Moderated by Erik Sandewall.

## Causality in Action Theories

The article mentioned above has been submitted to the Electronic Transactions on Artificial Intelligence, and the present page contains the review discussion. Click here for more explanations and for the webpage of theauthor, Camilla Schwind.

## Overview of interactions

N:o Question Answer(s) Continued discussion
1 15.7  Fangzhen Lin
30.7  Camilla Schwind

2 7.9  Hector Geffner
6.10  Camilla Schwind

3 21.9  Judea Pearl
21.9  Camilla Schwind
28.9  Judea Pearl
28.9  Camilla Schwind
28.9  Judea Pearl
28.9  Camilla Schwind
28.9  Judea Pearl

Q1. Fangzhen Lin (15.7):

Given that an online open review is written not just for the author and editors, but also for other readers, some of whom may not even have a chance to read the paper yet, it seems to me a good idea to summarize the paper first.

This paper considers statements of the form "A causes B", and proposes the following nine criteria to characterize them (in the following, I shall use   ·=>   to stand for "causes"):

1. Monotonicity: this concerns the question of whether from  A ·=> B , one can conclude that  A ^ C ·=> B  for any  C . This property should not hold, according to the author. In the following, all answers are those according to the author.

2. Transitivity: from  A ·=> B  and  B ·=> C , should one always conclude  A ·=> C ? No.

3. Contraposition: from  A ·=> B , should one conclude  ¬ B ·=> ¬ A ? No.

4. Reflexivity: Should  A ·=> A  always be true for any  A ? No.

5. Conjunction of preconditions: from  A ^ B ·=> C , should one conclude that either  A ·=> C  or  B ·=> C  must be true? No.

6. Conjunction: from  A ·=> B  and  A ·=> C , should one conclude  A ·=> B ^ C ? Yes.

7. Reasoning by cases: from  A ·=> C  and  B ·=> C , should one conclude  A v B ·=> C ? Yes.

8. Right weakening: from  A ·=> B  and that  C  logically follows from  B  (in classical logic), should one conclude that  A ·=> C ? Yes.

9. Left logical equivalence: from  A ·=> B  and that  A  and  C  are logically equivalent, should one conclude  C ·=> B ? Yes.

It then proceeds to consider five theories of causation found in the area of reasoning about actions and classify them using these nine criteria.

It seems to me that this is a reasonable approach to take for comparing various theories of causation, and the only paper that I know of that attempts to connect commonsense notion of causation with recent formal theories of causation proposed in the area of reasoning about actions.

I have no idea how fruitful it would be to connect causal theories of actions to commonsense notion of causation. My following comments and questions are only about issues that in my opinion are relevant to reasoning about actions.

1. First of all, why these nine conditions? In the paper, these conditions are often compared to the corresponding ones for material implication in classical logic. But there are many other properties of implication. For instance, if  p  implies  q , then  p  implies  q v r  for any  r . What is the corresponding property about causation? Is it a property worth investigating?

2. A related but more important question is how revealing these conditions are. For instance, the paper shows that some particular interpretations of McCain and Turner's causal logic (McCain & Turner AAAI'97) and Lin's causal action theory (Lin IJCAI'95) are monotonic and transitive, contrary to the proposed criteria 1 and 2 above. Now how bad this is for the two logics in question? Does it tell us anything about the range of applications of these two logics?

3. These nine conditions are about statements of the form "A causes B". But it seems to me that many theories of causation in the area of reasoning about actions are not directly about this kind of statements. For instance, Geffner (AAAI'91), Giordano et al (ECAI'98) and Turner (AIJ 1999) all use a modal operator, and Lin (IJCAI'95) uses a predicate that has only one argument for fluents. It makes a lot of difference whether to represent 'A causes B' as, for example,  boxA ·-> boxB ,  A ·-> boxB  (the one used in the paper), or  box(A ·-> B. It seems to me that there are good reasons for not going into a logic about the binary causal connective   ·=>  . On the one hand, the logic would be a lot more involved. One the other hand, it is not necessary for the purpose of reasoning about the effects of actions.

A1. Camilla Schwind (30.7):

The author's answer to this question is available in postscript.

Q2. Hector Geffner (7.9):

Erik has asked me to comment on Camilla Schwind's paper "Causality in Action Theories". Here are few comments that I hope Camilla and others may find useful.

The paper is an assessment of four theories of action that handle causal relations, in terms of a set of nine inference rules; namely, monotonicity, transitivity, contraposition, reflexivity, etc. The paper discusses (briefly) which rules should be valid and why, and then analyzes which inference rules are valid in each proposal.

This type of analysis has been common in the study of the logics of conditionals in philosophy (e.g, Lewis, Stalnaker, Adams, Nute, Pollock, ..) and more recently in the study of non-monotonic consequence relations and belief revision in AI (e.g., Gabbay, Makinson, Kraus et al; Gardenfors, Katsuno and Mendelson, ..).

