Saturday, May 03, 2008

Crossing Cultures in Free Will (by Guest Blogger Hagop Sarkissian)

Academic philosophers are a fairly homogeneous bunch--mostly male, mostly white. So, when a philosopher proposes a thought experiment and then goes on to make a general claim of the form "in this case, most people would surely say that P" or "in this case, it is clearly the case that P", one might wonder whether the philosopher's intuitions really are so obvious or widely shared. Perhaps only philosophers, or male philosophers, or Western male philosophers, or Western male philosophers who maintain theory x, would find it obvious (or even entertain the idea) "that P".

Indeed, in recent years, experimental philosophers interested in such intuitions about particular cases (and their role in philosophical theories) have discovered that it's not hard to find significant cross-cultural variation. For example, recent studies have shown that Americans, East Asians and Indians may differ considerably in their intuitions concerning key thought experiments in epistemology and philosophy of language.

Some colleagues and I wanted to see whether this phenomenon held true for beliefs concerning free will and moral responsibility. We asked participants in Colombia, Hong Kong, India, and the United States what they thought about the following case. (For a video presentation of these questions, click here.):

**********

Imagine a universe (Universe A) in which everything that happens is completely caused by whatever happened before it. This is true from the very beginning of the universe, so what happened in the beginning of the universe caused what happened next, and so on right up until the present. For example one day John decided to have French Fries at lunch. Like everything else, this decision was completely caused by what happened before it. So, if everything in this universe was exactly the same up until John made his decision, then it had to happen that John would decide to have French Fries.

Now imagine a universe (Universe B) in which almost everything that happens is completely caused by whatever happened before it. The one exception is human decision making. For example, one day Mary decided to have French Fries at lunch. Since a person’s decision in this universe is not completely caused by what happened before it, even if everything in the universe was exactly the same up until Mary made her decision, it did not have to happen that Mary would decide to have French Fries. She could have decided to have something different.

The key difference, then, is that in Universe A every decision is completely caused by what happened before the decision – given the past, each decision has to happen the way that it does. By contrast, in Universe B, decisions are not completely caused by the past, and each human decision does not have to happen the way that it does.

1. Which of these universes do you think is most like ours? (circle one)

Universe A-----Universe B

2. In Universe A, is it possible for a person to be fully morally responsible for their actions?

YES-----NO

**********

Previous work in cross-cultural psychology has shown that Westerners and non-Westerners differ in the way they think about moral responsibility, individual agency, and even the more fundamental notion of what it means to be a person; naturally, we expected to find some significant differences in the response patterns of these groups. Not so. In all four cultures, the majority of participants responded as indeterminists and incompatibilists! That is, the majority of participants believed our own universe to be indeterministic, and denied that moral responsibility could be compatible with determinism.





How is it, then, that individuals from such different cultural and religious backgrounds, with divergent ways of understanding the world, who have probably never been instructed on the topic of causal determinism, all tend to embrace the same two theses--indeterminism and incompatibilism? Such cross-cultural similarities in beliefs cry out for one of two kinds of explanation. The first would focus on innate endowment, such as some basic capacities concerning causal cognition or theory of mind. The second would focus on shared experience, such as the phenomenology of moral choice or the unpredictability of human action. Either approach seems worthy of exploration. It is also possible—and indeed quite likely—that the similarities here arise from a complex interaction between innate endowment and shared experience. Future research may shed light on the mechanisms involved. For now, it remains a puzzling finding in an area where one might expect some variation. 

21 comments:

Anonymous said...

Hi Hagop,

A very interesting project. Offhand I would expect a higher rate of libertarianism/incompatibilism in South Korea. But I don’t know if they have the right words.

I find that when I teach freewill/determinism/etc in service courses for nonmajors I have to introduce the concept of causation, and it isn’t always easy to get across. It tends to be confused with e.g. forcing. I wonder how “natural” a concept it is, and even how definite a concept it is. So I wonder what question the people taking the survey thought they were hearing. People who don’t hear what I hear in the word ‘causation’ are unlikely to hear something weaker. If they hear anything stronger, then all those wrong answers are probably not wrong at all.

