#+TITLE: Exercise Sheet 3 for: Adaptive Logics applied to the Philosophy of Science
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$\def\impl{\supset} \def\Dab{{\sf Dab}}$
$\newcommand{\vd}[1]{\vdash_{\bf #1}}$
* Task 1
Demonstrate that in system *P* strengthened with (*K*)
$$\vdash \mathsf{O}(A \rightarrow B) \rightarrow ({\sf O}A \rightarrow {\sf O} B)$$
the aggregation principle (*C*)
$$\vdash ({\sf O}A \wedge {\sf O}B) \rightarrow {\sf O}(A \wedge B)$$
is derivable.
* Task 2
Demonstrate that in system *P* strengthened with (*C*), (*D*)
$$\vdash {\sf O}A \rightarrow \neg{\sf O}\neg A$$
is derivable.
* Task 3
Check whether if we restrict ourselves to limited frames the following definition is equivalent to how $M,a \models_{P} {\sf O}A$ is defined in Def. (P-O) on page 386:
$M,a \models_{P} {\sf O}A$ iff there is a maximal$_{a}$ $b \in \mathcal{F}S_{a}$ such that $M,b \models_P A$.
* Task 4: Weak vs. Strong Completeness
The adequacy results in the paper concern what is sometimes called /weak completeness/.
It goes as follows: where a logic *L* is identified with its set of theorems ${\sf S}$, *L* is complete with respect to a set of frames $\mathcal{F}$ iff for every $A$ that is valid for $\mathcal{F}$, $A$ is a theorem of the logic *L*, ie., $A \in {\sf S}$.
The set of theorems of a logic *L* that is correlated with a set of axiom schemes and some inference rules is obtained as follows:
- take all instances of the axiom schemes (obtained by uniform substitution)
- apply the inference rules to this obtained set
Any formula that can be obtained by this procedure is a theorem of *L*.
/Strong completeness/ does not only concern theorems of a logic, but rather the consequence relation that is characterised by *L*.
The consequence set $C_{\bf L}(\Gamma)$ for a set of formulas $\Gamma$ is obtained by applying the inference rules of *L* to both $\Gamma$ and the instances of the axiom schemes. A logic *L* is strongly complete w.r.t. a set of frames $\mathcal{F}$ iff for all (possibly infinite) sets of premises $\Gamma$, if $A$ is valid in all models $\langle F, v\rangle$ of $\Gamma$ (where $F \in \mathcal{F}$) then $A \in C_{\bf L}(\Gamma)$.
One question that is not answered in Goble's article is the following:
Claim ($\dagger$): *SDL* also strongly complete w.r.t. limited standard preference frames.
Consider the following example, where $A > B$ is defined by $\neg(B \ge A)$:
$\Gamma = \left\{ \left(p_{i+1} \wedge \bigwedge_{j \le i} \neg p_j \right) > p_{i} \mid i \ge 1 \right\} \cup \{p_1 > \bot)\}$.
Does it establish a counterexample to the claim ($\dagger$)?