Session 1 - Summary of definitions

1. Formal Preliminaries: Formal Language, Truth at a World, and Logical Consequence

Language $\mathcal{L}$

A formal language $\mathcal{L}$ is defined as follows. Let $\Phi$ be a (countable) set of objects called “propositional variables”, that is $\Phi :=\{p_1,p_2,p_3,\dots \}$. Then, we define $\mathcal{L}$ via induction as follows.

1. Base: $\Phi \subseteq \mathcal{L}$.
2. Step: For all $\varphi ,\psi \in \mathcal{L}$:
3. If $\varphi \in \mathcal{L}$, then $\neg \varphi \in \mathcal{L}$;
4. If $\varphi \in \mathcal{L}$ and $\psi \in \mathcal{L}$ then $\varphi \wedge \psi \in \mathcal{L}$;
5. If $\varphi \in \mathcal{L}$ and $\psi \in \mathcal{L}$ then $\varphi \vee \psi \in \mathcal{L}$;
6. If $\varphi \in \mathcal{L}$ and $\psi \in \mathcal{L}$ then $\varphi \to \psi \in \mathcal{L}$.
7. Nothing else is in $\mathcal{L}$.

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2. Properties of Logical Consequence Relations and Operators

2.1. Logical Consequence

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2.2. Properties of Logical Consequence Relation and Operator

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3. Belief Revision Theory

Let $B\in \mathbf{B}$ be any belief set, $\varphi \in \mathcal{L}$ be any sentence, and $\ast :\mathbf{B}\times \mathcal{L}\to \mathbf{B}$ a candidate revision function. $\ast$ is a (AGM) (basic) belief revision function iff it satisfies the following postulates:

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

$\ast$ is a belief revision function iff it satisfies also the following postulates

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