Although there was much historic precedent, contemporary deontic logic began in the time of Alan Anderson’s explication of “it is forbidden that A” as “if A then there is a sanction”, taking the sanction to be a proposition h (“all hell breaks loose”, as he liked to say). The immediate problem, which resisted all efforts at solution, was to find a conditional “if … then” that would fit this idea.
The early problems, and the ‘classical’ response
When deontic logic was then popularly couched as a normal modal logic, with “it ought to be that” taken to be a non-factive ‘necessity’ operator, there were once again many problems about the conditional. For example, “You ought to make restitution if you have stolen” should be consistent with, but not logically implied by, “You ought not to steal”, The generally accepted new turn consisted in reading the “if” in such such examples not as a conditional connective but on the model of conditionalization in probability theory. Roughly speaking, OA is true if A is better (in some value scheme applying to propositions) than ~A, and O(A given B) is true if similarly (A & B) is better than (~A & B).
Let us call this sort of account, in terms of possible worlds, propositions equated to sets of worlds, and some form of evaluation of propositions, the classical semantics of deontic logic. It faced just one challenge: the submission that irresolvable moral conflicts are possible (whether absolutely, or under certain factual circumstances). After a slow increase in discussion of this putative problem, over a number of years, the truly new development in deontic logic was John Horty’s development of deontic logic as a non-monotonic logic, with moral imperatives represented as default rules. (Reasons as Defaults, 2012).
Horty’s critique
In a later paper (2014) Horty shows that the classical semantics as developed in greater generality by Angela Kratzer could handle a great deal of what could be done in non-monotonic logic. But Horty offered two quite simple examples that showed that there were still problems about conditionality which motivate the switch from the classical semantics to default theories.
What I want to do here is to describe Horty’s new examples, and then see whether they can be handled satisfactorily in the ‘hybrid’ semantics that I outlined in the previous post “Deontic logic: two paradoxes”.
Example 1. Etiquette requires me to follow two the two imperatives ‘Don’t eat with your fingers’ and ‘If you are served cold asparagus, eat it with your fingers.’ Of these two, the first is defeasible (priority less than the second).
(Note: in Horty’s form of deontic logic the imperatives have a priority ranking, which represents in this case something conveyed in the book of etiquette, in addition to the set of imperatives it presents.)
Example 2. Like Example 1, except that there is also a third imperative ‘Put your napkin on your lap’.
The original form of classical semantics has a problem with Example 1 because O(~F) and O(if A then F) together imply, on any ordinary reading of the conditional, O(~A). Surely etiquette does not forbid eating asparagus! The later form, which takes conditionalization seriously, does not have a difficulty with this. We can see this if we take the evaluation of propositions to be an additive function like probability or expectation value: ~F is better than F, but (F & A) is better than (~F & A). (Analogy: death from Covid 19 is not likely, but it is likely if you are an octogenarian.)
But in Example 2. we would like to infer that even if you eat asparagus, and do it with your fingers, you should put your napkin on your lap. After all, there is nothing in the situation to defeat or conflict with the napkin imperative. Now that later form of the classical semantics does badly, for the following inference is not valid:
~F is better than F; (F & A) is better than (~F & A); N is better than ~N; therefore (N & F & A) is better than (N & ~F & A).
This could be dealt with in a particular model by adjusting the evaluation of the propositions so as to make it come out OK. Miss Manners’ etiquette book does not say explicitly that even if you are eating asparagus you should put your napkin on your lap. This could be added (a footnote?) but then we are ignoring the fact that Miss Manners doesn’t have to say so: her imperatives should not be defeated if there isn’t any conflicting imperative in force.
Horty ends by showing clearly that the account in terms of default theories allows the correct handling of this situation.
The proposed hybrid deontic logic, with imperatives and values
So, how does it stand with the ‘hybrid’ version I proposed in the previous post? Here I will outline this version with the formality and details required for this discussion. Then I will reconstruct the examples in that framework.
We begin with a universe (the ‘worlds’, or as I shall say, more appropriately, the situations) and identify propositions subsets of this universe. Each situation is to be understood as situation in which an agent finds himself. A situation is identified with (represented by) an ordered pair Δ = <W, D> where W is a proposition (the agent’s knowledge) and D a set of imperatives (default rules).
(A note: in principle we allow a situation to have much more to it than this. It would be more accurate, but for present purposes not useful, to include in addition to W and D a number of parameters. These could represent such other characteristics as mass, hair color, surrounding flora, or whatever else the agent has.)
An imperative δ = <A, B> is an ordered pair of propositions; A is its antecedent and B its consequent. The truth conditions will be given for ordered pairs of worlds and scenarios,
An imperative is in force in Δ exactly if W implies its antecedent, and the intersection of W, the antecedent, and the consequent of this imperative is not empty: the agent knows that this imperative applies in his situation, and his knowledge does not imply that satisfying it is literally impossible. The satisfaction region of S(δ) of δ in Δ is the proposition W ∩ A ∩ B, and we say that δ is satisfied by all and only the situations in S(δ).
Imperatives are compatible if and only if their satisfaction region have a non-empty overlap. A scenario in Δ is a set of imperatives, a subset of D; its satisfaction region is the intersection of all the satisfaction regions of its members.
