The proposal that I sketched in the previous post seems to be require a weak logic. I do not know how the logic looks like, so I do not know how weak it is. Your response intends to show that the logic is too weak to handle various applications. I have to think more about the proposal and its range of applicability to see whether it is a serious alternative to other accounts. But I think that I disagree about the main points you made. The logic is less weak that it seems at first sight and the proposal seems to work fine in various particular applications.

Let’s start with the principle that you call Weakening the Consequent: ((A > B) & Necessarily (B then C)) then (A > C). The axiom in conditional logic is:

(Right Weakening) If |- B –> C, A > B entails A > C

I use a different name because the axiom is slightly different and it has a recognized name in non-monotonic logic and conditional logic (see the articles by Kraus, Lehmann and Magidor in AI and JSL). Right Weakening is trivially satisfied in the semantics sketched in the previous message. For take and belief state supporting A > B. Such state, let’s call it K, after being contracted with A, ~A, B and ~B is such that when we expand it with A, it entails B. But since B entails C, this contracted state will also entails C, so this state has to support A > C as well (given that it is closed under logical consequence).

So, the semantics does validate Right Weakening. The issue is how to apply the axiom. Assuming that Bet > Win & Bet is supported by a belief state of the sort presented in my previous post, I will certainly have:

Bet on Heads & Win the Bet –> Heads

as a true generalization in my belief state. But Bet & Win –> Heads is not a logical law. It is a material conditional that happens to be true. To apply Right Weakening I need a logical law, which I do not have.

Second you claim that I have to reject (Bet & Toss) > Win. I do not see why this is the case. The belief state used in my previous message supports this conditional and Morgenbesser conditional as well. To evaluate the conditional I have to contract Bet –> ~Toss and therefore I have to give up ~Bet. I have to give up as well ~Win and Toss. Then I expand with the antecedent of the conditional. This will give me Bet. I already have Heads in the belief state and the conditional:

Heads & Bet –> Win

So, I can detach Win. It seems that the belief state supports (Bet & Toss) > Win as well as Morgenbesser’s conditional. Various other claims in your post seem to depend in this.

Finally Kvart’s example. I think that I can handle this example easily. The belief state contains:

B: The body fights the poison successfully.
~A: She did not take the antidote
S: She survives.

In addition we have the law-like statement:

B&A –> S

So, we have to evaluate now: ‘If Jane had taken the antidote, she would (still) have lived’, i.e. we have to evaluate A > S.

We have to contract ~A and S, but the law remains. B remains as well. When one adds the A, since B is available, one can derive S via the law. So, A > S is supported by the belief state that seems relevant here, which is the belief state after knowing that B is the case.

Regarding the issue of London and Jupiter, the main point of this proposal was to guarantee some sort of relevant connection between antecedent and consequent. So, it is clear that the theory will not be applicable to these examples. I do not think that there is a universal theory of conditionals that solves all problems regarding conditionals. There are various types of conditionals and different theories that capture different aspects of the logic of each type. For example, Levi’s theory which requires only to contract the antecedent and its negation would handle the London-Jupiter examples without problems. The modification sketched in my previous post seems to offer a weaker theory that handles other aspects of conditional reasoning, including perhaps some forms of causal connection. I have some worries of my own about the tenability of the proposal, but I do not see the points that you have raised against it as problematic. Certainly I continue to see centering as more problematic than these objections (most of which can be handled by the proposal, as sketched above).

In any case, thanks for probing the idea a bit more.

You suspect that the theory of conditionals that you sketch that generates the consequences you desire is very weak. I concur. Not only do you have to reject Restricted Transitivity, but you also have to reject Weakening the consequent, which I also use to support (1)

Weakening the Consequent: ((A > B) & Necessarily (B then C)) then (A > C)

Trivially, from Bet > Win we have Bet > (Win & Bet). Necessarily, if you win and bet heads, the coin lands heads, so by Weakening the Consequent we have Bet > Heads. That is, if we reject (1), it is not the case that if you had bet heads the coin would have landed heads. But in that case, how can it be that if you had bet heads you would have won, as Morgenbesser states?

You would also have to reject (Bet & Toss) > Win on your proposal. But given that (Bet & ~Toss) > ~Win, you have to reject

Necessarily, ((Bet > Win) then (((Bet & Toss) > Win) v ((Bet & ~Toss) > Win)))

to avoid (Bet & Toss) > Win.

