## Modal Logic

## Modal Logic

# MODAL LOGIC

Traditionally, the modes implicit in modal logic are the modes of truth and ultimately the modes of being: necessary, possible, impossible, and contingent. While the study of the formal properties of those notions is still an important part of modal logic, other interpretations have been added over the years, such as temporal, epistemic, and deontic. Furthermore, more recently, other formal languages have been suggested, which, although not modal logic in a strict sense, are closely related to it, such as dynamic logic.

## Brief History

Modern modal logic began in 1912 when Clarence Irving Lewis published a paper in *Mind*, in which he recommended that the logic of *Principia Mathematica* be supplemented with what he called intensional connectives. Among the latter was a binary connective of strict implication for which he introduced a new symbol, a "fishhook" to distinguish it from the "horseshoe" of the material conditional. Thus, 𝜙 ⥽ ψ and 𝜙 ⊃ ψ would both be read "if 𝜙 then ψ," but Lewis specifically intended for the former to model the elusive notion of entailment. Other connectives were possibility, for which he used the symbol ♢ (a diamond), and necessity, for which F. B. Fitch would later suggest □ (a box): thus, ♢𝜙 and □𝜙 were read "it is possible that 𝜙" and "it is necessary that 𝜙," respectively. The interest in strict implication declined somewhat after it was discovered that there are paradoxes of strict implication in parallel with those of material implication in classical logic. Lewis's legacy was not lost, however. On the one hand, philosophers like Alan Ross Anderson and Nuel Belnap went on to develop logics of entailment and relevance, a tradition that has proved hardy. On the other hand, since necessity and possibility seem more interesting than strict implication—𝜙⥽ψ is in any case analyzable as either □(𝜙 ⊃ ψ) or ¬ ♢ (𝜙 ∧ ¬ψ)—later logicians preferred to do their modal logic in terms of those concepts.

Lewis's original ambition was to find the logic of strict implication. Much to his surprise he later found himself confronted by a veritable embarrassment of riches: an ever increasing number of modal logics—not only his own famous quintuple of systems S1, S2, S3, S4, and S5 (his own tentative favorite was S3, the so-called Survey system) but also many, in fact, infinitely many others. Since he never translated his semantic intuitions into a formal structure, the differentiation between different proposals became a problem. Some help in this regard arrived in the form of the concept of a matrix, essentially a set of truth-values (usually but not necessarily finitely many) plus a truth-value table for each connective. This idea, which was due to Jan Łukasiewicz, was then generalized into the notion of an algebra (essentially a set with operators) and taken into modal logic by Alfred Tarski and his collaborators. The advent of algebraic logic revitalized modal logic. Two works from this period are particularly noteworthy. One was the first formal result in modal logic worth the name, J.C.C. McKinsey's algebraic characterization of S2 and a proof that it is decidable. The other was a paper in 1951 by Bjarni Jónsson and Tarski foreshadowing the next major development: the era of possible-worlds semantics.

Since the term *possible-worlds semantics* is today used pretty much synonymously with the term *Kripke semantics*, it is germane to ask: Who invented Kripke semantics? In fact, this question has been the object of much discussion, some heated. When the new semantics emerged at the end of the 1950s, Rudolf Carnap had laid the ground work; his states really played the role that possible worlds would later play, even if he only worked with descriptions of them. What Carnap did not have, and which turned out to make all the difference, was the accessibility relation (this concept is explained later on). The accessibility relation did appear in the Jónsson-Tarski paper mentioned earlier, and it now seems likely that Arthur Prior and C. A. Meredith had also discovered it in the early 1950s. But it was Saul A. Kripke, along with Stig Kanger and Jaakko Hintikka, who first published accounts in which the accessibility relation was a central concept and its versatility recognized. That Kripke's work overshadowed the work by Kanger and Hintikka and proved so much more influential than theirs is perhaps not surprising, given the clarity and mathematical maturity Kripke's papers and the systematic development of his theory.

After Kripke's early work followed a period of increasingly formal concern. Not surprisingly, the philosophers have focused on the philosophy of modal logic, including modal metaphysics, while the mathematicians have pursued the mathematics of modal logic, including model theory, algebra, and even category theory. Another significant development has been the expanding use of modal logic in theoretical computer science: with energy and inventiveness—but of course guided by their own interests—computer scientists have, within a short time, transformed modal logic.

## Syntax

This entry only considers classical modal logic, that is, logic that extends classical logic. Historically, even though for a long time it is modal predicate logic that has been of particular interest to philosophers, propositional modal logic has received much more attention from formal logicians, probably because agreement on what constitutes a generally accepted conceptual framework for research was reached much earlier in the latter area.

### propositional logic

To the set of the usual truth-functional connectives, add two new connectives: a box operator □ and a diamond operator ♢. After Tarski, a theory, in a technical sense, is a set of formulas (called theses of the logic) that contains all classical two-valued tautologies and is closed under *modus ponens* (if 𝜙 and 𝜙 ⊃ ψ are theses of the logic, then so is ψ). Similarly, a logic, in the technical sense used here, is a theory that is closed under uniform substitution (if 𝜙(χ/P) results from a formula 𝜙 by replacing all occurrences of a certain propositional letter P with a formula χ, then 𝜙(χ/P) is a thesis of the logic if 𝜙 is). A normal modal logic is a logic that contains as theses all instances of the schema ♢𝜙 ≡ ¬□¬𝜙 as well as of the so-called Kripke schema □(𝜙 ⊃ ψ) ⊃ (□𝜙 ⊃ □ψ) and, in addition, is closed under the rule of necessitation (if 𝜙 is a thesis, then so is □𝜙). A great number of normal modal logics have been studied, many of them definable in terms of further schemata, for example,

(D) □𝜙 ⊃ ♢𝜙,

(T) □𝜙 ⊃ 𝜙,

(4) □𝜙 ⊃ □□𝜙,

(5) ¬□𝜙 ⊃ □¬□𝜙,

(G) ♢□𝜙 ⊃ □ ♢𝜙,

(H) (♢𝜙 ∧ ♢ψ) ⊃ (♢ (𝜙 ∧ ψ) ∨ ♢ (𝜙 ∧ ♢ψ) ∨ ♢ (ψ ∧ ♢𝜙)),

(W) □(□𝜙 ⊃ 𝜙) ⊃ □𝜙.

