Here is what I consider one of the biggest mistakes of all in modallogic: concentration on a system with just one modal operator. Theonly way to have any philosophically significant results in deonticlogic or epistemic logic is to combine these operators with: tenseoperators (otherwise how can you formulate principles of change?); thelogical operators (otherwise how can you compare the relative with theabsolute?); the operators likehistorical orphysical necessity (otherwise how can you relate the agent tohis environment?); and so on and so on.—DanaScott (1970: 161)

This entry focusses on a number of important aspects that appear inthe study of modal logics involving different modal operators, as theabove D. Scott’s quotation exemplifies. Indeed, when discussingdifferent modal operators, one requires a system that allows us todeal not only with the different attitudes they represent, but alsowith the complex relationship between them. To make formal sense ofthese situations, one requires amulti-modal system.

In such systems, each one of the relevant modal operators comesequipped with an interpretation that allows us to reason aboutdifferent possible worlds in which we vary the truth values of thepropositions under consideration. Such systems claim an importantplace in current logico-philosophical studies. First, they drive thediscussion on how certain modalities can be derived or defined interms of others, as the discussion in epistemology on the definitionof knowledge in terms of belief and vice versa. Moreover, they alsoillustrate how different basic operators interact with one another, aswell-known puzzles and paradoxes based on the interplay of modaloperators (e.g., Fitch’s paradox of knowability,Voorbraak’s puzzle and the Brandenburger-Kesler Paradox). Thisentry points the reader to both the logic-technical aspects ofcombining logical systems as well as to the philosophical aspects ofmulti-modal logic.

• 1. A Brief Presentation
• 2.Defining Concepts in Terms of Others
  • 2.1 Necessity and Possibility
  • 2.2 Knowledge and Beliefs of Groups
  • 2.3 A Simple Attempt at Relating Knowledge and Belief
  • 2.4 Distributed and Common Knowledge
  • 2.5 Beliefs in Terms of Evidence
  • 2.6 Plausibility Models
  • 2.7 Infinite Modalities via Syntactic Constructors
  • 2.8 TheDynamic Epistemic Logic Approach
• 3. General Strategies forCombining Modal Systems
  • 3.1 Fusion
  • 3.2 Product
  • 3.3 Modal Fibring
  • 3.4 Complexity Issues
• 4. Significant Interactions Between Modalities
  • 4.1 The Relationship Between Knowledge and Beliefs
  • 4.2 Time and Knowledge
  • 4.3 Knowledge and Questions
  • 4.4 Agents
  • 4.5 Modal First-Order Logic
  • 4.6 Dynamics of Intentions
  • 4.7 Obligations and Time
  • 4.8 Knowledge and Obligations
• 5. Multi-Modal Systems in Philosophical Discussions
  • 5.1 Abductive Reasoning
  • 5.2 Fitch Knowability Paradox
  • 5.3 The paradox of the perfect believer
  • 5.4 Truthmakers from a modal perspective
  • 5.5 The Brandenburger-Keisler Paradox
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. A Brief Presentation

Modal logics are particularly well suited to study a wide range ofphilosophical concepts, including different kinds of beliefs,obligations, knowledge, time, space, intentions, desires, obligations,evidence, preferences and diverse types of ontic and epistemicactions, among many others. Indeed, all these concepts have specificcontext-dependent features, which indicates that they can be beststudied using models that can express different modes of truth (e.g.,both global and local truth). Such an analysis provides us with thekey insights of the basic building blocks and principles that regulatethese different notions. This has proven to be useful in a wide rangeof academic disciplines, including Artificial Intelligence,Psychology, Social Sciences, Economics, Physics and so on. But, asScott’s quote above suggests, there is something missing whensuch philosophical concepts are studied in isolation. A big part ofwhat defines a concept lies in the way it interacts with others. Forinstance,rational beliefs are expected to rely on properarguments,justifications orevidence;disjunctions may not behave as they do in natural language inthe context ofobligations;knowledge is betterunderstood when looking for theactions that modify it;intentions may be understood as derived fromdesiresandbeliefs. What is required for this study are logicalsystems with more than one modal operator, commonly known asmulti-modal logics, describing not only the isolated properties of theindividual concepts, but also the way they relate to one another.Indeed, multi-modal logics have been designed for a wide range ofapplications, including reasoning about time, space, knowledge,beliefs, intentions, desires, obligations, actions such as public andprivate communication, observations, measurements, moves in a game andothers.

The present text intends to give a brief (but broad) overview of theinteraction between many different philosophical concepts, and to showhow the use of multi-modal logical systems can shed some light onthese concepts’ interaction. We start insection 2 (Defining concepts in terms of others) by discussing basicscenarios that, starting from existing systems, use a combination of‘syntactic’ and ‘semantic’ strategies fordefining further concepts. These cases are based on the idea that somenotions can be defined in terms of others, with the famousunderstanding of knowledge as justified true belief being one of themost notable examples. An alternative to this idea is to consider thatthe involved concepts emerge independently, but are still somehowrelated, as the case of the relationship between knowledge and time.From a formal perspective, this amounts to looking at the differentmodes in which two (or more) existing systems can be combined.Section 3 on General strategies forcombining modal systems presentsan overview of some of the most relevant strategies. After thisslightly technical excursion, the discussion takes a philosophicalperspective, describing first combinations of multiple modalities (section 4 onSignificant interactions between modalities), andfinishing with examples of cases where the interaction betweenmodalities sheds light on philosophical issues (section 5 on Multi-modal systems in philosophical discussions).

A note on notation and the level oftechnicality To discuss aspects of multi-modal logic, thisentry assumes basic knowledge ofmodal logic, specifically about its language and its relational ‘possible worlds’ semantics (though other semantic models will be mentioned too). Inparticular, a relational model is understood as a tuple containing aset of possible worlds, one or more (typically binary) relationsbetween them, and a valuation indicating what each possible worldactually represents. Such structures can be described by differentmodal languages. We will use \(\cL\) to denote the standardpropositional language, and \(\cL_{\left\{ O_1, \ldots, O_n\right\}}\) to denote its extension with modalities \(O_1\), …,\(O_n\). Given a relational modelM and a formula \(\varphi\),we will use \(\llbracket \varphi \rrbracket^{M}\) to denote the set ofworlds inM where \(\varphi\) holds. Readers can find moredetails about the basics of modal logic not only in the referred SEPentries, but also in the initial chapters of Blackburn, Rijke, &Venema (2001) and van Benthem (2010), and also in Blackburn & vanBenthem (2006).

Still, the goal of this text is not to provide a comprehensive studyof the topic, but rather to highlight the most interesting andintriguing aspects. Thus, although some level of formal discussionwill be used, most technical details will be restricted to theappendix.

2.Defining Concepts in Terms of Others

In order to use systems with multiple modalities, the question is howto build such settings. One of the most important points is to decidewhether one of the concepts to be studied is ‘morefundamental’ than the other, in the sense that the latter can bedefined in terms of the former. As mentioned, the famous understandingof knowledge as justified true belief is one of the most notableexamples. Others are equally relevant, as a definition of beliefs interms of the available arguments/evidence/justifications, or adefinition of epistemic notions for a group in terms of the epistemicnotions of its members. Yet, the basicalethic modal logic ofnecessity and possibility already provides a paradigmatic example ofhow to define the relationship between two concepts.

2.1 Necessity and Possibility

The basic alethic modal logic contains both apossibility\((\Diamond)\) and anecessity \((\Box)\) modality. Mostformal presentations of this system take one of these modalities asthe primitive syntactic operator (say, \(\Diamond)\), and then definethe other as itsmodal dual \((\oBox\varphi := \lnot\oDiamond \lnot \varphi)\). This is a seemingly harmlesssyntactic interdefinability, and comes from the fact that\(\Diamond\) and \(\Box\) are semantically interpreted in terms of theexistential and universalquantifiers, respectively. It is, in some sense, similar to the interdefinabilityof Boolean operators in classic propositional logic. Nevertheless, italready reflects important underlying assumptions. From aclassic point of view, something is necessary if and only ifit is not the case that its negation is possible \((\oBox\varphi\leftrightarrow \lnot \oDiamond \lnot \varphi)\), and something ispossible if and only if it is not the case that its negation isnecessary \((\oDiamond \varphi \leftrightarrow \lnot \oBox\lnot\varphi)\). However, this may not be the case in all settings. Forexample, while \(\oDiamond \varphi \rightarrow \lnot \Box\lnot\varphi\) isintuitionistically acceptable (the existence of a possibility where \(\varphi\) holds implies thatnot every possibility makes \(\varphi\) false), its converse \(\lnot\oBox \lnot\varphi \rightarrow \oDiamond \varphi\)is not (the fact that not every possibility makes \(\varphi\) false is notenough to guarantee theexistence of a possibility where\(\varphi\) is true). Thus, one should always be careful when defininga modality in terms of another.

2.2 Knowledge and Beliefs of Groups

Where the above examples started from a uni-modal logic, we providenow an example in which we start from a homogeneous multi-modal logic.Our setting is a logic consisting of a number of basic modalities ofthe same type, all being semantically interpreted via the same type ofrelation. Our example is the basicmulti-agent epistemiclogic. This setting is already multi-modal, as its language\(\cL_{\left\{ \oK{1}, \ldots,\oK{n} \right\}}\) has a knowledgemodality \(K_i\) for each agent \(i\) in the set \(\ttA = \{1, \ldots,n\}\). (In fact, this basic multi-agent epistemic logic is the fusion (section 3.1) of several single-agent epistemic logic systems, one for each agent\(i \in \ttA\).) Still, taking advantage of the finiteness of\(\ttA\), one can define a brand new modality for the group epistemicnotion ofeverybody knows:

In a similar way we can define a modality foreverybodybelieves in the logic with language \(\cL_{\left\{ \oB{1},\ldots, \oB{n} \right\}}\) as

These definitions assume that the knowledge/beliefs of a group ofagents corresponds to the conjunction of the agent’s individualknowledge/beliefs. However, in the context ofsocial epistemology, the reduction of group attitudes to the mere sum of those of theindividuals is contentious, especially when one focuses on group beliefs.^[1]

2.3 A Simple Attempt at Relating Knowledge and Belief

Another example of the interdefinability of modal concepts deals withthe relationship betweenknowledge andbelief. Inepistemology, researchers are searching for the correct characterization ofknowledge, and a common trend has been to view knowledge as a form ofjustified true belief (an idea that can be traced back to Plato’s dialogueTheaetetus). Gettier’sfamous counterexamples showed that such a simple characterization of knowledge is notsufficient: a further condition is required, such as safety,sensitivity, robustness or stability. In spite of this, acharacterization of knowledge as justified true belief is an importantfirst step. Classicepistemic logic does not explicitly deal with the notion ofjustification,^[2] so a starting point is a simpler understanding of knowledge as truebelief.

Take adoxastic relational model with its relation \(R_B\)being serial, transitive and Euclidean (aKD45 setting), anduse a modalityB semantically interpreted with respect to\(R_B\) in the standard way. In this setting, two options arise. Thefirst one issyntactic, as in the examples that have beendiscussed so far, and consists in defining a modality for knowledge as‘true belief’: \(K'\varphi := B\varphi \land \varphi\).The second issemantic, and consists in defining an epistemicequivalence relation \(R_K\) as the reflexive and symmetric closure ofthe doxastic relation, then using it in the standard way to give thesemantic interpretation of a modalityK.

It should be noted that the two approaches are not equivalent.Consider the following doxastic model (from Halpern, Samet, &Segev 2009a), with the serial, transitive and Euclideandoxastic relation \(R_B\) represented by dashed arrows, andits derived reflexive, transitive and symmetric (i.e., equivalence)epistemic relation \(R_K\) represented by solid ones.

Figure 1 [Anextended description of figure 1 is in the supplement.]

Note how the agent believesp on every world in the model,\(\llbracket \oB{}p \rrbracket^{M} = \left\{ {w_1, w_2, w_3}\right\}\); then, as the syntactic approach states that\({K'\varphi}\) holds in those worlds in which \(B\varphi \land\varphi\) is the case, we have

However, according to the semantic approach, \(K\varphi\) holds inthose worlds from which all epistemically accessible situationssatisfy \(\varphi\), so \(\llbracket K p \rrbracket^{M} = \left\{ w_1\right\}\). Thus, \({K'}\) andK are not equivalent. One of thereasons for this mismatch is that the two options do not enforce thesame properties on the derived notion of knowledge. For example, whilethe semantic approach enforces negative introspection (by making\(R_K\) an equivalence relation), the syntactic one does not. In fact,this property fails at \(w_3\), as \(\lnot {K'p}\) is true \(({B p}\land p\) fails, asp fails) but still \(K'\lnot{K'p}\)(unfolded as \({B (\lnot {B p} \lor \lnot p)} \land (\lnot B p \lor\lnot p))\) is false.^[3]

Section 4.1 comes back on the relationship between these two concepts, recallingalternative multi-modal accounts that relate knowledge and beliefwhile doing justice to the involved epistemological subtleties.

