Author: Andi

One of the aims of policy evaluation is to support policymakers in making decisions. In practice, policymaking is shaped by coalitions of actors (Sabatier, 1988), including ministers and civil servants, opposition parties, professional bodies, charities, think tanks, campaigners, academics and journalists. Evaluation should aim to inform the deliberations of all these actors, not only those of whoever commissioned the work – regardless of whether it is remotely feasible to include them in a participatory process as part of the evaluation. Since each actor may hold normative values that clash with others’ values (understatement of the century), they may draw different conclusions about what should be done even when presented with the same evidence.

If the evidence produced by evaluators is to be relevant to a range of actors, we potentially need to reason about conflicting values. Deontic logics, which make it possible to reason about oughts, provide one way of structuring this reasoning. Given the complexities of applying deontic logics formally, as we will see shortly, and the large number of normative values involved, it is unlikely that evaluators will use them in a fully formal way. However, just as process tracing is informed by Bayesian logic (Bennett, 2009), I am curious whether informal reasoning about values can be informed by deontic logics. This post is a first go to find out.

There are many systems of deontic logic. A previous post ruled out standard deontic logic (SDL), which, confusingly, ceased to be standard in the late 1960s (Parent & Torre, 2018, p. 20). An alternative I’d like to explore is Horty’s (2012) approach, a prioritised default logic, which has the following key characteristics:

1. Classical logic is monotonic, in the sense that adding more premises to an argument can never lead to the retraction of a conclusion: either the set of conclusions stays the same or grows (note the parallel with monotonic functions). Horty’s logic is nonmonotonic, meaning that it does allow conclusions to be withdrawn. The classic example concerns a bird named Tweety. All birds fly, so we conclude that Tweety flies. However, if we subsequently learn that Tweety is a penguin, we revise that conclusion and conclude that Tweety doesn’t fly.
2. Horty’s nonmonotonic logic is implemented using default rules (defaults for short), written as \(\phi \rightarrow \psi\), which are read as: if we have established \(\phi\), then we should conclude \(\psi\) by default (for example, that birds fly). This inference can be overruled if another default supports a contradictory conclusion (for example, that penguins do not fly).
3. Defaults represent reasons for believing things (e.g., Tweety can’t fly because Tweety is a Penguin) or reasons for doing things (e.g., meeting a friend for lunch because we promised to).
4. Some default rules have higher priority than others, e.g., if \(\delta_1\) and \(\delta_2\) are two defaults, then \(\delta_1 > \delta_2\) means that \(\delta_1\) has higher priority than \(\delta_2\), so can overrule it. This might be due to the rule being more specific, e.g., if we’re reasoning about penguins we should prioritise defaults about penguins rather than about birds more generally. When defaults refer to normative values, the priorities determine which values are more important than others.
5. The system therefore makes it possible to reason about moral conflicts. One of Horty’s examples involves a promise to meet a friend for lunch and a moral obligation to save a drowning child encountered en route to lunch. A reasonable ordering on defaults representing these norms is that saving a drowning child takes priority over fulfilling a lunch promise, so the logic concludes that the child should be rescued.

To illustrate how this works, let’s explore the following simplified example:

• An evaluation has shown that a programme leads to an outcome valued by a policymaker: \(P\).
• The same evaluation has shown an unanticipated harmful consequence of the programme that was not considered by policymakers or evaluators at the outset, but was highlighted by service users: \(H\).
• If the valued outcome is found, then we should roll out the programme nationally: \(d_1 = P \rightarrow R\).
• If harmful outcomes are found, then it should not be rolled out nationally: \(d_2 = H \rightarrow \neg R\).

We could complicate this further by introducing differences in the quality of evidence. For example, \(P\) might have been established through a rigorous impact evaluation, whereas \(H\) might have been identified through qualitative interviews with a small sample – a common way in which unintended consequences are discovered. However, to keep the discussion simple, let’s assume that there is no difference in the quality of evidence.

How do we evaluate evidence using Horty’s prioritised default logic? Since it is a logic, the rules are all formally defined, and applying them in practice can be challenging. Horty’s (2012) explanation spans several chapters. Here follows a concise summary; refer to the original text for fuller explanation and worked examples.

