Choice & Inference » Blog Archive » Causal Modeling and Counterfactuals

Causal Modeling and Counterfactuals

Apr 24th, 2009 by Rachael Briggs

I’ve always thought that causal modeling provided a neat way of cashing out the concept of the selection function in Stalnaker’s semantics for counterfactuals. Suppose the antecedent $A$ of a counterfactual is a value assignment to a variable, or a conjunction of value assignments to variables. Then relative to a causal model $M$ , the closest $A$ worlds are the ones correctly described by the model that results from performing the intervention $\textmd{DO}(A)$ on $M$ . So $A > B$ is true in $M$ just in case $B$ is true in the model that results from performing the intervention $\textmd{DO}(A)$ on $M$ .

But this isn’t a complete story about how the selection function works, because it doesn’t tell you what to do when $A$ is (say) a disjunction of value assignments to variables. At best, causal modeling provides a partial definition of the selection function.

If you’re a fan of causal modeling, there are a two obvious options:

Deny that counterfactuals with disjunctive antecedents are meaningful.
Find a way of extending the definition of the selection function.

Here’s a suggestion about how you might accomplish option 2, within the framework of causal models. First, require that the antecedent of any counterfactual be expressed as a disjunction of conjunctions of value assignments to variables. (I’ll write this disjunction as $D_1 \vee D_2\ldots \vee D_n$ . Intuitively, the $D_i$ s correspond to different ways of making the antecedent true.) Then let $A > C$ be true in a model $M$ just in case $C$ is true in every model that results from performing some intervention $\textmd{DO}(D_i)$ on $M$ , and false otherwise.

The suggestion ends up looking less like Stalnaker semantics than like the semantics suggested by Donald Nute in his 1976 paper “Counterfactuals” in the Notre Dame Journal of Philosophical Logic. Whereas Stalnaker uses a selection function that maps proposition-world pairs onto worlds, Nute uses a selection function that maps wff-world pairs onto sets of worlds. For Stalnaker, where $f$ is the selection function, $v_\alpha(A > B) = 1$ if $v_{f(A, \alpha)}(B) = 1)$ , and $v_\alpha(A > B) = 0$ otherwise. For Nute, where $f$ is the selection function, $v_\alpha(A > B) = 1$ if $\forall\beta(\beta \in f(A, \alpha) \supset v_\beta(B) = 1)$ , and $v_\alpha(A > B) = 0$ otherwise.

Some characteristics of the proposal, in no particular order:

Counterfactuals with logically equivalent antecedents are not themselves logically equivalent. This doesn’t seem so awful; it accords with some of my judgments about English-language examples. “If I were to eat cookies then I would be happy” is not equivalent to “If I were to eat cookies with broken glass or cookies without broken glass, then I would be happy”.
The schema $(A \gtrless B) \supset [(A > C) \supset (B > C)]$ is invalid.
The schema $(A \wedge B) \supset (A > B)$ is invalid for counterfactuals with disjunctive antecedents.
Conditional excluded middle, i.e. the schema $(A > B) \vee (A > \neg B)$ , is invalid. (If you don’t like this, you can take Stalnaker’s supervaluationist way out. Just say $A > C$ is true in a model $M$ if $C$ is true in every model that results from performing an intervention $\textmd{DO}(D_i)$ on $M$ , false in $M$ if $C$ is false in every model that results from performing an intervention $\textmd{DO}(D_i)$ on $M$ , and indeterminate otherwise.)
The schema $[(A \vee B) > C ]\supset (A > C) \wedge (B > C)$ is valid.
The counterfactuals I’ve discussed are still fairly restricted in their logical form. For instance, I haven’t said anything about how to deal with counterfactuals whose antecedents contain negations or other counterfactuals. Perhaps these could be translated into conditionals I whose antecedents are disjunctions of conjunctions of value assignments.

