Modal Logics

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# Modal Logics

## Idea

The term modal logic refers to an enrichment of standard formal logic, where the standard logical conjunctions (and, or, not, implication and, perhaps, for all, exists etc.) are accompanied by certain extra operations – called modal operators and traditionally denoted by symbols like “$\Box$”, “$\lozenge$”, , “$\bigcirc$” – such that for $p$ any proposition an expression like $\Box p$ is a(nother) proposition whose interpretation is intended to be:

• $p$ holds (only) in a certain way”,

where the “certain way” is (implicitly) specified by whatever axioms are satisfied by the given modal operator.

In other words, modal operators in modal logic are meant to express a certain mode of being true — and in generalization to modal type theory they express more generally a mode of being, whence the reference to modalities as in Kant‘s writings.

Beware however that there is no broad agreement on which further axioms exactly make a general modal operator — which in itself is any (co-)monad on the underlying universe of propositions (of types), see below — encode/reflect any such intended “mode of being (true)”. As a result:

‘Ask three modal logicians what modal logic is, and you are likely to get at least three different answers’. [Blackburn, deRijke & Venema (2001), Introduction]

Historically, the default example of a modal logic is known as S4 modal logic, originally motivated as a model for epistemic logic (with a modality for being “necessarily true”), but really just axiomatizing (propositional logic equipped with) any co-monad $\Box$ on the universe of propositions, without further specification. There are variants known as T modal logic and K modal logic which drop some of the axioms even on this comonad, but it seems good practice to agree that a modal operator is at least a (co-)monad on the universe of propositions (of types).

In this sense, modal logic is the counterpart in the computational trilogy to:

Among semantics/models of modal logics are the geometric models based on Kripke frames which are sets (of possible worlds) on which the propositions in the logic may dependend and equipped with relations which prescribe over which such worlds to quantify in interpreting the modal operators (eg. which worlds to inspect to conclude that a proposition holds possibly).

Specifically for epistemic logic this Kripke semantics is hence also known as possible worlds semantics. (As least for S5 modal logic this is closely related to categorical models of dependent types, as explained at necessity and possibilitypossible worlds via dependent types.)

Some accounts of modal logic essentially regard the theory as being the logical study of such Kripke frames – in the sense of sets equipped with relations, hence “relational structures” (for instance Blackburn, De Rijke & Venema (2001) p xi who start out saying that they do not know what modal logic is if not the study of relational structures).

There are also algebraic models for modal logic in terms of algebras with (co)-closure operators. For instance, temporal logics can have posets as models.

## Definition

### Modal Languages

Modal logics are built on modal languages, that is the usual propositional language plus those extra modalities. (Note that modalities may also be added to predicate logic, see first-order modal logic.) The way the modalities work has to be laid down in an axiom system for the logic in question, for instance, for the temporal logic we might require an axiom saying ‘If $F F\phi$ is true, then $F\phi$ is true’, which will read a ‘if it is going be true in the future that $\phi$ is going to be true in the future, then …’, see temporal logic. (Is this going to be something what we might want in ‘provability logic’; is it the case that we should expect that if it is provable that something is provable then that something must be itself provable. This concentrates the modelling process on exactly how we wish to have our ‘context’ to behave.) In this way the relational nature of a context that we are looking at can get encoded into the logic.

Modal languages add one or more modal operator, often denoted $\square$ or $\lozenge$ into the usual propositional logics. (For the moment, we will keep things fairly simple so assume these to be unary operators and we will not be considering operators that have more than one input, for the moment at least. The general case will be considered later on, but in any case is discussed in detail in some of the books on modal logic listed below.)

We will give a modal language with $n$ modal operators, $\lozenge_i$, $i = 1,\ldots, n$, which can be applied to propositions of the language to form new propositions. If $n=1$, we will refer to the language, defined below, as the basic modal language.

###### Definition

We suppose given a set of variables, the $n$-operator basic modal language, $\mathcal{L}_\omega(n)$, given by

$\phi ::= p_\lambda \mid \bot \mid \neg \phi \mid \phi_1 \vee \phi_2 \mid \lozenge_i\phi,$

where the $p_\lambda$ are the propositional variables ordered by finite ordinals, $\lambda$, and, as usual, $\lozenge_i$ is a modality for each $i=1, \ldots , n$.

###### Remark

This form of definition needs a bit of interpreting if you have not met it before. It gives a way of deciding if a formula is ‘well-formed’. The well formed formulae of $\mathcal{L}_\omega(n)$ are defined as follows: A formula is either

• a proposition variable,

• the propositional constant $\bot$, that is falsum,

• a negated formula,

• a disjunction of two formulae, or

• a formula prefixed by one of the diamonds / modal operators.

###### Remark

The basic modal language will be $\mathcal{L}_\omega(1)$. We will sometimes write $Prop$ for the set of propositional variables / atomic formulae or whatever other reasonable term is used in a context.

###### Remark

The interpretation of $\lozenge \phi$ depends on the context (to some extent), but in the initial form here it is usually said to mean ‘possibly $\phi$’.

###### Remark

Some authors use an equivalent generation rule using $\square \phi$, which is $\neg \lozenge \neg \phi$. Of course, this interprets as ‘necessarily $\phi$’ in this initial form. In epistemic logic the basic modal language interprets $\square \phi$ as saying ‘the agent knows that $\phi$’.

###### Remark

Other formulations replace $\vee$ and $\neg$ by implication $\phi_1\to \phi_2$, or by $\wedge$ and $\neg$.