2.4 Distributed and Common Knowledge

The second option in the previous case is, as described, semantic: ittakes the semantic counterpart of an existing modality(ies), and thenextracts from it (them) a further semantic component in terms of whicha new modality can be defined. Here are two further examples of thisstrategy.

Consider again the basic multi-agent epistemic logic with language\(\cL_{\left\{ \oK{1}, \ldots, \oK{n} \right\}}\). As mentioned above,this setting is multi-modal, as its language contains, for each agent\(i \in \{1, \ldots, n\}\), a knowledge modality \(K_i\) that issemantically interpreted in the standard way with respect to amatching epistemic relation \(R_i\). While a modality for the conceptofeverybody knows (E) is syntactically definable(because the set of agents is finite), other group epistemic notions,such as distributed knowledge and common knowledge are not.^[4]

Consider first the notion ofdistributed knowledge,intuitively understood as all the logical consequences of the union ofthe agents’ individual knowledge (alternatively, what themembers of the group can conclude by pooling all their informationtogether). From this intuitive definition, it is clear that thisconcept can be defined semantically in terms of the agent’sindividual epistemic relations. More precisely, a relation describingthe distributed knowledge modality should correspond to theintersection of the individual epistemic relations, \(R_D :=\bigcap_{i \in \ttA} R_i\). Thus, given an evaluation pointw,a worldu will be considered possible after the agents shareall they know if and only if all of them considered it possible beforethe communication (or, in other words,u will be consideredpossible if and only ifno one can discard it). One simplyextends the language with a modalityD, semanticallyinterpreted with respect to this new relation:

Another important notion, crucial in the study of social interaction,iscommon knowledge. This concept can be described as what everybody knows, everybodyknows that everybody knows, everybody knows that everybody knows thateverybody knows, and so on. Just as with distributed knowledge, thisnotion does not require the addition of further semantic components:the individual epistemic indistinguishability relations alreadyprovide everything that is needed to make the definition explicit. Ifone defines an epistemic relation for the“everybodyknows” modality in the natural way \((R_E := \bigcup_{i \in\ttA} R_i)\), and then define \(R_C\) as the transitive closure of\(R_E\),

one can simply extend the language with a modalityC,semantically interpreted in terms of \(R_C\):

At worldw a formula \(\varphi\) is commonly known among theagents if and only if \(\varphi\) is the case inevery world(the“for all” inC’s semanticinterpretation) that can be reached byany finite non-zerosequence of transitions in \(R_E\) (the fact that \(R_C\) is thetransitive closure of \(R_E)\). In other words, \(\varphi\) iscommonly known among the agents if and only if everybody knows\(\varphi\) (any sequence of length 1), everybody knows that everybodyknows \(\varphi\) (any sequence of length 2), and so on.^[5]

2.5 Beliefs in Terms of Evidence

There are more elaborated examples of frameworks extending a givensetting with modalities that ‘extract’ further informationfrom the semantic model. One of them isevidence logic,introduced in van Benthem & Pacuit (2011), and further developedin van Benthem, Fernández-Duque, & Pacuit (2014) andBaltag, Bezhanishvili, et al. (2016). It follows the idea ofrepresenting the evidence the agent has collected, and looks at howthis evidence gives support to further epistemic notions (e.g.,knowledge and beliefs). The semantics is given by a basic neighborhoodmodel (Montague 1970; Scott 1970): a tuple of the form \(M = {\langleW, N, V \rangle}\) whereW andV are a non-empty set ofpossible worlds and an atomic valuation, respectively (as in standardrelational models), and \(N:W \to {\wp(\wp(W))}\) is a neighborhoodfunction assigning, to every possible world, a set ofsets ofpossible worlds (so \(N(w) \subseteq {\wp(W)}\) isw’s neighborhood). In evidence logic, theneighborhood function is assumed to be constant (i.e., \(N(w) = N(u)\)for any \(w,u \in W)\), and thus the model can be simply understood asa tuple \({\langle W, E, V \rangle}\), with \(E \subseteq {\wp(W)}\)the (constant) neighborhood. This neighborhood, intuitively containingthe basic pieces of evidence the agent has collected, (a piece ofevidence \(U \subseteq W\) can be understood as indicating the agentthat the real world is in \(U\)) is required to satisfy two additionalproperties: evidenceper se is never contradictory\((\emptyset \not\in E)\), and the agent knows her ‘space’\((W \in E)\).

Syntactically, a neighborhood model can be described by a modallanguage \(\cL_{\left\{ \oBox \right\}}\), as is typically done instandard neighborhood models. There are at least two possibilities forthe semantic interpretation of the \({\oBox}\) modality (Areces &Figueira 2009), and the one chosen in evidence logic is thefollowing:

Thus, in this setting, \({\oBox\varphi}\) expresses that“the agent has evidence supporting \(\varphi\)”,corresponding semantically to the existence of a piece of evidencecontaining only \(\varphi\)-worlds.

What is the epistemic state of the agent that such a model entails? Inother words, given such a model, how can we define epistemic notionssuch as knowledge and belief?

In the case of knowledge, one can follow the traditional single-agentidea: all worlds in the model play a role in the agent’sepistemic state, and thus one can say that the agentknows agiven formula \(\varphi\) if and only if \(\varphi\) is true in everyworld of the model. For this, evidence logic uses aglobalmodalityA:

In the case of beliefs, there are more alternatives. A straightforwardidea says that the agent believes \(\varphi\) if and only if she hasevidence supporting \(\varphi\) (a syntactic definition of the form\(B\varphi := \oBox\varphi)\). However, this would allow the agent tohave contradictory beliefs, as two pieces of evidence might contradictthemselves (there may be \(X, Y \in E\) such that \(X \cap Y =\emptyset\), and thus \(Bp \land B{\lnot p}\) could be satisfiable).More importantly, this would be a ‘lazy’ approach, as theagent would be able to collect evidence (thus definingE), butnevertheless she would not be doing any ‘reasoning’ withit.

A more interesting idea is to define (semantically) a notion of beliefin terms ofcombinations of pieces of evidence. In vanBenthem and Pacuit (2011), the authors propose (roughly speaking) thatbeliefs should be given by themaximal consistent ways inwhich evidence can be combined, stating that the agent believes\(\varphi\) if and only ifall maximally consistentcombination of pieces of evidence support \(\varphi\). Moreprecisely,

Given these definitions, it is clear that knowledge implies bothbelief and evidence (i.e., both \(A\varphi \rightarrow B\varphi\) and\(A\varphi \rightarrow \oBox\varphi\) are valid). Still, it isinteresting to note not only that the agent might believe a given\(\varphi\) without having a basic piece of evidence supporting it\((B \varphi \rightarrow \oBox\varphi\) isNOT valid, asbeliefs are defined in terms ofcombined pieces of evidence),but also that she might have a basic piece of evidence supporting\(\varphi\) without believing \(\varphi\) \((\oBox\varphi \rightarrowB\varphi\) isNOT valid, as the basic evidence supporting\(\varphi\) might not be part of all maximally consistentcombinations).

In this setting, at least whenE is finite (and in many othercases), beliefs are consistent (i.e., \(\neg B \bot)\); still, thesetting also allows ‘bad’ models in which beliefs can turnout to be inconsistent. In Baltag, Bezhanishvili, et al. (2016), theauthors provide an example of such a model, and then solve the problemby extending the setting to a topological approach. Indeed, theauthors use thetopology generated byE, whichintuitively describes the different ways in which the available piecesof evidence can be combined.^[6] This is reasonable as, whileE can be understood as containingthe pieces of evidence the agent has received from external sources(observations, communication), the topology \(\tau_{E}\) can beunderstood as the different ways in which she can‘extract’ further information from them (i.e., the resultof her own reasoning processes). Given the topology, it is possible todefine (semantically) further epistemic notions, such as arguments,justifications, consistent beliefs, consistent conditional beliefs,and different forms of knowledge. For more on this, we refer toBaltag, Bezhanishvili, et al. (2016: Section 2).

2.6 Plausibility Models

As we have seen, a new modality can be introduced in syntactic terms(using the language to provide a formula defining the new concept),but also in a semantic way (using the semantic counterparts of theexisting modalities to define a further semantic notion, which in turnis used to interpret the new modality). Our examples so far have beenrestricted to the use of one of these two strategies, but theirinterplay is also possible. The case to be discussed here concerns theplausibility models of Board (2004); Baltag and Smets (2006, 2008); van Benthem (2007);here, the presentation of Baltag and Smets (2008) is used.

A plausibility model is a relational model \(M = {\langle W, \leq, V\rangle}\) in which the binary relation \(\leq\) is interpreted asdescribing theplausibility ordering the agent assigns to herepistemic possibilities \((w \leq u\) indicates that, for the agent,worldw is at least as plausible as worldu). In thesingle-agent case, the plausibility relation \(\leq\) isrequired to be awell-preorder: a total relation which isboth reflexive and transitive, and such that every non-empty subset ofthe domain has \(\leq\)-minimal elements. These minimal elements inW are then understood as the agent’smostplausible worlds. We see below that what is true in all the mostplausible worlds characterizes what an agent believes.

To start, take a modality \([\leq]\) semantically interpreted via theplausibility relation \(\leq\),

This modality has the properties of an S4 modal operator; hence, it isfactive, positively introspective but not negatively introspective. InBaltag and Smets (2008), it is argued that this modality is wellsuited to express a version of Lehrer’sindefeasible(“weak”, non-negatively-introspective) type ofknowledge (Lehrer 1990; Lehrer & Paxson 1969), and theauthors explain how it can be understood as belief that is persistentunder revision with anytrue piece of information. Using thismodality (also read assafe belief in Baltag & Smets2008), it is possible to define syntactically a notion of simplebelief astruth in the most plausible worlds:

As simple as a plausibility model is, it is powerful enough to encodea wide range of different epistemic concepts, all of which can bebrought to light by the proper semantic definitions. First, we definea relation ofepistemic possibility (or indistinguishability)\(\sim\) by taking it to be the universal relation,

thus understanding that two worlds are epistemically indistinguishableif and only if they can be compared via \(\leq\).^[7] Then, a notion of S5-knowledge can be expressed by introducing amodalityK semantically interpreted via \(\sim\):

With this new modalityK it is possible to define,syntactically, the finer notion ofconditional belief\(B^{\psi}\), intuitively describing what the agent would havebelieved was true had she learnt that a certain condition \(\psi\) isthe case. Indeed,

for \({\hK}\) the modal dual ofK (i.e., \(\hK\psi := \lnotK\lnot \psi)\). This extended language \(\cL_{\left\{ [{\leq}], K\right\}}\) can also express a notion ofstrong belief, \(Sb\varphi\), semantically understood as true whenever all\(\varphi\)-worlds are strictly more plausible than all\(\lnot\varphi\)-worlds, and syntactically defined as

Finally, note how a plausibility relation defines, semantically,layers of equally-plausible worlds, with the layers themselves orderedaccording to their plausibility. Then, one can see this orderedsequence of layers as a collection of concentric spheres. Indeed, thecontents of the innermost sphere, sphere \#0, are exactly those worldsin the most plausible layer, layer \#0; then, the contents of sphere\#1 are those worlds in layer \#0 together with those in the layerimmediately below it, layer \#1. The process continues, thus definingthe worlds in sphere \#\(i\) as the union of the contents of layers\#0 to layer \#\(i\). In this way, every plausibility corresponds to asphere model (Spohn 1988; Grove 1988), making it perfectly fit tomodelbelief revision. Still, even though in \(\cL_{\left\{ [{\leq}], K \right\}}\) thereare formulas expressing that \(\varphi\) holds inthe mostplausible sphere (the mentioned \(B\varphi\), given by\(\langle{\leq}\rangle[{\leq}]\varphi)\), no formula can express,e.g., that \(\varphi\) holds in thenext to most plausibleworlds. One way to fix this ‘problem’ is to define (nowsemantically) thestrict plausibility relation \({<} :={\leq} \cup {\not\geq}\) (with \(\geq\) theconverse of\(\leq\), defined in the standard way, \({\geq} := \left\{ (u,w) \in W\times W \mid w \leq u \right\})\), and then introduce a standardmodality for it:

With this new modality, one can provide syntactic definitions for theconcepts described above. Indeed, while the formula \(\lambda_0 :=[<]\bot\) characterizes the most plausible worlds (so \(K(\lambda_0 \rightarrow\varphi)\) expresses that the most plausibleworlds satisfy \(\varphi\), just as \(B\varphi\) does), the formula\(\lambda_1 := \lnot \lambda_0 \land [<]\lambda_0\) characterizesthe next to most plausible worlds (so \(K(\lambda_1\rightarrow\varphi)\) expresses that the next to most plausible worldssatisfy \(\varphi)\). This procedure can be repeated, producingformulas \(\lambda_i\) characterizing each layer, and thus it ispossible to deal syntactically with a qualitativedegree ofbeliefs (Grove 1988; Spohn 1988), looking for what holds‘from some level up’. (See Velázquez-Quesada 2017for more on the use of this modality within plausibility models, andAndersen, Bolander, van Ditmarsch, & Jensen 2017 for a comparisonbetween different languages describing these structures.)