As somebody who takes causal models seriously, I’m no longer sure whether $>$ is really a binary connective in the sense that $\wedge$ is a binary connective. It’s not obvious to me that for any propositions $A$ and $B$ , there is a proposition $A > B$ . Perhaps this is an under-explored way around the triviality theorems for conditionals, which state that no appropriate propositional connective $\rightarrow$ can satisfy the following notorious thesis for all appropriate probability functions $P$ :

$P(A \rightarrow B) = P(B|A)$

(The appropriateness conditions for both $\rightarrow$ and $P$ vary depending on the triviality theorem. And for some of the triviality theorems, the quantifier order is reversed.)

One could conceivably hold that $>$ is the sort of conditional that figures in the notorious Thesis, and accept that some sentences of the form $A > B$ express propositions, but still deny that $>$ is a propositional connective.

Posted in Causation, Counterfactuals

33 Responses to “Causal Modeling and Counterfactuals”

Lee Walters says:

April 24, 2009 at 10:13 am

It seems that you have provided excellent reasons not to endorse your proposed cashing out of the selection function! Simplification of disjunctive antecedents is, on reflection, completely implausible. The following exchange seems absolutely fine to me, but the conclusion on your account is false

A: If I were to eat cookies with broken glass or cookies without broken glass, then I would be happy.

B: Really?

A: Yes. Because If I were to eat cookies with broken glass or cookies without broken glass, then I would eat cookies without broken glass.

I presume that you would treat existential antecedents in a similar fashion? If so do you license the following transition?

If an animal escaped from the zoo, it would be a monkey. Therefore, if Ellie the Elephant had escaped from the zoo, she would have been a monkey!

Also are you happy to endorse the corresponding or-to-and inferences in modal contexts and the consequent trouble they make? For example, you may have salad or soup, therefore you may have salad and you may have soup.

Reply
Christopher Hitchcock says:

April 24, 2009 at 10:51 pm

Hi Rachael, hi everyone,

Great to see a blog on these topics.

I agree that this is the natural way to construct a semantics for cfs from causal models.

Note that the resulting logic won’t satisfy centering. E.g. suppose that X = 1; then the cf ‘X = 0 or 1 > X = 1′ comes out false. This is really at root what’s at issue in the disjunction case. The idea is that the antecedent is always made true by an intervention (or Lewis-style ‘miracle’) on the value of the variables mentioned in the antecedent, even if no intervention (or miracle) is needed to make the antecedent true. So changing the logical form of the antecedent will change which interventions are done. Intervening on whether I eat cookies, and on whether or not there is glass in the cookies is not the same as intervening on whether I eat cookies, even if the result of the second intervention is not specified.

Reply
- Lee Walters says:
  
  April 25, 2009 at 2:05 pm
  
  Christopher,
  
  Is there any reason why a causal modelling approach *needs* to reject centering? I can see that the way you specify it, centering does fail. But is there anything wrong with an approach that intervenes only when needed to bring about A? Of course this would mean rejecting simplification of disjunctive antecedents in the case when the disjunction obtains, but would be compatible with validating SDA when the antecedent is false (not that I would want to).
  
  Does anyone in the causal modelling literature accept centering – I’ve only read Hiddlestone’s Nous piece and a couple of pieces by you and Woodward.
  
  Reply
  - Christopher Hitchcock says:
    
    April 25, 2009 at 7:14 pm
    
    I don’t see why you couldn’t preserve centering, but you would have to be very careful how you formulate it. In particular, how do you handle counterfactuals with ‘mixed antecedents’ where some variable settings are actual, and some are counterfactual? Suppose, for example, we have the following model:
    
    A = 0
    B = 1
    C = A + B
    D = 1 if A = 1 and C = 2
    = 0 otherwise
    
    Now consider the counterfactual ‘If A were 1 and C were > 0, then D would be 1′. Is this true? We set A = 1. Now, do we reason: since C was actually > 0, we keep it at its actual value of 1? Or do we reason: once we have set A to 1, we don’t need to interfere any further to ensure that C > 0, so C will take the value 2? If we reason the first way, D would be 0. If we reason the second way, D would be 1. Or should we reason in this case in the way Rachael suggests: consider both the intervention that sets A to 1 and C to 1, and the intervention that sets A to 1 and C to 2. In this case, there is no value of D that *would* occur.
    
