# 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?

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

- 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 \(\Gamma'\).

### Exercise Sheet 2

To open this sheet in a dedicated window go here.

#### Task 1

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

- Is \(q\) a consequence with the reliability strategy?
- 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 (\(\dagger\))?