To bring some order into the bewildering multiplicity of modal logics, E. J. Lemmon suggested KX_{1}, … , X* _{n}* as a code name for the smallest normal modal logic that contains all substitution instances of schemata X

_{1}, … , X

*. In this notation, one may identify K as the smallest normal logic, KT as the Gödel/Feys/von Wright logic, and KT4 and KT45 as the logics S4 and S5, respectively. The logics KD, KD4, and KD45, of special interest to deontic and doxastic logic, are sometimes called weak T, weak S4, and weak S5, respectively. The logics KT4G and KT4H are better known as S4.2 and S4.3, respectively, and the logic K4W as the Gödel/Löb logic GL. The set of all normal logics, ordered by set inclusion, forms a lattice of immense complexity, as do sets of more inclusive classes of nonnormal modal logics. The efforts to explore these structures continue but are increasingly a concern for mathematicians rather than for philosophers.*

_{n}### predicate logic

Modal predicate logic does not exhibit the relative orderliness or maturity of its propositional relative. Philosophical questions such as the proper treatment of individuals persist. Quantification, in particular into opaque contexts—that is, contexts within the scope of modal operators—has been a main problem, as evidenced by Quine's unrelenting criticism over a lifetime. A formal beginning was made by Ruth Barcan Marcus, after whom two central formulas have been named—the Barcan formula (BF) and the converse Barcan formula (CBF):

(BF) ∀

x□𝜙 ⊃ □∀x𝜙,(CBF) □∀

x𝜙 ⊃ ∀x□𝜙

Other examples of formulas that were much discussed in early literature are

∀*x* ∀*y* (*x* = *y* ⊃ □(*x* = *y* )),

∀*x* ∀*y* (*x ≠ y* ⊃ □(*x ≠ y* )),

a = b ⊃ □(a = b),

a ≠ b ⊃ □(a ≠ b),

where a and b are individual constants. Various authors have held different views on which of these, if any, are valid. It would seem that to take a stand in such matters is to rely on implicit semantic ideas, however sketchy. It was accordingly an important step when at last, thanks to Kripke and others, formal semantics were articulated.

## Semantics

The development of modal logic, both material and formal, preceded in steps. Propositional logics were studied extensively before predicate logicians had been able to work out a generally accepted common ground. Till this day, the area of modal propositional logic is more definitive than the relatively more unsettled area of modal predicate logic.

### propositional logic

The possible-worlds semantics, introduced by Kripke in the early 1960s, may be cast in the following form (which differs from Kripke's original formulation in terminology and, to some extent, in substance). A frame is a pair (*U, R* ), where *U*, the universe of discourse or simply the universe of the frame, is a nonempty set of elements that are often called possible worlds but that may more neutrally be called points, and *R* is a binary relation in *U*, called the accessibility relation or sometimes the alternativeness relation or even the alternative relation. If two points *u, v* of *U* are related by *R* (i.e., if (*u, v* ) ∈ *R* ), then one says that *v* is accessible from *u* or that *v* is an alternative to *u*. A valuation in *U* is a function *V* assigning to each propositional letter P a subset *V* (P) of *U*. A model is a structure (*U, R, V* ) where (*U, R* ) is a frame and *V* is a valuation in *U*. Truth in modal logic is doubly relative: to a model and to a point in the model. Thus, if 𝔐 = (*U, R, V* ) is a model, *u* a point in *U* and 𝜙 a formula, one may inductively define the notion of 𝜙 being true at *u* in 𝔐, schematically *u* ⊧^{𝔐} 𝜙, as follows:*u* ⊧^{𝔐} P iff *u* ∈ *V* (P), if P is a propositional letter;*u* ⊧^{𝔐} ¬𝜙 iff not *u* ⊧^{𝔐} 𝜙,*u* ⊧^{𝔐} 𝜙 ∧ ψ iff *u* ⊧^{𝔐} 𝜙 and *u* ⊧^{𝔐} ψ,*u* ⊧^{𝔐} 𝜙 ∨ ψ iff *u* ⊧^{𝔐} 𝜙 or *u* ⊧^{𝔐} ψ,

and similar conditions for other truth-functional connectives:*u* ⊧^{𝔐} □𝜙 iff, for all points *v*, if (*u,v* ) ∈ *R* then *v* ⊧^{𝔐} 𝜙,*u* ⊧^{𝔐} ♢𝜙 iff there is some point *v* such that (*u, v* ) ∈ *R* and *v* ⊧^{𝔐} 𝜙.

(Readers may note the roles played in this definition by *R* and *V* : The latter is needed to get the definition started, the former to evaluate formulas beginning with a modal operator; the truth-functional connectives are taken care of by the usual truth tables.) A formula is valid in a frame if it is true at every point in every model definable on that frame; and it is valid in a class of frames if it is valid in each one of the frames of the class.