    Reply
    - Rachael Briggs says:
      
      April 26, 2009 at 3:20 am
      
      I think the way that to preserve centering is to say that $(D_1 \vee D_2 \vee \ldots D_n) > C$ is true in $M$ just in case either for some $D_i$ , $D_i \wedge C$ is true in $M$ or $C$ is true in every model that results from performing some intervention $\textmd{DO}(D_i)$ on $M$ , and false in $M$ otherwise. (That’s disjunctive, but it gets the job done.)
      
      Nice points about centering and the Prince of Wales, Chris.
      
      Reply
Christopher Hitchcock says:

April 24, 2009 at 11:16 pm

Just a follow-up to my earlier post. There is another way in which this semantics will yield results that are counter to the way some people think about counterfactuals. Suppose that I go to the story to buy ice cream. My two favorite flavors are, in order, chunky monkey and cherry garcia. Both are in stock, so I buy the chunky monkey. It might seem natural to say that if I had not bought the chunky monkey, I would have bought cherry garcia. But the semantics don’t license this. The idea is that an intervention or miracle is overriding the normal causal processes that determine which ice cream I buy.

Thinking about counterfactuals in this way allows a counterfactual theory of causation avoid some difficulties raised by Carolina Sartoria in “The Prince of Wales Problem for Counterfactual Theories of Causation” (available online at http://philosophy.wisc.edu/sartorio/pw.pdf )

Reply
Rachael Briggs says:

April 24, 2009 at 11:23 pm

Lee: Two quick points about “If I were to eat cookies with broken glass or cookies without broken glass, then I would eat cookies without broken glass”. First, on the analysis I’m suggesting, the truth values of counterfactuals are model relative. Second, translating from English to formalism is not at all straightforward, and there’s a bit of wiggle room.

Putting those two points together, I think I can create plausible models and translations from English to formalism in which “If I were to eat cookies with broken glass or cookies without broken glass, then I would eat cookies without broken glass” comes out true. Suppose I’m ordering dessert, and the menu lists cookies with broken glass and cookies without. We might represent the situation using the following causal model.

Interpretation of variables:
$D = \begin{tabular}{ll} 1 & \textmd{if you decide to order something}\\ 0 & \textmd{else} \end{tabular}$

$I = \begin{tabular}{ll} 1 & \textmd{if you're insane}\\ 0 & \textmd{else} \end{tabular}$

$C = \begin{tabular}{ll} 2 & \textmd{if you order cookies with glass}\\ 1 & \textmd{if you order cookies without glass}\\ 0 & \textmd{if you ask for the check} \end{tabular}$

$H = \begin{tabular}{ll} 1 & \textmd{if you're happy}\\ 0 &\textmd{ else} \end{tabular}$

Structural equations:
$C = (D + I) \times D$

$H = \begin{tabular}{ll} 0 & \textmd{If } C = 2\\ 1 & \textmd{else} \end{tabular}$

I’m inclined to interpret your conditional as $(D = 1) > C \neq 2$ , which comes out true so long as you are not insane.

Another possibility (which Nute endorses) is that there are multiple kinds of counterfactuals, some of which satisfy SDA and some of which don’t. That seems plausible, and compatible with everything I said above. I’m really talking about non-backtracking causal counterfactuals, but there are backtracking counterfactuals and non-causal counterfactuals too. So saying SDA is valid for all counterfactuals is probably overkill, and I should have been more careful.

I presume that you would treat existential antecedents in a similar fashion?

I’m not sure what to do with existential antecedents. The formalism I’m using doesn’t map completely comfortably onto English, which your objections bring out nicely

Also are you happy to endorse the corresponding or-to-and inferences in modal contexts and the consequent trouble they make?

Probably, as long as I don’t have to endorse the inference from $\textmd{may}(\phi)$ to $\textmd{may}(\phi \vee \psi)$ . I think treating permissibility as a normal modal operator was a bad idea in the first place. (I don’t think that follows from anything I say in this post, though, it’s just what I happen to think about permissibility operators.)

