# 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
- adaptive strategy: minimal abnormality

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

## Task 1

Let \(\Gamma = \{\circ p, \circ q, \circ r, !p \vee {!}r, !q \vee {}!r, (q\wedge p) \impl s, r \impl s\}\).

- Is \(s\) derivable by means of the minimal abnormality strategy? Prove it.
- Specify \(\Phi(\Gamma)\).
- Specify \(U(\Gamma)\).
- 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 \(\Gamma' = \Gamma \cup \{!p \vee {!}q\}\).

- Is \(s\) derivable? Give a reason for your answer.
- Specify \(U(\Gamma')\).
- 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 \(\Gamma'\).