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

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

## 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.

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

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 $$\Gamma'$$.

### Exercise Sheet 2

To open this sheet in a dedicated window go here.

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?

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 $$\{!A\} = \Delta_A \cap \phi$$.

(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.

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

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.

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.

Demonstrate that in system P strengthened with (C), (D) $$\vdash {\sf O}A \rightarrow \neg{\sf O}\neg A$$ is derivable.

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 ($$\dagger$$)?

Created: 2015-06-10 Wed 16:18

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