None of this strikes me as entirely satisfactory, but I don’t find the going alternatives very satisfactory either. Stalnaker doesn’t tell you what the selection function is. Lewis does, but his account can’t handle “If Nixon had pressed the button, there would have been a nuclear explosion” without resorting to ad hocery. Are there brilliant possibilities that I’m just missing?

Reply
Lee Walters says:

April 25, 2009 at 1:05 am

Hi Rachael,

The free choice permission problem arises for other modalities as well where we do want to endorse that from M(P) we can conclude M(PvQ), where M is the appropriate modal. Also, even in the deontic case, the inference is cancellable: you may have salad or soup, but i can’t remember which, I’ll go check. And it seems the same is true for SDA and counterfactuals.

I not sure what the problem for Lewis is that causal modelling avoids – this may well be because I’m ignorant of the causal modelling approach, so any help here would be appreciated.

I take it the issue is that assuming indeterminism, there are lawful worlds where Nixon presses the button but there is no explosion. the lesson then is that not all lawful A-worlds that are like @ up to near the time of A are amongst the closest worlds. lewis then appeals to quasi-miracles and that is unsatisfactory. See Robbie Williams’ PPR 2008 article for a Lewisean response.

Ok but what does your approach say? You have to limit your attention to models where DO (A) iff C. But what justifies that limited focus given that you are modelling the actual indeterministic causal networks? Well whatever it is, why can’t Lewis help himself to restricting his attention to certain lawful worlds?

Reply
- Rachael Briggs says:
  
  April 25, 2009 at 8:05 am
  
  I get worried about Lewis before the epicycle involving indeterminism comes in. In the original example, the point is just that worlds with button-pushing and no nuclear holocaust sure look more similar to the actual world than ones with button-pushing plus a nuclear holocaust. Lewis’s response is that similarity is to be gauged by the following three criteria, listed from most to least important.
  1. Avoid large miracles
  2. Maximize perfect spacetime matching
  3. Avoid small miracles
  4. Maybe maximize approximate similarity of particular fact.
  
  This already seems a bit ad hoc to me. Why those criteria? They get around the counterexample (unless you think the laws of the universe are time-symmetric, or indeterministic), and Lewis makes them sound plausible. But I’m not convinced that when people imagine what the world would be like if $A$ , they go about imagining it using anything like Lewis’s criteria. I’d say something like Igal Kvart’s theory or the causal modeling picture gets a better grip on the intuitive picture: hold fixed everything that gets settled prior to $A$ , change $A$ , and then let the situation evolve according to laws or local true lawlike generalizations or something.
  
  Now, what about the version of the example where the button is indeterministic? I thought that for Lewis, the problem wasn’t just that “If Nixon had pushed the button, there would have been a nuclear holocaust” comes out as not true, but that “If Nixon had pushed the button, there would not have been a nuclear holocaust” comes out as true. You get strictly more exact spacetime match in worlds where Nixon pushes the button but (as chance would have it) there’s no holocaust. Since the laws are indeterministic, the extra matching doesn’t cost you anything in terms of law violations.
  
  I’m not sure what the causal modeling framework says in the indeterministic cases. Pearl doesn’t allow structural equations to be indeterministic: all apparent indeterminism in his model results from deterministic laws plus unknown hidden variables. (I believe SGS take the same approach.) My inclination is to say that if your structural equations are indeterministic and $\textmd{DO}(A)$ gives you a probability rather than a truth value for $B$ , then $A > B$ is not true, but either false or indeterminate. So it’s not true that if Nixon had pushed the indeterministic button, there would have been a nuclear holocaust, but at least it’s not true that if Nixon had pushed the indeterministic button, there would not have been a nuclear holocaust. (I want to say that it’s got some sort of intermediate truth value equal to the probability of the consequent in the closest antecedent model, and that high truth values are good enough for assertability, but all that is highly controversial.)
  
  I’ll check out the Williams article and get back to you on that, though. (What you say about the free choice permission problem is interesting, but I just don’t have a general account to give you. I have a general methodological presumption in favor of fudging the translation between English and formal languages, but that’s not an account of anything.)
  