A default theory consists of three elements, which for the example above would be:

1. \(\mathcal{W} = \{ P, H \}\) is the starting point for our inferences, representing the evidence found by the evaluation and any background assumptions. This would include relationships between the variables, which we don’t have in our simplified policy example. For the Tweety example, it would include that penguins are birds.
2. \(\mathcal{D} = \{ d_1, d_2 \}\) is the set of defaults described above. For a default \(\phi \rightarrow \psi\), \(\phi\) is the premise and \(\psi\) the conclusion. We can refer to the premises and conclusions of a default or set of defaults, \(\Delta\), using \(\text{premise}(\Delta)\) and \(\text{conclusion}(\Delta)\).
3. \(<\) is the ordering on the defaults, which specifies which take precedence over others. This is a partial ordering, in the sense that some or all defaults may have the same precedence. Let’s begin with no ordering, so the defaults concerning desired (\(d_1\)) and harmful (\(d_2\)) outcomes have equal importance.

To draw inferences, we need to consider one or more scenarios: these are a subset of the defaults, \(S \subseteq \mathcal{D}\). A scenario could include some, all, or none of the defaults. Binding defaults are defaults in the full theory such that the following conditions hold for a scenario:

1. The default \(\delta \in \mathcal{D}\) is triggered, in the sense that its premise logically follows from the background theory and conclusions of the defaults in the scenario: \(\mathcal{W} \cup \text{conclusion}(S) \vdash \text{premise}(\delta)\).
2. The default \(\delta \in \mathcal{D}\) is not conflicted, i.e., its conclusion does not contradict the background theory and scenario: it is not the case that \(\mathcal{W} \cup \text{conclusion}(S) \vdash \neg \text{conclusion}(\delta)\).
3. The default \(\delta \in \mathcal{D}\) is not defeated by another default in the theory, i.e., there is no triggered default \(\delta^\prime > \delta\) such that \(\delta^\prime\) and \(\delta\) arrive at contradictory conclusions: it is not the case that \(\mathcal{W} \cup \text{conclusion}(\delta^\prime) \vdash \neg \text{conclusion}(\delta)\).

A scenario, \(S \subseteq \mathcal{D}\), is a proper scenario if it doesn’t include any extra defaults in \(\mathcal{D}\) that aren’t binding and doesn’t miss out any defaults that are binding. A proper scenario includes the good reasons for drawing an inference, and is what we (or an algorithm) are trying to find. More than one scenario may be proper.

We want to know what logically follows from each proper scenario, known as the extension, \(\mathcal{E}\), since it extends beyond the background theory and default rules to what they logically imply. This is a set of conclusions, defined

\(\mathcal{E} = \text{Th}\bigl(\mathcal{W} \cup \text{conclusion(S)}\bigr)\),

where \(\text{Th}(\Gamma)\) is the set of propositions, \(\phi\), such that \(\Gamma \vdash \phi\), i.e., every proposition that logically follows from \(\Gamma\).

Finally, how do we conclude that some proposition \(\phi\) ought to be the case, \(\mathsf{O} \phi\)? There are two ways to do this, depending on how we deal with conflicts:

1. Conflict account: \(\mathsf{O} \phi\) follows if and only if \(\phi\) is included in the extension of at least one proper scenario.
2. Disjunctive account: \(\mathsf{O} \phi\) follows if and only if \(\phi\) is included in extensions of all proper scenarios.

It is straightforward to find the proper scenarios for our example. We know both defaults are triggered, since the evidence \(\mathcal{W} = \{ P, H \}\), and the defaults are:

\(\displaystyle \begin{aligned}
d_1 &= P \rightarrow R \\
d_2 &= H \rightarrow \neg R
\end{aligned}\)

The conclusions of the two defaults contradict each other (\(R\) and \(\neg R\)), so if we included them both in a scenario they would be conflicted. Neither of the defaults could be defeated by the other since we have not put an ordering on them, and defeat requires an ordering. So there are two proper scenarios: \(\{ d_1 \}\) and \(\{ d_2 \}\), and two sets of extensions, call them \(\mathcal{E}_1\) and \(\mathcal{E}_2\).

• \(\mathcal{E}_1\) includes \(\{ P, H, R \}\), so we can conclude \(\mathsf{O} R\).
• \(\mathcal{E}_2\) includes \(\{ P, H, \neg R \}\), so we can conclude \(\mathsf{O} \neg R\).

Under the conflict account, this means we have two oughts: \(\mathsf{O} R\) and \(\mathsf{O} \neg R\). Note the extensions \(\mathcal{E}_1\) and \(\mathcal{E}_2\) both also include \(R \lor \neg R\) (using disjunction introduction), so by the disjunctive account we can conclude \(\mathsf{O}(R \lor \neg R)\). Therefore, with no ordering on the defaults, we can only say that we either ought to roll out the programme or not roll it out. Not particularly helpful!