There is a sense in which this semantics fits modal logic. The set of formulas that are valid in a given class of frames will always be a normal modal logic and can be called the logic determined by that class of frames. A logic is sound with respect to a class of frames if every thesis of the logic is valid in that class, and it is complete with respect to the class if every formula that is valid in the class is a thesis of the logic; hence a logic is determined by a class of frames if and only if it is both sound and complete with respect to that class. It is an interesting fact, and no doubt one reason for the popularity of Kripke semantics, that many of the logics defined in the philosophical literature are determined by simply defined classes of frames. For example, T, S4, and S5 are determined by the class of frames whose accessibility relations are reflexive, reflexive and transitive, and reflexive, symmetric, and transitive, respectively. Similarly, KD, KD4, and KD45 are determined by the class of all frames whose accessibility relations are serial, serial and transitive, and serial, transitive and euclidean, respectively. (A binary relation *R* is serial if, for every element *u* in its field there is some element *v*, not necessarily distinct from *u*, such that (*u, v* ) ∈ *R*, euclidean if, for all elements *u, v, w* in its field, if (*u, v* ) ∈ *R* and (*u, w* ) ∈ *R* then (*v, w* ) ∈ *R*.) At the extremes are the smallest normal modal logic K and the inconsistent logic, which are determined by, respectively, the class of all frames and the empty class of frames.

The way in which Kripke's semantics seems to fit modal logic led some authors, for example, Lemmon, to conjecture that all normal modal logics are complete, that is, determined by some class of frames. However, that the fit is less than perfect was proved in 1971 by Kit Fine and S. K. Thomason, who exhibited, independently of one another, instances of incomplete normal modal logics.

### predicate logic

Among several possible versions of semantics for modal predicate logic, the following is essentially a modified version of Kripke's semantics for first-order modal logic from 1963. For simplicity, assume a formal language for predicate logic containing predicate letters and individual constants (but, for example, no descriptions or functional operators); thus, the terms of this language are individual variables or individual constants. To generalize the central concepts *frame* and *model* used in propositional modal logic, several new notions must be introduced. To begin with, besides a universe *U* of points (possible worlds) and an accessibility relation *R*, as before, one needs a nonempty set *D* of objects and a function *E* defined on *U* that takes values in the set of subsets of *D*. One can refer to *D* as the domain and to *E* as the existence function, to the elements of *D* as possible individuals and to the elements of *E _{u}* as individuals existing at

*u*or individuals actual at

*u*(where

*u*is a point in

*U*). Altogether, a structure (

*U, R, D, E*), where

*U, R, D, E*are as specified, is a frame. Next, one can say that

*I*is an interpretation (in

*D*with respect to

*U*) if it is a family of functions

*I*, where

_{u}*u*ranges over

*U*, such that

*I*assigns a set of

_{u}*n*-tuples of elements of

*D*to each

*n*-ary predicate letter and an element of

*D*to each individual constant. If 𝔉 = (

*U,R,D,E*) is a frame, then 𝔐 = (𝔉,

*I*) = (

*U,R,D,E,I*) is a model (on 𝔉) if

*I*is an interpretation in

*D*with respect to

*U*.

The following observation shows the sense in which the present concept of model is a generalization over that of propositional semantics: Nullary predicate letters behave in the present setting as propositional letters do in the propositional case. To see this, let P be a nullary predicate letter. By the definition, the interpretation of P is a set of 0-tuples, hence *I _{u}* (P) is either ∅ (the empty set) or {∅} (the singleton set whose only member is the one and only 0-tuple). If one arbitrarily identifies {∅} with truth and ∅ with falsity, one thereby also in effect identifies the set {

*u*∈

*U : I*(P) = {∅}} with the proposition expressed by P in 𝔐. Thus, the interpretation plays a role in the predicate case similar to that of the valuation in the propositional case, albeit a much bigger role.

_{u}Besides all this, one needs yet another concept to define truth-conditions: something to take care of the quantifiers. An assignment (in a set *D* ) is a function from the set of individual variables of one's formal language to *D*. Notice that if *A* is an assignment in *D* and *x* is a variable, then *A* (*x* ) is an element of *D* but perhaps not of *E _{u}*, if

*u*is an arbitrary point in

*U*. If 𝔐 = (

*U, R, D, E, I*) is a model and

*A*is an assignment in

*D*, then the denotation of

*t*in 𝔐 under

*A*is a function ||

*t*||

^{𝔐}

*defined on*

_{A}*U*as follows:

The truth of a formula 𝜙 in a model 𝔐 under an assignment

*A*at a point

*u*, in symbols

*u*⊧

^{𝔐}

*A*𝜙, may now be defined:

,

if P is an

*n*-ary predicate letter,

The remaining clauses of the definition (for the truth-functional connectives and the modal operators) are as before. In particular,

.

As in the propositional case, one associates truth with models and validity with frames. Thus, one can say that a formula is true in a model if it is true under all assignments at all points in the model. By the same token, one can say that a formula is valid in a frame if it is true in all models on the frame.

Some object languages contain constant predicates besides predicate letters. Common examples of such predicates are the unary E (the existence predicate) and the binary = (the identity predicate) with corresponding truth-conditions:

The meaning of E and = depends neither on the interpretation *I* nor the assignment *A* ; for this reason E and = may be called logical constants. Notice that if the identity predicate is available, the existence predicate is definable: Provided that *t* is distinct from *x*, if *t* is a variable, E(*t* ) ≡ ∃*x* (*x* = *t* ) is a valid schema.

The following remarks apply to this particular modeling. All instances of the Barcan formula (BF) are valid in all and only frames satisfying the condition of decreasing domains, that is,

for all *u* and *v*, if (*u, v* ) ∈ *R* then *E _{u}* ⊇

*E*.

_{v}Similarly, all instances of the converse Barcan formula (CBF) are valid in all and only frames that satisfy the condition of increasing domains, that is

for all

*u*and

*v*, if (

*u, v*) ∈

*R*then

*E*⊆

_{u}*E*.