Another possible problem is that the description of Universe A can seem to *intend* to describe a universe different from ours, simply because it starts out “Imagine a universe where …” – and that can influence reasonable approaches to interpreting the paragraph.

One might also have liked a control question about whether people can in our world be wholly morally responsible for their actions.

Anonymous said...

I'd be interested to know what percentage of people endorsed both determinism and incompatibilism. If that number if relatively high (it's at least 40%, given that the mean across countries is ~70% for each question), then we'd have good reason to be suspicious of these results, as it's safe to assume that the vast majority in each of these countries make a regular practice of holding people morally responsible for their actions.

(Of course, it could be that people are endorsing detached theoretical claims here that would systematically come apart from their more concrete judgments -- a phenomenon that Shaun Nichols and Joshua Knobe have previously documented (see here: http://dingo.sbs.arizona.edu/~snichols/Papers/moralresponsibilityFinal.pdf).)

Also, I'm assuming you've done all possible pairwise chi-square tests on the proportions to check whether any of the differences were significant (otherwise, you wouldn't be justified in making the claim that there are no differences). If not, I'd be interested to know the sample sizes. Just plugging in some numbers, if there are 100 subjects from each country, then the differences between the Hong Kong subjects and all other countries would be significant in the case of the determinism question (at p<.05, marginally at p=.06 for Hong Kong vs. Columbia).

And just a couple additional comments regarding statistics: even if the differences aren't significant, it's possible that your survey was underpowered (i.e., it's possible that you didn't have a large enough sample to detect differences that actually do exist). More generally, keep in mind that you technically can't accept the null hypothesis (that there is no difference between the groups, in this case), you can only fail to reject it (i.e., "absence of evidence isn't evidence of absence").

Finally, while it's laudable that you managed to obtain such an interesting sample, I wonder if the questions were presented in English to even the non-US samples. Granted, presenting the questions only in English would control for confounds that are purely a result of language differences. And it is perhaps unreasonable to expect you to do have accurate translations done for each language, especially given that this appears to be your first experiment on this topic. But, in this case, it would ultimately be far more telling to test people who have had minimal exposure to Western thought. And people who are fluent enough in English to understand your scenarios surely wouldn't qualify on this score.

Nevertheless, this is some thought-provoking data. Thanks for posting it!

Anonymous said...

Dear Anonymous,

This is really a very impressive comment. If you ever come out of your anonymity, you could make quite a name for yourself!

Just as you say, there are a number of different ways of analyzing this sort of data. One approach is to do a single analysis that looks at all four cultures and checks to see if there is any significant effect of culture on responses (which is what we did here), while another approach would be to use pairwise comparisons to look at differences between each pair of different cultures. However, if one wanted to use pairwise comparisons, one would have to apply the Bonferroni correction, dividing alpha by the number of tests. None of the comparisons would then come out significant.

Still, I think the basic point of your post is right on target. It is almost inconceivable in a case like this that the null hypothesis could turn out to be literally correct. That is, there is no real chance that the true frequencies in the populations from which we are sampling are *exactly* the same. The key result of the study, then, is that the patterns of intuitions in these different cultures are so shockingly similar and that the majority of subjects in all four cultures are indeterminists and incompatibilists.

Edouard Machery said...

Hagop,

Great data set.

If I were you, however, I'd be even more cautious, in asserting the universality of this pattern of intuitions.

Research by Joe Henrich and colleagues (I am sure you are familiar with this work) has shown that judgments about fairness vary cross-culturally as a function of the integration of the relevant cultures in capitalist markets. So, people have similar intuitions about fairness whenever the society they belong to is integrated to a capitalist market. Thse days, this means about everybody.