This new modality \([<]\) allows us to define even more epistemicnotions. For example, a formula \(\varphi\) isweakly safelybelieved (a belief which might be lost but is never reversed whenrevising with true information) if and only if \(\varphi \land[{<}] \varphi\) holds. More details can be found in Baltag andSmets (2008: Subsection 2.4).

2.7 Infinite Modalities via Syntactic Constructors

Just as some multi-modal systems are created by extending existingones, some others are born with multiple modalities in mind. Amongthem,propositional dynamic logic (Harel, Kozen, & Tiuryn 2000) andBoolean modal logic(Gargov & Passy 1990; Gargov, Passy, & Tinchev 1987) deserve aspecial mention. The reason is that they both define, within thelanguage, operators for building new modalities from a collection ofbasic ones. As a consequence, both systems contain aninfinite number of modalities.

Following earlier approaches to reason about programs in Engeler(1967) and Hoare (1969), Propositional dynamic logic (PDL),the logic of programs (Harel, Kozen, & Tiuryn 2000), intends todescribe what programs can achieve. Semantically, programs areinterpreted in standard relational models, with one binary relation\(R_a\) for everybasic programa; syntactically, thelanguage contains a modality \([a]\) for each sucha.

So far,PDL is technically similar to a multi-agent epistemiclogic (the difference being, besides the symbols used for themodalities, the fact that there are no restrictions on the relationsfor the basic programs).^[8] The crucial insight is, however, that basic programs can becomposed in order to create more complex ones: one can thinkof executing one program after another, or repeating some of them anumber of times. Thus, these basic modalities are not enough. Forthis, a new syntactic entity is created: besides formulas, thelanguage ofPDL contains a set ofbasic programstogether withprogram constructors representing those forregular expressions (Kleene 1956). Formally,formulas\(\varphi\) andprograms \(\alpha\) of thePDL-language \(\cL_{\textit{PDL}}\) are definedsimultaneously via mutual recursion as

withp an atomic proposition coming from a given set, anda a basic program coming from a given set. For formulas, theintended reading of the Boolean operators is standard, and formulas ofthe form \([\alpha]\varphi\) express that“every executionof program \(\alpha\) from the current state leads to a statesatisfying \(\varphi\)”. For programs, while the basicprograms simply represent themselves, “\(\varphi \qbin\)”is a program that ‘does nothing’ when \(\varphi\) is thecase but ‘fails’ otherwise (essentially, atestfor \(\varphi)\), “\(\alpha \scbin \beta\)” represents theprogram that results from executing \(\alpha\) and then executing\(\beta\) (theirsequential composition), “\(\alpha\bcup \beta\)” represents the program that results fromexecuting either \(\alpha\) or else \(\beta\) (theirnon-deterministic choice), and“\({\alpha^{\ast}}\)” represents the program that resultsfrom repeating \(\alpha\) a finite number of times \((\alpha\)’siteration).

With these program constructors it is possible to build more complexprograms. Famous examples are

Then, it is possible to build formulas as \(p \rightarrow [(q \qbin\scbin a) \bcup (\lnot q \qbin \scbin b)]r\) (“ifpholds, thenr will be achieved by choosing between actionsa andb according to whetherq holds”)and \(\lnot p \rightarrow \langle a \scbin (\lnot q \qbin \scbina)^{\ast} \scbin q \qbin \rangle p\) (“if the desiredrequirementp is not true yet, it is possible to achieve it bya repeated execution ofa”).

For the semantic interpretation, a relation \(R_\alpha\) is requiredfor each program \(\alpha\). However, while the relations \(R_a\) forbasic programs are arbitrary, those for complex programs should behaveaccording to their intended meaning. The simplest way to obtain thisis to take the relations for the basic programs, and thendefine those for complex programs in an inductive way. Thisand further details aboutPDL can be found inSection 2 of the SEP entry ondynamic logic.

TheBoolean modal logic of Gargov and Passy 1990 and Gargov,Passy, and Tinchev 1987) follows a similar strategy. The difference isthat, whilePDL focuses on constructors for regularexpressions (sequential composition, non-deterministic choice, finiteiteration), Boolean modal logic focuses on constructors for theBoolean algebra over relations: complement \((\bdash)\), union\((\bcup)\) and intersection \((\bcap)\), together with a‘global’ constant \((\boldsymbol{1})\). Moreprecisely,

The semantic interpretation follows the same steps as inPDL:relations \(R_a\) for the basic modalitiesa are assumed, andrelations for complex ones are defined in the expected way (with\({\boldsymbol{1}}\) being interpreted with respect to the globalrelation \(W \times W)\).

Interestingly, by combining the negation over formulas and the Booleancomplement over relations, it is possible to define the followingoperator (often calledwindow; see Goldblatt 1974; vanBenthem 1979; Gargov, Passy, & Tinchev 1987):

Window is an extremely natural operator that complements the standarduniversal modality. Indeed, while formulas of the form\([\alpha]\varphi\) express thatall executions of \(\alpha\)reach a \(\varphi\)-state,

formulas of the form \(\oubracket{.7em}{\alpha}\varphi\) express thatall \(\varphi\)-states are reachable by an execution of\(\alpha\):

Not only that: window allows a smooth interaction between theconstructors \(\bcup\) and \(\bcap\). As discussed in Blackburn,Rijke, and Venema (2001: 427),

[i]n a sense, the relations are divided into two kingdoms: theordinary \([\alpha]\) modalities govern relations built with\(\bcup\), the window modalities \(\oubracket{.7em}{\alpha}\) governthe relations built with \(\bcap\), and the \(\bdash\) constructoracts as a bridge between the two realms:

\[ \begin{align} \Vdash {[\alpha \bcup\beta]\varphi}&\leftrightarrow ([\alpha] \varphi \land [\beta]\varphi), &\Vdash {[\bdash\alpha]\varphi} &\leftrightarrow\oubracket{.7em}{\alpha} \lnot\varphi \\ \Vdash\oubracket{3.1em}{\alpha \bcap\beta} \varphi &\leftrightarrow\left(\oubracket{1em}{\alpha} \varphi \land \oubracket{.8em}{\beta}\varphi\right), & \Vdash [\alpha]\lnot\varphi &\leftrightarrow\oubracket{1.8em}{\bdash\alpha}\varphi.\\ \end{align} \]

Of course, many other program constructors can be used. Among them,one worthy of mention is that for theconverse of a given relation. Modalities for the converse of a relation havebeen used in, e.g.,tense logic, with the ‘past’ modalities (H andP, theuniversal and existential versions, respectively) interpretedsemantically in terms of the converse of the relation used forinterpreting the ‘future’ modalities (G andF, respectively).

2.8 TheDynamic Epistemic Logic Approach

The case ofdynamic epistemic logic, the study of modallogics of model change, is of particular interest. In these systems,the relationship between their modalities is special. Here we willonly recall the basic notions, referring the reader to theSEP entry by Baltag and Renne (2016) for an in-depth discussion

In a nutshell, adynamic epistemic logic (DEL)framework has two components. The ‘static’ part consistsof a ‘standard’ modal system: a language including one ormore modalities for the one or more concepts under study, togetherwith the semantic model on which the formulas are interpreted. The‘dynamic’ part consists of modalities expressing differentways in which the studied concept(s) might change, with the crucialinsight being that these modalities are semantically interpreted noton the given model, but rather on one that results fromtransforming the given one in an appropriate way.

The discussion here will focus on the paradigmaticDEL case,public announcement logic (PAL), which studies the interaction of knowledge and public communication.^[9] Syntactically, its language extends the basic epistemic language\(\cL_{\left\{ K \right\}}\) with a modality \([\chi{!}]\) (for\(\chi\) a formula of the language), thanks to which it is possible tobuild formulas of the form \([\chi{!}]\varphi\):“after\(\chi\) is publicly announced, \(\varphi\) will be thecase”. Within this new language \(\cL_{\left\{ K, {!}\right\}}\) it is possible to build formulas describing theknowledge the agent will have after a public communicationaction; one example is \([(p \land q){!}] Kq\), expressing that“after \(p \land q\) is publicly announced, the agent willknowq”. For the semantic interpretation, the publicannouncement of any given \(\chi\) is taken to be completelytrustworthy; thus, the agent reacts to it by eliminating all\(\lnot\chi\) possibilities from consideration. More precisely, givena model \(M = {\langle W, R, V \rangle}\) and a formula \(\chi \in\cL_{\left\{ K, {!} \right\}}\), the model \(M_{\chi{!}} = \langleW_{\chi{!}}, R_{\chi{!}}, V_{\chi{!}} \rangle\) is defined as

Note how, while \(W_{\chi{!}}\) is the set of worlds of the originalmodel where \(\chi\) holds, \(R_{\chi{!}}\) is the restriction of theoriginal epistemic relation to the new domain, and so is the newvaluation function \(V_{\chi{!}}\). Then,

Thus, \(\varphi\) is the case after \(\chi\) is publicly announced atw inM (in symbols, \((M, w) \Vdash [\chi{!}]\varphi)\)if and only if \(\varphi\) is true atw in the situation thatresults from \(\chi\)’s announcement (in symbols,\((M_{\chi{!}}, w) \Vdash \varphi)\) whenever \(\chi\) can actually beannounced (in symbols, \((M, w) \Vdash \chi)\).^[10] Note that the public announcement modality \([\chi{!}]\) isintroducedsemantically, as its semantic interpretationrequires ‘extracting’ further information from the initialmodel, just as the intersection of individual epistemic relations isused to create the relation for distributed knowledge. Still, it usesa ‘more advanced’ version of such a strategy: it performsan operationover the full model, thus creating a new one inorder to evaluate formulas that fall inside the scope of the newmodality.

With the semantic interpretation of \([\chi{!}]\) given, it is nowpossible to answer the crucial question in this setting: what is theeffect of a public announcement on an agent’s knowledge? Or,more precisely, how is the agent’s knowledgeafter anannouncement related to her knowledgebefore it? Here is theanswer:

This validity characterizes the agent’s knowledge after theaction in terms of the knowledge she had, before the action,aboutthe effects of the action. It tells us that after the publicannouncement of \(\chi\) the agent will know \(\varphi\),\([\chi{!}]K\varphi\), if and only if, provided \(\chi\) could beannounced, ‘\(\chi \rightarrow \)’, she knew that itstruthful public announcement would make \(\varphi\) true, \(K(\chi\rightarrow [\chi{!}]\varphi)\). Note how thisbridgeprinciple, relating the two involved modalities, is not‘chosen’: it arises as a consequence of the givendefinition of what knowledge is (truth in all epistemic possibilities)and the given understanding of what a public announcement does(discard all possibilities where the announcement fails).