_{v}Of the other predicate logical formulas discussed earlier, ∀

*x*∀

*y*(

*x*=

*y*⊃ □(

*x*=

*y*)) and ∀

*x*∀

*y*(

*x ≠ y*⊃ □(

*x ≠ y*)) are valid, while neither a = b ⊃ □(a = b) nor a ≠ b ⊃ □(a ≠ b) is valid. This reflects an important difference between how individual variables and individual constants are treated in this modeling: In spite of their name, the denotation of individual constants may vary from point to point in the universe, whereas the denotation of variables, their name notwithstanding, remains fixed throughout the universe. Here is obviously a niche to be filled! Suppose one introduces a new syntactic category of names and requires that the interpretation of a name n be constant over the universe of points; formally,

*I*(n) =

_{u}*I*(n), for all

_{v}*u, v*∈

*U*. Then, if m and n are any names, m = n ⊃ □(m = n) and m ≠ n ⊃ □(m ≠ n) are both valid. The proposed modification amounts to treating the elements of the new category of names as what is now known, after Kripke, as rigid designators.

Among other modelings for modal predicate logic, David Lewis's counterpart theory should be mentioned. According to the Kripke paradigm, an individual may exist in more than one possible world (with respect to the formal modeling defined above, it is possible that *E _{u}* and

*E*should overlap, in a model, even if

_{v}*u ≠ v*). For Lewis, however, each individual inhabits its own possible world; but it may have counterparts in other possible worlds. This approach has also been influential, both in philosophical and in mathematical quarters.

## Interpretations

The original interpretation of modal logic—the official interpretation, if one prefers—was of course the one that led to its construction: the interpretation in terms of necessity and possibility. But over time there have been many others.

### the alethic interpretation

In formal philosophy, as in formal conceptual analysis generally, there is a constant interplay between intuition and formalism. Efforts to explicate pretheoretical notions lead to a formalism, for example, an axiom system in a formal language or a set theoretical modeling. Once a formalism is in place, it takes on a life of its own: Not only may it undergo a formal development but it can also be interpreted, sometimes in ways that are not foreseen. Reflections on such interpretations lead to refined, sometimes revised, intuitions. The latter in turn may inspire more sophisticated formalisms. And so it goes. The formalism described earlier in this entry is a product of such interplay, having arisen principally as a result of efforts to understand what Georg Henrik von Wright called the alethic modalities necessity and possibility. Not surprisingly, questions persist about to what extent this formalism is a successful explication of one's informal understanding of necessity and possibility.

Formal semantics for modal logic is, by itself, philosophically neutral. The elements of the universe of a modal logical frame, which from a formal point of view are just points in a logical space, must be given a substantial meaning by philosophers who wish to use them outside the realm of pure abstraction. In tense logic the points will be points of time, in epistemic logic perhaps epistemic situations, and so on. Under the alethic interpretation they are often referred to as possible worlds, an ordinary language word with no clear content. Indeed, the question as to what a possible world is has exercised philosophers since the beginning of the Kripke era. Answers—besides those rejecting the entire modal logical enterprise—have been numerous. Lewis argued for an extreme modal realism according to which possible worlds are concrete alternative universes existing in parallel with the actual world. Other philosophers, like Kripke, Alvin Plantinga, Robert Stalnaker, and David M. Armstrong also argued for one kind of modal realism or other but have taken them to be abstract entities. Still other philosophers regarded possible-worlds talk as a kind of convenient fiction or refer to linguistic conventions. The debate continues.

An exact and expressive formalism has the advantage that old informal questions falling within its range of interpretation can be addressed anew. One such question is the venerable distinction between *de dicto* and *de re*. To take Willard Van Orman Quine's well-worn example, consider the claim that the number of planets is necessarily greater than seven. Is it true? There seem to be two different ways of understanding this claim. To bring them out, one can translate them into an ad hoc, quasi-formal language:

(1) ∃

x((x= the number of planets) ∧ □(x> 7)),(2) □∃

x((x= the number of planets) ∧ (x> 7)).

Statement (1) is said to be *de re*, statement (2) *de dicto*. It may be argued that they say different things (presumably, most would agree that the former is true but that the latter is false). The former seems to "say of an object" (the *res*, the number of planets) that, by necessity, it has a certain property ("being greater than seven"). By contrast, the latter statement says that a certain statement is necessarily true (the *dictum*, namely, that the number of planets, whatever that number may be, is greater than seven). This example illustrates the important interaction between quantifying and modalizing: It is one thing to put a modal operator in front of a closed sentence, as in (2), it is another, arguably more problematic, to quantify into the scope of a modal operator, as in (1). The old topic of essences is obviously not far away.

Another distinction, which has been argued by Kripke, is that between logical modalities and metaphysical modalities (there may also be others, such as physical modalities). Logical necessity implies metaphysical necessity, but the converse is not true. For example, "Phosphorus is identical with Hesperus" (assuming the names *Phosphorus* and *Hesperus* are regarded as rigid designators) and "The chemical composition of water is H_{2}O" (again assuming that *water* and H_{2}O are rigid designators) have been offered as examples of statements that are metaphysically, but not logically, necessary.

The (epistemological) distinction between *a priori* and *a posteriori* also comes in here. In Kripke's theory, the two examples given in the preceding paragraph exemplify statements that, although metaphysically necessary, are nevertheless *a posteriori*. By contrast, given certain assumptions, "The Paris meter is one meter long" may be an example of a statement that is true *a priori* but is not metaphysically necessary.

