Modal Logic: Fundamentals and Applications

By Fouad Sabry

What Is Modal Logic

Statements regarding necessity and possibility can be represented with the use of a type of logic known as modal logic. As a method for gaining a grasp of ideas like knowledge, obligation, and causality, it is an essential component of philosophy and other subjects that are closely related to it. For instance, the formula can be used to describe the statement that is known in the epistemic modal logic. Using the same formula, one can express that which is a moral responsibility within the framework of deontic modal logic. The conclusions that can be drawn from modal assertions are taken into consideration by modal logic. For instance, the majority of epistemic logics consider the formula to be a tautology, which is a representation of the concept that the only assertions that may be considered to have knowledge are those that are true.

How You Will Benefit

(I) Insights, and validations about the following topics:

Chapter 1: Modal Logic

Chapter 2: First-order Logic

Chapter 3: Propositional Calculus

Chapter 4: Saul Kripke

Chapter 5: Kripke Semantics

Chapter 6: Temporal Logic

Chapter 7: Epistemic Modal Logic

Chapter 8: Accessibility Relation

Chapter 9: S5 (Modal Logic)

Chapter 10: Dynamic Logic (Modal Logic)

(II) Answering the public top questions about modal logic.

(III) Real world examples for the usage of modal logic in many fields.

(IV) 17 appendices to explain, briefly, 266 emerging technologies in each industry to have 360-degree full understanding of modal logic' technologies.

Who This Book Is For

Professionals, undergraduate and graduate students, enthusiasts, hobbyists, and those who want to go beyond basic knowledge or information for any kind of modal logic.

• Language: English
• Publisher: One Billion Knowledgeable
• Release date: Jun 30, 2023

Book preview

Chapter 1: Modal logic

Statements regarding need and possibility are represented using a type of reasoning known as modal logic.

It is a crucial component of philosophy and related fields as a tool for comprehending ideas like knowing, obligation, and causation.

For instance, in modal epistemic logic, the formula \Box P can be used to represent the statement that P is known.

Deontological modal logic, that same formula can represent that P is a moral obligation.

The inferences that modal statements lead to are taken into account by modal logic.

For instance, most epistemic logics treat the formula {\displaystyle \Box P\rightarrow P} as a tautology, representing the idea that knowledge may only be derived from true statements.

Modal logics are formal systems that include unary operators such as \Diamond and \Box , indicating both a possibility and a requirement.

For instance the modal formula \Diamond P can be read as possibly P while \Box P can be read as necessarily P .

Relational semantics for modal logic, the norm, Formulas are given truth values based on a hypothetical world.

The truth values of other formulas at other accessible possible worlds can influence the truth value of a formula at one possible world.

In particular, \Diamond P is true at a world if P is true at some accessible possible world, while \Box P is true at a world if P is true at every accessible possible world.

There are numerous proof systems that are valid and comprehensive with regard to the semantics obtained by limiting the accessibility relation.

For instance, modal logic deontic If one needs the accessibility relation to be serial, D is sound and complete.

Although the idea of modal logic has existed since antiquity, C. I. Lewis created the first modal axiomatic systems in 1912. The work of Arthur Prior, Jaakko Hintikka, and Saul Kripke in the middle of the 20th century gave rise to the now-standard relational semantics. Alternative topological semantics, like neighborhood semantics, and relational semantics applications that go beyond their philosophical roots are recent advances.

Modal logic differs from other kinds of logic in that it uses modal operators such as \Box and \Diamond .

The first one is typically said aloud as necessarily, and can be employed to symbolize ideas like moral or legal obligation, knowledge, historical inevitability, among others.

The latter can be used to denote ideas such as permission and is often read as perhaps., ability, consistency with the evidence.

While well formed formulas of modal logic include non-modal formulas such as P\land Q , it also contains modal ones such as {\displaystyle \Box (P\land Q)} , {\displaystyle P\land \Box Q} , {\displaystyle \Box (\Diamond P\land \Diamond Q)} , and so forth.

Thus, the language {\mathcal {L}} of basic propositional logic can be defined recursively as follows.

If \phi is an atomic formula, then \phi is a formula of {\mathcal {L}} .

If \phi is a formula of {\mathcal {L}} , then \neg \phi is too.

If \phi and \psi are formulas of {\mathcal {L}} , then \phi \land \psi is too.

If \phi is a formula of {\mathcal {L}} , then \Diamond \phi is too.

If \phi is a formula of {\mathcal {L}} , then \Box \phi is too.

By implementing rules similar to #4 and #5 above, modal operators can be extended to different types of logic.

Modal predicate logic is one widely used variant which includes formulas such as {\displaystyle \forall x\Diamond P(x)} .