I think this is a principled and meaningul approach to assess entailment relations, in particular in an area where a lot of the discussion is often based on individual examples.

Yet when applying this type of analysis to causal reasoning, it's natural to ask questions like:

1. are the inference rules considered particularly relevant to causal reasoning?
2. are there significant inference rules that are not in the list?

I'm not less sure about this, as the paper does not make a convincing argument about the choice of the rules. Indeed, to me, it looks that very few of the rules have to do with "causality", and most have to do with "plausible (non-monotonic) reasoning" in general. Indeed, monotonicity, transitivity, conjunction, cases, right weakening, and left logical equivalence, have been discussed before in the general non-mon setting. The arguments supporting the validity or invalidity of each of these rules in the general setting applies in the specific setting of non-monotonic causal reasoning. So they have little to do with "causal" reasoning in particular. The same could be said about the other three rules left.

I think for this analysis to be more insightful, the principles have to appeal to the distinction between causal and non-causal relations.

For example, something that could be called "Pearl's principle" (see his papers) says something like this:

Let  T  be a theory including two causal relations:
C1 = "A causes B to have (truth) value  v ", and
C2 = "C causes B to have (truth) value  v' ",
where  v  and  v'  can be any truth values (or non-truth values). Furthermore, let C1 be a "strict" causal relation (no exception). Then "Pearl's principle" says that the theory [ T ^ A  is true] should be equivalent to the theory [ T' ^ A  is true], where  T'  is  T  with the causal relation C2 removed.

The interesting point is that the same principle is NOT supposed to hold when "A causes B" is replaced by "A implies B". So, if reasonable, the principle says something that is true for "A causes B" that is NOT true for "A (logically or defeasibly) implies B".

Other "causality" principles could be formulated in a similar way (again, I'd suggest to look at Pearl's work).

I think that one could get further by analyzing and building on principles such as these -- that are based on the distinction between causal and non-causal relations -- than on more general principles of plausible reasoning. I've attempted something like this in an IJCAI-97 paper but with limited success.

Hector Geffner

A2. Camilla Schwind (6.10):

Answer to Hector Geffner, who raised the following questions:

 1. are the inference rules considered particularly relevant to causal reasoning? 2. are there significant inference rules that are not in the list?

First, let me point out that my main purpose has been to compare existing approaches and not to define postulates. I prefer the term "comparison criteria". This is what I formulated and NOT postulates.

ad 1. The criteria have been formulated with respect to the comparison task of several action theories which integrated causality; in this respect they are relevant as I think.

ad 2. It happened to me to use other criteria, which are not in the list and which are more related to causality and update. Since NO one of the formalisms I reported could be compared along those criteria, I found it moderately interesting to include them.

I think indeed that causality is a nonmonotonic inference relation (see my criterion 1). But there are properties of nonmonotonic inference which I reject for causlity (e.g. reflexivity).

Camilla Schwind

Q3. Judea Pearl (21.9):

I have some comments on recent ENRAC discussions of Camilla's paper, "Causality in Action Theories".

Axioms are good for detecting and summarizing differences among theories, but not so good as means of evaluating adequacy of theories. Camilla's list of axioms illustrates this observation relative to theories of causation. Even some of the most innocently-looking axioms on her list are not free from objections.

Take for example Right Weakening, with  A ·=> B  standing for "A causes B". The axiom says:
 8. Right weakening: from  A ·=> B  and that  C  logically follows from  B  (in classical logic), one concludes that  A ·=> C .
I have not heard anyone challenge this axiom; Fangzhen Lin approved of it, Hector Geffner let it stand, and, indeed, the axiom appears immaculate and harmless. But here is a counterexample.

Let  C  stand for some true and irrelevant fact, say  C  is "my eyes are brown". Being true,  C  logically follows from any  B . Yet, from " A  causes  B ", we would NOT wish to conclude that " A  causes my eyes to be brown".

The same goes for transitivity. If we ask a person on the street if causality is transitive, the answer would be: Of course. If we ask any causal analyst, the answer would be: Of course not. (e.g., my whistle causes Joe to shoot. When Joe shoots he kills birds. However, my whistle frightened the birds away).

So what assumptions must a person on the street be making, when asked to judge if an axion is plausible or not?

Here is a suggestion (for transitivity). Our person-on-the-street interprets transitivity to mean: If

1.  A  causes  B  and
2.  B  causes  C  regardless of  A , then
3.  A  causes  C

The suggestion is that questions about transitivity bring to mind chain-like processes, where  A  influences  B  and  B  influences  C , but  A  does not have direct influence over  C . Under such qualification, we have indeed a theorem that transitivity holds. This theorem is sound in both the counterfactual and manipulative interpretations of " A  causes  B " (that is " A  and  B  are true, and  B  would be false if  A  were made false") as well as in every semantically clear theory of causation that I have seen.