### two early mathematical interpretations

In the 1930s two technical interpretations of modal logic were made by the two greatest logicians of the twentieth century. One was the so-called provability interpretation, due to Kurt Gödel, according to which □𝜙 is interpreted as "𝜙 is provable" or "𝜙 is provable in *S*," where *S* is a certain formal system. This interpretation was never forgotten, but it attracted major attention only relatively recently. The other interpretation, due to Tarski, is in terms of topology: Let C and I denote the closure C*X* and the interior I*X*, respectively, of any subset *X* of a topological space *U*. Tarski noted that the closure operator and the interior operator behave in a way analogous to the way the possibility operator and the necessity operator behave in S4. For example, if 𝜙 and ψ correspond to *X* and *Y*, respectively, then the formulas ♢(𝜙 ∨ ψ) ≡ (♢𝜙 ∨ ♢ψ), ♢𝜙 ≡ ♢♢𝜙, 𝜙 ≡ ⊤, and 𝜙 ≡ ⟂ correspond to the equations C(*X* ∪ *Y* ) ≡ C*X* ∪ C*Y*, C*X* = CC*X, X* = *U* and *X* = Ø. More generally, Tarski proved that an equation in topological terms is true in all topological spaces if and only if the corresponding formula is a thesis of S4. Like Gödel's interpretation, Tarski's interpretation, which is related to the development of the theory of closure algebras, was seminal.

### the temporal interpretation

A long-standing interest in the work of early Greek logicians combined with a passion for modal logic led Arthur Norman Prior, in the 1950s, to the idea of a modal logic of time. He dubbed his creation tense logic since one of his original motivations was to throw light on the grammatical notion of tense. In the beginning Prior was led to study frames (*U, R* ) in which *R* is a linear relation on *U* (i.e., reflexive, transitive, and connected). Under that interpretation, the interpretation of the modal operators □ and ♢ in effect becomes "always in the future" and "some time in the future." One focus for his early interest was the frame (ℕ, ≤), where ℕ is the set of natural numbers, which he associated with Diodorus Cronus. Trying to axiomatize the set of formulas valid in this frame—the Diodorean logic, as he called it—Prior successively made three conjectures. The first was that it is S4. This conjecture was disproved by Hintikka, who pointed out that all instances of the schema

(H) (♢𝜙 ∧ ♢ψ) ⊃ (♢(𝜙 ∧ ψ) ∨ ♢(𝜙 ∧ ♢ψ) ∨ ♢(ψ ∧ ♢𝜙))

are theses of the Diodorean logic but not all of S4. Prior's response was the new conjecture that it is S4.3, that is, the logic whose Lemmon code is KT4H. However, Michael Anthony Eardley Dummett showed that all instances of the schema

(Dum) □(□(𝜙 ⊃ □𝜙) ⊃ □𝜙) ⊃ (♢□𝜙 ⊃ □𝜙)

are theses of the Diodorean logic but not all of S4.3. Prior's third conjecture was that the Diodorean logic is S4.3Dum. This final conjecture turned out to be correct, proved by R. A. Bull and, independently, by Kripke.

In general, Prior allowed the temporal ordering to be irreflexive. He also introduced operators for past time as well as for future time. Thus, the basic operators of tense-logic are the diamond operators F and P, with readings "it will be the case (some time in the future) that" and "it was the case (some time in the past) that," and the box operators G and H with the reading "always in the future" and "always in the past." Their truth-conditions in a frame (*U*, <), where < is at least a strict partial ordering (i.e., irreflexive and transitive), are:*u* ⊧^{𝔐} F𝜙 iff *v* ⊧^{𝔐} 𝜙, for some point *v* such that *u* ≤ *v*,*u* ⊧^{𝔐} P𝜙 iff *v* ⊧^{𝔐} 𝜙, for some point *v* such that *v* ≤ *u*.*u* ⊧^{𝔐} G𝜙 iff *v* ⊧^{𝔐} 𝜙, for all points *v* such that *u* ≤ *v*,*u* ⊧^{𝔐} H𝜙 iff *v* ⊧^{𝔐} 𝜙, for all points *v* such that *v* ≤ *u*.

Tense logic is in effect a kind of bimodal logic: It is natural to think of a tense-logical frame as a frame with two accessibility relations, one for the future and one for the past. What is special to tense logic is that those two relations are inverses of one another (and, consequently, all instances of the schemata PG𝜙 ⊃ 𝜙 and FH𝜙 ⊃ 𝜙 are valid).

The temporal operators mentioned are not the only ones possible. A particularly important pair of operators studied by Hans Kamp are since and until:*u* ⊧^{𝔐} 𝜙 since θ iff there is some *w* ∈ *U* such that *w* < *u* and *w* ⊧^{𝔐} θ and, for all *x* ∈ *U*, if *w* < *x* < *u* then *x* ⊧^{𝔐} 𝜙, *u* ⊧^{𝔐} 𝜙 until θ iff there is some *w* ∈ *U* such that *u* < *w* and *w* ⊧^{𝔐} θ and, for all *x* ∈ *U*, if *u* < *x* < *w* then *x* ⊧^{𝔐} 𝜙.

(In the literature, 𝜙 since θ and 𝜙 until θ are often written S(θ, 𝜙) and U(θ, 𝜙), respectively.) Kamp proved that in certain contexts, for example, over (ℝ, <) (where ℝ is the set of reals and < is the natural strict linear order) his operators suffice for temporal completeness; that is, in those contexts, all operators corresponding to first-order conditions on the temporal relation can be defined in terms of since and until and truth-functional connectives. But in general there is no temporal completeness in this sense.

Still another important tense-logical operator is now, which refers to a designated, fixed point of reference. A language involving that operator requires a somewhat modified truth-definition: Where before the definition is with respect to a model and a point, it will now be with respect to a model and two points, which one might call the current point and the point of reference—the former is variable, the latter is fixed throughout the definition. The clauses pertaining to the old operators, which only involve the current point, are obvious. The novel clause is

(*u, t* )⊧^{𝔐} now 𝜙 iff (*t, t* )⊧^{𝔐} 𝜙.

