RUB | Fakultät für Philosophie und Erziehungswissenschaft | Institut für Philosophie II
    Workgroup Home | Christian tiny_christian.stra_er.jpg | Dunja tiny_dunja._e_elja.jpg | Jesse tiny_jesse.heyninck.jpg | Mathieu tiny_mathieu.beirlaen.jpg | FAQ for members

Course: Adaptive Logics applied to the Philosophy of Science @UGENT 2015

Table of Contents

\(\def\impl{\supset} \def\Dab{{\sf Dab}}\) \(\newcommand{\vd}[1]{\vdash_{\bf #1}}\)

Course Outline

1. Adaptive logics: what are they, how do they work?

Exercises 3.1 and 3.2.

Literature:

  • Batens, D. (2004). The need for adaptive logics in epistemology. In (Eds.), Logic, epistemology, and the unity of science (pp. 459–485). Springer.
  • Batens, D. (2007). A universal logic approach to adaptive logics. Logica Universalis, 1, 221–242.
  • Christian Stra\sser (2014). Adaptive logic and defeasible reasoning. applications in argumentation, normative reasoning and default reasoning. Springer.

2. Normative reasoning: what are deontic logics?

Exercises 3.3.

We discuss the following articles:

  • Goble, L. (2003). Preference semantics for deontic logic. Part I: simple models. Logique et Analyse, 183–184, 383–418.
  • Makinson, D., & Torre, L. v. d. (2000). Input/Output logics. Journal of Philosophical Logic, (29), 383–408.
  • Goble, L. (2013). Deontic logic (adapted) for normative conflicts. Logic Journal of the IGPL.

Final Exam

This is here.

Exercises

Exercise Sheet 1

To open this sheet in a dedicated window go here.

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 \(\Gamma'\).

Exercise Sheet 2

To open this sheet in a dedicated window go here.

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 \(\{\circ B \wedge \neg B\}\)

Exercise Sheet 3: Goble_LA_2003

To open this sheet in a dedicated window go here.

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 (\(\dagger\))?

Author: Christian Straßer

Created: 2015-06-10 Wed 16:18

Emacs 24.5.1 (Org mode 8.2.10)

Validate