A feasible scenario for this situation Δ is a set of mutually compatible imperatives, all of which are in force. A proper scenario is a maximal feasible scenario. The satisfaction regions of the proper scenarios I will call ideal propositions for Δ.
The proposition OA is true in this situation Δ if and only if there is a proper scenario whose corresponding ideal proposition implies (is a subset of) A. The connector O can be read as “It is a primary obligation that A”. Primary obligations can conflict with each other, because there may be more than one proper scenario.
(A note: in general a proposition is true in a situation if and only if that situation is one of its members. So the above should be understood as meaning that OA is the set of situations Δ such that … etcetera. But speaking of truth and satisfaction is a bit more user-friendly.)
The connector ⊙, on the other hand, can be read as “It ought to be, all things considered, that A”. To introduce it we have to add something to the model (as described so far), namely a value-ranking of the propositions. The source for such a ranking may be other factors, not explicitly listed here, but a typical example could be expectation value (thinking of the ideal propositions as outcomes of actions determined by decisions that obey or honor some primary obligation). The value-ranking defines a “better than” relation on the propositions. An ideal proposition E will be called value ideal for Δ exactly if, among the ideal propositions for Δ, none are better than E.
The proposition ⊙A is true in situation Δ if and only if there is a proper scenario in Δ, , whose corresponding ideal proposition is value ideal and implies A.
Both O and ⊙ can be conditionalized in the way Horty does. The situation Δ[A] = <W ∩ A, D>, and O(B|A), , is true in Δ exactly if OB is true in scenario Δ[A]. Similarly for ⊙(B|A)
Reconstructing Horty’s examples
‘Don’t eat with your fingers’ δ1 = <T, ~F> where T is the tautology
‘If you are served cold asparagus, eat it with your fingers.’ δ2 = <A, F>
Δ = <α, W, D> with W = T, the tautology, and D = {δ1, δ2}. We do not have priority rankings on imperatives.
There are two proper scenarios, for the satisfaction regions of δ1 and δ2 do not overlap. Only δ1 is in force, so both O(~F) and ⊙(~F) are both true.
We choose as value ranking one by which the satisfaction region of δ2 is better that that of δ1. The justification for this is precisely the same content or aspect of the book of etiquette that was the source for the priority ranking in Horty’s approach to the example.
In the scenario Δ[A] both imperatives are in force, and the satisfaction regions of both δ1 and δ2 are ideal propositions for Δ[A]. Hence O(~F|T) and O(F|A) are both true in scenario Δ. This is a case of conflicting primary obligations.
However, only ⊙(F|A), and not ⊙(~F|T), is true in scenario Δ, since only the satisfaction region of δ2 is value ideal for Δ[A].
Conclusion: ⊙(~F) and ⊙(F|A) are both true in this world, for scenario Δ.
Now we add to D the imperative δ3 = <T, N> : “Place your napkin on your lap”. We require that the set of situations in which you place your napkin on your lap is large: it has a non-empty intersection with both ~F and (A ∩ F). We also require that the intersection with N does not change the value ranking: (A ∩ F ∩ N) is better than (A ∩ ~F ∩N).
(A note: in Horty’s discussion of the examples the place of δ3 in the priority ranking is not specified. It is tacitly understood that there is no conflict, so it does not need to be specified. That there is no conflict between N, F, and A, however is important.)
This gives us scenario Δ* = <W, {δ1, δ2, δ3}>. So now there are two proper scenarios, {δ1, δ3} and {δ2, δ3}. Only the first is in force, as before. So we still have ⊙(~F), but we also have ⊙N as well as ⊙(~F ∩ N).
In Δ*[A] now all three imperatives are in force, so we have two ideal propositions to consider, (A ∩ ~F ∩ N) and (A ∩ F ∩ N). Of these only the latter is value ideal. So in scenario Δ*[A] it is ⊙(F ∩ N) that is the case, and it is not the case that ⊙(~F). Consequently, in Δ* it is the case that ⊙(F|A), as well as ⊙(N|A) and ⊙(F ∩ N|A)
Conclusion: in scenario Δ*, ⊙(~F), ⊙N, and ⊙(N ∩ F | A) are all true.
And this is how it should be.
NOTES
Thanks to Ali Farjami for alerting me to an omission in an earlier version of this post. I had mistakenly allowed for imperatives that were literally impossible to satisfy. While an agent may have conflicting obligations, which cannot be jointly satisfied, there cannot be an obligation which it is impossible to satisfy.
I said that in Alan Anderson’s deontic logic, the conditional resisted all efforts at explication. In the classical framework, with all the machinery of possible world modeling also for various sorts of conditionals, things looked better for the conditional. There is recent work on this, for example:
Catherine Saint-Croix and Richmond H. Thomason (2020) “Chisholm’s Paradox and Conditional Oughts”. http://web.eecs.umich.edu/~rthomaso/documents/deontic-logic/ctd.pdf
The new examples are in Horty, John (2014) “Deontic modals: why abandon the classical semantics?” Pacific Philosophical Quarterly 95: 424-460.
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