Conclusion: testing axioms against intuition is a dangerous enterprise; formal semantics is safer.

Judea

PS. Structural semantics of cause and counterfactuals is proposed in my IJCAI-99 paper; posted on

```  http://bayes.cs.ucla.edu/jp_home.html
```
Slides and (part of) the lecture transcript are on
```  http://bayes.cs.ucla.edu/IJCAI99/ijcai99_files/v3_document.htm>http://bay
```

A3. Camilla Schwind (21.9):

Dear Judea,

Thank you for your remarks on my paper.

I am not sure that in my paper I have been sufficiently clear. I do not think that I discussed axioms, but something weaker: postulates, or just criteria, i.e. properties of a causal relation. I find this approach useful for studying different theories for causality and actions and I beliefe that it is useful to question whether a theory for causal reasoning has these properties or not. As I have shown in my paper, and partly in my answer to Fangzhen Lin, this study allows to exhibit general properties for actions and causality and to show the advantages and limits of several approaches.

In more detail:

I do not agree with your counter-example concerning right weakening.

Let  C  stand for "my eyes are brown". Obviously,  C  is not logically true, but just true in some model (and can be false in some other model, namely in every model where my eyes are not brown). Then it is not true that any  B  follows from  C .

I have not advanced a special opinion on transitivity. As I pointed out, I think there are arguments for having transitivity as well as for not having transitivity. I think your example is interesting, because it illustrates, that the presence of transitivity may relate in some way to two different readings of causality:

1.  A  causes always  B
2.  A  causes sometimes  B
In the interpretation of (1), causality is probably transitive ? in (2) most certainly, causality is not transitive ?

C3-1. Judea Pearl (28.9):

Dear Camilla,

Perhaps, we can better study the problem with Right-Weakening if we replace C by any tautology, say C = "2+1 = 3". C is true in all worlds, hence it logically follows from B = smoke, which is caused by A=fire. We do not wish to conclude that fire causes the equality 2+1 = 3.

C3-2. Camilla Schwind (28.9):

Dear Judea,

"2+1 = 3" is not a tautology neither. "+" is a function and its semantics in natural number arithmetic is such that it is true in this particular world! We can of course find a world and a meaning for "+" (and for the 0-ary finction symbols "2", "3" and "1") such that "2+1 = 3" is not true.

But take the tautology "C = it rains or it does not rain". C follows from B = smoke which is caused by A = fire. Hence we conclude that fire causes the tautology "it rains or it does not rain". For me this is ok.

More generally, every sentence can cause every tautology!

C3-3. Judea Pearl (28.9):

Dear Camilla,

You wrote
 More generally, every sentence can cause every tautology!

Glad this discussion has led to such crisp summary of the difference between the postulate-based and the semantical approaches to causation.

To students of the latter, only propositions that can be false (in at least some imaginable worlds) can be caused by other things (events or actions).

C3-4. Camilla Schwind (28.9):

 More generally, every sentence can cause every tautology!
 Glad this discussion has led to such crisp summary of the difference between the postulate-based and the semantical approaches to causation.
This has nothing to do with the "difference between the postulate-based and the semantical approaches to causation". It is of course possible to have a postulate-based approach to causation which does not have the above property!

(For my postulates, it would be sufficient to skip or to weaken RW).

But I want to point out that therre is an enormous misunderstanding. The focus of my paper is not to present postulates for causation, neither to present a particular approach to causation. I have used criteria in order to compare and to study different approaches to action and causation!

C3-5. Judea Pearl (28.9):

 Glad this discussion has led to such crisp summary of the difference between the postulate-based and the semantical approaches to causation.
 This has nothing to do with the "difference between the postulate-based and the semantical approaches to causation". It is of course possible to have a postulate-based approach to causation which does not have the above property! (For my postulates, it would be sufficient to skip or to weaken RW).
Agree. But I still think it has something to do with the approach: It is often too easy for us to inspect a list of postulates and conclude "Yes, these sound reasonable". I have seen this happening many times in postulate-based approaches to beliefs and probability. (Recall, it took over a decade for people to discover that the celebrated AGM postulates are inadequate for belief update or even for belief revision).

 But I want to point out that therre is an enormous misunderstanding. The focus of my paper is not to present postulates for causation, neither to present a particular approach to causation. I have used criteria in order to compare and to study different approaches to action and causation!
I understand the goal, but the question remains: Do we have a set of postulates that is suffiently refined for discriminating among the different approaches to action and causation! Suppose (taking an extreme case) that only one of your postulates truly characterizes causation, would it be instructive then to use this one postulate as a comparison gauge? What if all approaches comply with that one postulate, are they equivalent? Or should we use the entire list, regardless of whether it applies to causation? What I am questioning is whether the language of postulates is suffienctly rich to characterize things, and various approaches to things.

I believe Hector Geffner expressed the same kind of concerns.

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