Similarly, given that (Bet & ~Toss) > ~Win, you have to reject the conjunction of

Necessarily, ((Bet > Win) iff (((Bet & Toss) v (Bet & ~Toss)) > Win))

and

(((A v B) > C) & (A > ~C)) then (B > C)

To avoid (Bet & Toss) > Win.

Perhaps, given that your approach is more epistemic than mine, you welcome the likely resultant hyperintensionality?

You reject Bet > Heads because there is no causal correlation between my betting and the coin landing heads. But this is precisely the reason many accept it, on the proviso that the coin did land heads (see below).

Penczek (1997 Erkenntnis) adopts rationale similar to yours for rejecting centering: it should not count in favour of A>B that A or B happen to be true. However, as he rightly notes, such a proposal is subject to counterexample from semi-factual conditionals where the antecedent is irrelevant to the consequent. If you are prepared to accept that the *only* true counterfactuals are those where the antecedent ‘brings about’ the consequent, then this is not a problem for you, but I think this is too much to accept. Bennett, Edgington, Penczek, McDermott, Kvart and others who reject centering do not want to reject the truth of irrelevant semi-factuals. As McDermott (1998) notes the conjunction of ‘If London were a large city, Jupiter would have twelve moons’ and ‘If London were not a large city, Jupiter would have twelve moons’ is a natural way to say that Jupiter’s having twelve moons is not dependent on the size of London.

Moreover, on your proposal, we would have to reject what Kvart calls cases of pure-positive effect. For example, Jane is bitten by a poisonous snake. There is an antidote to the snake’s venom that is 50% effective. That is, there remains a 50% chance of death among people exposed to both the poison and the antidote. In any case, Jane does not have any of the antidote with her. Somewhat improbably, however, Jane’s body fights off the poison and she survives. It is true that ‘If Jane had taken the antidote, she would (still) have lived’. But on the contraction view you are proposing it comes out as false.

So to maintain your semantics and avoid centering in the way you suggest, we have to reject that many useful counterfactuals are true and cripple our counterfactual logic. I think the appropriate response is to maintain our non-centering commitments and accept that centering is a consequence of these commitments, even if counterfactuals made true solely in virtue of the truth of their components are assertable.

Morgenbesser: If you had bet heads, you would have won. (Bet > Win)

Lee runs then the following argument:

Morgenbesser: Bet > Win
(1) (Bet > Win) –> (Bet > Heads)
(2) (Bet > Heads) –> ((Bet ? Toss) > Heads)
(3) ((Bet ? Toss) > Heads) –> (Toss > Heads)
Therefore
Coin: Toss > Heads

Notice that the last conditional says:

Coin: If I had tossed the coin at t, the coin would have landed heads

In a situation where is known that the coin was tossed and landed heads. Many think that this conditional is false. But many think as well that Morgenbesser’s conditional is true. Lee shows that if we accept the latter, and one accepts some basic principles of conditional reasoning, one must accept that Coin is true as well. Lee takes this as a defense of centering. I think that the argument is interesting but I would draw quite different conclusions from it. My point of departure is that Coin is indeed false and that Morgenbesser is true. So, there are only a few options open to us to maintain both intuitions. Some of the principles used in the derivation have to fail. So, my question is: is there any step in the derivation that is suspect? If so, is it possible to offer an account of supposition that accommodates the facts (i.e. makes Morgenbesser true, Coin false and shows that some step in Lee’s proof is wrong)? I think that this is possible, but one needs to modify some of the existing theories of supposition to achieve this goal.

Let’s first look at the justifications on the steps of Lee’s proof. The first step is justified in terms of an instance of Restricted Transitivity:

(1) (Bet > Win) & [(Bet & Win) > Heads] –> (Bet > Heads)

Since [(Bet & Win) > Heads] seems acceptable (true) one has (Bet > Heads). Restricted Transitivity is an axiom of many conditional logics. In spite of this it seems to me that if one wants to reject Coin one should reject as well (Bet > Heads): If I were to bet on heads, the coin would have landed heads. Under the point of view of causation, for example, there is no causal correlation between my betting on heads and the outcome of the coin toss.

Here it is an epistemic theory that might explain the failure of Restricted Transitivity and might preserve as well the truth (acceptability) of Morgenbesser and the falsity of Coin.

