness (logic) * Complete theory, Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, ...
wrt ''C'' if ''L'' ⊇ Thm(''C'').

## Correspondence and completeness

Semantics is useful for investigating a logic (i.e. a derivation system) only if the
semantic consequence Logical consequence (also entailment) is a fundamental concept Concepts are defined as abstract ideas A mental representation (or cognitive representation), in philosophy of mind Philosophy of mind is a branch of philosophy that studies ...
relation reflects its syntactical counterpart, the '' syntactic consequence'' relation (''derivability''). 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 logic, a normal modal logic is a set ''L'' of modal formulas such that ''L'' contains: * All propositional tautology (logic), tautologies; * All instances of the Saul_Kripke, Kripke schema: \Box(A\to B)\to(\Box A\to\Box B) and it is closed under: ...
(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 logicJaparidze's polymodal logic (GLP) is a system of provability logic with infinitely many provability modal logic, modalities. This system has played an important role in some applications of provability algebras in proof theory, and has been extensiv ...
. 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 : $\Box A\to A$. T is valid in any reflexive frame $\langle W,R\rangle$: if $w\Vdash \Box A$, then $w\Vdash A$ 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: $u\Vdash p$ if and only if ''w'' ''R'' ''u''. Then $w\Vdash \Box p$, thus $w\Vdash p$ by T, which means ''w'' ''R'' ''w'' using the definition of $\Vdash$. 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 $\Box\left(A\leftrightarrow\Box A\right)\to\Box A$ 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 $\Box A\to\Box\Box A$.

### Common modal axiom schemata

The following table lists common modal axioms together with their corresponding classes. The naming of the axioms often varies.

## 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 sets 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 model theory#Definition, logical theory ''T'' consists of the equivalence classes of Sentence (mathematical logic), sentences of the theory (i.e., the quotient, unde ...
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 $\langle W,R,\Vdash\rangle$, where ''W'' is the set of all ''L''-''MCS'', and the relations ''R'' and $\Vdash$ are as follows: : $X\;R\;Y$ if and only if for every formula $A$, if $\Box A\in X$ then $A\in Y$, : $X\Vdash A$ if and only if $A\in X$. The canonical model is a model of ''L'', as every ''L''-''MCS'' contains all theorems of ''L''. By
Zorn's lemma Zorn's lemma, also known as the Kuratowski–Zorn lemma, after mathematicians Max August Zorn, Max Zorn and Kazimierz Kuratowski, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (or ...
, 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
compact Compact as used in politics may refer broadly to a pact A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations International relations (IR), international affairs (IA) or internationa ...
. 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: Henrik 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 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 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.

## Multimodal logics

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 ''Ri'' for each ''i'' ∈ ''I''. The definition of a satisfaction relation is modified as follows: : $w\Vdash\Box_i A$ if and only if $\forall u\,\left(w\;R_i\;u\Rightarrow u\Vdash A\right).$ A simplified semantics, discovered by Tim Carlson, is often used for polymodal provability logics. A Carlson model is a structure $\langle W,R,\_,\Vdash\rangle$ with a single accessibility relation ''R'', and subsets ''Di'' ⊆ ''W'' for each modality. Satisfaction is defined as : $w\Vdash\Box_i A$ if and only if $\forall u\in D_i\,\left(w\;R\;u\Rightarrow u\Vdash A\right).$ 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 Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic Mathematical logic, also called formal logic, is a subfield of mathematics exploring the formal applications of logic to mathematics. I ...
follows the same principles as the semantics of modal logic, but it uses a different definition of satisfaction. An intuitionistic Kripke model is a triple $\langle W,\le,\Vdash\rangle$, where $\langle W,\le\rangle$ is a preordered Kripke frame, and $\Vdash$ satisfies the following conditions: * if ''p'' is a propositional variable, $w\le u$, and $w\Vdash p$, then $u\Vdash p$ (''persistency'' condition (cf. monotonicity)), * $w\Vdash A\land B$ if and only if $w\Vdash A$ and $w\Vdash B$, * $w\Vdash A\lor B$ if and only if $w\Vdash A$ or $w\Vdash B$, * $w\Vdash A\to B$ if and only if for all $u\ge w$, $u\Vdash A$ implies $u\Vdash B$, * not $w\Vdash\bot$. The negation of ''A'', ¬''A'', could be defined as an abbreviation for ''A'' → ⊥. If for all ''u'' such that ''w'' ≤ ''u'', not ''u'' ''A'', then ''w'' ''A'' → ⊥ is vacuous truth, vacuously true, so ''w'' ¬''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 $\langle W,\le,\_\rangle$, where $\langle W,\le\rangle$ is an intuitionistic Kripke frame, ''Mw'' is a (classical) ''L''-structure for each node ''w'' ∈ ''W'', and the following compatibility conditions hold whenever ''u'' ≤ ''v'': * the domain of ''Mu'' is included in the domain of ''Mv'', * realizations of function symbols in ''Mu'' and ''Mv'' agree on elements of ''Mu'', * for each ''n''-ary predicate ''P'' and elements ''a''1,...,''an'' ∈ ''Mu'': if ''P''(''a''1,...,''an'') holds in ''Mu'', then it holds in ''Mv''. Given an evaluation ''e'' of variables by elements of ''Mw'', we define the satisfaction relation $w\Vdash A\left[e\right]$: * $w\Vdash P\left(t_1,\dots,t_n\right)\left[e\right]$ if and only if $P\left(t_1\left[e\right],\dots,t_n\left[e\right]\right)$ holds in ''Mw'', * $w\Vdash\left(A\land B\right)\left[e\right]$ if and only if $w\Vdash A\left[e\right]$ and $w\Vdash B\left[e\right]$, * $w\Vdash\left(A\lor B\right)\left[e\right]$ if and only if $w\Vdash A\left[e\right]$ or $w\Vdash B\left[e\right]$, * $w\Vdash\left(A\to B\right)\left[e\right]$ if and only if for all $u\ge w$, $u\Vdash A\left[e\right]$ implies $u\Vdash B\left[e\right]$, * not $w\Vdash\bot\left[e\right]$, * $w\Vdash\left(\exists x\,A\right)\left[e\right]$ if and only if there exists an $a\in M_w$ such that $w\Vdash A\left[e\left(x\to a\right)\right]$, * $w\Vdash\left(\forall x\,A\right)\left[e\right]$ if and only if for every $u\ge w$ and every $a\in M_u$ , $u\Vdash A\left[e\left(x\to a\right)\right]$. Here ''e''(''x''→''a'') is the evaluation which gives ''x'' the value ''a'', and otherwise agrees with ''e''. See a slightly different formalization in.