The given semantic interpretation of \([\chi{!}]\) also gives rise toother validities. Among them, consider the following:

These validities, together with the previous one characterizing\([\chi{!}]K\varphi\), are known as thereduction axioms.Here is our first twist: a careful look at these formulas reveals thateach one of them characterizes the truth of an announcement formula\([\chi{!}]\varphi\) (the left-hand side of \(\leftrightarrow\)) interms of formulas (the right-hand side of \(\leftrightarrow\)) whosesub-formulas appearing under the scope of \([\chi{!}]\) arelesscomplex. Moreover: the formula dealing with atoms eliminates\([\chi{!}]\). Thus, given any concrete formula in \(\cL_{\left\{ K,{!} \right\}}\), successive applications of these axioms willeventually produce asemantically equivalent formula where no\([\chi{!}]\) modality appears. This indicates that,expressivity-wise, the public announcement modalities\([\chi{!}]\) are not really needed: anything that can be expressedwith them can be also expressed by a formula without them. Moreprecisely, for any formula \(\varphi\) in \(\cL_{\left\{ K, {!}\right\}}\), there is a formula \({\operatorname{tr}(\varphi)}\) in\(\cL_{\left\{ K \right\}}\) such that, for any \((M, w)\),

Thistruth-preserving translation, whose precise definitioncan be found in van Ditmarsch, van der Hoek, and Kooi (2008: Section7.4), shows that the public announcement modality can also be seen ashaving asyntactic definition: any formula involving\([\chi{!}]\) can be rewritten within \(\cL_{\left\{ K \right\}}\).^[11] Nevertheless, this is not a ‘one line’ definition, as itis the case, e.g., for the ‘everybody knows’ modalityE. The translation is given by arecursive approach,with the modality defined in a different way depending on the formulaone needs to place under its scope. This leads us to the second twist:because of this recursive definition, even though adding \([\chi{!}]\)does not increases the language’s expressivity, its additiondoes change the properties of the logical system. Indeed, in\(\cL_{\left\{ K, {!} \right\}}\), the rule ofuniformsubstitution of atomic propositions by arbitrary formulas is notvalidity-preserving anymore. Consider the following formula, statingthat“after the public announcement ofp, the agentwill know thatp is the case”: \[\Vdash [p{!}]Kp.\] Theformula is valid: a truthful public announcement ofp discardsworlds from the original modelM wherep was not thecase. Hence, the resulting \(M_{p{!}}\) will have only worldssatisfyingp, thus making \(Kp\) true Now consider the formulabelow, which results from substitutingp by \(p \land \lnotKp\) in the previous validity: \[[(p \land \lnot Kp){!}] K(p \land \lnot Kp).\] The above formula statesnow that

after the public announcement of “p is true and theagent does not know it”, she will know that “p istrue and she does not know it”.

This formula can be equivalently stated (by distributingK over\(\land\) in the sub-formula under the scope of \([(p \land \lnotKp){!}]\)) as

after the public announcement of “p is true and youdo not know it”, the agent will know both thatp is trueand that she does not know it.

But now something is odd: after hearing \(p \land \lnot Kp\), theagent surely should know thatp is the case \((Kp)\). But then,how is it possible that, at the same time, she knows that she does notknow it \((K\lnot Kp)\)?

The suspicions are correct: the formula is not valid, and the modelbelow on the left provides a counter-example.

Figure 2 [Anextended description of figure 2 is in the supplement.]

In \((M, w)\), the atomic propositionp is the case, but theagent does not know it: \((M, w) \Vdash p \land \lnot Kp\). Thus, \(p\land \lnot Kp\) can be truthfully announced, which produces thepointed model \((M_{(p \land \lnot Kp){!}}, w)\) on the right. Notehoww has survived the operation (it satisfies \(p \land \lnotKp)\), butu has not (it does not satisfy \(p \land \lnot Kp\),as it makesp false). In the resulting pointed model, the agentindeed knows thatp is the case: \((M_{(p \land \lnot Kp){!}},w) \Vdash Kp\). Nevertheless, she does not know that she does not knowp: \((M_{(p \land \lnot Kp){!}}, w) \not\Vdash K\lnot Kp\); infact, she knows that she knowsp: \((M_{(p \land \lnot Kp){!}},w) \Vdash KKp\).^[12]

Recapitulating, dynamic epistemic logics deal with modal operators formodel operations, thus allowing the explicit representation of actionsand the way they affect the concept under study. The particularrelationship between the ‘static’ concept and the‘dynamic’ action can be described by bridge principlesthat arise naturally, and yet this does not come with an additionalcost, as the model-operation and dynamic-modality machinery can beembedded into the static base logic. This has important repercussions,particularlycomplexity-wise, as will be discussed insection 3.4.

3. General Strategies forCombining Modal Systems

The previous section focused on some of the ways one can take a systemwith a single modality and create a system with multiple modalitiesfrom it. Another alternative to build a multi-modal system is to takeexisting uni-modal systems, and then put them together by using aparticular strategy. This section contains a brief description of someof the possible techniques; for a deeper discussion, the reader isreferred to the SEP entry oncombining logics by Carnielli and Coniglio (2016).

3.1 Fusion

The method offusion ofmodal logics (introduced in Thomason 1984) was developedwith the idea ofcombining relation-based (hencenormal) modal logics in both a syntactic way (by puttingtogether their respective Hilbert-style axiom systems) and a semanticway (by taking the relations corresponding to the modality of eachsystem, and putting them together in a single model).^[13] Although the fusion of modal systems is fairly simple, thetransference results that guarantee that properties arepreserved (e.g., whether the combination of the sound and completeaxiomatization of the existing systems is indeed sound and completefor the resulting one) are not straightforward (see, e.g., Kracht& Wolter 1991, 1997; Fine & Schurz 1996; Schurz 2011).

When this strategy is followed, and leaving technical details aside,the most important decision is the possible introduction ofbridge principles that link the main modalities of thesystems to be combined. Paraphrasing Schurz (1991), a schema\(\varphi\) is abridge principle if and only if it containsat least one schematic letter which has at least one occurrence withinthe scope of the modality of one system and at least one occurrencewithin the scope of the modality of the other. (This definition wasgiven in the context of the David Hume’s discussion on whetherought can be derived fromis; seeSection 1 ofcombining logics.)

In order to provide a better explanation of this technique, here wewill discuss the construction of a simple temporal epistemic logic. Onthe epistemic side, recall that the basicepistemic logic system is given, syntactically, by the language \(\cL_{\left\{ K\right\}}\), and semantically, by a relational model. In it, themodalityK is semantically interpreted in terms of a binaryrelation \(R_K\). On the temporal side, define the‘future’ fragment of the basictemporal (tense) logic as a system which is syntactically specified by the language\(\cL_{\left\{ G \right\}}\) (withG auniversalquantification on the future, andF itsexistentialcounterpart given by \(F\varphi := \lnot G\lnot \varphi)\), andsemantically, by a relational model with \(R_G\) as the crucialrelation.

Thefusion of these systems is syntactically specified by thelanguage \(\cL_{\left\{ K, G \right\}}\) (i.e., a language freelygenerated by the union of the modalities of \(\cL_{\left\{ K\right\}}\) and \(\cL_{\left\{ G \right\}})\). Formulas of thislanguage are semantically interpreted inrelational models ofthe form \({\langle W, R_K, R_G, V \rangle}\) such that \({\langle W,R_K, V \rangle}\) is a model for \(\cL_{\left\{ K \right\}}\) and\({\langle W, R_G, V \rangle}\) is a model for \(\cL_{\left\{ G\right\}}\). For the semantic interpretation, formulas of the newlanguage are interpreted in the standard way, with each modality usingits correspondent relation. With respect to axiom systems, it isenough to put together those of the individual logics.

But we are not done yet. As mentioned before, the most interestingpart is the possible inclusion of bridge principles. So, which is theproper interaction betweentime and knowledge? One might requireperfect recall: the agent’sknowledge is not decreased over time or, in other words, uncertaintyat any moment should have been ‘inherited’ fromuncertainty from the past. This corresponds to the following bridgeprinciple:

This is clearly an idealization, and as such it makes sense only undercertain interpretations; still, it might imply more than meets theeye. Assuming that the agent never forgets the truth-value of anatomic propositionp might be reasonable; but, what if\(\varphi\) is a more complex formula, in particular one involving theepistemic modality? For example, take the formula \(\lnot Kp \land\lnot K\lnot p\) (“the agent does not know whetherp”), yielding the instance \(K(\lnot Kp \land \lnotK\lnot p) \rightarrow GK(\lnot Kp \land \lnot K\lnot p)\)(“if the agent knows of her ignorance about whetherp, then she will always know about such ignorance”).Is this within the expected consequences?

A related property is that ofno learning (the agent’sknowledge is not increased over time; in other words, any currentuncertainty will be preserved). This property corresponds to

A slight elaboration on these and related properties can be found onthe discussion on time and knowledge (section 4.2; for a deeper study, see Halpern & Vardi 1989; van Benthem &Pacuit 2006, albeit in a different semantic setting).

3.1.1 ‘Temporalizing’ a System

The fusion method provides a simple yet powerful strategy for adding afurther aspect to an existing modal system by using another alreadyexisting system dealing with this further aspect independently.Indeed, the discussed example can be seen as adding atemporal aspect to the standard study of the properties ofknowledge. Besides the traditional epistemic questions (e.g., whetherknowledge should be positively/negatively introspective), one can alsodiscuss not only whether knowledge should change, but also thedifferent ways in which it might do so.

This idea of adding a temporal aspect makes sense not only forknowledge, but also for other modal concepts. For example, one canthink of adding a temporal feature to a modal system for preferences,thus discussing different ways in which the preferences of an agentmight change over time (and alsowhy they do so, if furtherdynamic machinery is added to talk aboutactions andtheir consequences; Grüne-Yanoff & Hansson 2009; Liu 2011).Similarly, adding a temporal aspect to modal systems ofdeontic logic raises interesting concepts and questions, an example being thenotion ofdeontic deadlines, discussed insection 4.7 on obligations and time.

3.1.2 ‘Epistemizing’ a System

The study of some concepts such as preferences or obligations givesraise to an epistemic concern: how much do the involved agentsknow about such preferences and obligations? In most exampleswhere such concepts play a role, whether or not the agent knows aboutthe involved preferences or obligations makes an important difference.In case of the first, should an agent act according to her and otheragents’ preferences, even when she does not know what thesepreferences are? In case of the second, is an agent compelled to obeya duty even when she does not know what the duty is?

The fusion of a basic preference/deontic setting and epistemic logicprovides basic formal tools to discuss the epistemic aspects ofpreferences and obligations. For example, consider theparadox of epistemic obligation: a bank is being robbed (r), and the guards ought to know aboutthe robbery \((OKr)\). But knowledge is factive \((Kr \rightarrowr)\), so then the bank ought to be robbed \((Or)\)! More on theseconcerns (under different formal systems) can be found within thediscussion on knowledge and obligations (section 4.8).

For a deeper study on the fusion of modal logics, the reader isreferred to Wolter (1998), Gabbay, Kurucz, Wolter, and Zakharyaschev(2003: Chapter 4), and Kurucz (2006: Section 2). Further examples offusion can be foundin the SEP entry discussing methods forcombining logics. Still, before closing this subsection, we add a word of caution. Oneneeds to be careful when building the fusion of modal systems. This isbecause, in the system that results from the fusion, there are alreadyformulas combining the modalities of its different fragments. Then,even if no particular bridge (valid) principles are enforced, thelanguage might gain inexpressive power, which might increaseitscomplexity profile.Section 3.4 on complexity issues elaborates on this important but often forgottenaspect.

3.2 Product

The fusion of modal systems produces a rich language that allows us toexpress the different ways in which the involved modalities interact(the bridge principles). Still, from a semantic perspective, there isstill justone point of reference, as all formulas of thisricher language are still evaluated on a single possible world.