One might hypothesize that intuitions about responsibility are linked to fairness judgments. If so, one would expect the former to also vary as a function of integration to capitalist markets.

Now, you could not have found evidence against this hypothesis, because, I suppose, your subjects were students, thus were members of a culture used to capitalist exchanges. To find relevance evidence, you'd have to go to small-scale societies (or maybe to survey members of recluse religious orders!), as Joe and colleagues did.

Edouard

Chris said...

I'd be interested to know the sample size, too, because that's always important when claiming that there are no differences between groups. I'd also want to know a little bit more about the subjects. Were they college students, for example?

However, the bigger problem, I think, is methodological. That is, it's strange to infer either incompatibilism or indeterminism from one global, fairly abstract question. As one of the previous commenters put it (I think), one can imagine that people answer these fairly strange (from a lay person's perspective) questions one way, but act or treat individual cases quite differently. There's no real reason to assume that people are consistently either indeterminists or determinists, in fact, or that their intuitions of compatibilism/incompatibilism in ethical situations are independent of the individual case.

For the compatibilism/incompatibilism questions, these are probably easy problems to solve. You could use specific scenarios, manipulate whether actors' actions were determined or not, etc. You might also ask for ratings rather than using forced choice questions.

As for the determinism question, that might be a bit more difficult. Asking people about their global metaphysical beliefs seems a bit tricky as it is, but asking them global metaphysical questions involving difficult concepts like causality seems even trickier. Nisbett and his colleagues have found cultural differences in causal reasoning, and causal reasoning related to people's behavior in particular, which seems pretty important in interpreting people's responses to your determinism question. Then there's the question of context. One could imagine (consistent with Knobe's work, for example) that people are perfectly willing to be determinists about good or neutral actions, but if they really are global incompatibilists, are less likely to be determinists about bad actions.

Hagop said...

Thanks for all the comments, which have given me much thought. I hope to respond to them all, but this might have to wait until tomorrow. For now...

Hi Bill,

Thanks for your input. I agree that causation probably isn’t a folk concept, and it’s always tricky translating philosophical discourse into folk discourse for purposes of experimentation. The vignettes (used originally in the Nichols and Knobe paper mentioned by Anonymous) tried to present the issues without getting too 'techy', but I suppose there's always room to improve.

I might add, though, that in previous studies the incompatibilist response has come about even when the concept of causation was not deployed. For example, in one study, determinism was introduced by characterizing it in terms of perfect predictability (and not complete causation), and the effect still came about.


Anonymous,

These are all excellent points—my thanks for sharing them. I can’t improve much on what Joshua said. I might just add that one can be a hard determinist (i.e. one who embraces the two theses you mention in your first paragraph—determinism and incompatibilism) and still, in practice, hold people morally responsible for their actions owing to the positive consequences that might result. For example, if holding people responsible leads them to modify their behavior in beneficial ways, then this can be enough (for some) to justify upholding our practices of praising and blaming in spite of embracing hard determinism. Of course, we have no way to know whether the folk who embraced hard determinism actually think this way, but it suggests a quite plausible way one might justify maintaining practices of morally responsibility (in the way you suggest). So I don’t think the hard determinist responses need lead us to suspect the results.

(Also, to answer your question: all participants except the Colombia sample were given the vignettes in English.)

Edouard and Chris—I hope to get to your comments soon!

Hagop

Hagop said...

Hi Edouard,

I find these suggestions very interesting. I'm familiar with Henrich's work with ultimatum games, but have not looked into it in detail. I'll be reviwing it this summer and maybe I'll have more to say then. Thanks for the advice.

Perhaps, as you say, our population isn't quite as diverse as it may seem on the surface, given that all our participants were college students. We mention this worry in the paper. Still, it's interesting to note that even within cultures we can often find variation in moral attitudes from students of different backgrounds (such as Nisbett's work with college students from the north vs. the south in the US).

