I've just started looking into epistemic logic and belief revision approaches, and I'm struggling already with proving some basic properties.
I am given the following definition:
Let $\mathcal{L}_0$ be the set of propositional formulas, generated by some set of atoms $P$, and the classical connectives. Let $Cn(.)$ be classical consequence operator, i.e. $Cn(\Sigma) = \{ \sigma \in \mathcal{L}_0 | \Sigma \vdash \sigma \}$. A belief set $\mathcal{K}$ is a set of propositional formulas closed under $Cn(.)$, i.e. $Cn(\mathcal{K}) = \mathcal{K}$.
I'm looking at the expansion postulates:
Type: $(\mathcal{K} \oplus 1): \mathcal{K} \oplus \varphi$ is a belief set
Success: $(\mathcal{K} \oplus 2): \varphi \in \mathcal{K} \oplus \varphi$.
Expansion: $(\mathcal{K} \oplus 3): \mathcal{K} \subseteq \mathcal{K} \oplus \varphi$
Minimal action: $(\mathcal{K} \oplus 4):$ if $\varphi \in \mathcal{K}$ then $\mathcal{K} = \mathcal{K} \oplus \varphi$
Monotony: $(\mathcal{K} \oplus 5):$ For all $\mathcal{H}$, if $\mathcal{K} \subseteq \mathcal{K}$ then $\mathcal{K} \oplus \varphi \subseteq \mathcal{H} \oplus \varphi$.
Minimal change: $(\mathcal{K} \oplus 6): \mathcal{K} \oplus \varphi$ is the minimal set satisfying $(\mathcal{K} \oplus 1 )- (\mathcal{K} \oplus 5)$
These postulates I assume are meant to be intuitive representations of adding new beliefs to our sets.
I am now trying to prove the following:
A function $\oplus$ satisfies $\mathcal{K} \oplus 1- \mathcal{K} \oplus 6$ iff $\mathcal{K} \oplus \varphi = Cn(\mathcal{K} \cup \{\varphi\})$.
One difficulty I'm having is about what I should assume about $\vdash$. I'm a bit confused in terms of whether I am meant to assume the ordinary natural deduction properties, or a specific system. Would really appreciate someone taking a look at my reasoning and helping with the bits I can't solve.
My attempt so far:
$\Rightarrow$. Assume postulates 1-6. I first want to show that $Cn(K \cup \varphi) \subseteq K \oplus \varphi$. By postulate 2 and 3 I have that $\varphi \in \mathcal{K} \oplus \varphi$ and $\mathcal{K} \subseteq \mathcal{K} \oplus \varphi$ which means that $\mathcal{K} \cup \{ \varphi\} \subseteq \mathcal{K} \oplus \varphi$. Now I'm not sure if I'm allowed to do this, but I take the consequence set of each to get:
$Cn(K \cup \{\varphi\} \subseteq Cn(K \oplus \varphi) = K \oplus \varphi$ by postulate 1.
That's all I've got to so far, I couldn't manage to prove the other subset inclusion.
$\Leftarrow$
Postulate 1 is trivial by the equality. Postulate 2 is straightforward since $\varphi \vdash \varphi$ and so $\mathcal{K} \cup \varphi \vdash \varphi$. Postulate 3 holds by similar reasoning: $\mathcal{K} \subseteq \mathcal{K} \cup \varphi \subseteq Cn(\mathcal{K} \cup \varphi))$. Postulate 4 because if $\varphi \in \mathcal{K}$ then we get $\mathcal{K} \oplus \varphi = Cn(\mathcal{K} \cup \varphi = \mathcal{K})$ which is true by definition of belief set (and 1). Postulate 5 I think must be because of some sort of transitive property of deductive systems but couldn't flesh out the proof exactly. Property 6 I don't really know.
So just to recap: I need help in proving the set inclusion: $K \oplus \varphi \subseteq Cn(K \cup \varphi)$ as well as proving postulate 5 and 6.
Thank you
$Cn$ is the classical consequence operator, so $\vdash$ is the classical consequence relation.
As for the other inclusion, you should try to prove that $Cn(K\cup\{\varphi\})$ satisfies axiomes 1 through 5 and appeal to the minimality of $\oplus$.
To prove property 6, use what you've done in the previous part to show minimality.
Property 5 for $Cn(K\cup\{\varphi\})$ follows from the fact that $Cn$ is increasing, because $\vdash$ is: you can add hypotheses, it cannot (in classical logic) reduce the set of theorems you can prove. Essentially this comes from "$\Sigma\vdash \sigma_0 \iff $ some finite subset $\Sigma_0\subset \Sigma$ satisfies $\Sigma_0\vdash\sigma_0$"