The strategy of defining aproduct of modal logics (introduced in Segerberg 1973 and Šehtman1978) shares the idea of using a language that is freely generated bythe union of the modalities of the original languages. But, on thesemantic side, the approach is quite different: instead of working ona one-dimension domain, it works on amulti-dimensionaldomain that has one dimension for each one of the involved aspects(i.e., modalities). More precisely, if the semantic models of theto-be-combined systems are \(M_1 = {\langle W_1, R_1, V_1 \rangle}\)and \(M_2 = {\langle W_2, R_2, V_2 \rangle}\), the models whereformulas of the resulting language are evaluated are now of the form\(M' = {\langle W_1 \times W_2, R'_1, R'_2, V_1 \times V_2 \rangle}\).The domain \(W_1 \times W_2\) is, then, the standard Cartesian productof the original domains, and the valuation \(V_1 \times V_2\) is suchthan an atomp is true in a world \((w_1, w_2)\) if and only ifp was true at \(w_1\) in \(M_1\) and also true at \(w_2\) in\(M_2\) (i.e., \((V_1 \times V_2)(p) := V_1(p) \times V_2(p))\). Forthe relations, each one of them is given as in their original models,restricted now to their respective dimensions:

The product of modal logics is amany-dimensional modal logic(Gabbay et al. 2003; Marx & Venema 1997; Venema 1992). Withinthese models, formulas are now evaluated in pairs \((w_1, w_2)\), witheach modality semantically interpreted in the standard way (but nowwith respect the new version of its matching relation):

This specific way of interpreting each one of the original modalitiesyields another crucial difference between the fusion and the productof modal systems: the latter enforces, by its own nature, certainbridge principles (on top of those that might be added). Indeed,because of the definition of the relations and the modalities’semantic interpretation, the following schemas are valid:

The product of modal systems, with itsn-dimensional nature, isa very useful tool and, in particular, it has been of help for dealingwith the philosophical semantics technique oftwo-dimensionalism (see also Chalmers 2006; Stalnaker 1978). This technique has beenapplied in different fields. In linguistics, it is the basis of DavidKaplan’s semantic framework forindexicals (Kaplan 1989), which in turn has been used to explain conventionalsemantic rules governing context-dependent expressions as‘I’ and‘now’. Consider, forexample, a setting built to talk about the features of a group offriends at different times; the context in which formulas will beevaluated can be defined as a tuple \((w, a, m)\), with \(w \in W\) apossible world, \(a \in \ttA\) an agent within that world and \(m \inT\) a moment in time when the agent exists in that world. Then, asentence of the form“I am tired now” correspondssimply to an atom “tired”, with its truth-valuebeing potentially different in different contexts, depending onwhether, in the given worldw, the given agenta istired at the given momentm. Moreover: suppose the settingcontains an alethic possibility relation between worlds \((R \subseteqW \times W)\), afriendship relation between agents\(({\asymp} \subseteq \ttA\times \ttA)\) and a temporalfuture relation between moments \((R_G \subseteq T \timesT)\). Then, one can use matching modalities (the universal ones,\(\oBox\) and \([\asymp]\), for the first two; the existential one\({F}\) for the third) to express sentences as“I have afriend who is playing right now and necessarily will be tired at somemoment later” \((\langle \asymp\rangle(\textit{playing}\land \oBox F\textit{ tired}))\) and“if one of my friendsis playing now, all of them might be playing later”\((\langle\asymp\rangle \textit{playing} \rightarrow \DiamondF[\asymp]\textit{playing})\).

In philosophy of mind, two-dimensional semantics has been used byDavid Chalmers (combining both epistemic and modal domains) to providearguments againstmaterialism in philosophy of mind (detailscan be found in Chalmers 2009).

A final example of the product of two modal logics (though notoriginally conceived as such, and presented in a slightly differentway), is theFacebook Logic of Seligman, Liu, & Girard(2011, 2013), useful for talking about friends and social informationflow. The setting can be seen as the combination of a standardsingle-agent epistemic logic and a modal logic for social networks(cf. Baltag, Christoff, Rendsvig, & Smets 2019; Smets &Velázquez-Quesada 2017). Its semantic model consists on twodomains (possible worldsW, agents \(\ttA)\) and two relations:a binary epistemic relation \({\sim_a} \subseteq (W \times W)\) foreach agent \(a \in \ttA\), and a binary friendship relation\({\asymp_w} \subseteq (\ttA\times \ttA)\) for each world \(w \in W\).On the syntactic side, the language is freely generated by thestandard Boolean operators and two universal modalities,K(knowledge) and \([\asymp]\) (friendship). These formulas areevaluated in pairs (world,agent), with the keyclauses being the following:

Note that the modalities areindexical in both the worldand the agent. Thus, while formulas of the form \(K\varphi\)are read as“I know \(\varphi\)”, formulas of theform \([\asymp]\varphi\) are read as“all my friends satisfy\(\varphi\)”.

The given examples show how the product strategy can be used to‘temporalize’ and/or ‘epistemize’ a givenmodal system (Kaplan’s semantic framework can be understood asthe temporalization of an alethic system, and the describedFacebook Logic can be understood as the‘epistemization’ of a social network setting). For more onthe products of modal logics, the reader is referred to thealready-mentionedcombining logics, and also to Gabbay et al. (2003: Chapter 5), Kurucz (2006: Section 3)and van Benthem, Bezhanishvili, et al. (2006).

3.3 Modal Fibring

The strategies of fusion and product for combining modal logics relyin merging both the languages and the semantic models of the modallogics to be combined. In thefibring strategy (calledfibring by functions in Carnielli,Coniglio, et al. 2008), the languages are also merged, but thesemantic models remain separated. Formulas can be evaluated in pointedmodels of any of the original systems, in the following way. When themodality to be semantically evaluated ‘matches’ the chosensemantic model, the evaluation is done as in the original system; whenthe modality comes from the other system, the fibring strategy uses atransfer mapping to obtain a model and an evaluation point inthe class of models for the modality under evaluation, and then theevaluation proceeds as in the original system. Thus, modal fibringrequires a correspondence between the class of models of each one ofthe systems, and uses it to move between them when the modality underevaluation requires it.

More precisely, let \(\cL_{\left\{ \Box_{1} \right\}}\) and\(\cL_{\left\{ \Box_{2} \right\}}\) be the languages of the system tobe combined, and let \(\cM_1\) and \(\cM_2\) be their correspondentclasses of models. Let \(h_1\) be atransfermapping, taking a world of any model in \(\cM_1\), and returninga pair consisting of a model \(M_2\) in \(\cM_2\) and a world \(w_2\)in \(M_2\); let \(h_2\) be atransfer mapping in the otherdirection, taking a world of any model in \(\cM_2\), and returning apair consisting of a model \(M_1\) in \(\cM_1\) and a world \(w_1\) in\(M_1\). Formulas of the language \(\cL_{\left\{ \Box_{1}, \Box_{2}\right\}}\) can be evaluated in either tuples of the form \(\langleh_1, h_2, M_1, w_1 \rangle\) (with \(w_1\) in \(M_1 = \langle W_1,R_1, V_1 \rangle\) and \(M_1\) in \(\cM_1)\) or else tuples of theform \(\langle h_1, h_2, M_2, w_2 \rangle\) (with \(w_2\) in \(M_2 =\langle W_2, R_2, V_2 \rangle\) and \(M_2\) in \(\cM_2)\). In thefirst case, the semantic interpretation of Boolean operators is asusual; for the modality \(\Box_{1}\),

Thus, modalities ‘matching’ the model are evaluated as intheir original systems. Then, for the modality \(\Box_{2}\) of theother system, the transfer mapping \(h_1\) is used:

In other words, when \(\Box_{2}\) needs to be evaluated, the transfermapping uses current evaluation point \(w_1\) to obtain a pointedsemantic model \(h_1(w_1) = (M_2, w_2)\) where the modality will beevaluated. When the analogous situation arises, facing the evaluationof \(\Box_{1}\) on a tuple \(\langle h_1, h_2, M_2, w_2 \rangle\), itis the turn of the second transfer function \(h_2\) to make itsappearance, taking us then from \(\langle h_1, h_2, M_2, w_2 \rangle\Vdash \Box_{2}\varphi\) to \(\langle h_1, h_2, h_2(w_2) \rangle\Vdash \Box_{2}\varphi\).

For more details on the fibring of modal logics, the reader isreferred to Gabbay (1999: Chapter 3). For other forms of fibring, seeSection 4.3 ofcombining logics.

3.4 Complexity Issues

So far, this text has described different ways in which a multi-modalsystem can emerge. We have briefly discussed how a single modal systemcan give rise to a multi-modal one by providing either syntactic orsemantic definitions of new concepts (section 2), and also how two or more modal systems can be combined in order toproduce a multi-modal one (section 3). As the provided examples have shown (and the examples insection 4 will continue to do), the addition of modalities allows us toestablish and/or find significant relationships between the involvedconcepts, thus providing a better understanding of what each one ofthem is.

But there is also another side to this coin. By adding modalities to asystem, one increases its expressivity, and this may also have theundesirable consequence of raising itscomputational complexity. Indeed, a modal language allows us to describe a certain class ofmodels. If the language is fairly simple, then deciding whether agiven formula is true in a given world of a given model (themodel-checking problem) and deciding whether a given formulais true in all worlds of all models in a given class (thevalidity problem) are simple tasks. Now suppose a moreexpressive language is used. It is then possible to distinguish modelsthat were, from the first language’s perspective, the same (see,e.g., the appendix on the non-definability of distributed and commonknowledge within \(\cL_{\left\{ \oK{1}, \ldots, \oK{n} \right\}}\)).However, intuitively, we might then need to make a stronger effort tosee those differences: we might need more time to make thecalculations, and we might need more space to save intermediateresults. In a single sentence, expressivity and complexity go hand inhand, and an increase in the first typically produces an increase inthe second.

The simplest example of this phenomenon is given by the relationshipbetween the two best-known logical languages, the propositional andthe first-order predicate one, when used to describe first-ordermodels. The validity problem for the propositional language, which canbe understood as one that only allows us to talk about the properties(i.e., monadic predicates) of a single object and their Booleancombination, isdecidable: there are effective proceduresthat can answer the validity question for any given propositionalformula. The first-order predicate language can see much more (allobjects of the domain, together with their properties and theirn-ary relations, among others), but this comes at a price: itsvalidity problem isundecidable, as there is no effectiveprocedure that can answer the validity question for all itsformulas.

In the modal realm there are also such cases, some of them in whichseemingly harmless combinations produce dramatic results. An exampleof this can be found in Blackburn, Rijke, and Venema (2001: Section6.5), where it is shown that the fusion of two decidable systems, aPDL-like system (with sequential composition and intersectionas the syntactic constructors) and a system with the global modality(Goranko & Passy 1992), crosses the border into undecidability.Even if the new multi-agent system turns out not to be undecidable,its complexity might be such that solving its validity problem forrelatively small instances is, for all practical purposes, impossible.An example of such case is the basic multi-agent epistemic logic, withno requirements on the accessibility relations. The validity problemfor formulas in \(\cL_{\left\{ \oK{1}, \ldots, \oK{n} \right\}}\) isPSPACE: the space (and thus time) required to decide whether any given\(\varphi\) is valid is given by a polynomial function. However,adding the common knowledge operator makes the validity problem forformulas in \(\cL_{\left\{ \oK{1}, \ldots, \oK{n}, C \right\}}\)EXPTIME (sometimes also calledEXP): the required time is now given by an exponential function.^[14]

A major methodological issue is then to strike a proper balancebetween expressive power and computational complexity, with the bestmulti-modal systems being those that manage to achieve a goodcompromise in this sense. We end this section noticing briefly that,in the case of thecombination of modal logics, complexitydepends deeply on the assumed bridge axioms. A famous example of thisis the landmark paper by Halpern & Vardi (1989) on the complexityof (96!) epistemic and temporal logics overinterpretedsystems, all of them differing on the used language (single ormultiple-agents, common knowledge or lack of) and the assumed bridgeprinciples (the aforementionedperfect recall andnolearning,synchronicity, unique or multiple initialstate).

4. Significant Interactions Between Modalities

The previous sections have described several ways of obtainingmulti-modal systems. The current one presents some of the mostinteresting examples, together with the discussions that arise fromthe interplay of the modalities involved.

4.1 The Relationship Between Knowledge and Beliefs

The interplay between knowledge and belief is an important topic inepistemology. Historically, one of the most important proposals isPlato’s characterization of knowledge asjustified true belief, which has been one of the motivations used in the development ofjustification logic. However, can knowledge be truly defined as justified true belief? Theexamples provided (among others) in Gettier (1963) seem to go against thisidea. Gettier describes situations in which an agent believes a given\(\varphi\) and has a justification for it; moreover, \(\varphi\) isindeed the case. Nevertheless, the justification is not an appropriateone: \(\varphi\) happens to be the case because of some other luckyunrelated circumstances. This has lead to proposals that focus on therequirement of acorrect justification (theno falselemma: Clark 1963; Armstrong 1973; Shope 1983). Some others haveused a strongerindefeasibility requirement, stating thatknowledge is justified true belief that cannot be defeated by trueinformation, i.e., there is no true proposition \(\psi\) such that, ifthe agent were to learn that \(\psi\) was the case, would lead her togive up her belief, or to be no longer justified in holding it (Klein1971; Lehrer & Paxson 1969; Swain 1974). The aforementionedtopological modal approach of Baltag, Bezhanishvili, et al. (2016)relates this idea with other epistemic concepts, and a deeperdiscussion on what it means to know something can be found inthe analysis of knowledge by Ichikawa and Steup (2018).

