Session 1: Introduction
Today’s session served as a general introduction to basic ideas and concepts underlying deontic logic. What is deontic logic? To answer this question it was useful to remind ourselves of the general aim of formal logic. It is to provide theories (or the (?) theory) of valid inference. It helps us to understand rules underlying correct inferences and to identify faulty reasoning. Take the following example:
- Either he is at the Mensa or he is at Q-West.
- He’s not at the Mensa.
- Thus, he is at Q-West.
This inference follows a general rule, disjunctive syllogism. In order to phrase general laws we have to abstract from the concrete content of inferences and reduce them to their logical form. For this, in turn, we need an abstract, formal language. The language of propositional logic, which will accompany us in this course, has the following ingredients:
- logical constants: \(\wedge\) (“and”: conjunction), \(\vee\) (“or”: disjunction), \(\neg\) (“not”: negation), \(\rightarrow\) (“implies”: implication), and \(\equiv\) (“equivalent/co-implies”: equivalence/co-implication).
- atoms (or “sentential letters”): \(p, q, s, r\) …
Having this linguistic tools in our repertoire, we can phrase the rule of disjunctive syllogism in its general form:
- \(p \vee q\)
- \(\neg p\)
- Thus, \(q\).
Our concrete inference above is an instance of this if we interpret \(p\) with “He is at the Mensa.” and \(q\) with “He is at Q-West.”.
Now, deontic logic has it’s focus on a specific kind of inference for which the vocabulary of propositional logic is too poor: reasoning with norms. In particular this concerns inferences featuring the following norm types (which come with typical verbs and which will be formally expressed with logical constants on the right):
| norm type | typical verbs | logical constant |
|---|---|---|
| obligations | ought, must | \(\mathsf{O}\) |
| permission | may, permitted to, can | \(\mathsf{P}\) |
| prohibition | forbidden, prohibited | \(\mathsf{F}\) |
(We will at some point discuss others, such as supererogation, etc.)
Here are some examples of statements featuring obligations, that will help us to make some more fine-grained distinctions:
- There ought to be an exam at the end of the term.
- You ought to bring about that the dishes are washed.
- Peter ought to wash the dishes.
- Wash the dishes!
- There is an obligation to write an exam at the end of the term.
When phrasing normative statements with ought to be (statement 1), the ought applies to a proposition: it ought to be that \(A\) is true (like in 1 that “there is an exam at the end of the term”). In contrast, when phrasing a normative statement with ought to do (statement 3), our obligation applies to an action and it concerns an agent. Statement 2 is interesting in that it seems to offer a compromise between the two: while the ought still applies to a —rather generic— action, namely to bring something about, what is to be brought about is a state of affairs which is described by a proposition (“The dishes are washed”). So, if we let \(\mathsf{O}\) be interpreted as ought-to-bring-about or as ought-to-be, it takes as arguments propositions, which we are able to express in propositional logic. However, if we let it express ought-to-do, we need to enrich our language with ways to express atomic and complex actions, such as washing the dishes. There are ways to do so, but in the first part of the course we will stick to interpretations of \(\mathsf{O}\) that apply to propositions.
Statement 4 is an imperative: by uttering it, a norm is “brought to life” (i.e., if certain contextual factors are fulfilled such as: the one uttering the imperative has an authority, is not acting or joking, etc.). Imperatives are prescriptive statements: they express norms and do not just report on their existence. Prescriptive statements need not take the form of an imperative, though. E.g., statement 2 may have a prescriptive reading under which you are given the obligation to wash the dishes. In contrast, there are also descriptive statements reporting on the existence of norms. Clearly, statement 5 falls into this category: it does not itself express the norm or bring it into existence, but rather reports on it. Often, without contextual information, statements are ambiguous: they can be read prescriptively or descriptively. (E.g., take statement 1.) It is not clear, and even quite dubious, whether prescriptive statements have truth values. Given the centrality of truth-values in logical theories of valid inference, this has been considered a problem. Can we not have a logical theory about reasoning with “norms” (so, prescriptive statements), but only about “normative propositions” (so, descriptive statements about norms)?