## 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. 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 In 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 ...
, there are methods for constructing a new Kripke model from other models. The natural homomorphisms 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 $\langle W,R\rangle$ and $\langle W\text{'},R\text{'}\rangle$ is a mapping $f\colon W\to W\text{'}$ such that * ''f'' preserves the accessibility relation, i.e., ''u R v'' implies ''f''(''u'') ''R’'' ''f''(''v''), * whenever ''f''(''u'') ''R’'' ''v''’, there is a ''v'' ∈ ''W'' such that ''u R v'' and ''f''(''v'') = ''v''’. A p-morphism of Kripke models $\langle W,R,\Vdash\rangle$ and $\langle W\text{'},R\text{'},\Vdash\text{'}\rangle$ is a p-morphism of their underlying frames $f\colon W\to W\text{'}$, which satisfies : $w\Vdash p$ if and only if $f\left(w\right)\Vdash\text{'}p$, for any propositional variable ''p''. P-morphisms are a special kind of bisimulations. In general, a bisimulation between frames $\langle W,R\rangle$ and $\langle W\text{'},R\text{'}\rangle$ is a relation ''B ⊆ W × W’'', which satisfies the following “zig-zag” property: * if ''u B u’'' and ''u R v'', there exists ''v’'' ∈ ''W’'' such that ''v B v’'' and ''u’ R’ v’'', * if ''u B u’'' and ''u’ R’ v’'', there exists ''v'' ∈ ''W'' such that ''v B v’'' and ''u R v''. A bisimulation of models is additionally required to preserve forcing of atomic formulas: : if ''w B w’'', then $w\Vdash p$ if and only if $w\text{'}\Vdash\text{'}p$, 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 $\langle W,R,\Vdash\rangle$ and a fixed node ''w''0 ∈ ''W'', we define a model $\langle W\text{'},R\text{'},\Vdash\text{'}\rangle$, where ''W’'' is the set of all finite sequences $s=\langle w_0,w_1,\dots,w_n\rangle$ such that ''wi R wi+1'' for all ''i'' < ''n'', and $s\Vdash p$ if and only if $w_n\Vdash p$ for a propositional variable ''p''. The definition of the accessibility relation ''R’'' varies; in the simplest case we put :$\langle w_0,w_1,\dots,w_n\rangle\;R\text{'}\;\langle w_0,w_1,\dots,w_n,w_\rangle$, but many applications need the reflexive and/or transitive closure of this relation, or similar modifications. Filtration is a useful construction which uses 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 $\langle W,R,\Vdash\rangle$ is a mapping ''f'' from ''W'' to a model $\langle W\text{'},R\text{'},\Vdash\text{'}\rangle$ such that * ''f'' is a surjection, * ''f'' preserves the accessibility relation, and (in both directions) satisfaction of variables ''p'' ∈ ''X'', * if ''f''(''u'') ''R’'' ''f''(''v'') and $u\Vdash\Box A$, where $\Box A\in X$, then $v\Vdash A$. 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 ≡X v'' if and only if for all ''A'' ∈ ''X'', $u\Vdash A$ if and only if $v\Vdash A$. 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

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 systems, 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

Similar work that predated Kripke's revolutionary semantic breakthroughs:preprint
(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.

* Alexandrov topology * Normal modal logic * Two-dimensionalism * Induction_puzzles#Muddy_Children_Puzzle, Muddy Children Puzzle

# Notes

:aAfter Andrzej Grzegorczyk.

# References

* * * * * * * * *