There are also other alternatives. An interesting proposal, discussedin Lenzen (1978) and Williamson (2002), follows the other direction:start from a chosen notion of knowledge, and then weaken it in orderto obtain a ‘good’ (e.g., consistent, introspective,possibly false) notion of belief. These ideas have been discussed informal settings. In Stalnaker (2006), the author argues that the“true” logic of knowledge is the modal logic S4.2, givenby the standardK axiom \((K(\varphi \rightarrow\psi)\rightarrow (K\varphi \rightarrow K\psi))\) and the generalizationrule \((K\varphi\) for every validity \(\varphi)\), together withveridicality \((K\varphi \rightarrow \varphi\): knowledge istruthful), positive introspection \((K\varphi \rightarrow KK\varphi\):if the agent knows \(\varphi\), then she knows that she knows it) andthe ‘convergence’ principle \((\hK K\varphi \rightarrowK\hK\varphi\): if the agent considers it possible to know \(\varphi\),then she knows that she considers \(\varphi\) a possibility). In thissetting, Stalnaker (2006) argues that belief can be defined as theepistemic possibility of knowledge, that is,

Note how this is exactly what the definition of belief in thepreviously discussed plausibility models entails: if the modality\([\leq]\) is understood as indefeasible knowledge (Lehrer 1990;Lehrer & Paxson 1969), then \(B\varphi :=\langle{\leq}\rangle[{\leq}]\varphi\) states that belief is thepossibility of knowledge. In this context, it is a small step to movefrom studying simple beliefs to conditional beliefs. A completeaxiomatization of the logic of indefeasible knowledge and conditionalbelief, first posed as an open question in Board (2004), was providedwith a solution in Baltag and Smets (2008).

A further proposal that uses knowledge as the basic notion is that ofBaltag, Bezhanishvili, Özgün, and Smets (2013), whichgeneralizes Stalnaker’s (2006) formalization by using atopological (neighborhood) semantics. An important feature ofthe notion of belief that arises in this setting is that it issubjectively indistinguishable from knowledge: an agent believes\(\varphi\) \((B\varphi)\) if and only if she believes that she knowsit \((BK\varphi)\).

4.2 Time and Knowledge

Temporal-epistemic approaches have been briefly mentioned in thistext. Indeed, many logical systems have been used to describe the wayin which the knowledge of agents changes over time. The proposalsinclude not onlyinterpreted systems (IS; Fagin etal. 1995) but alsoepistemic-temporal logic (ETL;Parikh & Ramanujam 2003), logics ofagency (e.g.,see to it that logic,STIT; Belnap, Perloff, & Xu 2001) and theDEL approachmentioned before (section 2.8). In all of them, an important point of discussion is the interactionbetween the temporal and epistemic modalities.

As mentioned before, two famous requirements have been those ofperfect recall (the agent’s knowledge is not decreasedover time) andno learning (the agent’s knowledge isnot increased over time). In the simple fusion of epistemic logic andthe future fragment of tense logic described above, these tworequirements can be expressed, respectively, as

For some, theno learning condition might be too harsh, as itseems to say that the passage of time never helps to increaseknowledge. A related but more reasonable condition is that ofnomiracles, introduced in a slightly richer setting in van Benthemand Pacuit (2006), which states that the uncertainty of the agentscannot be erasedby the same event.^[15] A further interaction property is that ofsynchronicity,which states that epistemic uncertainty only happens among epistemicsituations that occur at the same moment of time. For example, theagent always knows ‘what time it is’, as she might notknow which action was executed, but she always knows that some actionhas taken place.

For more information on the interaction of time and knowledge, thereader is referred, among others, to Halpern, van der Meyden, andVardi (2004); van Benthem, Gerbrandy, et al. (2009). See also vanBenthem and Dégremont (2010) and Dégremont (2010) foranalogous interactions between time andbeliefs, with thelatter represented by plausibility preorders similar to theplausibility models described before.

4.3 Knowledge and Questions

The interplay between questions and propositions is an importantfactor in driving reasoning, communication, and general processes ofinvestigation (Hintikka 2007; Hintikka, Halonen, & Mutanen 2002).Indeed,

[s]cientific investigation and explanation proceed in part through theposing and answering of questions, and human-computer interaction isoften structured in terms of queries and answers. (from the SEP entryonquestions by Cross & Roelofsen 2018)

But then, what is the relationship between an agent’s knowledgeand her questions? Maybe more important: given that different agentsmight be posing different questions (i.e., they might be interested indifferent issues), what is the relationship between the knowledge ofdifferent agents?

The proposal of Boddy (2014) and Baltag, Boddy, and Smets (2018)studies these concerns (then also studying what the ‘real’common and distributed knowledge of a group is). Their model (based ontheepistemic issue model introduced in van Benthem &Minica 2012) assumes that agents have not only their individualknowledge, but also their individualissues: the topics thateach one of them has put on the table, which determine theirindividual agenda on what is currently under investigation. On thesyntactic side, besides the standard knowledge modality \((K_i\) foreach agenti), there is also a modality \(Q_{i}\varphi\), readas“\(\varphi\) can be known solely based on learnableanswers toi’s questions”. In other words,\(Q_a\) describes the maximum knowledge agenta can acquire,given her questions and the answers that are learnable for her. Thus,as a principle, ifa knows \(\varphi\), she can know it solelybased on answers to her question(s):

More interesting is the relationship between agenta’sknowledge and that of other agentb: in order for an agent toconsider any potential knowledge, such knowledge must be relevant forher in the sense that she can distinguish it as a possible answer toone of her questions. In other words,

[a]gents are therefore only able to coherently represent the knowledgeof others […] if the fact that they (the others) possess thisknowledge […] is relevant to them. (Boddy 2014: 28)

Thus, “ifb knows something that is relevant toa,then it is relevant toa thatb knows this” and

ifb can know (given her issue) anything that is relevant toa, then this fact (thatb’s potential knowledgeincludes potential knowledge ofa) is itself relevant toa.

In symbols,

A more in-depth discussion about the consequences of adding theagent’s issues to the picture (including alternative definitionsfor the group’s distributed and common knowledge) can be foundin the above references.

4.4 Agents

In the context of the design and implementation of autonomous agents,one of the most famous architectures is thebelief-desire-intention (BDI) model (Bratman 1987;Herzig, Lorini, Perrussel, & Xiao 2017).

Developed initially as a model of human practical reasoning (Bratman1987), theBDI model proposes an explanation of practicalreasoning involving action, intention, belief, goal, will,deliberation and several other concepts. Thus, it is natural to thinkabout combining simple modal logics for some of these notions in orderto define logics for such richer settings. Indeed, several formalsemantics for such models have been proposed, some of them based ondiverse temporal logics (Cohen & Levesque 1990; Governatori,Padmanabhan, & Sattar 2002; Rao & Georgeff 1991), some othersbased on dynamic logics (van der Hoek, van Linder, & Meyer 1999;Singh 1998), and some based on both (Wooldridge 2000). The crucialpart in most of them is the interaction between these differentattitudes. For example, on the one hand, if an agent intends toachieve something (say, \(I\varphi)\), one would expect for her todesire that something \((D\varphi)\); otherwise, it does not makesense to devote resources to achieve it. On the other hand, desiringsomething should not imply an intention to achieve it: it does notseem reasonable to commit resources to all our desires, even theunrealistic ones (and, perhaps more importantly, intention and desireswould collapse into a single notion). Moreover, it seems clear that anagent who desires to be rich \((Dr)\) does not necessarily believethat she is rich \((Br)\). Finally, if an agent has an intention towrite a book \((Ib)\), should she believe that she will write it\((BFb)\), thus ruling out all possible unforeseen circumstances thatcould prevent her from doing it?

A concise description of this interaction in some of these proposalscan be found in the first part ofSubsection 4.2 in the SEP entry onthe logic of action written by Segerberg, Meyer, and Kracht (2016).

4.5 Modal First-Order Logic

Modal first-order (i.e., quantified modal) logic is perhaps one of themost intriguing multi-modal systems, as the combination of quantifiersand modalities raises several interesting questions. Here we willdiscuss briefly two important points; readers interested in a furtherdiscussion are referred to the SEP discussion onquantified modal logic by Garson (2018).

The modal first-order language is built in a straightforward way:simply take the classicalfirst-order language, with its universal \((\forall)\) and existential \((\exists)\)operators indicating quantification over objects, and add the twobasic modal alethic operators, necessity \((\Box)\) and possibility\((\Diamond)\), usually interpreted as quantifying over possiblesituations. The resulting language turns out to be very expressive,allowing us to distinguish between thede dicto and thede re readings of natural language sentences (a contrast thatcan be traced back to Aristotle; see Nortmann 2002). For example,assume that an individualf has exactly 3 sisters, and considerthe sentence“the number of sisters off isnecessarily greater than 2”. The claim can be understood intwo different ways. Under ade dicto interpretation, itstates thatthe number of sisters thatf has isnecessarily greater than 2, but this is clearly questionable: underdifferent circumstances,f might have had two or fewer sisters.However, under ade re interpretation, the claim states thatthenumber of sisters thatf has, the number 3, isnecessarily greater than 2: this is definitely true, at least whenrestricting ourselves to the standard understanding of numbers.

In modal first-order logic, the difference betweende re andde dicto is given bythe scope of the involved quantifiers and modal operators. On the one hand, ade dicto (“of theproposition”) sentence indicates a property of a proposition,with the involved quantifier occurring under the scope of modalities.For example, thede dicto reading of the previous sentence isgiven by the formula \(\oBox\left(\exists x (x = s(f) \land x > 2)\right)\) (withs a function returning its parameter’snumber of sisters). On the other hand, ade re (“of thething”) sentence indicates a property of an object, with theinvolved modality occurring under the scope of quantifiers. Forexample, thede re reading of the previous sentence is givenby the formula \(\exists x \left( x = s(f) \land \oBox(x > 2)\right)\). This crucial distinction can be exemplified by thedifference between some agenti knowing that there is someonethat makes herHappy (but maybe without knowing who this personis), \(K_i\exists x H(x, i)\), and the always preferred existence ofsomeone whoi knows makes her happy \((\exists x K_i H(x,i))\).

But the expressivity comes with a cost. As usual in multi-modalsystems, the crucial question is the interaction between the involvedmodalities, and in this case, the discussion typically centers on thefollowing two properties: theBarcan formula, \(\forall x\oBox Px \rightarrow \oBox\forall x Px\), and its converse,\(\oBox\forall x Px \rightarrow \forall x \oBox Px\) (see Barcan1946). The reason for the controversy becomes clear when the genericpredicateP is replaced by, say, the formula \(\exists x(x=y)\), which can be read as“x exists”.Then, the first formula becomes

stating that if everything exists necessarily then it is necessarythat everything exists. In terms of a possible worlds semantic model,this boils down to stating that every object existing in analternative possible world should also exist in the current one: whenone moves to alternative scenarios, the domain does not grow.Analogously, the second formula becomes

stating that if it is necessary that everything exists then everythingexists necessarily. In terms of a possible worlds semantic model, thisboils down to stating that every object existing in the current worldshould also exist in an alternative possible one: when one moves toalternative scenarios, the domain does not shrink.

Thus, a decision about whether such principles hold corresponds toanswering a crucial question when building a model for the modalfirst-order language: what is the relationship between the domains ofthe different possible worlds? On the one hand, from the perspectiveof actualism, everything there is (everything that can in any sense besaid to be) is actual, that is, itexists; hence, there is afixed domain across all possibilities. On the other hand, from theperspective of possibilism, ‘the things that exist’include possible but non-actual objects; hence, different possibleworlds might have different domains. There is a large literature onthe discussion between these two positions, as shown in the SEP entryon thepossibilism-actualism debate.

4.6 Dynamics of Intentions

The notion ofintention is crucial inBDI systems, as it in some sense defines thechoices the agent will make, thus affecting her behavior. Thus, thedynamics of intention is also a crucial subject, as itdescribes the way intentions are generated, preserved, modified ordiscarded.