  Reply
Gregory Wheeler says:

April 25, 2009 at 8:00 am

Nice idea, Rachael. So, since $(A\vee B>C)\rightarrow (A>C)\wedge (B>C)$ is valid, I take it that you are thinking that disjunctive antecedents refer to cases of over-determination. Is that right?

So, if ‘A or B were true, C’ is unpacked as expressing that A and B each are (counterfactually) sufficient for C.

If this is correct, why think that the proposal extends the selection function for disjunction simpliciter rather than provide an account of disjunctive antecedents which express overdetermined causes? If the latter, do you think there are syntactic or semantic markers in natural language sentences to distinguish between over-determined and (let’s call it) Boolean disjunction counter-factuals? And (realizing, of course, that this reed of mine might be planted in a too-small and cracked pot) if you cannot find linguistic features in natural language to distinguish these two cases, how would you motivate the analysis?

Reply
Jesús Zamora says:

April 25, 2009 at 3:57 pm

Perhaps I am too positivistic, but I have never understood the fascination people have with exploring the ‘logical analysis’ of counterfactuals, without paying almost any attention to the following question: once you have proposed a particular analysis of the meaning of counterfactuals, does this analysis illuminate in some way the question of how can we KNOW whether a counterfactual is true or not? We make thousands of counterfactuals in our everyday life, as well as in science, and it seems that it SHOULD not be too difficult to learn whether some of are true or false. I think that, from the point of view of philosophy of SCIENCE (as different from logic or phil. of language), the important question is the empirical testing of counterfactuals, and any logical analysis should give a clear hint of how this testing runs. This is the reason I have always suspected of possible-world semantics analysis: how can one KNOW whether a non-tautological statement about a possible world is true or not (not to say one about a possibly infinite set of worlds), given that we only observe the ‘real’ world.

Reply
- Rachael Briggs says:
  
  April 26, 2009 at 2:12 am
  
  Jesús, this looks like a positive reason to prefer the causal modeling approach to the Lewis/Stalnaker approach. Advocates of causal modeling do devote a lot of attention to the question of how one performs statistical tests to match situations with causal models. Of course, you need causal information going into these tests: you must be able to identify a repeatable, causally homogenous situation, and you must be able to identify manipulable variables. But that’s not an instance of the problem you’re worried about (I don’t think).
  
  In Lewis’s defense, he gives at least an implicit account of how to ascertain whether particular counterfactuals are true. First, one ascertains the laws of nature. One does this (according to Lewis’s best-system analysis) by observing how properties are instantiated over as large a swath of spacetime as possible, and coming up with the best deductive system to summarize your data. (We have to add in some assumptions about the swath of spacetime you observe being representative of spacetime in general, but I’ve yet to see any solution to the problem of induction that doesn’t require some kind of uniformity of nature principle.) If a sentence is a theorem of the best system, then it’s a law of nature. You then use the list of priorities I cited in response to Lee to construct a closeness relation among possible worlds. I don’t buy any of this, but it is an account.
  
  I’m also curious: how general is your worry? Is it an objection to modal logics as well as logics of counterfactuals?
  
  Reply
  - Jesús Zamora says:
    
    April 26, 2009 at 12:19 pm
    
    Thanks so much, Rachael, you clarify a lot the question.
    Regarding your las point, I am certainly very skeptic about modalities in general; modal logics and modal concepts are useful, of course, but, as a philosopher of science, I’m mostly concerned with the question of how do we learn that something is necessary (nor accidental) or contingent.
    
    Reply
Horacio Arló-Costa says:

April 26, 2009 at 2:37 am

Does anyone know of recent or not so recent work on the interpretation of iterated conditionals in the causal setting? Pearl shows in his book a result indicating that the logic of his causal conditionals is the logic of Lewis’ system VC (Lewis’s `official’ system of ontic conditionals). But after a second look it seems that what Pearl really proved is that the logic of his causal conditionals is axiomatized by the non-nested fragment of VC (not VC). Perhaps iterated conditionals can be interpreted with respect to conditions on sub-graphs.

