A look at violations of three logical principles (Weakening, Transitivity, Import-Export) for analogies with imperatives, with a side-long glance at counterfactual conditionals and Chisholm’s paradox.
Arrows and imperatives are both binary functions on propositions. We even use practically the same wording for both:
If the weather clears, we go on a picnic
If the weather clears, [do] go on a picnic!
So analogies may be illuminating– perhaps the difficulties we have often seen with arrows will point us to ideas about imperatives.
[1] Weakening
First, a good and very familiar reason for rejecting the material conditional as capturing conditionals in actual usage is the problem with the law of Weakening. It has a precisely similar problem for imperatives:
[For conditionals] If it bursts into flame, we will go for the fire extinguisher.
Infer:
If it bursts into flame, and is immediately doused with water, we will go for the fire extinguisher.
[For imperatives] If it bursts into flame, [do] go for the fire extinguisher!
Infer:
If it bursts into flame, and is immediately doused with water, [do] go for the fire extinguisher!
In neither case is the inference correct.
[2] Transitivity
There is a similar problem with the law of Transitivity which holds both for material and strict conditionals. Again, I see an analogy to imperatives:
If Hoover had been a Communist, he would have been a highly effective spy.
If Hoover been Russian, he would have been a Communist
Infer:
If Hoover been Russian, he would have been a highly effective spy.
We don’t accept the inference because we think that fanatical loyalty to his country was part of Hoover’s character. But he might well have had sufficient epistemic distance to see what he would have to do in each case, so that for him the corresponding imperatives would be in force:
If you are a Communist, be a highly effective spy!
If you are Russian, be a Communist!
But he would not recognize or accept the imperative:
If you are Russian, be a highly effective spy!
Both the original example and the analogue for imperatives can be contested, in the way counterfactuals are in general. The objection is that the sort of language is context-dependent, and the two premises would not be true in the same context of discussion. The example loses all force if we spell out what was kept constant in each case:for the first premise, but not for the second, that Hoover was American.
Nevertheless, the logic of conditionals that was prompted by such examples, which involve counterfactuals, lacks Weakening and Transitivity. So this may harbor suggestions for reasoning with imperatives.
[3] Import-Export
The next familiar logical law to look at is Import-Export, which is the basic principle of Intuitionistic logic:
(A & B) –> C if and only if A –> (B –>C)
This fails in counterfactual reasoning. I’ll give the example that Branden Fitelson recently sent me. A die is about to be cast, and we consider three propositions:
A: the outcome will be 1, 3, 5, or 6
B: the outcome will be even
C: the outcome will be 6
(A & B) is the same as C, so clearly (A & B) –> C, no matter what. The other side would be:
A –> (B –>C): If the outcome were to be 1, 3, 5, or 6 then (if the outcome were to be even, then the outcome would be 6)
That may seem right at first blush, but it is refutable. Take a model in which there are only two possible worlds, α and β. Of course (A & B) –> C is true in both. Imagine we are in α, where A is true but B and C are false, because the outcome is 1. Then to evaluate the conditional in α, we go to a possible situation β where B is true because the outcome is 2. But then C is false in β. So we conclude that (B –> C) is false in α, where A is true.
What about imperatives, is there an analogy? There are two ways to construe this question.
[ONE] Suppose that (A & B) implies C, and also that the imperative (Do A!) is in force. Does it follow that then, by implication, (if B do C!) is in force?
Counterexample: take C to be just (A & B). If (Do A!) is in force, does it follow, for any proposition B at all, that (If B, see to it that A & B!) is in force? But that is equivalent to (if B, see to it that A!), regardless of what B is. So no, definitely not!
[TWO] For the second way to think about this, let’s leave the example, which has the very special feature that C is equivalent to (A & B). Let A, B, C be arbitrary factual propositions and suppose that (If A and B, see to it that C!) is in force. Suppose A is true. Does it follow that the imperative (If B, see to it that C!) is now in force?
One question is whether a factual proposition can be such that, necessarily, if it is true then a certain imperative is in force. This opens up a new subject, that requires much more comples modeling of decision situations than I have been thinking about so far.
An example might be the factual proposition that the captain has ordered the soldier to stand guard. As a result, the imperative (Stand guard!) is in force in the soldier’s situation. And the soldier’s knowledge could include “If the captain has ordered me to stand guard then the imperative (Stand guard!) is in force for me.”
But the Import-Export principle, if it is a logical law, then it must hold for any propositions, not just for such special ones that establish authority. Suppose now that (If A and B, see to it that C!) is in force. If the person knows that A is true, there are two possibilities: he also knows that B is true (in which case the imperative kicks in) or he does not know that B is true (so the imperative does not kick in). What he knows then includes that:
if he were to come to know that B is true, he should see to it that C, if all the present imperatives are still in force
which reminds us of the previous discussion, about imperatives, time, and the Moebius strip. We cannot assume in general that imperatives stay in force over time, that is a special case. In our present reflection then, we have to say that even on this second way of taking the question, the answer is that the analogue to Import-Export fails.