For an initial point, how do intentions change after the agent learnsa new piece of information? According to Roy (2008: Chapter 5), if theoriginal intentions are compatible with the new information, then theyare ‘reshaped’; otherwise, the agent discards them withoutcreating any new intention (or, analogously, generating an intentionfor something that has been already achieved). Thus, after anannouncement of \(\chi\) the agent intends to do \(\varphi\) if andonly if \(\chi\) is compatible with her intentions and \(\varphi\) isa restricted consequence of the agent’s initial intentions, orelse \(\varphi\) is a ‘known’ consequence of\(\chi\)’s announcement. In a formula,

There are other proposals. In van der Hoek, Jamroga, and Wooldridge(2007), intentions are defined, roughly speaking, as plans the agentbelieves have not yet been fulfilled. As a consequence of this,changes in the agent’s beliefs lead to changes on herintentions. For example, after any observation, the agent will dropintentions that she believes have been accomplished:

Moreover, she will drop any intention she believes it is impossible toachieve:

But, just as changes in the agent’s information (knowledge,beliefs) should trigger changes in her intentions, changes in herintentions may also trigger changes in (some of) her beliefs.Intuitively, having the intention to achieve a given \(\varphi\)reduces the actions that the agent ‘can’ perform, from theones she can actually carry on, to those that will still allow (andmaybe assure) that \(\varphi\) will be achieved. In other words, achange in the agent’s intentions triggers also a change in herbeliefs about the (sequence of) actions that will be available in thefuture. This is the idea followed in Icard, Pacuit, and Shoham (2010),which studies the interaction between intention revision and beliefrevision by introducing postulates for both actions, with thesepostulates describing the two processes’ interplay. In theinteresting case of intention revision, the postulates state that (i)a new intention will take precedence over previous ones (and thus oldones should be eliminated when in conflict), (ii) modulo coherence, nofurther change should be made on the agent’s intentions (inparticular, no extraneous intentions should be added), and (iii)non-contingent beliefs do not change with intention revision.^[16]

4.7 Obligations and Time

As the reader might guess, adding a temporal dimension is typically agood idea, as in most cases it enriches the initial system by allowingus to talk about how the conceptchanges. Besides epistemicsettings, others that benefit from this are systems ofdeontic logic, which study the properties of concepts aspermissions (e.g.,“\(\varphi\) is allowed”) andobligations (e.g.,“\(\varphi\) is required”). Such systems are extremelyuseful, as they involve topics such as law, social and businessorganizations, and even security systems.

One of the interesting concepts that arise when time and obligationsinteract with each other is the notion ofdeontic deadlines:obligations that need to be fulfilledonly once, at a time ofone’s choosing, as long as it is before certain condition becometrue. Indeed,

[…] deontic deadlines are interactions between two dimensions:a deontic (normative) dimension and a temporal dimension. So, to study[them], it makes sense to take a […] temporal logic […]and a standard deontic logic […], and combine the two in onesystem. (Broersen, Dignum, Dignum, & Meyer 2004: 43)

Such formal systems help to provide a proper understanding of what adeadline is: as the aforementioned reference asks,

is a deadline (1) an obligation at a certain point in time to achievesomething before another point in time, or (2) is a deadline simply anobligation that persists in time until a deadline is reached, or (3)is it both?

Then, the formal setting also allows the possibility to make furtherfiner distinctions, as the one between an obligation to always satisfya given \(\varphi\) (in symbols, and withO a modality forobligation, \(OG\varphi)\) and an obligation for \(\varphi\) thatshould be always fulfilled \((GO\varphi)\). Further and deeperdiscussions on deontic deadlines can be found in Broersen et al.(2004); Broersen (2006); Brunel, Bodeveix, & Filali (2006);Demolombe, Bretier, & Louis (2006); Governatori, Hulstijn,Riveret, & Rotolo (2007); and Demolombe (2014), among others.

4.8 Knowledge and Obligations

Equally important is the relationship between knowledge andobligations, as it is shown by the aforementionedparadox of epistemic obligation, which arises within the fusion (section 3.1) of astandard deontic logic and a standard epistemic logic. But the relationship between theseconcepts goes beyond their interaction in such a basic system. Forexample, if an agent does not know about the existence of anobligation, should she be expected to fulfill it? In some cases theanswer seems to be “no”: a physician whose neighbor ishaving a heart attack has no obligation to provide assistance unlessshe knows about the emergency. Still, in some other cases, the answerseems to be “yes”: the juridical principle“ignorantia juris non excusat” (roughly,ignorance of the law is not excuse) is an example of this.

There have been proposals dealing with these issues. One of them is byPacuit, Parikh, & Cogan (2006), which uses a setting in whichactions can be considered “good” or “bad”. Itintroduces a notion ofknowledge-based obligation under whichan agent is obliged to perform an action \(\alpha\) if and only if\(\alpha\) is an action which the agent can perform and sheknows that it is good to perform \(\alpha\). This is then aform ofabsolute obligation which remains until the agentperforms the required action.

Interestingly, the involvement of knowledge gives raise to forms of‘defeasible’ obligations that can disappear in the lightof new information. For example, having being informed about herneighbor’s illness, the physician could have the obligation toadminister a certain drug; however, this obligation would disappear ifshe were to learn that the neighbor is allergic to this medication.This ‘weaker’ form of obligation can also be capturedwithin the setting discussed in Pacuit et al. (2006).

The interaction between knowledge and obligations is not limited tothe way knowledge ‘defines’ obligations. An important roleis also played by whether the agent consciously violates hercommitments. In fact, most juridical systems contain the principlethat an act is only unlawful if the agent conducting it has a‘guilty mind’ (mens rea): for the agent to beguilty, she must have committed the act intentionally/purposely. Ofcourse, there are different levels of ‘guilty minds’, andsome legal systems distinguish between them in order to assign‘degrees of culpability’ (e.g., an homicide is consideredmore severe if done intentionally rather than accidentally). Forexample, on the one hand, stating that it is illegal to do \(\alpha\)negligently means that it is illegal to do \(\alpha\) whilebeing aware that the action carries a substantial and unjustifiablerisk. On the other hand, stating that it is illegal to do \(\alpha\)knowingly means that it is illegal to do \(\alpha\) whilebeing certain that this conduct will lead to the result.^[17] These and other modes ofmens rea are formalized in Broersen(2011) within theSTIT logic framework.

5. Multi-Modal Systems in Philosophical Discussions

As the previous sections indicate, the specific interplay betweendifferent modalities (the way they are combined and which bridgeprinciples hold) is crucial to provide an accurate representation andanalysis of different philosophical concepts. In fact, in severaloccasions, the combination of different modalities have shed light onphilosophical issues. We will illustrate this for the concepts ofabduction, knowability, ‘believing to know’, truthmakersand the interplay between assumptions and beliefs while keeping inmind an endless list of other philosophical paradoxes and problemsthat all arise in a multi-modal setting. (Among many others, see theYablo paradox in Yablo (1985, 1993) as well as the SEP discussions ondeontic paradoxes and onparadoxes without self-reference. See also the SEP analyses on theknower paradox, ondynamic epistemic paradoxes and on thesurprise examination paradox; for the latter, see also a proposed solution in Baltag and Smets(2010Other Internet Resources).)

5.1 Abductive Reasoning

The termabduction has been used in related but sometimes different senses. Roughlyspeaking, abductive reasoning (also called [or closely connected toconcepts as]inference to the best explanation,retroduction, andhypothetical,adductiveorpresumptive reasoning, among many other terms) can beunderstood as theprocess through which an agent (or a groupof them) looks for an explanation of a surprising observation. Manyforms of intellectual tasks, such as medical and fault diagnosis,legal reasoning, natural language understanding, and (last but notleast) scientific discovery, belong to this category, thus makingabduction one of the most important reasoning processes.

In its simplest form, abduction can best be described withPeirce’s 1903 schema (Hartshorne & Weiss 1934: CP 5.189):

This is the understanding that has been most frequently cited and usedwhen providing formal approaches to abductive reasoning. Still,typical definitions of an abductive problem and its solution(s) havebeen given in terms of a (propositional, first-order) theory and aformula, leaving the attitudes of the involved agents out of thepicture.

However, there have been also proposals that formalize (parts of) theabductive process in terms of diverse epistemic concepts (e.g.,Levesque 1989; Boutilier & Becher 1995). Among them,Velázquez-Quesada, Soler-Toscano, &Nepomuceno-Fernández (2013) understand abductive reasoning as aprocess of belief change that is triggered by an observation andguided by the knowledge the agent has. In symbols(Velázquez-Quesada 2015), abductive reasoning from a surprisingobservation \(\psi\) to a belief \(\varphi\) can be described as

stating thus that if the agent knows \(\varphi \rightarrow \psi\) andan announcement of \(\psi\) (“\([\psi{!}]\)”) makes herknow it \((\psi)\), then she can perform an act ofbeliefrevision with \(\varphi\) (“\([\varphi\Uparrow]\)”)in order to believe it. This formalization emphasizes not only thatthe agent’s initial knowledge plays a crucial role in thegeneration of possible abductive solutions, but also that the chosensolution can only be accepted in a weak way (so the agent believes itbut is not certain about it), therefore making it a candidate forbeing revised/discarded in the light of further information.

Other proposals have incorporated further aspects into the picture.One of them, Ma & Pietarinen (2016), follows Peirce’s latterunderstanding of abductive reasoning (called thenretroduction: “given a (surprising) factC, ifA impliesC, then it is to be inquired whetherAplausibly holds”; Peirce 1967: 856) as a form of reasoning fromsurprise toinquiry. This can be connected to the notions ofissues and questions described insection 4.3. As the authors mention,

[t]he important discovery is that [, in the new formulation,] theconclusion is presented in a kind of interrogative mood. But theinterrogative mood does not merely mean that a question is raised. Infact, it means that the possible conjectureA becomes thesubject of inquiry: the purpose is to determine whether thatAis indeed plausible or not. Peirce termed such mood “theinvestigand mood”. Hence abduction can be viewed as the dynamicprocess toward a plausible conjecture and, ultimately, toward alimited set of the most plausible conjectures.

5.2 Fitch Knowability Paradox

Thisparadox emerges from what is commonly known as the verificationist thesis(VT), which claims that all truths are verifiable.Formalizing this thesis in a multi-modal logic that combines aknowledge operator with a possibility operator would yield

with \(\Diamond K\varphi\) read as“it is possible to know\(\varphi\)”. In this context, the paradox refers toFitch’s argument containing an idea conveyed to him in 1945which shows that, if all truths are knowable, then all truths arealready known. As the argument goes, we clearly do not know all truths(as we are not omniscient!); hence, the premise has to be false: notall truths are knowable. The paradox can be summarized by thederivation

which poses a problem for the non-omniscient verificationist. Thederivation that leads to the paradox as we have stated it here isbased on a multi-modal logical system in which at least the followingprinciples hold: (i) the principle of non-contradiction, to capturethat contradictions cannot be true and thus are not possible, (ii) theclassical laws of double negation, transitivity of the materialimplication and substitution, (iii) normality of the modal logicoperatorK, the modal logic principleT stating thatknowledge is truthful, and the normality of the modal possibilityoperator \(\Diamond\). One of the simplest presentations of theparadox, which shows how it leads to the unwanted equivalence betweentruth and knowledge, can be found in van Benthem (2004). Start withthe formula stating the verificationist thesis, \(\varphi \to \DiamondK\varphi\), and substitute \(\varphi\) with \((p \land \lnotKp)\):

This paradox gave rise to an active debate in the philosophicalliterature, leading us to signal out two main types of proposedsolutions: those proposing a weakening of our logical principles (asin paraconsistent, intuitionistic or weaker modal logics) whilekeeping the verification thesis, and those that in contrast do notchange/restrict the underlying logic but propose a specificformalization or reading of the verificationist thesis.^[18] While we refer the reader to the SEP entry onepistemic paradoxes for an overview of these proposed solutions, here it is illustrativeto highlight how this paradox, which in some sense arises fromcombining a modal logic for knowledge (K) with a modal logicfor possibility \((\Diamond)\), can be ‘demystified’ bythe further introduction of a modality for communication (related tothe public announcement modality ofPAL). Indeed, accordingto van Benthem (2004), what the verificationist thesis expresses isnot about ‘static’knowability, but rather abouta form oflearnability: “what is true may come to beknown” (van Benthem 2004). This statement can be formally statedin a suitablearbitrary announcement framework as:

which is read as“if \(\varphi\) is the case, then there isa formula \(\psi\) after whose announcement \(\varphi\) will be known”.^[19] This reading of the verificationist thesis brings us in touch with anumber of results in dynamic epistemic logic onunsuccessfulformulas (those that become false after being truthfully announced;van Ditmarsch & Kooi 2006; van Benthem 2011; Holliday & Icard2010), which indicate that indeed not all sentences are learnable. Infact, this solution shows us that

[…] there is no saving VT—but there is also no suchgloom. For in losing a principle, we gain a general logical study ofknowledge and learning actions, and their subtle properties. Thefailure of naive verificationism just highlights the intriguing waysin which human communication works. (van Benthem 2004: 105)

5.3 The paradox of the perfect believer

On first sight it seems natural to say that one can believe to‘know’ something, even when in fact one does not actuallyknow it. So believing to know something is philosophically conceivedto be different from claiming true knowledge. Yet, under theassumption that what is known is also believed, this specificinterplay between modalities for knowledge and belief can lead us intotrouble if we identify belief with aKD45-modalityB, and knowledge with aS5-modality forK. For suppose \(BK\varphi \land \lnot K\varphi\) is assumed.Then, by negative introspection of the second conjunct, we derive\(K\lnot K\varphi\). But as knowledge implies belief, we derive\(B\lnot K\varphi\). This together with the first conjunct\(BK\varphi\) will give us, by additivity of belief, \(B(K\varphi\land \lnot K\varphi)\). Hence we derive the belief in acontradiction, which is not compatible with the assumption ofconsistency of beliefs (axiomD) inKD45. This problem is known as the paradox of theperfect believer (and also as the Voorbraak paradox), as itwas originally (but equivalently) described (Voorbraak 1993) as thederivability of thebridge principle \(BK\varphi \rightarrowK\varphi\), which states that a belief in knowing a given \(\varphi\)is enough to know \(\varphi\). (The derivation of \(BK\varphi\rightarrow K\varphi\) also relies on the negative introspection ofknowledge, the normality and consistency of beliefs, and the bridgeprinciple stating that knowledge implies belief; Gochet &Gribomont 2006: 114.)