To arrive at a decision, we need to put an ordering on the defaults. We could reason that an important normative value is first do no harm, which could be formalised as \(d_2 > d_1\). In this case, the only proper scenario is \(\{ d_2 \}\). This is because a proper scenario cannot have any defeated defaults, and in scenarios \(\{ d_1 \}\) and \(\{ d_1, d_2 \}\), \(d_1\) would be defeated by \(d_2\). We can’t use the empty scenario since it excludes binding defaults, so it is not a proper scenario. The extension of the proper scenario leads to the conclusion \(\mathsf{O} \neg R\): we ought not roll out the programme.

I noted earlier that SDL is ruled out as an appropriate logic for deontic reasoning. It does not, for example, handle conflicts in obligations. However, there’s something slightly puzzling about how Horty (2012) derives oughts: if \(\phi\) is in the extension of a scenario, we can conclude \(\mathsf{O} \phi\) for that scenario. But defaults can include non-deontic conclusions, such as the inference that penguins do not fly. It seems odd to move from this belief to the conclusion that penguins ought not fly. Fuhrmann (2017) suggests instead ditching Horty’s move from extensions to oughts and supplementing the system with SDL. Our original set of defaults would then be:

\(\displaystyle \begin{aligned}
d_1 &= P \rightarrow \mathsf{O} R \\
d_2 &= H \rightarrow \mathsf{O} \neg R
\end{aligned}\)

The idea is that instead of using classical logical consequence for building extensions, we use SDL. An extension of a scenario can then include an ought “for free”, since default conclusions can include oughts. The conflict and disjunctive accounts still work for examples Fuhrmann tries, yielding reasonable inferences, and we also still have the very helpful machinery of default logic. I don’t know whether adding SDL introduces other problems, though.

Does Horty’s prioritised default logic, perhaps supplemented with SDL, help evaluation? Firstly, it offers a deontic logic that formalises some elements of evaluative thinking. Even if we do not use it in a fully formal way, it provides clues about what evaluative thinking needs to include and highlights some of the issues that may arise. For example, we need to anticipate what the potential findings might be and how different normative values may lead to different evaluative judgements.

Secondly, it suggests that one way to do this is through default rules – formally or informally specified – with orderings where possible to reduce the number of conflicting conclusions. This allows some defaults to take priority over others. The justifications for that ordering lie outside the logic itself and form part of what coalitions of actors will debate.

Finally, as alluded to above, we also need an ordering on the evidence, and we need to ensure evaluations are designed so that, for example, if harms are identified, evidence of those harms cannot simply be dismissed on the grounds that the evidence is of insufficient quality. For example, suppose we have evidence from an impact evaluation (e.g., a quasi-experiment) of no harms (\(\neg H_I\)) and evidence from a process evaluation (e.g., qualitative interviews) of harms (\(H_P\)). We could setup defaults to interpret this evidence as:

\(\displaystyle \begin{aligned}
i_1 &= H_I \rightarrow H \\
i_2 &= \neg H_I \rightarrow \neg H \\
p_1 &= H_P \rightarrow H \\
p_2 &= \neg H_P \rightarrow \neg H
\end{aligned}\)

where \(H\) means we have inferred the programme has caused harm. Order the defaults so that \(i_x > p_x\) for \(x = 1,2\). This would mean that with conflicting evidence of harms, the impact evaluation evidence would always trump the process evaluation evidence. So, given the evidence \(\{ \neg H_I, H_P \}\), the conclusion would be no harm, \(\neg H\). This mirrors how process evaluation evidence of perceived impact is often, in practice, defeated by impact evaluation evidence of no impact. These interpretations can be anticipated before any data is collected and feed into the design of an evaluation.

I think it is unrealistic to expect an evaluation to settle debates about normative values, for the reasons explored in a previous post. However, deontic logic could help us think through what evidence needs to be produced to support others when they have those debates, making explicit both the assumed priorities among different forms of evidence and the ways in which this evidence is taken to justify particular policy decisions.

References

Bennett, A. (2009). Process Tracing: A Bayesian Perspective. In J. M. Box-Steffensmeier, H. E. Brady, & D. Collier (Eds), The Oxford Handbook of Political Methodology (pp. 702–721). Oxford University Press.

Fuhrmann, A. (2017). Deontic Modals: Why Abandon the Default Approach. Erkenntnis, 82(6), 1351–1365.

Horty, J. F. (2012). Reasons as defaults. Oxford University Press. (Final version preprint available here.)

Parent, X., & Torre, L. van der. (2018). Introduction to Deontic Logic  and Normative Systems. College Publications.

Sabatier, P. A. (1988). An advocacy coalition framework of policy change and the role of policy-oriented learning therein. Policy Sciences, 21, 129–168.