Now, is there at all some logical regularity governing normative reasoning? If not, deontic logic may be an idle business. Let us look at some examples of intuitively sound normative inferences:
| Anne may write an exam or give a talk. | \(\mathsf{P}(p \vee q)\) |
| Thus, Anne may write an exam or she may give a talk. | \(\mathsf{P} p \vee \mathsf{P} q\) |
One may even argue we can conclude \(\mathsf{P}p \wedge \mathsf{P}q\) from \(\mathsf{P}(p \vee q)\). This is called “free choice permission”: if you are permitted to \(p\) or \(q\), whatever your choice, \(p\) or \(q\), you have the permission to do so. This principle is surprisingly hard to model in deontic logic and will be discussed thoroughly in another session.
| Anne is obliged to write an exam | \(\mathsf{O} p\) |
| Thus, Anne is permitted to write an exam. | \(\mathsf{P}p\) |
It seems reasonable that if you’re obliged to bring about \(p\), you’re also permitted to do so.
Now here are two faulty inferences:
| Anne is obliged to write an exam or give a talk. | \(\mathsf{O}(p \vee q)\) |
| * Thus, Anne is obliged to write an exam or she is obliged to give a talk. | \(\mathsf{O} p \vee \mathsf{O} q\) |
and
| Anne is obliged to write an exam or give a talk. | \(\mathsf{O}(p \vee q)\) |
| She’s allowed to not write an exam. | \(\mathsf{P}\neg p\) |
| * Thus, she’s obliged to give a talk. | \(\mathsf{O} q\) |
Clearly, we expect deontic logic to give us an understanding of what goes wrong with these inferences and to help us identify other faulty inferences.
So, what if we replace \(\mathsf{P}\neg p\) in the last inference with \(\mathsf{O}\neg p\)? If this doesn’t make you scratch your head yet, things can get arbitrarily complicated. A deontic logic should give us a precise framework and methods to solve such puzzles.
But how to do so? So far we only learned how to formalize our language, but how to identify general laws of reasoning? Let us recall one way to do so in propositional logic. There, it was useful to formulate a theory of the meaning of logical constants. Such a theory was given in terms of truth tables, such as
| \(\wedge\) | 0 | 1 | \(\vee\) | 0 | 1 | \(\neg\) | ||
| 0 | 0 | 0 | 0 | 1 | 1 | |||
| 1 | 0 | 1 | 1 | 1 | 0 |
A formula was tautological, if it was true in all possible interpretations of the atoms occurring in it. E.g., in the example at the beginning of this entry we have:
| \(p\) | \(q\) | \(((p \vee q))\) | \(\wedge\) | \(\neg p)\) | \(\rightarrow q\) |
|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 1 | 1 | 1 | 1 | 1 |
| 1 | 0 | 1 | 0 | 0 | 1 |
| 1 | 1 | 1 | 0 | 0 | 1 |
So, a natural question is, whether truth tables help us also to express the meaning of \(\mathsf{O}\) and \(\mathsf{P}\). Unfortunately, this approach doesn’t work. The reason is that the truth-value of \(\mathsf{O}p\) is independent of the truth-value of \(p\). (\(\mathsf{O}\) and \(\mathsf{P}\) are not truth-functional, unlike the propositional connectives \(\wedge, \neg, \vee, \rightarrow\) and \(\equiv\).)
One way to think about the meaning of obligations is that their arguments should be true in ideal worlds: after all, ideally we fulfill our obligations. So, it is not sufficient to only observe the current factual situation and what is true in it (given by the truth-values of atoms), but rather we need to consider ideal situations.
This idea is made formally precise in so-called possible worlds semantics. A logical model \(M\) is given by a quadruple \(( W, R, @, v)\) where
- \(W\) is a set of worlds
- \(R\) is an accessibility relation between worlds
- \(@\) is the actual world, our reference point
- \(v\) is a function that assigns truth values 0 and 1 to pairs \((w, p)\) of worlds in \(W\) and atoms. It expresses whether \(p\) holds at world \(w\) or not.
The basic understanding is that worlds reachable from the actual world \(@\) via the accessibility relation \(R\) are ideal counterparts of \(@\). We expect all of the given obligations to be fulfilled in these ideal worlds. Also other worlds than \(@\) may have ideal counterparts. (If these are really “ideal” worlds, you may expect them to be ideal counterparts of themselves, but we don’t demand this at this point.)
The model is not informative yet, since we still need to determine when a formula is valid at a given world in the model. For this we use our assignment function \(v\) in a bottom-up way. Where \(w \in W\), we have:
- \(M,w \models p\) iff \(v(w,p) = 1\).
- \(M,w \models \neg A\) iff \(M, w \not\models A\).
- \(M,w \models A \wedge B\) iff \(M,w \models A\) and \(M,w \models B\).
- \(M,w \models A \vee B\) iff \(M,w \models A\) or \(M,w \models B\).
- \(M,w \models A \rightarrow B\) iff \(M,w \not\models A\) or \(M,w\models B\).