In systems of modal logic where \Box and \Diamond are duals, \Box \phi can be taken as an abbreviation for {\displaystyle \neg \Diamond \neg \phi } , so obviating the requirement for an additional syntactic rule to introduce it.

However, In systems where the two operators are not interdefinable, distinct syntactic rules are required.

Common notational variants include symbols such as {\displaystyle [K]} and {\displaystyle \langle K\rangle } in systems of modal logic used to represent knowledge and [B] and \langle B\rangle in those used to represent belief.

These notations are particularly prevalent in systems that employ several modal operators at once.

For instance, a combined epistemic-deontic logic could use the formula {\displaystyle [K]\langle D\rangle P} read as I know P is permitted.

There are an infinite number of modal operators distinguishable by indices in modal logic systems, i.e.

{\displaystyle \Box _{1}} , {\displaystyle \Box _{2}} , {\displaystyle \Box _{3}} , and so forth.

The relational semantics is the recognized semantics for modal logic. With this method, the veracity of a formula is assessed in relation to a point that is frequently referred to as a possible world. A modal operator's truth value can change depending on what is true in other accessible worlds. As a result, relational semantics uses the models described below to interpret modal logic formulations.

A relational model is a tuple {\displaystyle {\mathfrak {M}}=\langle W,R,V\rangle } where:

W is a set of possible worlds

R is a binary relation on W

V is a valuation function which assigns a truth value to each pair of an atomic formula and a world, (i.e.

{\displaystyle V:W\times F\to \{0,1\}} where F is the set of atomic formulae)

The set W is often called the universe.

The binary relation R is called an accessibility relation, and it regulates which worlds can see each other in order to establish what is real.

For example, {\displaystyle wRu} means that the world u is accessible from world w .

Thus, to sum up, the state of affairs known as u is a live possibility for w .

Finally, the function V is known as a valuation function.

It establishes which worlds have valid atomic formulas.

Then we recursively define the truth of a formula at a world w in a model \mathfrak{M} :

{\displaystyle {\mathfrak {M}},w\models P} iff {\displaystyle V(w,P)=1}

{\displaystyle {\mathfrak {M}},w\models \neg P} iff w\not \models P

{\displaystyle {\mathfrak {M}},w\models (P\wedge Q)} iff w\models P and w\models Q

{\displaystyle {\mathfrak {M}},w\models \Box P} iff for every element u of W , if {\displaystyle wRu} then u\models P

{\displaystyle {\mathfrak {M}},w\models \Diamond P} iff for some element u of W , it holds that {\displaystyle wRu} and u\models P

In light of this semantics, a formula is necessary with respect to a world w if it holds at every world that is accessible from w .

It is possible if it holds at some world that is accessible from w .

Possibility thereby depends upon the accessibility relation R , It enables us to express how relative possibility is.

For example, We could claim that, based on our physical rules, humans cannot go faster than the speed of light, but that under different conditions, it might have been able to accomplish that.

In order to translate this situation, we can use the accessibility relation as follows: All accessible worlds, including the one we live in, Contrary to popular belief, people cannot move at the speed of light, However, at one of these reachable planets, there is a world that is reachable from those worlds but not from our own, and where people can move at the speed of light.

Sometimes the choice of accessibility relation alone is enough to determine whether a formula is true or false.

For instance, consider a model \mathfrak{M} whose accessibility relation is reflexive.

due to the relationship's reflexivity, we will have that {\displaystyle {\mathfrak {M}},w\models P\rightarrow \Diamond P} for any {\displaystyle w\in G} regardless of which valuation function is used.

Because of this, Sometimes, modal logicians will discuss frames, which is the part of a relational model that does not include the valuation function.

A relational frame is a pair {\displaystyle {\mathfrak {M}}=\langle G,R\rangle } where G is a set of possible worlds, R is a binary relation on G .

Utilizing frame conditions, modal logic's various systems are defined. A frame is known as:

If w R w, then every w in G is reflexive.

symmetric if for any w and u in G, w R u implies u R w

If all w, u, and q in G are transitive, then w R u and u R q together entail w R q.

For each w in G, there must be some u in G such that w R u.

Euclidean if implies that for any u, t, and w, w R u and w R t (by symmetry, it also implies t R u, as well as t R t and u R u)

These frame conditions' underlying logics are:

K:= no requirements

D := serial

T := reflexive

B: = symmetrical and reflexive

S4:= Transitive and reflexive

S5: = Euclidean and reflexive

Symmetry and transitivity are produced by the Euclidean property and reflexivity. (Symmetry and transitivity can also be used to derive the Euclidean property.) The accessibility relation R is therefore provably symmetric and transitive if it is reflexive and Euclidean. R is therefore an equivalence relation for models of S5 because it is reflexive, symmetric, and transitive.

