Session 2 - Summary of definitions

AGM Belief Revision Postulates

Basic postulates:

$$\begin{array}{cl}B\ast \varphi \text{ is a belief set}& \text{(Closure)}\\ & \\ \varphi \in B\ast \varphi & \text{(Success)}\\ & \\ \text{If }\varphi ⊬\perp \text{, then }Cn(B\ast \varphi )\ne \mathcal{L}& \text{(Consistency)}\\ & \\ B\ast \varphi \subseteq Cn(B\cup \{\varphi \})& \text{(Inclusion)}\\ & \\ \text{If }B⊬\neg \varphi \text{, then }B\ast \varphi =Cn(B\cup \{\varphi \})& \text{(Vacuity)}\\ & \\ \text{If }Cn(\{\varphi \})=Cn(\{\psi \})\text{, then }B\ast \varphi =B\ast \psi & \text{(Congruence)}\end{array}$$

Additional postulates:

$$\begin{array}{cl}B\ast (\varphi \wedge \psi )\subseteq (B\ast \varphi )+\psi & \text{(Superexpansion)}\\ & \\ \text{If }B\ast \varphi ⊬\neg \psi \text{, then }(B\ast \varphi )+\psi \subseteq B\ast (\varphi \wedge \psi )& \text{(Subexpansion)}\end{array}$$

Truth-Sets

Truth-Set

Let $\varphi \in \mathcal{L}$. The truth-set function $[\cdot ]:\mathcal{L}\to \mathcal{P}(W)$ maps a sentence to the set of its models:

$$[\varphi ]:=\{w\in W:w\models \varphi \}$$

We can extend the domain of $[\cdot ]$ to include sets of sentences. Let $\Gamma \subseteq \mathcal{L}$ be a set of sentences. Its truth-set is the intersection of the truth-sets of all its members:

$$[\Gamma ]:=\bigcap _{\varphi \in \Gamma }[\varphi ]=\{w\in W:w\models \varphi \text{ for all }\varphi \in \Gamma \}$$

Link to original

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Properties of Theory Function and Truth-Set Function

Theory Function $T(\cdot )$: For all $V\subseteq W$:

$$T(V):=\{\varphi \in \mathcal{L}:V\subseteq [\varphi ]\}$$

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Properties of ${⪯}_B$:

1. Connectedness: For any worlds $w_1,w_2\in W$: Either $w_1{⪯}_B{w}_2$ or $w_2{⪯}_B{w}_1$.
2. Transitivity: For any worlds $w_1,w_2,w_3\in W$: If $w_1{⪯}_B{w}_2$ and $w_2{⪯}_B{w}_3$, then $w_1{⪯}_B{w}_3$.
3. Centeredness:
   1. If $w_1,w_2\in [B]$, then $w_1{⪯}_B{w}_2$.
   2. If $w_1\in [B]$ and $w_2\notin [B]$, then $w_1{\prec }_B{w}_2$.
4. Limit Assumption: For any sentence $\varphi \in \mathcal{L}$: If $[\varphi ]\ne \varnothing$, then there exists at least one world $w_1\in [\varphi ]$ such that $w_1{⪯}_B{w}_2$ for all $w_2\in [\varphi ]$.

Definition of $B\ast \varphi$:

$$B\ast \varphi :=T(\underset{B}{\min }([\varphi ]))$$

where ${\min }_B([\varphi ]):=\{w\in [\varphi ]:\text{For all }{w}^{\prime }\in [\varphi ],w{⪯}_B{w}^{\prime }\}$.