After presenting this problem, Voorbraak (1993) proposed to deal withit by discarding the bridge principle \(K\varphi \rightarrow{B\varphi}\). Another option is to allow inconsistent beliefs (Gochet& Gribomont 2006: Section 2.6). Still, a further possiblesolution, closer to the spirit of these notes, is to consider anintermediate notion of ’knowledge’ that is not as strongas the absolute irrevisable (i.e., irrevocable) notion given by theS5 modal operatorK. More precisely, theproposal in Baltag and Smets (2008) looks at Lehrer’sdefeasibility theory of knowledge (Lehrer 1990; Lehrer &Paxson 1969), and works with theindefeasible (“weak”,non-negatively-introspective) knowledge given, in theplausibility models discussed above, by the modality \([\leq]\) (read before also assafe belief). Indeed, Lehrer and Stalnaker call this conceptdefeasible knowledge, a form of knowledge that might bedefeated byfalse evidence, but cannot be defeated bytrue evidence. The concept satisfies both the truth axiom\(([\leq]\varphi \rightarrow \varphi)\) and positive introspection\(([\leq]\varphi \rightarrow [\leq][\leq]\varphi)\), but it lacksnegative introspection; thus, the previous derivation of inconsistentbeliefs from an agent mistakenly believing that she (defeasibly) knows\(\varphi\) \(({B[\leq]\varphi} \land \lnot [\leq]\varphi)\) is nolonger possible. Instead, it can be easily shown how a belief indefeasible knowledge, \({B[\leq]\varphi}\), is equivalent to a simplebelief, \({B\varphi}\).

5.4 Truthmakers from a modal perspective

Paraphrasing Fine (2017), atruthmaker is something on theside of the world, as a fact or a state of affairs, making truesomething on the side of language or thought, as a statement or aproposition. Truthmaking has been an important topic in bothmetaphysics and semantics. For the first, “truthmaking serves asa conduit taking us from language or thought to an understanding ofthe world” (Fine 2017: 556); for the second, it providesadequate semantics for a given language by establishing how the worldmakes sentences of the language true.

In Fine (2017), the author explains the basic framework of truthmaker(‘exact’) semantics for propositional logic. It is basednot onpossible worlds, but rather onstates orsituations; the crucial difference is that, while a possibleworld settles the truth-value of any possible statement (i.e., given aformula and a possible world, the formula is either true or elsefalse), a situation might not be enough to decide whether a givensentence holds.

Formally, astate space is a tuple \({\langle S, \sqsubseteq\rangle}\) whereS is a non-empty set of states and\({\sqsubseteq} \subseteq (S \times S)\) is a partial order (i.e., areflexive, transitive and antisymmetric relation), with \(s_1\sqsubseteq s_2\) understood as“state \(s_2\) extends state\(s_1\)”. It is assumed that any pair of states has aleast upper bound (i.e., asupremum); formally, forany \(s_1, s_2 \in S\) there is \(t_1 \sqcup t_2 \in S\)satisfying

1. \(t_1 \sqsubseteq (t_1 \sqcup t_2)\) and \(t_2 \sqsubseteq (t_1\sqcup t_2)\) (so \(t_1 \sqcup t_2\) is an upper bound of both \(t_1\)and \(t_2)\), and
2. ift is an upper bound of both \(t_1\) and \(t_2\), then\((t_1 \sqcup t_2) \sqsubseteq t\) (so \(t_1 \sqcup t_2\) is theleast upper bound).

This supremum \(t_1 \sqcup t_2\) (its uniqueness follows from\(\sqsubseteq\)’s antisymmetry) can be understood as the‘sum’, ‘merge’ or ‘fusion’ ofstates \(t_1\) and \(t_2\), and it provides the crucial tool fordeciding whether a ‘conjunction’ is the case, as shownbelow.

Astate model is a tuple \({\langle S, \sqsubseteq, V\rangle}\) with \({\langle S, \sqsubseteq \rangle}\) a state space and\(V:\mathtt{P}\to (\wp(S) \times \wp(S))\) a valuation functionreturning not only set of states that make a given atomp true(abbreviated as \(V^+(p))\), but also the set of states that make itfalse (abbreviated as \(V^-(p))\). In principle, given an atomp, there needs to be no relation between the two sets. Theymight be overlapping \((V^+(p) \cap V^-(p) \neq \emptyset)\), thusyielding a state that makesp both true and false; they mightbe limited \((V^+(p) \cup V^-(p) \neq S)\), thus yielding a state thatmakesp neither true nor false; they might be neither, thusbeing exclusive \((V^+(p) \cap V^-(p) = \emptyset)\) and exhaustive\((V^+(p) \cup V^-(p) = S)\), and making the states behave as possibleworlds with respect top.

Given a state model, the relations \(\Vvdash_v\) (verified bya state) and \(\Vvdash_f\) (falsified by a state) are definedas follows.

Note the clauses for verifying a conjunction and falsifying adisjunction. A state makes a conjunction true if and only if it is thefusion of states that verify the respective conjuncts \(\varphi\) and\(\psi\). Analogously, a state makes a disjunction false if and onlyif it is the fusion of states that falsify the respective disjuncts\(\varphi\) and \(\psi\).

Truthmaker semantics can be seen from a (multi)modal perspective (vanBenthem 1989), since a state model \({\langle S, \sqsubseteq, V\rangle}\) can be understood as a modal information logic, and thuscan be described by modal languages. One interesting possibility (vanBenthem 2019: Section 12) starts by taking two modalities,\(\langle\sqsubseteq\rangle\varphi\) and\(\langle\sqsupseteq\rangle\varphi\), whose semantic interpretation isgiven in the standard modal way, with the first modality relying onthe partial order \(\sqsubseteq\), and the second relying on itsconverse \(\sqsupseteq\).^[20] Then, one can add a (binary) modality describing least upperbound

and a ‘dual’ one describing theinfimum (thegreatest lower bound)^[21]

With these tools, it is possible to define a faithful translation fromtruthmaker logic into modal information logic (see van Benthem 2018:Section 13 for details). This translation brings methods from modallogic to the study of truthmaking. More important is the fact that itmakes truthmaker semantics, a framework that works by providing a newmeaning to Boolean connectives, completely compatible with classical(modal) logic, which keeps standard definitions but extends theframework’s expressivity by studying much richer languages.

5.5 The Brandenburger-Keisler Paradox

Consider the following situation (Brandenburger & Keisler 2006)involving two epistemic attitudes,belief andassumption.

Ann believes that Bob assumes that \(\underbrace{\textit{Annbelieves that Bob’s assumption is wrong}.}_{\varphi}\)

Given this, the question is the following: is \(\varphi\)(“Ann believes that Bob’s assumption iswrong”) true or false?

Paraphrasing Pacuit and Roy (2017:Section 6), suppose \(\varphi\) is true. So, what \(\varphi\) represents is true,that is, Ann believes that Bob’s assumption is wrong. Moreover,by belief introspection, she believes that“she believesBob’s assumption is wrong”, that is, she believesBob’s assumption. But the description of the situation tells usthat Ann believes that Bob assumes \(\varphi\); then, in fact, Annbelieves that Bob’s assumption is correct. Thus, \(\varphi\),“Ann believes that Bob’s assumption iswrong”, is false.

But now suppose \(\varphi\) is false. Then, following Pacuit and Roy (2017:Section 6) again, Ann believes that Bob’s assumption is correct, that is,Ann believes \(\varphi\) is correct. Furthermore, the description ofthe situation states that“Ann believes that Bob assumesthat Ann believes that Bob’s assumption is wrong”,which, given that \(\varphi\) is Bob’s assumption, can berewritten as“Ann believes that Bob assumes that Annbelieves that \(\varphi\) is wrong”. But then, not only Annbelieves that she believes that \(\varphi\) is correct; she alsobelieves that Bob assumption is that she believes that \(\varphi\) iswrong. Thus, it is the case that she believes Bob’s assumptionis wrong (Ann believes that Bob’s assumption is that shebelieves that \(\varphi\) is wrong, but she believes that is wrong:she believes that \(\varphi\) is correct). So, \(\varphi\) istrue.

This paradox is interesting because it suggests that not everydescription of beliefs can be ‘represented’ (just asRussell’s paradox suggests that not every collection can constitute a set). Asexplained in Pacuit (2007), in order to show that this situationcannot be ‘represented’, the original paper (Brandenburger& Keisler 2006) introduces abelief model. This structurerepresents each agent’s beliefs about the beliefs of the otheragent. More precisely, a belief model is a two-sorted structure, onesort for each agent, with each sort representing an epistemic statethat its agent might have. The model’s first component is itsdomain, given by the union of \(W_a\) and \(W_b\), the disjoint setsof states of Ann and Bob, respectively. The model also has a relationfor each agent, \(R_a\) and \(R_b\), with \(R_auv\) (restricted to \(u\in W_a\) and \(v \in W_b)\) read as“in stateu, Annconsidersv possible, and analogous for \(R_bvu\). Notehow each collection \(\cU_b\) of subsets of \(W_b\) can be understoodas a language for Ann (the beliefs she might have about Bob’sbeliefs), and analogous for Bob. A full language is then defined asthe union of a language for each agent. The epistemic attitudesfeatured in the situation under discussion can be defined as follows.On the one hand, beliefs have a somehow standard interpretation: Annbelieves a given \(U \in \U_b\) if and only if the set ofstates that she considers possible isa subset of \(U\). Onthe other hand, an assumption is understood as the strongest belief:Annassumesa given \(U \in \U_b\) if and only if the set ofstates she considers possible is \emph{exactly} \(U\).

With these tools, it is now possible to make precise the claim thatthe situation described above cannot be represented. A language issaid to becomplete for a belief model if and only if everypossible statement in a player’s language (i.e., every statementin her language that is true in at least one state) can be assumed bythe player. It is then possible to show, using a diagonal argument,that no belief model is complete for ‘its first-orderlanguage’, i.e., the language containing all first-orderdefinable subsets of the model’s domain.

Closing Words

The study of multi-modal logics and their applications is an on-goingfield of study. While we have illustrated some of the maindevelopments in this area, many more combinations of modal operatorshave emerged as tools for the formal analysis of different interestingphenomena, with many of them being still currently further explored.Some examples are related to ‘old’ questions. An instanceis the famous logical omniscience problem (Hintikka 1962; Stalnaker1991), whose multi-modal discussion involves not only the concepts ofimplicit and explicit knowledge/beliefs (e.g., Levesque 1984;Lakemeyer 1986; Velázquez-Quesada 2013; Lorini 2020) but alsothe relationship between knowledge andawareness (see, e.g.,Fagin and Halpern 1988; Halpern & Rêgo 2009, van Benthem& Velázquez-Quesada 2010; Belardinelli & Schipper2023). Some other examples have emerged from new developments. Aninstance is the combination of modal operators to reason about theflow of quantum information (Baltag and Smets 2022). A very fruitfularea has been that ofgames (see, e.g., Baltag, Li, &Pedersen 2022; van Benthem, Pacuit, & Roy 2011, van Benthem 2014),where many different attitudes and concepts (knowledge, beliefs,preferences, intentions, awareness, actions and so on) converge. Theapplications of multi-modal logics are numerous and reach well beyondthe aspects of studies connected to the area of multi-agent systems inAI and Philosophical Logic that have been covered in this entry.

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