frame definability

PHIL 452 · Modal Logic

Gabriel Uzquiano

frame definability

What emerged from our discussion of frames is that different classes of frames are characterized by distinctive formulas of modal propositional logic.

modal definability

A formula \(\varphi\) of \(\mathcal{L}^\Box\) modally defines a class of frames \(\mathcal{C}\) iff for all frames \((W, R)\),

\[\begin{array}{lll} (W, R) \models \varphi & \text{iff} & (W, R) \in \mathcal{C} \end{array}\]

• \(\textsf{T}\) modally defines the class of reflexive frames

• \(\textsf{B}\) modally defines the class of symmetric frames

• \(\textsf{4}\) modally defines the class of transitive frames

• \(\textsf{5}\) modally defines the class of euclidean frames

first-order definability

Let \(\mathcal{L}\) be a first-order language with a binary predicate \(R\) as part of the non-logical vocabulary.

A formula \(\varphi\) of first-order \(\mathcal{L}\) first-order defines a class of frames \(\mathcal{C}\) iff for all frames \((W, R)\),

\[\begin{array}{lll} (W, R) \models \varphi & \text{iff} & (W, R) \in \mathcal{C} \end{array}\]

• \(\forall x Rxx\) first-order defines the class of reflexive frames

• \(\forall x \forall y (Rxy \to Ryx)\) first-order defines the class of symmetric frames

• \(\forall x \forall y \forall z (Rxy \wedge Ryz \to Rxz)\) first-order defines the class of transitive frames

• \(\forall x \forall y \forall z (Rxy \wedge Rxz \to Ryz)\) first-order defines the class of euclidean frames

• \(\forall x \forall y \ (Rxy \leftrightarrow x =y)\) first-order defines the class of identity frames

• \(\forall x \forall y \ Rxy\) first-order defines the class of universal frames

are all first-order definable classes of frames modally definable?

Some negative results:

• \(\forall x \ \neg Rxx\) first-order defines the class of irreflexive frames.

• no formula of \(\mathcal{L}^\Box\) modally defines the class of irreflexive frames.

• \(\forall x \forall y (Rxy \to \neg Ryx)\) first-order defines the class of asymmetric frames.

• no formula of \(\mathcal{L}^\Box\) modally defines the class of asymmetric frames.

• \(\forall x \forall y \forall z (Rxy \wedge Ryz \to \neg Rxz)\) first-order defines the class of intransitive frames.

• no formula of \(\mathcal{L}^\Box\) modally defines the class of intransitive frames.

• \(\forall x \forall y (Rxy \leftrightarrow x \neq y)\) first-order defines the class of difference frames.

• no formula of \(\mathcal{L}^\Box\) modally defines the class of difference frames.

• \(\forall x \ Rxx\) first-order defines the class of universal frames.

• no formula of \(\mathcal{L}^\Box\) modally defines the class of universal frames.

are all modally definable classes of frames first-order definable

Some negative results:

• \(\Box (\Box p \to p) \to \Box p\) (Löb) modally defines the class of transitive converse well-founded frames.

• no first-order formula defines the class of transitive converse well-founded frames.

• \(\Box \Diamond p \to \Diamond \Box p\) (McKinsey) modally defines the class of unsplittable frames.

• no first-order formula defines the class of unsplittable frames.

  A coloring of a frame \((W, R)\) is a function \(c: W \to \{ \text{red}, \text{blue}\}\). \(R(u)\) is the set of \(R\)-successors of each member \(u \in W\), that is, \(R(u) = \{v \in W: Ruv\}\). Given a coloring \(c\) for a frame \((W, R)\), call a world \(u \in W\) split by \(c\) iff \(R(v)\) contains both a red world and a blue world.

  Call a frame unsplittable iff no coloring splits every successor of every world. Given the McKinsey formula, if every world accessible from a given world \(u\) sees a red world, then some successor world of \(u\) sees only red.

• \(\Box (\Box (p \to \Box p) \to p) \to p\) (Grzegorczyk) modally defines the class of partial orders with no infinite strictly ascending \(R\)-chains.

• no first-order formula defines the class of partial orders with no infinite strictly ascending \(R\)-chains.

example: transitive converse well-founded frames

Proposition 1 No first-order formula \(\psi\) is true in all and only the transitive converse well-founded frames.

A relation \(R\) on a set \(W\) converse well-founded iff there are no infinite ascending \(R\)-chains of the form \(x_1Rx_2 \cdots x_nRx_{n+1} \cdots\).

This is a consequence of Compactness: if a set of first-order formulas \(\Gamma\) is finitely satisfiable, then \(\Gamma\) is satisfiable.

