Modal/Alethic Logic - Introduction

Introduction

Modal logic is a family of different related systems that provide a variety of different operators that have their own formation rules and rules of inference.

There are two operators:

  1. Possibility: ⬦, It is possible that.
  2. Necessity: □, it is necessary that.

Modal logic has been relatively useful in expanding the range of expression used in propositional logic.

The Distribution Axiom

If we have statements like this, “It is necessary that Q if P.”. Then we can deduce, “If it is necessary that P, then it is necessary that Q.”

This is called the distribution axiom.

Distribution Axiom: □( □(P→Q))⊢  □□P→ □Q.

For instance, if it is necessary that a square is a rectangle, then if it is necessary a shape is a square, then it is necessary that the shape is a rectangle.

Negation Rules

The following rules also hold:

□( □P)⊢¬⬦¬P 

□( ⬦P)⊢¬□¬P 

These rules tell us something like the following:

It is necessary that 1=1. This implies that it is not possible that 1 does not equal 1.

It is possible the moon is old. This implies it is not necessary that the moon is not old.

Axiom M, S4, S5

Axiom M says that if P is necessary, then P.

Axiom M: □( □P)⊢P 

This axiom is important because it allows us to produce some iterations:

□( □P)⊢□□P, if P is necessary, then P is necessarily necessary.

□( ⬦P)⊢□⬦P, if P is possible, then P is necessarily possible.

The following is an attempt to simplify:

S4: □( □P)⊢□(□(□□P)) ,□(□(□(□P) ) ) ⊢□(□P) ,□( ⬦P)⊢⬦⬦P ,⬦⬦P⊢⬦P, you can collapse redundant modal operators.

We also have a rule for the last operator in a proposition:

S5: □( □⬦P)⊢⬦P,□( ⬦□P)⊢□P, only the last operator in a proposition matters.

Some Other Classifications

There are other ways to classify statements which use modal operators:

Necessity: □( □P),□(□(¬⬦¬P)) 

Possibility: □( ⬦P),¬□¬P 

Analyticity: □P∨□¬P, ¬(⬦P∧¬⬦P)

Contingency: ⬦P∧¬⬦P, ¬(□□P∨□¬P)

Impossibility: □( ¬⬦P),□¬P


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