### the epistemic interpretation

The possibility of epistemic logic (the logic of knowledge) and doxastic logic (the logic of belief) was realized by von Wright, who coined the terms, but it was Hintikka who set the field going. Hintikka associated, with each agent *a*, two operators **K** * _{a}* and

**B**

*, reading "*

_{a}*a*knows that 𝜙" for

**K**

*𝜙 and "*

_{a}*a*believes that 𝜙" for

**B**

*𝜙. By the same token, the formal counterparts of "for all that*

_{a}*a*knows, 𝜙" and "𝜙 is consistent with everything

*a*believes" are ¬

**K**

*¬𝜙 and ¬*

_{a}**B**

*¬𝜙. Already Hintikka's new notation was useful. To know that someone Qs is not the same as knowing someone who Qs, but Hintikka's notation makes this patent—*

_{a}**K**

*∃*

_{a}*x*Q

*x*has to mean something different from ∃

*x*

**K**

*Q*

_{a}*x*(compare the distinction between

*de dicto*and

*de re*mentioned earlier). Discussion about logical relationships was also facilitated. For example, is it reasonable to regard the type (4) schema

**K**

*𝜙 ⊃*

_{a}**K**

_{a}**K**

*𝜙 (positive introspection, the KK-thesis) and the type (5) schema ¬*

_{a}**K**

*𝜙 ⊃*

_{a}**K**

*¬*

_{a}**K**

*𝜙 (negative introspection) as valid for rational knowledge? (Hintikka's own inclination was to accept the former but reject the latter.) Another example of the applicability of Hintikka's logic was to the puzzle known after George Edward Moore as Moore's paradox. Suppose I am ignorant of the fact, say, that it is currently raining in Cambridge, England, but that I am sufficiently informed of my own beliefs to be aware of my ignorance. Then someone who knows me may say, truly, "It is raining, but you don't believe it." But, as observed by Moore, it would be distinctly odd of me to agree, saying, "Yes, it is raining, but I don't believe it." Hintikka accounts for the oddness by suggesting that a belief operator*

_{a}**B**

*must satisfy certain minimum conditions to count as an operator expressing rational belief. For example, it would be enough if the logic of*

_{a}**B**

*was at least as strong as the normal modal logic KD4, for in that logic a sentence 𝜙 ∧ ¬*

_{a}**B**

*𝜙 may be consistent, but a sentence*

_{a}**B**

*(𝜙 ∧ ¬*

_{a}**B**

*𝜙) is always inconsistent (or, in Hintikka's terminology, doxastically indefensible).*

_{a}Knowledge and belief about knowledge and belief has been an issue of late, of interest not only to philosophers but also to computer scientists and game theorists. It may be that everyone in a group of agents knows that 𝜙, but this does not mean that 𝜙 is common knowledge in the group (a concept first studied by David Lewis); for that to be the case it is also required that everyone knows that everyone knows that 𝜙, knows that everyone knows that everyone knows that 𝜙, and so on. Interestingly, this concept can be axiomatized. If *G* is a nonempty, finite set of agents—for simplicity, assume that *G* = {1, … , *n* }—write **E** * _{G}* 𝜙 for "every member of

*G*knows that 𝜙" and

**C**

*𝜙 for "it is common knowledge among the members of*

_{G}*G*that 𝜙." Assuming that

**K**

*is an S4-operator, for each*

_{i}*i*∈

*G*, the logic of the two new operators may be characterized by requiring

**C**

*also to be an S4-operator and adding the following conditions:*

_{G}**E**

*𝜙 ≡ (*

_{G}**K**

_{1}𝜙 ∧ · · · ∧

**K**

*𝜙),*

_{n}**C**

*𝜙 ⊃*

_{G}**E**

*𝜙,*

_{G}(𝜙 ∧

**C**

*(𝜙 ⊃*

_{G}**E**

*𝜙)) ⊃*

_{G}**C**

*𝜙.*

_{G}### the deontic interpretation

When von Wright published his seminal paper "Deontic Logic" in 1951, he in effect delivered a discipline just waiting to be born. The next decades saw a great number of papers and books written on this topic, but it is probably fair to say that the results are less definitive than those of several other subfields of modal logic. The basic idea is to study operators **O, P** , and **F** with the informal readings "it is obligatory that 𝜙" for **O** 𝜙, "it is permitted that 𝜙" for **P** 𝜙, and "it is forbidden that 𝜙" for **F** 𝜙. In so-called standard deontic logic (STD), **O** is treated as the box operator and **P** as the diamond operator of a normal logic; **F** may then be defined by a condition such as **F** 𝜙 ≡ **O** ¬𝜙 or **F** 𝜙 ≡ ¬**P** 𝜙 (to be compared with the validities **P** 𝜙 ≡ ¬**O** ¬𝜙 and **O** 𝜙 ≡ ¬**P** ¬𝜙). STD—not a precise concept—provides the schema (D) **O** 𝜙 ⊃ **P** 𝜙. One schema that for obvious reasons would be inappropriate in a deontic logic is (T), but weaker schemata such as **OO** 𝜙 ⊃ **O** 𝜙 and **O** (**O** 𝜙 ⊃ 𝜙) are sometimes included in STD.

Efforts to apply STD to even fairly simple everyday situations will often fail, as shown by the existence of so-called paradoxes, a topic much discussed in the literature. Best known among the latter are perhaps the paradoxes of William David Ross, Roderick Chisholm, and James W. Forrester. (Ross's paradox was originally formulated within the logic of imperatives, but it is equally relevant for deontic logic.) A person is under an obligation to see to it that (𝜙) a letter is posted. Should he or she do it by seeing to it that (ψ) the letter is burned? Since 𝜙 ⊃ (𝜙 ∨ ψ) is a tautology, **O** 𝜙 ⊃ **O** (𝜙 ∨ ψ) is a thesis of STD. Evidently, according to STD the person should see to it that the letter is posted or burned; Ross found this conclusion bizarre. In Chisholm's paradox there are two things A and B that you may or may not do: Whether (𝜙) you do A is logically independent of whether (ψ) you do B. On the one hand, it ought to be the case that you do B if you do A (**O** (𝜙 ⊃ ψ)). On the other hand, if you do not do A, then neither ought you to do B (¬𝜙 ⊃ ¬**O** ψ). Furthermore, even though A is something you ought to do (**O** 𝜙), you will not do it (¬𝜙). In STD this description of a situation, regrettable perhaps but otherwise unremarkable, leads to contradiction. Forrester's paradox is subtler: suppose there is something one must not do, but that if one nevertheless does it, then one should do it in such and such a way. Again, STD comes to grief.