We can demonstrate that these frames provide the same collection of true sentences as those that allow all worlds to view all other worlds of W. (i.e., where R is a total relation). As a result, the relevant modal graph is entirely finished (i.e., no additional edges or relationships can be added). In any modal logic dependent on frame conditions, for instance:

w\models \Diamond P if and only if for some element u of G, it holds that u\models P and w R u.

If we take into account frames depending on the overall relation, we may simply state that.

w\models \Diamond P if and only if for some element u of G, it holds that u\models P .

Because it is trivially true of every w and u that w R u in such total frames, we can remove the accessibility requirement from the later stipulation. However, keep in mind that this is not always the case in S5 frames, since some of them may still have numerous components that are fully related to one another but not to each other.

Axiomatic definitions of each of these logical systems are also possible, as demonstrated in the section below.

For example, in S5, the axioms {\displaystyle P\implies \Box \Diamond P} , {\displaystyle \Box P\implies \Box \Box P} and {\displaystyle \Box P\implies P} (corresponding to symmetry, Continuity and reflection, correspondingly) hold, while one of these axioms does not apply in each of the other situations,, weaker logics.

Topological structures have also been used to interpret modal logic. For instance, the Interior Semantics reads the modal logic formulations as follows.

A topological model is a tuple {\displaystyle \mathrm {X} =\langle X,\tau ,V\rangle } where {\displaystyle \langle X,\tau \rangle } is a topological space and V is a valuation function which maps each atomic formula to some subset of X .

The fundamental interior semantics reads modal logic's formulations as follows::

{\displaystyle \mathrm {X} ,x\models P} iff {\displaystyle x\in V(P)}

{\displaystyle \mathrm {X} ,x\models \neg \phi } iff {\displaystyle \mathrm {X} ,x\not \models \phi }

{\displaystyle \mathrm {X} ,x\models \phi \land \chi } iff {\displaystyle \mathrm {X} ,x\models \phi } and {\displaystyle \mathrm {X} ,x\models \chi }

{\displaystyle \mathrm {X} ,x\models \Box \phi } iff for some {\displaystyle U\in \tau } we have both that {\displaystyle x\in U} and also that {\displaystyle \mathrm {X} ,y\models \phi } for all {\displaystyle y\in U}

Relational techniques are absorbed by topological ones, enabling atypical modal logics. A clear approach of representing some notions, such as the support or justification one has for their beliefs, is also made possible by the additional structure they offer. Topological semantics has roots in older work like David Lewis and Angelika Kratzer's logics for counterfactuals and has been used extensively in formal epistemology in recent years.

Axiomatic rules guided the earliest formalizations of modal logic. Since C. I. Lewis started researching the area in 1912, other versions with extremely diverse characteristics have been presented. For instance, Hughes and Cresswell (1996) define 42 regular and 25 irregular modal logics. Hughes and Cresswell omit some systems that Zeman (1973) discusses.

Modern presentations of modal logic start by adding two unary operations to the propositional calculus, the first signifying necessity and the second possibility.

the C notation.

I.

Lewis, several jobs since, denotes necessarily p by a prefixed box (□p) whose scope is established by parentheses.

Likewise, a prefixed diamond (◇p) denotes possibly p.

Similar to first-order logic's quantifiers, necessarily p (□p) does not assume the range of quantification (the set of accessible possible worlds in Kripke semantics) to be non-empty, whereas possibly p (◇p) often implicitly assumes \Diamond \top (viz.

The universes that are reachable are not empty.).

Whatever the notation, In traditional modal logic, each of these operators can be defined in terms of the other:

□p (necessarily p) is equivalent to ¬◇¬p (not possible that not-p)

◇p (possibly p) is equivalent to ¬□¬p (not necessarily not-p)

Hence □ and ◇ form a dual pair of operators.

The following equivalents of de Morgan's laws from boolean algebra are satisfied by the necessity and possibility operators in numerous modal logics:

The logical equivalent of It is possible that not X is It is not necessary that X..

It is required that not X and It is not feasible that X are logically identical.

It is a matter of philosophical opinion, frequently influenced by the theorems one wishes to prove; or, in computer science, it is a matter of what kind of computational or deductive system one wishes to model, as to precisely what axioms and rules must be added to the propositional calculus to create a usable system of modal logic. The following rule and axiom are common to many modal logics, also referred to as normal modal logics:

N, Rule of Necessity: If p is a theorem or tautology (of any model or system invoking N), then □p is likewise a theorem (i.e.

{\displaystyle (\models p)\implies (\models \Box p)} ).

K, Distribution Axiom: □(p → q) → (□p → □q).

The modal logic's weakest syllogism, called K in Saul Kripke's honor, is simply the propositional calculus augmented by □, the N rule, and K, an axiom.

K is weak since it can't tell whether a statement can be required; it can only tell whether it's contingently necessary.

That is, it is not a theorem of K that if