## FANDOM

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Modal Logic, also known as Alethic Modal Logic is a term refering to a number of formal systems that deal with necessity and possibility. These systems usually are a Propositional logic that has two new symbols: $\Box$, which denotes necessity; and $\Diamond$, which denotes possibility.

## SyntaxEdit

The syntax of Modal logic is usually the syntax of propositional logic, with a new rule: If $\phi$ is a well-formed formula, then $\Box \phi$ and $\Diamond \phi$ are well-formed formulas.

## SemanticsEdit

The semantics of Modal Logic commonly is given by kripke semantics . Let $\mathcal {M}=(G,R,v)$ be a kripke model, where $(G,R)$ is a kripke frame, $G$ is a set of worlds, $R$ is the accessibility relation, and $v:G \times A \to \{t,f\}$ is a mapping from the Cartesian product of $G$ and the set of propositional variables $A$ to a set of truth values. Equivalently, $v$ is a binary relation on $G$ and the set of modal logic propositional variables.

Let $w^*$ be an element of $G$ that denotes the Actual world.

Let $\mathcal{M},w\models \phi$ denote the truth value of $\phi$ in the possible world $w$ under the model $\mathcal{M}$.

Then, the T-schema of Modal logic could be defined recursively in the following way:

• $\mathcal{M},w\models P$ if and only if $v(w,P)=t$
• $\mathcal{M},w\models \lnot \phi$ if and only if it is not the case that $\mathcal{M},w\models \phi$.
• $\mathcal{M},w\models \phi \land \psi$ if and only if $\mathcal{M},w\models \phi$ and $\mathcal{M},w\models \psi$.
• $\mathcal{M},w\models \Box \phi$ if and only if for all $x \in G$, if $wRx$ then $\mathcal{M},x\models \phi$.
• $\mathcal{M},w\models \Diamond \phi$ if and only if there is a $x \in G$ such that $wRx$ and $\mathcal{M},x\models \phi$.
• $\models \phi$ if and only if $\mathcal{M},w^*\models \phi$

## Necessitation RuleEdit

The necesitation rule is a rule of inference that states if a well-formed formula is a theorem under a system K, then the necessitation of the well-formed formula is also a theorem:

$\vdash \phi \Rightarrow \vdash \Box \phi$.

## Table of axiomsEdit

Here is a list of axioms that are commonly used in Modal Logic systems. Some of these axioms are controversial, others are not.

Name: Axiom: Condition on Frames
Distribution Axiom

$\Box(P \to Q) \to (\Box P \to \Box Q)$

$D$ $\Box P \to \Diamond P$ $\forall x \exists y(xRx)$ (Serial)
$M$ $\Box P \to P$ $\forall x (xRx)$ (Reflexive)
$4$ $\Box P \to \Box \Box P$ $\forall x,y,z (xRy \land yRz \Rightarrow xRz)$ (Transitive)
$B$ $P \to \Box \Diamond P$ $\forall x,y (xRy \Rightarrow yRx)$ (Symmetric)
$5$ $\Diamond P \to \Box P$ $\forall x,y,z (xRy \land xRz \Rightarrow yRx)$ (Eucledian)
$CD$ $\Diamond P \to \Box P$ $\forall x,y,z (xRy \land xRz \Rightarrow y=z )$(Functional
$\Box M$ $\Box(\Box P \to$ $\forall x,y (xRy \Rightarrow yRy)$ (Shift Reflexive)
$C4$ $\Box \Box P \to \Box P$ $\forall x,y (xRy \Rightarrow \exists z(xRz \land zRy)$ (Dense)
$C$ $\Diamond \Box P \to \Box \Diamond P$ $\forall x,y,z (xRy \land xRz \Rightarrow \exists w(yRw \land zRw))$ (Convergent)

Where $R$ is the accessiblity relation in the kripkle frame $(W,R)$

## List of systemsEdit

• System K: Necessitation rule and axiom N.
• System T: System K and axiom T.
• System S4: System T and axiom 4
• System D?: System K and axiom D.
• System S5: System S4 and axiom 5.

## TerminologyEdit

• True propositions: Propositions true in the actual world.
• False propositions: Propositions false in the actual world.
• Possible propositions: Propositions that are true in at least one possible world.
• Impossible propositions: Propositions that are true in no possible world.
• Necessarily true propositions: Propositions that are true in all possible worlds.
• Contingent propositions: Propositions that are true in some possible worlds and false in others.