In particular I wonder whether the so-called Export-Import laws hold for causal conditionals or not. It is tempting to interpret $A > (B > C)$ as $(A \wedge B) > C$ but as a matter of fact Lewis’ semantics does not validate this inference. Bayesian accounts of conditionals, in contrast, do impose Export-Import.

Regarding centering Menzies’ account of conditionals in his PSA article seems to offer an axiomatization in terms of Lewis’ system V, which does not obey centering. He gives some philosophical reasons to abandon centering. So, centering has been abandoned in some recent accounts, which are independently motivated.

Reply
- Lee Walters says:
  
  April 26, 2009 at 11:05 am
  
  Centering has itself been motivated too. For a defence of centering see my forthcoming paper Morgenbesser’s Coin and Counterfactuals with True Components: http://www.ucl.ac.uk/~uctylwa/papers/Morgenbesser%27s%20Coin.pdf
  
  Reply
  - Horacio Arló-Costa says:
    
    April 26, 2009 at 10:50 pm
    
    Lee: Thanks for the reference to your article. I read some of the main arguments which are interesting. There are some variants of the Coin example you mention that seem more robust (and difficult to relate to Morgenbesser’s example). One of them was offered by Levi in his 1996 book:
    
    EXAMPLE (Levi, 1996): Suppose agent X is offered a gamble on the outcome of
    a toss of a fair coin where he wins $ 1,000 if the coin lands heads and loses
    nothing if the coin lands tails. Let utility be linear in dollars. The expected
    value is $ 500. X has to choose between this gamble and receiving $ 700 for
    sure. X has foolishly (given his beliefs and values) accepted the gamble and
    won $ 1,000. Y points out to him that his choice was foolish. X denies this.
    He says: ‘If I had accepted the gamble, I would have won $ 1,000.’
    
    It is clear though that if one has (A&B) –> (A > B) one would be committed to the last conditional, presumably against intuition. At least prima facie it seems that it is more difficult to infer this conditional from Morgenbessor, but perhaps you have an idea how to do it (one would have two different and somewhat conflicting bets interacting — in one case one bets in favor of heads and in the other case one does not).
    
    In any case, there are epistemic accounts of conditionals (like the one proposed by Levi in the aforementioned book) that would not recommend to accept the conditional in the Coin example on normative grounds. The main philosophical problem for these accounts is to articulate the idea of supposing A when A is believed. The recommendation in this cases is to open one’s mind with respect to A (contract both A and its negation) and then expand with A. This account of supposition is not universally accepted but it manages to explain interesting aspects of the logic of epistemic conditionals.
    
    Reply
    - Lee Walters says:
      
      April 29, 2009 at 12:25 pm
      
      Horacio,
      
      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.
      
      Reply
      - Horacio Arló-Costa says:
        
        May 2, 2009 at 6:30 am
        
        Lee: Thanks for your response. I see that the two betting arguments can be combined successfully. So, to keep things simple, I will return to your original argument. In order to make this discussion accessible to other readers I will present some background. We have first Morgenbesser’s conditional. I toss an indeterministic coin, and whilst the coin is in mid-air, I offer you good odds that it will come up heads. You decline the bet and the coin lands heads. The following is then true
        
        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.
      - Lee Walters says:
        
        May 3, 2009 at 2:59 pm
        
        Horacio, thanks for your comments. It had never occurred to me that someone would reject my first premise, so your thoughts are most welcome.
        
        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.
      - Horacio Arló-Costa says:
        
        May 3, 2009 at 10:00 pm
        
        Lee:
        
        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.
- Christopher Hitchcock says:
  
  April 26, 2009 at 6:48 pm
  
  There is a natural way to interpret nested conditionals of the form A > (B > C). Perform the action Do(A), which gives you a new causal model. Then see if B > C is true in the new model. It seems pretty clear to me that it won’t satisfy import-export. Let the model have two variables X and Y. The only equation is Y = X, and both take the actual value 1. Let A be X = 1, B be X = 0, and C be Y = 1. Then A > (B > C) is false while B > (A > C) is true. (A & B) > C is also trivially true, since A & B is impossible.
  
