9.1 Introduction to modal logic

Modal logic adds a new dimension to classical logic, introducing operators for necessity and possibility. It allows us to reason about what must be true, what could be true, and what's impossible across different possible worlds.

This powerful framework lets us analyze complex statements about knowledge, belief, and time. By formalizing these concepts, modal logic provides tools for tackling philosophical questions and practical problems in computer science and artificial intelligence.

Modal Logic Basics

Key Concepts and Syntax

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• Modal logic extends classical propositional and predicate logic by introducing modal operators to express modalities (necessarily, possibly)
• The basic syntax of modal logic includes:
  • Propositional variables ($p$, $q$, $r$, etc.)
  • Logical connectives ($\neg$, $\land$, $\lor$, $\to$, $\leftrightarrow$)
  • Modal operators ($\square$ for necessity, $\lozenge$ for possibility)
• In modal logic, a well-formed formula (wff) is defined recursively:
  • Propositional variables are wffs
  • If $\phi$ and $\psi$ are wffs, then $\neg\phi$, $(\phi \land \psi)$, $(\phi \lor \psi)$, $(\phi \to \psi)$, $(\phi \leftrightarrow \psi)$, $\square\phi$, and $\lozenge\phi$ are also wffs

Semantics and Accessibility

• The semantics of modal logic is based on the concept of possible worlds where a proposition can be true in some worlds and false in others
• The accessibility relation between possible worlds determines which worlds are "accessible" from a given world and plays a crucial role in the interpretation of modal formulas
• For example, in a model with two worlds $w_1$ and $w_2$, if $w_1$ is accessible from $w_2$, then any proposition that is necessarily true in $w_2$ must also be true in $w_1$
• Different properties of the accessibility relation (reflexivity, symmetry, transitivity) give rise to different modal logic systems with varying sets of valid formulas

Modalities: Necessity, Possibility, Contingency

Defining Modalities

• Necessity ($\square\phi$) means that a proposition $\phi$ is true in all accessible possible worlds
• Possibility ($\lozenge\phi$) means that a proposition $\phi$ is true in at least one accessible possible world
• Contingency means that a proposition is neither necessarily true nor necessarily false, being true in some accessible worlds and false in others
• Impossibility ($\neg\lozenge\phi$) and non-necessity ($\neg\square\phi$) can be derived from the basic modalities of necessity and possibility

Relationships between Modalities

• The relationships between modalities can be expressed using logical equivalences:
  • $\lozenge\phi \equiv \neg\square\neg\phi$ (possibility is the negation of the necessity of the negation)
  • $\square\phi \equiv \neg\lozenge\neg\phi$ (necessity is the negation of the possibility of the negation)
• These equivalences allow for the conversion between different modalities and the simplification of modal formulas
• For example, the formula $\neg\square\neg p$ can be simplified to $\lozenge p$ using the equivalence $\lozenge\phi \equiv \neg\square\neg\phi$

Possible World Semantics for Modal Logic

Kripke Models

• A Kripke model for modal logic consists of:
  • A set of possible worlds
  • An accessibility relation between worlds
  • A valuation function that assigns truth values to propositional variables in each world
• The truth value of a modal formula $\phi$ in a world $w$ is determined by the truth values of its subformulas in the worlds accessible from $w$
• A formula $\square\phi$ is true in a world $w$ if and only if $\phi$ is true in all worlds accessible from $w$
• A formula $\lozenge\phi$ is true in a world $w$ if and only if $\phi$ is true in at least one world accessible from $w$

Properties of Accessibility Relations

• The properties of the accessibility relation determine the specific modal logic system and the valid formulas within that system
• Some common properties include:
  • Reflexivity: Every world is accessible from itself ($\forall w: wRw$)
  • Symmetry: If a world $w_1$ is accessible from $w_2$, then $w_2$ is also accessible from $w_1$ ($\forall w_1, w_2: w_1Rw_2 \to w_2Rw_1$)
  • Transitivity: If $w_1$ is accessible from $w_2$ and $w_2$ is accessible from $w_3$, then $w_1$ is accessible from $w_3$ ($\forall w_1, w_2, w_3: (w_1Rw_2 \land w_2Rw_3) \to w_1Rw_3$)
• Different combinations of these properties give rise to different modal logic systems, such as K, T, S4, and S5, each with its own set of valid formulas and axioms

Modal Logic Argumentation

Constructing Arguments

• Modal logic can be used to formalize and analyze arguments involving necessity, possibility, and other modalities
• The basic principles of modal logic, such as the distribution axiom ($\square(\phi \to \psi) \to (\square\phi \to \square\psi)$) and the necessitation rule (if $\vdash \phi$, then $\vdash \square\phi$), can be used to derive new modal formulas from given premises
• For example, given the premises $\square(p \to q)$ and $\square p$, one can derive the conclusion $\square q$ using the distribution axiom:
  1. $\square(p \to q)$ (premise)
  2. $\square p$ (premise)
  3. $\square(p \to q) \to (\square p \to \square q)$ (instance of the distribution axiom)
  4. $\square p \to \square q$ (modus ponens from 1 and 3)
  5. $\square q$ (modus ponens from 2 and 4)

Evaluating Arguments

• The validity of a modal argument can be evaluated by constructing a counterexample using possible world semantics, i.e., finding a Kripke model where the premises are true, but the conclusion is false
• For example, to show that the argument $\square p \to \square q, \lozenge p \vdash \lozenge q$ is invalid, one can construct a model with two worlds $w_1$ and $w_2$ such that:
  • $p$ is true in $w_1$ and false in $w_2$
  • $q$ is false in both $w_1$ and $w_2$
  • $w_1$ is accessible from $w_2$
• In this model, the premises $\square p \to \square q$ and $\lozenge p$ are true, but the conclusion $\lozenge q$ is false, demonstrating the invalidity of the argument
• The choice of the appropriate modal logic system depends on the specific context and the intended interpretation of the modalities involved in the argument