- \(M,w \models A \equiv B\) iff (\(M,w \models A\) iff \(M,w \models B\)).
Finally, and most interestingly, when is a formula \(\mathsf{O}A\) true at some world \(w\). Well, we follow our initial motivation: exactly when \(A\) holds in all ideal counterparts of \(w’\)!
- \(M,w \models \mathsf{O}A\) iff for all \(w’\) which are \(R\)-reachable from \(w\), \(M,w’ \models A\).
- \(M,w \models \mathsf{P}A\) iff there is a \(w’\) which is \(R\)-reachable from \(w\) and \(M,w’ \models A\).
For permission, it is sufficient to find one ideal counterpart of the given world where the argument of \(\mathsf{P}\) holds.
Let us test this approach with our previous example:
| Anne may write an exam or give a talk. | \(\mathsf{P}(p \vee q)\) |
|---|---|
| Thus, Anne may write an exam or the may give a talk. | \(\mathsf{P} p \vee \mathsf{P} q\) |
If this is a valid inference, it should be true in any model in any world. Is this so. Let’s see:
- Suppose \(M,w \models \mathsf{P}(p \vee q)\)
- Thus, there must be an ideal counterpart \(w’\) of \(w\) for which \(M,w’ \models p \vee q\).
- Thus, \(M,w’ \models p\) or \(M,w’ \models q\).
- Hence, \(M,w \models \mathsf{P}p\) or \(M,w \models \mathsf{P}q\).
- Thus, \(M,w \models \mathsf{P}p \vee \mathsf{P}q\).
We have to be careful however with the inference from \(\mathsf{O}p\) to \(\mathsf{P}p\). According to our definition of a model above, it could happen that a world has no ideal counterpart! In such a world for any formula \(A\), \(\mathsf{O} A\) and \(\neg \mathsf{P} A\) will hold. (See for yourself why this is so!)
In order to disallow such situations, we require that every world has at least one ideal counterpart. In order words, the accessibility relation should be serial:
- for all \(w \in W\) it holds that \(wRw’\) for some \(w’ \in W\). (This \(w’\) could be but need not be identical to \(w\).)
The logic which uses possible world models with serial accessibility relations is called KD or SDL (for Standard Deontic Logic). We will see that it has serious limitations, but it is a good starting point for gaining a precise formal understanding of normative reasoning.
In SDL also our other example
| Anne is obliged to write an exam | \(\mathsf{O} p\) |
|---|---|
| Anne is permitted to write an exam. | \(\mathsf{P}p\) |
becomes a valid inference. Let us show this:
- Suppose for some world \(w\) in a model \(M\) (with serial accessibility relation \(R\)) we have \(M,w \models \mathsf{O}p\).
- Hence, in all ideal counterparts \(w’\) of \(w\) we have \(M,w’ \models p\).
- Since in view of the seriality of $R$e there is at least one such \(w’\), we also have \(M,w\models \mathsf{P} p\).
Exercises
Exercise 1
We have the following model \(M = (W, R, @, v)\) where
- \(W = \{@, w\}\)
- \(R = \{(@,w), (w,w)\}\)
and \(v\) is given by (to keep things simple we work with a language with only two atoms \(p,q\)):
| \(@\) | \(w\) | |
|---|---|---|
| \(p\) | 1 | 1 |
| \(q\) | 0 | 0 |
Which of the following statements is true:
- \(M,@ \models \mathsf{O}p\)
- \(M,@ \models \mathsf{O} \mathsf{O}p\)
- \(M,@ \models \mathsf{P} q\)
- \(M,@ \models \mathsf{O}p \rightarrow q\).
- \(M,@ \models \mathsf{O}p \rightarrow p\).
Exercise 2
Can you find a model with only two worlds and a serial accessibility relation for which it doesn’t hold that \(M,@ \models \mathsf{O} \mathsf{O} p \rightarrow \mathsf{O}p\)?
Exercise 3
How many worlds does a model with serial accessibility relation have, whose accessibility relation is transitive and irreflexive. Recall:
- transitive means that for all \(w_1, w_2, w_3\) we have that if \(w_1 R w_2\) and \(w_2 R w_3\) then \(w_1 R w_3\).
- irreflexive means that for all \(w\) we don’t have \(wRw\).
Exercise 4
Construct a model \(M = (W, R, @, v)\) in which the following are true:
- \(M,@ \models \mathsf{O}(p \vee q)\)
- \(M,@ \models \neg\mathsf{O}p \wedge \neg \mathsf{O}q\)
- \(M,@ \models \mathsf{P}p\)
- \(M,@ \models \mathsf{P}q\)