###### Definition

A modal logic in $\mathcal{L}_\omega(n)$ is any set $\Lambda$ of $\mathcal{L}_\omega(n)$-formulae such that

• $\Lambda$ includes all $\mathcal{L}_\omega(n)$-formulae that are instances of tautologies,

and

• $\Lambda$ is closed under the inference rule if $\phi$, $\phi\to \psi \in \Lambda$ then $\psi \in \Lambda$ , i.e. detachment or modus ponens.

One of the basic axiom systems leads to normal modal logics.

###### Definition

(deducibility)

Suppose given a normal modal logic, $\Lambda$ in $\mathcal{L}_\omega(n)$. A formula $\phi$ is $\Lambda$-deducible from a set, $\Gamma$ of formulae, if there are finitely many formulae $\psi_0,\ldots,\psi_m\in \Gamma$ such that

$\vdash_\Lambda (\psi_0\to (\psi_1\to (\ldots \to (\psi_m\to \phi)\ldots)$

that is the formula $(\psi_0\to (\psi_1\to (\ldots \to (\psi_m\to \phi)\ldots)$ is in $\Lambda$.

###### Remark

As the set of all formulae in $\mathcal{L}_\omega(n)$ satisfies the conditions for a logic and any intersection of logics is itself also a logic, we have that given a set of formulae decreed to be ‘axioms’ for a logic, there is a smallest modal logic containing them.

### Common modal axioms

• (K) $\Box(A \to B) \to \Box A \to \Box B$
• (M) $\Box A \to A$
• (4) $\Box A \to \Box \Box A$
• (5) $\lozenge A \to \Box \lozenge A$
• (B) $A \to \Box \lozenge A$

## Examples

Here is a brief list of flavors of modal logic. More details are discussed below.

• epistemic logics: it is known to $X$ that, it is common knowledge that;

• temporal logic: henceforth, eventually, hitherto, previously, now, tomorrow, yesterday, since, until, inevitably, finally, ultimately, endlessly, it will have been, it is being …

• dynamic logic?: after the program/computation/action finishes, the program enables, throughout the computation;

• doxastic logic: it is believed that

• deontic logic: it is obligatory/forbidden/permitted/unlawful that

• geometric logic: it is locally the case that

• metalogic: it is valid/satisfiable/provable/consistent that (Goldblatt).

The metalogical version was important in the history of modern modal logic. Gödel played an important role there.

• provability logics

## Semantics

We discuss the semantics of modal logics, its models.

### In Kripke frames / relational structures

The usual semantics of modal languages is in terms of frames, and this is where the link with relational structures comes in. (These are quite often called ‘Kripke frames’ as Kripke was one of the first to use relational semantics in this context.

A discussion of the history can be found in (BlackburnDeRijkeVenema, page 42).

(As there is another sense to frame as the dual of a locale, we need to consider the terminology here and where necessary will use frame (modal logic) as the entry name.) A more detailed discussion of frames, models and the whole question of the semantics of modal logics is to be found at that entry.

Modal logics are thus also the logics of relational structures, in fact, Blackburn et al (see references) have as their first slogan: Modal languages are simple but expressive languages for talking about relational structures. The temporal logic that satisfies the axiom $(4)$ has models that are posets, for instance, whilst many of the epistemic logics have models which are sets with equivalence relations on them. In each case, the idea is that the relational structure gives all the possible states of some system and the modal logic describes that system. This explains the great interest in computer science and artificial intelligence of applications of modal logics.

### Algebraic semantics

The usual algebraic semantics of modal logic is in terms of Boolean algebras with operators and is described in the entry algebraic models for modal logic.

### Coalgebraic semantics

The geometric / relational /Kripke semantics of modal logics are instances of coalgebraic semantics. This type of semantics also provides an excellent motivation and intuition for various types of modal logic that arise naturally in computer science.

### Categorical semantics

Every category comes with its internal logic. Modal operators in this internal logic can at least sometimes be identified with (co)reflectors into specified (co)reflective subcategories. See at modal type theory for more on this.

Examples for this are local toposes and cohesive toposes. See there for more details.

## Theorems

One of the earliest texts that exhibits the intuitionist context is

translated in:

• Stefania Centrone, Pierluigi Minari, Oskar Becker, On the Logic of Modalities (1930): Translation, Commentary and Analysis, Synthese Library 444, Springer (2022) [doi:10.1007/978-3-030-87548-0]

The modern formalization of modal logic (and their enumeration by S1, …, S4, S5) originates with

(who say “modal functions” for modal operators).

Early discussion making explicit epistemic modal logic and variants:

Historical overview and introduction:

Textbook accounts:

Other texts:

• Dana Scott, Advice on Modal Logic, in Karel Lambert (ed.) Philosophical problems in Logic – Some recent developments, Reidel 1970

Here is what I consider one of the biggest mistakes in all of modal logic: concentration on a system with just one modal operator. (p. 161)

• Michael Makkai, Gonzalo Reyes, Completeness results for intuitionistic and modal logic in a categorical setting, Annals of Pure and Applied Logic 72 (1995) 25-101

(on de dicto and de re phenomena in hyperdoctrinal modal logic)

A formulation of modal logic in terms of typing judgements and type formation rules is in

• Frank Pfenning, Rowan Davies, A judgemental reconstruction of modal logic, Mathematical Structures in Comp. Sci. 11 4 (2001) 511-540 [pdf]