Among the many problems still not resolved in modern deontic logic—Hector–Neri Castañeda's work and his distinction between propositions and practitions notwithstanding—is the age-old question about the relationship between Seinsollen (ought to be) and Tunsollen (ought to do). It is interesting that von Wright, the father of the discipline, originally had intended for his deontic operators to take as arguments, not propositions, but actions; he seems to have changed his mind for technical reasons. With the advent of dynamic logic, it is nowadays possible to reconsider this option.

### other interpretations

The techniques of modal logic have been applied to a number of other areas of philosophical interest: imperatives, action, preference, place, even questions. Many of the more interesting applications make use of several modalities. For example, Kanger's theory of rights, which builds on Wesley Newcomb Hohfeld's famous analysis, combines concepts from deontic logic and the logic of action.

## Extensions of Modal Logic

### conditional logic

The analysis of conditionals has occupied philosophers for generations. Not all the resulting analyses belong to the field of modal logic, but there is a natural sense in which the conditional logics of Robert Stalnaker and David Lewis may be seen as generalizations of classical modal logic. This is obvious if one employs a notation suggested by Brian Chellas: writing [𝜙]ψ and 〈𝜙〉ψ where Lewis had 𝜙 □→ ψ ("if it were the case that 𝜙, then it would be the case that ψ") and 𝜙 ♢→ ψ ("if it were the case that 𝜙, then it might be that ψ"), respectively. By this device, one moves from the language of traditional modal logic, where there is one box operator □, to a language in which there are as many box operators [𝜙] as there are well-formed formulas 𝜙. Corresponding to the minimal normal modal logic K is the minimal normal conditional logic in which every box operator satisfies the Kripke schema and the rule of necessitation, and which is also closed under the rule of congruence (if θ* is the result of replacing all occurrences of 𝜙 in θ by an occurrence of ψ, then θ ≡ θ* is a thesis if 𝜙 ≡ ψ is). Lewis's logic VC of counterfactuals is the smallest normal conditional logic that contains all instances of the schemata:

[𝜙]𝜙,

〈𝜙〉ψ ⊃ 〈ψ〉⊤,

𝜙 ⊃ (ψ ⊃ [𝜙]ψ),

𝜙 ⊃ ([𝜙]ψ ⊃ ψ),

[𝜙 ∧ ψ]θ ⊃ [𝜙](ψ ⊃ θ),

〈𝜙〉ψ ⊃ ([𝜙](ψ ⊃ θ) ⊃ [𝜙 ∧ ψ]θ).

Stalnaker's logic is obtained by requiring that also all instances of the schema

〈𝜙〉θ ≡ [𝜙]θ

be theses.

### dynamic logic

Looking for a useful way to formalize reasoning about programs, Vaughan Pratt, a computer scientist, arrived at what is nowadays known as dynamic logic, a formalism similar to modal logic; in fact, dynamic logic may be viewed as a generalization of modal logic in the same way as Chellas-formulated conditional logic may be seen as a generalization of modal logic. With each program α Pratt associated a box operator [α] and a diamond operator 〈α〉, reading [α]𝜙 as "after every terminating computation according to α, 𝜙" and 〈α〉𝜙 as "after some terminating computation according to α, 𝜙." The resulting logic, originally called the modal logic of programs, evidently contains two basic categories of expressions, terms (for programs), and formulas (for propositions). A further complication over modal logic is the existence of term operators for the so-called regular operations. Thus, if α and β are programs, then α + β is the program consisting of α or β (the latter concept is of course of interest only in the context of nondeterministic automata) while α ; β is the program consisting of α immediately followed by β, and α* is the program consisting of α some finite number of times, possibly 0 (again, of interest only in a nondeterministic context). Finally, Pratt allowed a test program: ?𝜙 is a program that, if run, fails if 𝜙 is false but otherwise returns to status quo. An axiomatization of PDL (propositional dynamic logic) is obtained by requiring each box operator [α] to be a normal modal operator and adding the following axiom schemata:

[α + β]𝜙 ≡ ([α]𝜙 ∧ [β]𝜙),

[α ;β]𝜙 ≡ [α][β]𝜙,

[α*]𝜙 ⊃ 𝜙,

[α*]𝜙 ⊃ [α]𝜙,

[α*]𝜙 ⊃ [α*][α*]𝜙,

(𝜙 ∧ [α*](𝜙 ⊃ [α]𝜙)) ⊃ [α*]𝜙,

[?𝜙]χ ≡ (𝜙 ⊃ χ).

### other interpretations

Some of the generalizations of modal logic that have been made over the last few decades have an origin far from modal logic. Dynamic logic is one example that has already been mentioned. Another example is description logic, which is a family of formalisms used by computer scientists to represent knowledge that is already expressed in a certain regimented form; only after extensive work did those practitioners realize that what they were doing could be seen as a version of multimodal logic, that is, modal logic with several normal operators.