Proof. Suppose, for contradiction, that \(\psi\) is true in all and only the transitive converse well-founded frames.

• Let \(\Sigma\) be a set of first-order formulas that require \(R\) to be transitive.

• Add infinitely many new constants \(c_1, c_2, c_3, \dots , c_n, \dots\) and let \[ \Gamma_0 := \{Rc_ic_{i+1} : i \geq 1\}. \]

• Claim. \(\Sigma \cup \Gamma_0 \cup \{\psi\}\) is finitely satisfiable.

step 1: finite satisfiability

Let \(\Delta\) be a finite subset of \(\Sigma \cup \Gamma_0 \cup \{\psi\}\).

• Let \(n\) be the largest index of a \(c_i\) in \(\Delta\) (or \(1\) if there is none).

• Let \(W = \{w_1, \dots, w_n\}\), let \(R\) be the transitive closure of a strict order \(w_1 R w_2 R \cdots R w_n\), and interpret \(c_i\) as \(w_i\).

• \((W, R)\) is transitive and converse well-founded

• By assumption, \((W, R) \models \psi\).

• \((W, R)\) satisfies \(\Sigma\) and every \(Rc_ic_{i+1}\) in \(\Delta\).

So, \(\Delta\) is satisfiable.

step 2: compactness

By Compactness, \(\Sigma \cup \Gamma_0 \cup \{\psi\}\) has a model \((W, R)\):

• \(R\) is transitive and \((W, R) \models \psi\).

• \(c_1 R c_2 R c_3 R \dots\) is now an infinite ascending \(R\)-chain.

• \((W, R)\) is not converse well-founded.

• But \(\psi\) is true only in converse well-founded frames. Contradiction.

We conclude that the class of transitive converse well-founded frames is not first-order definable. \(\blacksquare\)

transitive converse well-founded frames are modally definable

Proposition 2 For every frame \((W, R)\):

\[ (W, R) \models \Box(\Box p \to p) \to \Box p \quad\text{iff}\quad R \text{ is transitive and converse well-founded on } W. \]

Proof. Fix a frame \((W, R)\). We now proceed in two stages:

• (\(\Rightarrow\)) We target the contrapositive.

  If \(R\) is not transitive, or not converse well-founded, then we will find a valuation \(V\) and a world where Löb fails.

• (\(\Leftarrow\)) If \(R\) is transitive and converse well-founded, then, given a valuation \(V\), we will argue that each world \(w \in W\) verifies Löb: if \(w\) falsifies \(\Box p\), then \(w\) falsifies \(\Box(\Box p \to p)\).

(\(\Rightarrow\)) \(R\) not transitive

Suppose \(wRu\) and \(uRv\) but not \(wRv\). Let \(V(p) = W \setminus \{u, v\}\).

• \(w \nVdash \Box p\), since \(wRu\) and \(u \notin V(p)\).

• \(u \nVdash \Box p\), since \(uRv\) and \(v \notin V(p)\). So \(u \Vdash \Box p \to p\).

• If \(wRt\), then \(t = u\) or \(t \in V(p)\) (as \(t \neq v\)). Either way, \(t \Vdash \Box p \to p\).

• So \(w \Vdash \Box(\Box p \to p)\) but \(w \nVdash \Box p\).

(\(\Rightarrow\)) \(R\) not converse well-founded

Suppose \(u_1 R u_2 R u_3 R \cdots\) is an infinite ascending chain. Let \(V(p) = W \setminus \{u_1, u_2, \dots\}\).

• Each \(u_n\) sees \(u_{n+1} \notin V(p)\), so \(u_n \nVdash \Box p\) and \(u_n \Vdash \Box p \to p\). All worlds off the chain satisfy \(p\). So \(\Box p \to p\) is true at all worlds.

• Hence \(u_1 \Vdash \Box(\Box p \to p)\), but \(u_1 \nVdash \Box p\).

(\(\Leftarrow\)) \(R\) transitive and converse well-founded

Fix \(V\) and \(w\), and suppose \(w \nVdash \Box p\). Let \[ X = \{u \in W : wRu \text{ and } u \notin V(p)\} \neq \varnothing. \]

• Since \(R\) is converse well-founded, \(X\) has an \(R\)-maximal \(v\): no \(t \in X\) with \(vRt\).

• \(v \nVdash p\), since \(v \in X\).

• If \(vRt\), then \(wRt\) by transitivity, and \(t \notin X\), so \(t \Vdash p\). Thus \(v \Vdash \Box p\).

• So \(v \nVdash \Box p \to p\). As \(wRv\), \(w \nVdash \Box(\Box p \to p)\). \(\blacksquare\)