  Reply
  - Horacio Arló-Costa says:
    
    April 26, 2009 at 10:20 pm
    
    Thanks. This looks right. One needs a different counterexample for the inverse inference from (A > (B > C)) to (A&B) > C. But I suppose that this fails as well.
    
    Perhaps with this proposal one can reconstruct all the fragment of VC (or V depending on ideas about centering) that admits iteration to the right.
    
    It seems that this proposal would guarantee immediately inferences from A and (A > B) to B, when B is a conditional. Such inferences are guaranteed in Lewis’ framework but they have been questioned by McGee and others in probabilistic and epistemic settings. It seems that the notion of iterated causal conditional that is validated is quite different from these types of conditionals.
    
    Reply
- Christopher Hitchcock says:
  
  April 26, 2009 at 7:01 pm
  
  I should add, though, that other sorts of nested counterfactuals are harder to construe. E.g., (A > B) > C. It’s not clear what we’re meant to do to the causal model to make the antecedent true.
  
  Reply
Jake Chandler says:

April 27, 2009 at 1:59 pm

Hi there: Just a quick question. I am not at all familiar with the causal modeling literature, but I take it that $do(.)$ cannot take a disjunction as an argument (e.g. $do(X\in \{ 1, 2, 3\})$ ). Is that correct, and if so, why can it not?

Reply
- Rachael Briggs says:
  
  April 28, 2009 at 10:49 am
  
  Jake, that’s right. $\textmd{DO}(\cdot)$ is supposed to uniquely pick out, for each argument, a single way of changing a causal model. There would be no unique way of performing a disjunctive intervention. It seems like a good idea to require that the names of interventions be unambiguous. One could presumably extend the definition of $\textmd{DO}(\cdot)$ so that it took disjunctive arguments, but I don’t really see why that would be a useful thing to do.
  
  Reply
  - Jake Chandler says:
    
    April 28, 2009 at 12:12 pm
    
    Ok, I see. Thanks for clarifying!
    
    (1) With respect to the usefulness of the generalisation to disjunctions, it seems that if $do(.)$ is supposed to model intervention/action, then the move seems desirable: I can presumably intervene to make it the case that $A\vee B$ without its being true that that I specifically make it the case that $A$ or that I specifically make it the case that $B$ , no? In terms of contrastive causation, one might say that I can cause, say $A\vee B$ rather than $\neg A\wedge \neg B$ , without causing $A$ rather than $B$ or vice versa. I might for instance, by firing Simon, bring about his suicide (sorry for the gruesome example), without causing him to commit suicide by overdose rather than commit suicide by gunshot or vice versa. Perhaps I’m missing the point about what $do(.)$ is meant to represent. Am I? (Also, allowing for $do(.)$ to take disjunctive arguments then obviously simplifies your treatment of counterfactuals, I guess)
    
    (2) Regarding the issue of uniqueness, I agree that no single change would be mandated, unless, we added more structure, like a linear ordering of variable assignments.
    
    Reply
    - Rachael Briggs says:
      
      April 28, 2009 at 12:30 pm
      
      The interventions are meant to be instances of direct manipulation. If you fire Simon and thereby cause his suicide, it makes sense to represent you as intervening on a firing variable. Your intervention may make a difference to whether Simon kills himself or not, but if you only make a difference by changing the value of some other variable, then you are not intervening on the suicide variable. So not every way of affecting a variable’s value is a way of intervening on that variable.
      
      I should add that the concept of an intervention is always relative to a model, and that there’s probably not a unique right model for every situation. If you’re pretty sure that you can get Simon to kill himself by firing him, you might represent Simon’s suicide as a binary variable, and treat it as something on which you can intervene. Or you might treat Simon’s suicide as a variable with multiple values on which you cannot perform an intervention, but which you can manipulate by intervening on other variables over which you have greater control. So “direct” control is really just control that’s direct enough for the purposes of a model. Part of getting a good model for your situation is getting a model on which you can represent your actions as interventions on variables.
      
