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: , which denotes necessity; and , which denotes possibility.
The syntax of Modal logic is usually the syntax of propositional logic, with a new rule: If is a well-formed formula, then and are well-formed formulas.
The semantics of Modal Logic commonly is given by kripke semantics . Let be a kripke model, where is a kripke frame, is a set of worlds, is the accessibility relation, and is a mapping from the Cartesian product of and the set of propositional variables to a set of truth values. Equivalently, is a binary relation on and the set of modal logic propositional variables.
Let be an element of that denotes the Actual world.
Let denote the truth value of in the possible world under the model .
Then, the T-schema of Modal logic could be defined recursively in the following way:
- if and only if
- if and only if it is not the case that .
- if and only if and .
- if and only if for all , if then .
- if and only if there is a such that and .
- if and only if
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:
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|
Where is the accessiblity relation in the kripkle frame
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.
- 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.