10.1 Syntax and semantics of modal logic

Modal logic adds necessity and possibility operators to propositional logic, allowing us to reason about what must be true and what could be true. This powerful extension lets us model and analyze concepts like knowledge, belief, and time in formal systems.

Possible worlds semantics provides the foundation for interpreting modal logic formulas. By considering different possible scenarios and their relationships, we can define precise truth conditions for modal statements and explore their logical properties.

Modal Operators and Formulas

Modal Operators and Propositional Variables

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• Modal operators express modality in modal logic
  • Modality refers to the way in which a proposition can be true or false
• Two main modal operators necessity (□) and possibility (◇)
  • Necessity (□) asserts that a proposition is true in all accessible possible worlds
  • Possibility (◇) asserts that a proposition is true in at least one accessible possible world
• Propositional variables (p, q, r) represent atomic propositions in modal logic formulas
  • Atomic propositions are the basic building blocks and cannot be further broken down

Well-formed Formulas in Modal Logic

• Well-formed formulas (wffs) are syntactically correct expressions in modal logic
  • Follow specific rules for combining propositional variables, logical connectives, and modal operators
• Formation rules for wffs in modal logic:
  1. Every propositional variable (p, q, r) is a wff
  2. If φ is a wff, then ¬φ is also a wff (negation)
  3. If φ and ψ are wffs, then (φ ∧ ψ), (φ ∨ ψ), (φ → ψ), and (φ ↔ ψ) are also wffs (conjunction, disjunction, implication, and equivalence)
  4. If φ is a wff, then □φ and ◇φ are also wffs (modal operators)
• Examples of well-formed formulas:
  • □p (it is necessary that p)
  • ◇(p ∨ q) (it is possible that either p or q)

Possible Worlds Semantics

Possible Worlds and Accessibility Relation

• Possible worlds semantics provides a framework for interpreting modal logic formulas
• A possible world represents a complete and consistent way the world could be
  • Different possible worlds can have different truth values for the same proposition
• Accessibility relation (R) is a binary relation between possible worlds
  • If world w1 is accessible from world w0 (w0Rw1), then the truth values of propositions in w1 are relevant to the truth values of modal formulas in w0
  • Accessibility relation determines which worlds are considered when evaluating modal formulas

Truth Conditions for Modal Formulas

• Truth conditions define when a modal formula is true or false in a given world
• For necessity (□φ):
  • □φ is true in a world w if and only if φ is true in all worlds accessible from w
  • Formally: w ⊨ □φ iff for all w' such that wRw', w' ⊨ φ
• For possibility (◇φ):
  • ◇φ is true in a world w if and only if φ is true in at least one world accessible from w
  • Formally: w ⊨ ◇φ iff there exists a w' such that wRw' and w' ⊨ φ
• Examples:
  • If □p is true in world w0, then p must be true in all worlds accessible from w0
  • If ◇q is true in world w0, then q must be true in at least one world accessible from w0

Logical Properties

Validity and Satisfiability in Modal Logic

• Validity and satisfiability are important logical properties in modal logic
• A modal formula φ is valid if it is true in all possible worlds under any interpretation
  • Formally: ⊨ φ iff for all models M and all worlds w in M, M, w ⊨ φ
  • Valid formulas are tautologies in modal logic (e.g., □(p ∨ ¬p))
• A modal formula φ is satisfiable if there exists at least one model and one world where it is true
  • Formally: φ is satisfiable iff there exists a model M and a world w in M such that M, w ⊨ φ
  • Satisfiable formulas are consistent and can be true in some models (e.g., ◇p)
• Validity and satisfiability are related concepts:
  • A formula is valid if and only if its negation is not satisfiable
  • A formula is satisfiable if and only if its negation is not valid
• Examples:
  • □(p → p) is a valid formula because it is true in all possible worlds
  • ◇(p ∧ ¬p) is not satisfiable because it cannot be true in any possible world