      Reply
      - Jake Chandler says:
        
        April 28, 2009 at 11:06 pm
        
        Thanks Rachel. Ok, so my example is objectionable on grounds of the intervention being indirect. Still, I don’t see why there couldn’t be direct interventions that make true a disjunction of variable assignments. For some reason, I can’t seem to think of any real-life examples of direct interventions full stop, let alone of direct interventions of this variety, so I don’t have an example at hand. It just struck me that this extension could be easily implemented and would allow us to model a wider range of phenomena. I don’t know, as I said, I have only a very sketchy grasp of this literature…
      - Rachael Briggs says:
        
        May 2, 2009 at 1:15 am
        
        Jake, I don’t think interventions have to be “direct” in the sense of being instances of agent causation or anything; they just have to be direct in the sense that relative to a model, they pick out unique settings of variables. I this is more a naming convention than anything else. I concede that you could define “extended interventions” that were disjunctions of regular interventions, or regular interventions that were orderings of regular interventions. I’m just still not convinced that this would be useful.
        
        Suppose you’re contemplating an action that controls whether $A \vee B$ gets made true, but doesn’t control which disjunct is made true. This looks like a useful general way of reasoning. Which of $\{A, B, \neg(A \vee B)\}$ 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 $\{A, B, \neg(A \vee B)\}$ is true, you should not represent your action as an intervention on $X$ . Instead, you should represent $X$ as having two parents: $Y$ , which you control, and $W$ , which the world controls. The value of $W$ is unknown, but you can perform an intervention on $Y$ .
        
        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.
Christopher Hitchcock says:

April 29, 2009 at 8:33 pm

Rachael et al.,

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.

Reply
- Rachael Briggs says:
  
  May 2, 2009 at 1:27 am
  
  I hear your conditional (“If either Assassin had poisoned Victim’s drink, Victim would have died”) as true. If there’s a reading on which it is not true, one possible way of accommodating that reading is to fudge the translation between the formalism and the English.
  
  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 $\vee$ 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.
  
  Reply
  - Christopher Hitchcock says:
    
    May 2, 2009 at 3:06 am
    
    Rachael wrote:
    
    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.
    
    Reply
Rachael Briggs says:

April 25, 2009 at 8:23 am

So, if ‘A or B were true, C’ is unpacked as expressing that A and B each are (counterfactually) sufficient for C.

Yes. (It’s a somewhat odd sense of “overdetermination”, since there’s no implication that A and B occur in the very same world. But sure, let’s say “overdetermination” anyhow.)

If this is correct, why think that the proposal extends the selection function for disjunction simpliciter rather than provide an account of disjunctive antecedents which express overdetermined causes?

Since I backtracked in response to Lee, I’d better go with your second option. But I don’t know of any syntactic markers for counterfactuals that express overdetermined causes. “If I ate cookies with glass or cookies without glass, I would be happy” seems to have both readings available. So, how do I motivate this account? I guess the story is something like this:

Causal models give a fruitful, natural-seeming story about the kind of non-backtracking conditionals that people use when they’re deciding how to manipulate the world to get what they want. From this point of view, it makes sense that the range of permissible antecedents should be limited: the relevant antecedents are ones that you can manipulate, or could manipulate in principle. But sometimes it’s useful to make disjunctive plans about what to manipulate: I might say “I’ll either go hiking in Blackheath or at Echo Point. Either way I’ll get what I want, and either way I’ll need to remember to pack snacks and drinking water and a subway pass, and I don’t need to decide which of those two things I’ll do right now.” So it’s useful to extend the analysis to conditionals with disjunctive antecedents.

(This is a bit impressionistic and I’m supposed to run off and meet a friend in a couple of minutes, but I think it gets the general idea across.)

Reply

Causal Modeling and Counterfactuals

33 Responses to “Causal Modeling and Counterfactuals”

Leave a Reply

External Affiliates

Recent Comments

Categories

Blogroll

Archives

Subscribe