Modal operators extend classical logic by introducing concepts of necessity and possibility. These operators, symbolized as $\square$ and $\diamond$, allow us to reason about what must be true and what could be true across different possible worlds.

Possible world semantics provide a framework for evaluating modal propositions. By considering the truth values of propositions in accessible worlds, we can determine if statements are necessary, possible, or contingent. This approach enriches our logical toolkit beyond traditional propositional logic.

Modal Operators and Their Semantics

Modal operators and symbols

• Modal operators extend classical propositional logic by expressing concepts of necessity ($\square$) and possibility ($\diamond$)
  • $\square P$ reads as "It is necessary that P" or "P is necessarily true"
  • $\diamond P$ reads as "It is possible that P" or "P is possibly true"
• Modal operators are unary connectives that operate on a single proposition (P)

Semantics of modal vs propositional logic

• Modal logic introduces the concept of possible worlds to evaluate the truth of propositions
  • Possible world: a complete description of a way the world could be
• In modal logic, a proposition is true or false relative to a particular world
  • Classical propositional logic only considers the actual world
• Accessibility relation: a binary relation between possible worlds
  • If world w1 is accessible from world w0, then any proposition necessary in w0 must be true in w1

Possible world semantics

• To evaluate the truth of a modal proposition, consider the truth values of its subpropositions across accessible worlds
• Example: $\square (P \rightarrow Q)$
  • True in a world w if and only if $P \rightarrow Q$ is true in all worlds accessible from w
    • In every accessible world where P is true, Q must also be true
• Example: $\diamond (P \land Q)$
  • True in a world w if and only if there exists at least one accessible world where both P and Q are true

Types of modal propositions

• Necessary proposition: true in all possible worlds
  • "2 + 2 = 4"
  • Symbolically: $\square P$, where P is the proposition
• Possible proposition: true in at least one possible world
  • "It is raining outside"
  • Symbolically: $\diamond P$
• Contingent proposition: true in some possible worlds and false in others
  • Neither necessarily true nor necessarily false
  • "The coin lands on heads"
• Relationship between modal operators:
  • $\square P \equiv \lnot \diamond \lnot P$
    • If P is necessary, then not-P is not possible
  • $\diamond P \equiv \lnot \square \lnot P$
    • If P is possible, then not-P is not necessary