Levi has proposed that in order to evaluate A > B one should open one’s mind first with respect to A first, i.e. one should contract both A and its negation from the current view. Then one has to expand this open view with A and check whether B holds. I would extend this view here requiring that one open one’s mind with respect to B as well. The current view with respect to which (Bet > Heads) is evaluated contains Heads (and the negation of Bet). The proposal is that the evaluation of the conditional requires to check whether supposing Bet induces the acceptance of Heads, with respect to a state where one is in suspense with respect to both antecedent and consequent (but preserves other parts of background knowledge). This might check the conditional connection between Bet and Heads.

There are a few technical issues that are needed before we consider an example. The first step is to distinguish a belief base in the current view. For example, in the example we are considering the belief base is:

~Bet, ~ Win, Toss
Heads
Heads & Bet –> Win
Win –> Heads

Let’s call this base B. It contains basic information plus an entrenchment relation. In this case, for example, the two conditionals are considered as generalizations that are better entrenched than the other facts. Cn(B) = K is the present view containing all the commitments of the agent (where Cn(B) indicates the logical consequences of B).

When one revises this base with new information one has the following equation:

K*A = Cn(B*A)

In other words, one revises the base first and then closes under logical consequence (for a presentation of base contraction see Hansson’s book on Belief Change). So, the base has a privileged status under an epistemological point of view. Let me illustrate with an example. If one has the base:

B’ = {Heads, Heads & Bet –> Win}

K’ = Cn(B’) contains Bet –> Win. But this is derived belief that is in K’ only because the two components of B’ are there. If I contract Heads from B’, Bet –> Win is also contracted from the new view. On the other hand if I contract directly the theory K’, Bet –> Win would remain in the resulting contraction.

So, with these elements let’s focus on the example. Let’s first consider Morgenbesser: Bet > Win. I have to contract B with Bet, ~Bet, Win and ~Win. So, given the current entrenchment, I have to eliminate ~Bet and ~Win from the view. Then I add Bet. But I still have Heads and Heads & Bet –> Win, so I have Win. So, the conditional is acceptable.

Let’s consider now [(Bet & Win) > Heads]. In this case the contraction that matters is the contraction with the negation of the antecedent: Bet –> ~Win. This is entailed by ~Bet and by ~Win, that we have to contract. We contract Heads as well. But when we expand with Bet & Win, we have still Win –> Heads that yields the desired result making the conditional acceptable.

Let’s focus now on Bet > Heads. Given the current entrenchment I have to contract ~Bet, as well as Heads. The other two generalizations remain. But now when I add Bet, Heads is not derivable. Here is where I use the idea that one operates on bases. The current base entails Bet –> Win but this is not a basic belief that can be used in the contraction. It is only a derived belief.

Finally let’s look at Coin: Toss > Heads. One has to contract Toss and Heads and their negations. But then when Toss is added Heads is not derivable. So, Coin is not acceptable.

I do not know at the moment other details of the view of conditionals that go with this analysis. I suspect that the theory of conditionals that thus arises is very weak. Some basic principles of conditionals are verified, like Conditional Modus Ponens, for example. But I think that the notion of supposition sketched here might be adequate for many applications.

I considered causal models for the inference via Restricted Transitivity and it seems that the inference is supported by this kind of models. I think that probably this is defect of causal models (although I might be missing something — I am less familiar with this type of models). My criticism of the inference was in part based on causal considerations. There is no causal connection between betting on heads and the outcome heads.

In any case, I think that the problem that you proposed is very interesting. In a way it inspired me to rethink some basic ideas about supposition. Finally a note in passing: In the previous analysis I used a belief revision model which is my favorite model of supposition, but I think that one can run a similar argument with other models of supposition.

Thanks again for the nice example. In a way this is a first reaction to it. Perhaps there is a way of dealing with it that is compatible with a stronger theory of conditionals. In this case it would be nice to see how this theory looks like. But I continue to be skeptic about the tenability of centering.

The conditional might mean “If either assassin had poisoned Victim’s drink, or if both had, Victim would have died.” (This looks plausible if you think about the truth table for in disjunctive normal form.) In that case, it will come out as indeterminate.

That’s interesting. If that’s right, and if the disjunction in the original counterfactual was inclusive, this is another kind of case in which counterfactuals with logically equivalent antecedents are not equivalent.

The conditional might mean “If either assassin had poisoned Victim’s drink, or if both had, Victim would have died.” (This looks plausible if you think about the truth table for in disjunctive normal form.) In that case, it will come out as indeterminate.

