Possible worlds and accessibility relations expand classical logic, introducing necessity and possibility. These concepts allow us to evaluate propositions across different scenarios, providing a framework for analyzing modal statements and their truth values.

Accessibility relations determine which worlds are reachable from others, impacting the validity of modal formulas. Properties like reflexivity, symmetry, and transitivity shape how we interpret necessity and possibility in different models, influencing our understanding of modal logic.

Possible Worlds and Accessibility Relations

Concept of possible worlds

• Represent different states or scenarios in which propositions can be evaluated
• Allow a proposition to be true in one possible world and false in another (raining in London, sunny in Paris)
• Extend classical logic by incorporating the concepts of necessity and possibility
• Necessity ($\square$) means a proposition is true in all accessible possible worlds (all humans are mortal)
• Possibility ($\diamond$) means a proposition is true in at least one accessible possible world (it is possible to win the lottery)
• Enable the analysis of modal statements and their truth values across different scenarios (in some possible worlds, pigs can fly)

Properties of accessibility relations

• Determine which possible worlds are reachable from a given world
• If world w1 is accessible from world w0, then the propositions that are necessary in w0 must be true in w1 (if it is necessary that all birds have wings in w0, then in all accessible worlds like w1, all birds must have wings)
• Reflexivity means a world is always accessible from itself (w0 is accessible from w0)
• Symmetry means if world w1 is accessible from world w0, then w0 is also accessible from w1 (if w0 is accessible from w1 and vice versa)
• Transitivity means if world w1 is accessible from w0, and world w2 is accessible from w1, then w2 is also accessible from w0 (if w0 is accessible from w1, w1 is accessible from w2, and w0 is accessible from w2)

Impact of accessibility on validity

• The properties of accessibility relations affect the validity of modal formulas
• In a reflexive model, the formula $\square p \rightarrow p$ is always valid because if a proposition is necessary, it must also be true in the current world (if it is necessary that 2+2=4, then 2+2 must equal 4 in the current world)
• In a symmetric model, the formula $p \rightarrow \square \diamond p$ is always valid because if a proposition is true, then it is necessary that it is possible (if it is raining, then it must be possible that it is raining)
• In a transitive model, the formula $\square p \rightarrow \square \square p$ is always valid because if a proposition is necessary, it is also necessary that it is necessary (if it is necessary that all humans are mortal, then it is necessary that it is necessary that all humans are mortal)

Models for modal propositions

• Create a model with possible worlds and accessibility relations to evaluate the truth value of a modal proposition
• Assign truth values to propositional variables in each possible world (p is true in w0, false in w1)
• Use the accessibility relations to determine the truth values of modal formulas
  1. A formula $\square p$ is true in a world if $p$ is true in all accessible worlds (if p is true in all worlds accessible from w0, then $\square p$ is true in w0)
  2. A formula $\diamond p$ is true in a world if $p$ is true in at least one accessible world (if p is true in at least one world accessible from w0, then $\diamond p$ is true in w0)
• A modal formula is valid if it is true in all possible worlds of a model (in all worlds of the model, $\square p \rightarrow p$ holds)
• A modal formula is satisfiable if it is true in at least one possible world of a model (in at least one world of the model, $\square p$ holds)