But let me say that I wholeheartedly embrace your call to go out and do the fieldwork in small-scale societies. Care to join in on a proposal? ;)


Chris,

Thanks for these remarks. I think the most interesting finding is that that the majority in each of these cultures answered as incompatibilists and as indeterminists. Our study just couldn't show that there are no differences between the cultures; indeed, raising the n's would probably show differences. It's the pattern of results that we found striking.

I see your point about drawing general conclusions about folk beliefs based on an abstract question alone. There's still much work to be done. But I don't think it the fact that folk might actually hold one another morally responsible in practice defeats the idea that they may find the thesis of incompatibilism to be intuitive. (Indeed, previous work has shown that concrete, affect laden vignettes can lead participants to give compatibilist answers. However, it's not clear to me that when philosophers claim, for example, that 'incompatibilism' is the intuitive view, they mean that it is reflected in our practices or that it will affect our practices on a concrete level. Rather, it seems they mean that when presented with the theses of free will and determinism on an abstract level, one would find the two to be incompatible in some rather obvious fashion.)

Finally, our total sample size was 231, broken down as follows:

US=66
India=55
Hong Kong=40
Colombia=70

Again, these comments have given me a lot to think about, especially regarding how to properly present these questions to the folk and the need for further research. Thanks for that.

Hagop

Matthew J. Brown said...

While this doesn't reflect on this research, which is super interesting and seems well done, I think it is an interesting if strange phenomenon that people often infer from a lack of any serious cross-cultural difference, a lack of any cultural cause whatsoever. It is as if people think that cultures vary stochastically and independently from each other. When cross-cultural variation is lacking, we must seek purely individualistic explanations: psychological or phenomenological. Why couldn't culture still be a crucial part of the equation? After all, these cultures are not isolated from each other now, at least some of them have a long history of interactions, and cultures do arise and evolve in response to something, rather than just being entirely unconstrained, right?

It is a common enough assumption, but one I find more and more bizarre the more cases I see of it.

Hagop said...

For those who may be interested, a draft of our paper discussing these findings is available here at my website, under the section titled "Under Review"

Anonymous said...

If any student were asked to define determinism and responded with Knobe and Nichols answer they would be lucky to get a passing grade.

The instructor would point out that "complete cause" is Bad English. After all, what is the difference between an event's having incomplete cause and being uncaused? If there is a difference it needs explanation and if there is no difference then "completely caused" is pleonastic: it just means "caused". But there can be indeterministic causes. The student would be told to go away to look up a proper definition of determinism and do some reading about causation.

And what is the instructor to make of "So, if everything in this universe was exactly the same up until John made his decision, then it had to happen that John would decide to have French Fries."

Like so many questions in the experimental literature this is semi-literate. "Exactly the same as" what? the instructor impatiently scribbles in the margin.

In any case even if the universe is deterministic it does not follow that it "had to happen that John would decide to have french fries". Determinism is compatible with contingency. What is being described is a Fatalist universe. The student would be sent away to learn the difference between Fatalism and Determinism.

The upshot of is that these studies cannot tell us anything about people's views about Free Will and Determinism as because neither scenario describes a Deterministic universe properly so called.

Matt Sigl said...

Very nice study. Though one point to note here is that the results are EXACTLY what we should expect. Philosophers have long noted that the experience of choice and the sense of an indeterminate future seem primary. The real shocking result would have been if people in other cultures thought of themselves as wholly caused and the future as wholly deterministic, so deeply ingrained is this intuition (though calling it merely an "intuition" does this bedrock conviction some disservice.) The "natural" connection between indeterminism and incompatibalism is something that a lot of compatabilists forget about. When arguing for compatabilism there are some very strong "knee-jerk" intuitions against the view and any satisfactory compatabilist account will have to address this forcefully. "Why is this intuition so universally accepted? Does a satisfactory compatabilist story wash it away?" It's a project I think doomed to failure, but the compatabilists are going to keep trying, of that I have little doubt.

Anonymous said...

