Exercise Sheet 2 for: Adaptive Logics applied to the Philosophy of Science

Table of Contents

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

Task 1

Let \(\Gamma = \{!p_1 \vee {!}p_2, !p_1 \supset {!}p_3, !p_2 \supset {!}p_4, !p_1 \vee q, !p_4 \vee q\}\).

  1. Is \(q\) a consequence with the reliability strategy?
  2. Is \(q\) a consequence with the minimal abnormality strategy?

Task 2

Show that the following holds:
\(\phi \in \Phi(\Gamma)\) iff \(\phi\) is a choice set of \(\Sigma(\Gamma)\) and for all \(!A \in \phi\) there is a \(\Delta_A \in \Sigma(\Gamma)\) for which \(\{!A\} = \Delta_A \cap \phi\).

Task 3

(This is a more difficult task: if you cannot solve it, no problem and we simply go through it together next time.)

Let \(\phi\) be a choice set of \(\Sigma(\Gamma)\). Show that there is a \(\psi \subseteq \phi\) such that \(\psi \in \Phi(\Gamma)\).

Tip: Let \(\phi = \{A_1, A_2, \ldots \}\). Use Task 1 to iteratively/recursively construct \(\psi\). For this, let \(\psi_0 = \phi\) and let \(\psi_{i+1}\) be the result of manipulating \(\psi_i\) in a way that is inspired by the result in Task 1. Then let \(\psi = \bigcap_{i \ge 1} \psi_i\). Now you show that \(\psi\) is a choice set and that it satisfies the property stated in Task 1.

Task 4

Show that the following holds:
\(\Gamma \vd{CL_\circ} \Dab(\Delta)\) iff \(\Gamma \vd{CL_\circ^r} \Dab(\Delta)\).

Task 5

Indicate mistakes in the following proof fragment from \(\Gamma = \{(\circ A \wedge \neg A) \vee (\circ B \wedge \neg B), (\circ A \wedge \neg A) \vee \neg (\circ B \wedge \neg B), \circ A, \circ B\}\):

1 \((\circ A \wedge \neg A) \vee (\circ B \wedge \neg B)\) PREM \(\emptyset\)
2 \((\circ A \wedge \neg A) \vee \neg (\circ B \wedge \neg B)\) PREM \(\emptyset\)
3 \((\circ A \wedge \neg A)\) 1,2; RU \(\emptyset\)
4 \(\circ A\) PREM \(\emptyset\)
5 \(A\) 4; RU \(\{\circ A \wedge \neg A\}\)
6 \(\circ B\) RU \(\emptyset\)
\(^\checkmark\) 7 \(B\) 6; RC \(\{\circ B \wedge \neg B\}\)

Author: Christian Straßer

Created: 2015-04-17 Fri 11:16

Emacs 24.4.1 (Org mode 8.2.10)