🤹🏼Formal Logic II Unit 9 Review

Kripke semantics gives modal logic a concrete, visual foundation. Instead of treating "necessity" and "possibility" as abstract notions, you work with a structured collection of possible worlds and the connections between them. Truth values for modal formulas then depend on what holds across those connected worlds.

Frame properties like reflexivity, symmetry, and transitivity constrain how worlds relate to each other. Each property corresponds to a specific modal axiom, and these correspondences are what let us classify different modal logics (S4, S5, etc.) and understand exactly what they commit us to.

Kripke Frames and Models

Components and Structure

A Kripke frame is a pair $\langle W, R \rangle$ where:

$W$ is a non-empty set of possible worlds
$R$ is a binary accessibility relation on $W$

If $(w, v) \in R$ , we say world $v$ is accessible from world $w$ . Think of $R$ as encoding which worlds are "visible" or "reachable" from a given world.

A Kripke model adds a valuation function $V$ to a frame, forming a triple $\langle W, R, V \rangle$ . The valuation $V$ maps each propositional variable $p$ to a subset of $W$ , telling you at which worlds $p$ is true. So $w \in V(p)$ means $p$ is true at world $w$ .

Visualization and Examples

Kripke frames are naturally visualized as directed graphs: nodes are worlds, and a directed edge from $w$ to $v$ means $(w, v) \in R$ .

Example: Consider a model with $W = \{w_1, w_2, w_3\}$ , $R = \{(w_1, w_2), (w_2, w_3)\}$ , and $V(p) = \{w_1, w_3\}$ .

$w_2$ is accessible from $w_1$ , and $w_3$ is accessible from $w_2$
$p$ is true at $w_1$ and $w_3$ , but false at $w_2$
Notice that $w_3$ is not directly accessible from $w_1$ (there's no edge $(w_1, w_3)$ )

Truth Values in Kripke Models

The key recursive clauses for modal operators at a world $w$ in model $\mathcal{M}$ :

$\mathcal{M}, w \models \Box A$ iff for every $v$ such that $(w, v) \in R$ , $\mathcal{M}, v \models A$
$\mathcal{M}, w \models \Diamond A$ iff for some $v$ such that $(w, v) \in R$ , $\mathcal{M}, v \models A$

Non-modal formulas work as you'd expect: propositional variables get their truth value from $V$ , and Boolean connectives follow the standard truth-functional rules.

One edge case worth noting: if a world $w$ has no accessible worlds (it's a "dead end"), then $\Box A$ is vacuously true at $w$ for any $A$ , and $\Diamond A$ is false at $w$ for any $A$ .

Validity and Examples

A formula is valid in a model if it's true at every world in that model. A formula is valid on a frame if it's true at every world in every model based on that frame. This distinction matters: validity on a frame is the stronger condition.

Using the example model from above ( $W = \{w_1, w_2, w_3\}$ , $R = \{(w_1, w_2), (w_2, w_3)\}$ , $V(p) = \{w_1, w_3\}$ ):

$\Box p$ is false at $w_1$ : the only world accessible from $w_1$ is $w_2$ , and $p$ is false at $w_2$
$\Diamond p$ is true at $w_2$ : $w_3$ is accessible from $w_2$ , and $p$ is true at $w_3$
$\Box(p \lor \lnot p)$ is valid in any Kripke model, since $p \lor \lnot p$ is a tautology and therefore true at every world

Note on the original guide: It claimed $\Box p$ is true at $w_1$ because "p is true at the only accessible world (w2)." But $V(p) = \{w_1, w_3\}$ , so $p$ is false at $w_2$ . Therefore $\Box p$ is false at $w_1$ .

Frame Properties: Reflexivity, Symmetry, Transitivity

Definitions and Examples

These three properties describe structural constraints on the accessibility relation $R$ .

Reflexivity: A frame $\langle W, R \rangle$ is reflexive if for every $w \in W$ , $(w, w) \in R$ . Every world can access itself.

Example: $W = \{w_1, w_2\}$ , $R = \{(w_1, w_1), (w_2, w_2)\}$ . In the graph, every node has a self-loop.

Symmetry: A frame is symmetric if whenever $(w, v) \in R$ , then $(v, w) \in R$ . Accessibility always goes both ways.

Example: $W = \{w_1, w_2\}$ , $R = \{(w_1, w_2), (w_2, w_1)\}$ . Every edge has a matching reverse edge.

Transitivity: A frame is transitive if whenever $(w, v) \in R$ and $(v, u) \in R$ , then $(w, u) \in R$ . If you can get from $w$ to $v$ to $u$ , you can get directly from $w$ to $u$ .

Example: $W = \{w_1, w_2, w_3\}$ , $R = \{(w_1, w_2), (w_2, w_3), (w_1, w_3)\}$ . The "shortcut" edge $(w_1, w_3)$ is required by transitivity.

Proving Frame Properties

To prove a frame has a property, you show the defining condition holds for all relevant worlds in the frame.

State the property's definition clearly
Take arbitrary worlds satisfying the antecedent condition
Show the consequent follows from the structure of $R$

To disprove a property, you only need one counterexample.

To show a frame is not symmetric, find specific worlds $w, v$ where $(w, v) \in R$ but $(v, w) \notin R$
To show a frame is not transitive, find $w, v, u$ where $(w, v) \in R$ and $(v, u) \in R$ but $(w, u) \notin R$

Correspondence Between Properties and Axioms

This is one of the central results in Kripke semantics: specific frame properties correspond exactly to specific modal axioms. The main correspondences you need to know:

Frame Property	Modal Axiom	Axiom Schema
Reflexivity	T	$\Box A \to A$
Symmetry	B	$A \to \Box\Diamond A$
Transitivity	4	$\Box A \to \Box\Box A$

Each correspondence works in two directions:

Soundness direction: If a frame has the property, then the corresponding axiom is valid on that frame
Completeness direction: If the axiom is valid on a frame, then the frame must have the corresponding property

To build intuition for why these hold, consider the T axiom ( $\Box A \to A$ ). If $\Box A$ is true at $w$ , then $A$ is true at every accessible world. Reflexivity guarantees $w$ accesses itself, so $A$ must be true at $w$ too. Without reflexivity, $w$ might not access itself, and $A$ could fail at $w$ even though it holds everywhere $w$ can "see."

Named Systems and Applications

These correspondences let us classify modal logics by which axioms they adopt (on top of the base system K):

T = K + T axiom → reflexive frames
S4 = K + T + 4 → reflexive and transitive frames (preorders)
S5 = K + T + B + 4 → reflexive, symmetric, and transitive frames (equivalence relations)

S5 is particularly clean: because $R$ is an equivalence relation, the set of worlds partitions into clusters where every world in a cluster accesses every other world in that cluster. In S5, $\Box A$ is true at $w$ iff $A$ is true at every world in $w$ 's cluster.

These frame conditions also connect to different philosophical interpretations of modality. Epistemic logic (knowledge) typically uses S5 frames, since if you know something, you know that you know it (axiom 4), and if you don't know something, you know you don't know it (axiom 5/B). Deontic logic (obligation) often uses weaker frames, since not all of these introspection properties make sense for moral obligation.

🤹🏼Formal Logic II Unit 9 Review