10.2 Kripke frames and models

Kripke frames and models are the backbone of modal logic, providing a way to visualize and reason about possibility and necessity. They consist of possible worlds connected by accessibility relations, allowing us to interpret modal statements across different scenarios.

These structures help us understand various modal logics by defining different properties of accessibility relations. Reflexivity, symmetry, transitivity, and other characteristics correspond to specific axioms, shaping how we interpret necessity and possibility in different contexts.

Kripke Frames and Models

Defining Kripke Frames

• Kripke frame $F$ is a pair $(W,R)$ where $W$ is a non-empty set of possible worlds and $R$ is a binary relation on $W$ called the accessibility relation
• $W$ represents the set of all conceivable scenarios or states of affairs in the context of modal logic
• Each element $w \in W$ is a possible world that captures a specific configuration of propositions being true or false
• $R \subseteq W \times W$ determines which worlds are considered accessible from each world in the frame
• If $(w,v) \in R$, then world $v$ is accessible from world $w$, meaning that if a proposition is necessary at $w$, it must be true at $v$

Extending Frames to Models

• Kripke model $M$ is a triple $(W,R,V)$ where $(W,R)$ is a Kripke frame and $V$ is a valuation function
• $V: Prop \to 2^W$ assigns to each propositional variable $p \in Prop$ the set of worlds where $p$ is true
• For each $p \in Prop$ and $w \in W$, $w \in V(p)$ means that $p$ is true at world $w$ in the model $M$
• Truth of modal formulas at a world in a Kripke model depends on both the valuation and the accessibility relation
• Example: $M,w \models \square \varphi$ iff for all $v$ such that $(w,v) \in R$, $M,v \models \varphi$, meaning $\varphi$ is necessary at $w$ if it is true at all accessible worlds

Interpreting Accessibility

• Accessibility relation $R$ captures the notion of relative possibility between worlds in a Kripke frame or model
• Different interpretations of $R$ correspond to different modal logics and their associated axioms
• In epistemic logic, $(w,v) \in R$ means that world $v$ is considered possible from the perspective of an agent's knowledge at world $w$
• In deontic logic, $(w,v) \in R$ means that world $v$ is considered ideal or permissible according to the norms or obligations at world $w$
• In temporal logic, $(w,v) \in R$ means that world $v$ is a future state reachable from world $w$, capturing the flow of time

Properties of Accessibility Relations

Reflexivity and Symmetry

• Reflexivity: $\forall w \in W: (w,w) \in R$, meaning each world is accessible from itself
• Reflexivity corresponds to the modal axiom $\square \varphi \to \varphi$ (axiom T), capturing the idea that necessity implies truth
• Example: In epistemic logic, reflexivity means that if an agent knows $\varphi$, then $\varphi$ is actually true
• Symmetry: $\forall w,v \in W: (w,v) \in R \implies (v,w) \in R$, meaning accessibility is bidirectional
• Symmetry corresponds to the modal axiom $\varphi \to \square \Diamond \varphi$ (axiom B), capturing the idea that truth implies possibility of necessity
• Example: In epistemic logic, symmetry means that if $\varphi$ is considered possible by an agent, then the agent considers it possible to know $\varphi$

Transitivity and Euclideanness

• Transitivity: $\forall w,v,u \in W: (w,v) \in R \wedge (v,u) \in R \implies (w,u) \in R$, meaning accessibility is transitive
• Transitivity corresponds to the modal axiom $\square \varphi \to \square \square \varphi$ (axiom 4), capturing the idea that necessity is idempotent
• Example: In epistemic logic, transitivity means that if an agent knows that they know $\varphi$, then they know $\varphi$
• Euclidean property: $\forall w,v,u \in W: (w,v) \in R \wedge (w,u) \in R \implies (v,u) \in R$, meaning worlds accessible from the same world are accessible from each other
• Euclideanness corresponds to the modal axiom $\Diamond \varphi \to \square \Diamond \varphi$ (axiom 5), capturing the idea that possibility of necessity implies necessity of possibility
• Example: In epistemic logic, Euclideanness means that if an agent considers $\varphi$ possible and $\psi$ possible, then in situations where $\varphi$ is true, the agent must also consider $\psi$ possible

Seriality and Combinations

• Seriality: $\forall w \in W, \exists v \in W: (w,v) \in R$, meaning each world has at least one accessible world
• Seriality corresponds to the modal axiom $\square \varphi \to \Diamond \varphi$ (axiom D), capturing the idea that necessity implies possibility
• Example: In deontic logic, seriality means that for each situation, there is at least one ideal or permissible situation
• Combining properties of accessibility relations leads to different modal logics with specific axiomatizations
• Example: Modal logic S4 has a reflexive and transitive accessibility relation, characterized by axioms K, T, and 4
• Example: Modal logic S5 has a reflexive, symmetric, and transitive accessibility relation, characterized by axioms K, T, B, 4, and 5