Or it might mean “If someone had performed an intervention on some variable that was a joint parent of the Assassin #1 variable and the Assassin #2 variable (maybe the Assassin Stimulus Package variable or something), then Victim would have died.” If the connections between the stimulus package and the individual poisonings are indeterministic, it will then come out as indeterminate. If the connections between the stimulus package and the individual poisonings are deterministically controlled by hidden variables, it will come out as having a determinate but unknown truth value.

I can’t get any readings of your conditional that aren’t covered by what I’ve suggested above, but I’m also lousy at philosophical intuiting.

Suppose you’re contemplating an action that controls whether gets made true, but doesn’t control which disjunct is made true. This looks like a useful general way of reasoning. Which of gets to be true depends partly on your choice and partly on the world’s input. So if you’ve got a variable $X$ that determines which of is true, you should not represent your action as an intervention on . Instead, you should represent as having two parents: , which you control, and , which the world controls. The value of is unknown, but you can perform an intervention on .

The above is basically a way of turning models with disjunctive interventions into models without disjunctive interventions. It doesn’t require the idea of an action that is “direct” in any model-independent sense.

Is this helpful? Let me know if I start talking in circles.

A question. Consider a disjunction involving different variables: e.g. X = 1 or Y = 1. In a counterfactual of the form (X = 1 or Y = 1) > (Z = 1), must the consequent be true only in the models where we perform Do(X = 1) and Do(Y = 1), or must it also be true in the model where we perform Do(X = 1 & Y = 1)? As you define it, it would be the former, and that would get my vote. For example, suppose Assassin #1 has an acid poison, and Assassin #2 has an alkali poison. Each is lethal singly, but together they cancel out. Neither one is able to poison Victim’s drink. Consider the counterfactual: “If either Assassin had poisoned Victim’s drink, Victim would have died.” The semantics you propose makes this true. But if you think that one way to make the antecedent true is to have both administer the poison, it would come out false.

This is relevant to the discussion with Jake about disjunctive interventions. In the case where the disjunction is over values of a single variable, e.g. X = 1 or 2, I don’t think it much matters whether you think in terms of a disjunction of interventions, or an intervention to make the disjunction true. The important point is that when there is an intervention, the ordinary causes of the value of the variable are overridden. Thus if you could intervene to set X = 1 or 2, we still wouldn’t be able to reason that X would have been 2 because… That is, the actual value of X would be indeterminate, and you would have to evaluate the counterfactual in essentially the way you describe. On the other hand, if you could directly intervene to make the disjunction X = 1 or Y = 1 true, then it seems that one possible result of the intervention would be to make both true, and this could lead to a different result than the semantics you propose.

Thanks for reading my paper and for the Levi reference – I was not aware of this work, so I’ll have to take a look.

McDermott (2008 Acta Analytica) offers a gambling counterexample to what we have been calling centering. I would respond to both McDermott and Levi in the same way and only did not respond to Mcdermott because of limitations of space. In the case of Levi it goes as follows (with ‘then’ as the material conditional):

Let us add to Levi’s story that Peter and Paul witness’s your situation and that Peter offers Paul good odds that you won’t win the bet, Paul declines. The coin lands, you win and Peter says to Paul “if you had bet that he would win, then you would have won yourself” (Bet>Paul Win)

(1) Bet>Paul Win
(2) (Bet>Paul win) then (Bet>You Win)
(2) (Bet> You Win) then ((Bet & you had gambled)>You Win)
(3) ((Bet & you had gambled)>You Win) then (you had gambled >You Win)
Therefore
you had gambled > You Win

(If anyone other than Horacio is reading this and wants to know what is going on, please see my paper here)

Vessell (Phil Studies) also raises a similar case against centering. According to centering we can do things that are morally/epistemically irresponsible and yet seem in some sense justified by a counterfactual that happens to be true in virtue of the truth of its components. I’m not an expert on any of this, but this does not seem peculiar to conditionals. Often irresponsible actions lead to good consequences, and in such situations we can say things like ‘you shouldn’t have done that, but I’m glad you did’. Of course that things turned out well this time is no guarantee that they will next time – indeed that’s why we acted irresponsibly – but improbable things happen.

Similarly we can say things that we don’t know to be true and are unassertable, but which just happen to turn out to be true.