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

can be used to represent the statement that

is known.

Deontological modal logic, that same formula can represent that

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

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

and

,indicating both a possibility and a requirement.

For instance the modal formula

can be read as "possibly

"while

can be read as "necessarily

".

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,

is true at a world if

is true at some accessible possible world, while

is true at a world if

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

and

.

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

, it also contains modal ones such as

,

,

, and so forth.

Thus, the language

of basic propositional logic can be defined recursively as follows.

If

is an atomic formula, then

is a formula of

.

If

is a formula of

, then

is too.

If

and

are formulas of

, then

is too.

If

is a formula of

, then

is too.

If

is a formula of

, then

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

.

In systems of modal logic where

and

are duals,

can be taken as an abbreviation for

, 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

and

in systems of modal logic used to represent knowledge and

and

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

read as "I know P is permitted".

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

,

,

, 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

where:

is a set of possible worlds

is a binary relation on

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

where

is the set of atomic formulae)

The set

is often called the universe.

The binary relation

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

For example,

means that the world

is accessible from world

.

Thus, to sum up, the state of affairs known as

is a live possibility for

.

Finally, the function

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

in a model

:

iff

iff

iff

and

iff for every element

of

, if

then

iff for some element

of

, it holds that

and

In light of this semantics, a formula is necessary with respect to a world

if it holds at every world that is accessible from

.

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

.

Possibility thereby depends upon the accessibility relation

, 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

whose accessibility relation is reflexive.

due to the relationship's reflexivity, we will have that

for any

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

where

is a set of possible worlds,

is a binary relation on

.

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:

if and only if for some element u of G, it holds that

and w R u.

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

if and only if for some element u of G, it holds that

.

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...