An example closer to ordinary modal logic is hybrid logic, a way of doing modal logic actually anticipated by Prior. Here, the object language of traditional modal logic is augmented by the introduction of concepts belonging to semantics, a device that can greatly increase the expressive strength of the formal language. One such augmentation is to allow a new category of syntactic objects, called nominals, a special set of propositional constants whose semantic interpretation is as singleton sets; in other words, nominals represent propositions that are true at exactly one point in the universe of a model. If i is a nominal and 𝜙 an ordinary formula, then (i ⊃ 𝜙) ∨ (i ⊃ ¬𝜙) and ♢(i ∧ 𝜙) ⊃ □(i ⊃ 𝜙) exemplify formulas valid in every frame. By contrast i ⊃ □¬i is an example of a formula valid in exactly the class of frames (*U, R* ) in which *R* is irreflexive. This is a striking fact, for irreflexivity is notoriously not expressible in ordinary modal logic—the logic determined by the class of all frames with irreflexive accessibility relations is the same as the logic determined by the class of all frames, that is, K.

Like description logic, hybrid logic is actually a family of logics with different object languages. This proliferation of languages bears witness to the many different uses to which modal logic is nowadays being put. In this regard it is interesting to note a certain trade-off between more restrictive and more permissive options: in general, the more expressive a language is, the more endangered are desirable properties like completeness and decidability. Some philosophers may find the multifariousness of present-day computer science–driven modal logic bewildering. At any rate, we have come a long way from the beginning of modal logic when C. I. Lewis sought, and for a while thought he had found, the one and only logic of strict implication.

** See also ** A Priori and A Posteriori; Armstrong, David M.; Carnap, Rudolf; Chisholm, Roderick; Diodorus Cronus; Dummett, Michael Anthony Eardley; Gödel, Kurt; Hintikka, Jaakko; Kripke, Saul; Lewis, Clarence Irving; Lewis, David; Logic, History of; tukasiewicz, Jan; Marcus, Ruth Barcan; Mathematics, Foundations of; Modality, Philosophy and Metaphysics of; Moore, George Edward; Plantinga, Alvin; Prior, Arthur Norman; Provability Logic; Quine, Willard Van Orman; Ross, William David; Tarski, Alfred; Wright, Georg Henrik von.

## Bibliography

### original works

Carnap, Rudolf. *Meaning and Necessity: A Study in Semantics and Modal Logic*. Chicago: University of Chicago Press, 1947.

Hintikka, Jaakko. *Knowledge and Belief: An Introduction to the Logic of the Two Notions*. Ithaca, NY: Cornell University Press, 1962.

Jónsson, Bjarni, and Alfred Tarski. "Boolean Algebras with Operators: Part I." *American Journal of Mathematics* 73 (1951): 891–939.

Kanger, Stig. "Provability in Logic." PhD diss., Stockholm University, 1957.

Kripke, Saul A. "Semantical Analysis of Modal Logic: I. Normal Modal Propositional Calculi." *Zeitschrift für Mathematische Logik und Grundlagen der Mathematik* 9 (1963): 67–96.

Kripke, Saul A. "Semantical Analysis of Modal Logic: II. Non-normal Modal Propositional Calculi." In *The Theory of Models*, edited by J. W. Addison, L. Henkin, and A. Tarski, 206–220. Amsterdam, Netherlands: North-Holland, 1965.

Kripke, Saul A. "Semantical Considerations on Modal Logic." *Acta Philosophica Fennica* 16 (1963): 83–94.

Lewis, Clarence Irving. *A Survey of Symbolic Logic*. Berkeley: University of California Press, 1918.

Lewis, Clarence Irving, and Cooper Harold Langford. *Symbolic Logic*. New York: Century, 1932.

Lewis, David. *Counterfactuals*. Cambridge, MA: Harvard University Press, 1973.

McKinsey, J. C. C. "A Solution of the Decision Problem for the Lewis Systems S2 and S4 with an Application to Topology." *Journal of Symbolic Logic* (1941): 117–134.

Montague, Richard. "Logical Necessity, Physical Necessity, Ethics, and Quantifiers." *Inquiry* 4 (1960): 259–269.

Prior, Arthur. *Past, Present, and Future*. Oxford, U.K.: Clarendon Press, 1967.

Prior, Arthur. *Time and Modality*. Oxford, U.K.: Clarendon Press, 1957.

von Wright, Georg Henrik. "Deontic Logic." *Mind* 60 (1951a): 1–15.

von Wright, Georg Henrik. *An Essay in Modal Logic*. Amsterdam, Netherlands: North-Holland, 1951b.

### survey articles

Blackburn, Patrick, Frank Wolter, and Johan van Benthem. *Handbook of Modal Logic*. Forthcoming.

Gabbay, D. M., and F. Guenthner. *Handbook of Philosophical Logic. Vol. 2, Extensions of Classical Logic*. Dordrecht, Netherlands: D. Reidel, 1984.

Gabbay, D. M., and F. Guenthner. *Handbook of Philosophical Logic*. Vols. 3, 4, and 7. 2nd ed. Dordrecht, Netherlands: Kluwer Academic, 2001–2002.

Gochet, P., and P. Gribomont. "Epistemic Logic." In *Handbook of the History and Philosophy of Logic*, edited by D. M. Gabbay and John Woods. Forthcoming.

Goldblatt, Robert. "Mathematical Modal Logic: A View of Its Evolution." *Journal of Applied Logic* 1 (2003): 309–392.

### textbooks

Chellas, Brian F. *Modal Logic: An Introduction*. New York: Cambridge University Press, 1980.

Fitting, Melvin, and Richard L. Mendelsohn. *First-Order Modal Logic*. Dordrecht, Netherlands: Kluwer Academic, 1998.

Goldblatt, Robert. *Logics of Time and Computation*. CSLI Lecture Notes. Vol. 7. Chicago: Chicago University Press, 1987.

Hughes, G. E., and M. J. Cresswell. *A New Introduction to Modal Logic*. New York: Routledge, 1996.

*Krister Segerberg (2005)*