[4] Chisholm’s Paradox
So far then we have, in view of those examples, come to the point where the arrow is interpreted as Stalnaker and Lewis do, allowing for counterfactual conditionals. With a side-long glance at a recent paper by Saint-Croix and Thomason (see NOTES below), let’s see how that fares in the old Chisholm’s paradox. I will add a time element, so that we can evaluate it in a situation before it is settled whether or not Jones goes to his neighbor’s help.
- It ought to be that Jones will go to help his neighbor tomorrow.
- It ought to be that if Jones will go to help his neighbor tomorrow then Jones tells his neighbor that he is going to come.
- If Jones does not go to help his neighbor tomorrow then Jones ought not to tell his neighbor that he is going to come.
- Jones does not go to help his neighbor tomorrow.
Suppose that we take the “if … then” in this example to be the conditional of Lewis or Stalnaker logic. Saint-Croix and Thomason point out the following. Let us start with a situation α in which Jones has promised to to help his neighbor tomorrow; as a result, 1. is true. It is not determined yet whether he will go to help his neighbor: that will happen tomorrow or not at all. Is premise 3. true?
To answer this, we look at the ‘nearest’ world to α, call it β, in which Jones does not go to help his neighbor the next day. Being ‘nearest’, it is true there too that Jones has promised to go help his neighbor. Is it true that he ought not to tell his neighbor that he is going to come? Not at all: in view of his promise, he ought to go to his neighbor’s aid the next day, and to tell him that he will do so.
Saint-Croix and Thomason respond to this by contextualizing the ‘nearest’ world selection, and their view as a whole is very attractive (though I am not yet ready to give up the rival view of conditional obligation statements as analogous to conditional probability statements). But however that may be, let us see how the Chisholm story plays out with imperatives.
Corresponding to the first three premises we imagine Jones in a situation α* where three imperatives are in force (using obvious abbreviations):
(Do Help!), (If Help, do Tell!), (If ~Help, do ~Tell!)
There are two scenarios for Jones’ action: (A) he both goes to help, and tells that he is going to come, (B) he does not go to help and does not tell that he is going to come. The former has the greater value, so in α* it is the case that he ought to go to help and tell that he will.
Now let us look at the next day, and suppose that he does not help his neighbor. Could all three imperatives still be in force.? The first has now become impossible to carry out, and there can be no obligation to do the impossible (as opposed to having two obligations which cannot both be satisfied). So no, the first imperative is no longer in force. The second may be, but does not kick in. The third, presumably still in force, does kick in and we conclude that on that next day, in which Jones does not go to his neighbor’s aid, he ought not to tell him that he is coming. Precisely the conclusion that common sense would suggest.
[5] Moral Conflicts
This discussion walks so closely to questions about conflict that it is timely to say that construing the “if … then” as a Stalnaker or Lewis arrow, in such examples, will not do at all if we are to accommodate genuine moral conflicts.
Look at the first two premises in Chisholm’s paradox. It is typical when this is discussed to say that together they imply that Jones ought to tell that he is going to come. Well, in Jones’ situation, if we imagine him as subject to the corresponding imperatives, he ought to do so, as we saw. But the pattern of inference cannot be generally valid for Ought statements, without eliminating the possibility of genuine moral conflicts:
5. Ought(A)
6. Ought(B)
7. B implies [if A then (A & B)] …. with “if then” as material implication
8. Ought[if A then (A & B)]
9. Ought(A & B)
Here 5 and 6 are premises, and 9 the unwelcome conclusion. The inference I meant was the one from 5 and 8 to 9:
Ought(X), Ought (if X then Y), therefore Ought(Y)
That is a standard inference form, in the present case connected with the similar
Z –> X, Z –> (if X then Y), therefore Z –> Y
and which thus needs to be rejected if we are to accommodate genuine moral conflicts.
As to the inference form 6 and 7 to 8, that is valid even in the minimal deontic logic that allows for conflicts, for it is a single-premise inference (recall Farjami’s “Up” ).
NOTES
Catharine Saint-Croix and Richmond H. Thomason “Chisholm’s Paradox and Conditional Oughts”. In Fabrizio Cariani et al., eds, Deontic Logic and Normative Systems:2014a, DEON 2014. Springer- Verlag, Berlin, 2014, pp. 192–207. Downloaded from https://web.eecs.umich.edu/~rthomaso/documents/deontic-logic/ctd.pdf
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