# Exercise Sheet 1 for: Adaptive Logics applied to the Philosophy of Science

Back to the main course site $$\def\impl{\supset} \def\Dab{{\sf Dab}}$$ $$\newcommand{\vd}[1]{\vdash_{\bf #1}}$$

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

I use the following abbreviation: $$!A = \circ A \wedge \neg A$$.

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 $$\Gamma$$.

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 $$\Gamma'$$.