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Modal logic is a type of formal logic primarily developed in the 1960s that extends classical propositional and predicate logic to include operators expressing modality. A modal—a word that expresses a modality—qualifies a statement. For example, the statement "John is happy" might be qualified by saying that John is usually happy, in which case the term "usually" is functioning as a modal. The traditional alethic modalities, or modalities of truth, include possibility ("Possibly, p", "It is possible that p"), necessity ("Necessarily, p", "It is necessary that p"), and impossibility ("Impossibly, p", "It is impossible that p").[1] Other modalities that have been formalized in modal logic include temporal modalities, or modalities of time (notably, "It was the case that p", "It has always been that p", "It will be that p", "It will always be that p"),[2][3] deontic modalities (notably, "It is obligatory that p", and "It is permissible that p"), epistemic modalities, or modalities of knowledge ("It is known that p")[4] and doxastic modalities, or modalities of belief ("It is believed that p").[5]

A formal modal logic represents modalities using modal operators. For example, "It might rain today" and "It is possible that rain will fall today" both contain the notion of possibility. In a modal logic this is represented as an operator, "Possibly", attached to the sentence "It will rain today".

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 \Diamond 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.