#+TITLE: Exercise Sheet 1 for: Adaptive Logics applied to the Philosophy of Science
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$\def\impl{\supset} \def\Dab{{\sf Dab}}$
$\newcommand{\vd}[1]{\vdash_{\bf #1}}$
In the following tasks we use the adaptive logic that is characterized by the triple:
- lower limit logic: ${\bf CL_\circ}$
- set of abnormalities: is constituted by all formulas of the form $\circ A \wedge \neg A$ where $A$ is a well-formed formula
- adaptive strategy: minimal abnormality
I use the following abbreviation: $!A = \circ A \wedge \neg A$.
* Task 1
Let $\Gamma = \{\circ p, \circ q, \circ r, !p \vee {!}r, !q \vee {}!r, (q\wedge p) \impl s, r \impl s\}$.
1. Is $s$ derivable by means of the minimal abnormality strategy? Prove it.
2. Specify $\Phi(\Gamma)$.
3. Specify $U(\Gamma)$.
4. How many different types of reliable models of $\Gamma$ are there relative to their abnormal parts? Answer the question by naming all abnormal parts of reliable models of $\Gamma$.
* Task 2
Let $\Gamma' = \Gamma \cup \{!p \vee {!}q\}$.
1. Is $s$ derivable? Give a reason for your answer.
2. Specify $U(\Gamma')$.
3. How many different types of reliable models of $\Gamma'$ are there relative to their abnormal parts? Answer the question by naming all abnormal parts of reliable models of $\Gamma'$.