This study is pure bullshit. Why? Because it does not take into account any anthropological research and it seems to be woefully unaware of critiques of using surveys for cross cultural comparison. There is a third possibility that you ignored in your list of possible explanations. The third possibility is that the people that you interviewed have shared experiences because of their positions in the global economy and thus, have been influenced by Western ideologies of free will. Capitalist economies are predicated on the on the notion of "free will." I do not see how one can separate the idea "free will" from the economic and ideological context in which it exists.

Finally, it is entirely possible that if you had asked similar questions two hundred or more years ago that you would have gotten different answers. Cultures change. It seems strange that one would ignore the possibility of cultural diffusion and assume static culture.

I am sick of people from other fields asking "cultural" questions and completely ignoring theoretical debates within anthropology. Three is a huge body of literature that deals with issues related to cross culturel comparison and culture change. You really need to read it.

Eric Schwitzgebel said...

Pissed off anthropologist: Where would you recommend starting? I'm particularly interested in critiques of cross-cultural surveys.

Anonymous said...

I would suggest the Volume 25: Number 2 issue of ethnos which has various articles discussing the relationship between ethnographic data and quantitative methods. While not necessarily containing critiques of cross-cultural surveys, some of the articles do point out the problem of assuming that research subjects share the same conceptual categories as the researcher. I think that the emphasis on context that these researchers emphasize is sorely lacking from your survey as it has been constructed here.

I know there are others out there but at the moment I do not have access to academic resources.

I would also suggest that you define what you actually mean by "cross cultural" or for that matter "culture." There are many disagreements among anthropologists about what constitutes a culture or even if the very idea of culture is a viable concept. The best way to find out about these is to find a syllabus online for a basic Anthropological Theory, History of Anthropology or Contemporary Anthropological Theory course. I would particularly pay attention to more recent critiques of the idea of culture as a "bound" entity that would be located at the end of these syllabi.

By using units in your survey such as Hong Kong, India, Columbia and the United States you are implying that there is a monolithic culture that all individuals within those societies share. While the majority of people in this survey may have responded in one way, we have know way of knowing whether these people are all of a certain class, caste, religion, ethnic group, etc. There could be important differences between these groups that result in differences in response.

On another note, I apologize for the tone in my previous post. It was a short emotional response. Unfortunately, the internet makes it easier to fall into the trap of anonymous sniping.

Anonymous said...

Hey Hagop,

I found this a really interesting discussion.

Unfortunately I must be pedantic and add to the definitional criticisms above; Your description of Universe A, which you intend to be the deterministic universe, is actually a description of a universe where proposition 1

1 - 'All events are caused'

is the case, not a description of determinism;

d - All events are logically implied by past events/some past state of affairs

Proposition 1 is entirely compatible with indeterminism;

Id - Disjunctions of events( e.g (Av-A)) are logically implied by past events/past states of affairs, and one of either A or -A happens ('ramdomly/indeterminately).

It's quite poor that your accounts of key concepts in the debate are confused.

I think that more fruitful questions can be asked; all the above does is comply peoples beliefs regarding whether or not determinism is the case, and whether or not libertarianism is the case etc. Statistics on intuitive responses to thought experiments would be far more interesting e.g. whether all people respond in the same way to Frankfurt or Gettier cases.

Furthermore, I would take the anthropologists advice with a pinch of salt; in all experience I have found them to be absolute fruit cakes.

Yours

Anonymous said...

pissed off anthropologist writes:
"There are many disagreements among anthropologists about what constitutes a culture or even if the very idea of culture is a viable concept."

That's as maybe. There are also many disagreements among philosophers about what constitutes knowledge or even if the very idea of knowledge is a viable concept, and likewise, mutatis mutandis, for many other concepts. I'm sure this doesn't stop anthropologists using the concepts in question.

In short, the fact that a concept is disputed in a field devoted to the study of such concepts doesn't prohibit those outside the field making use of the concept, especially when the relevant disputes have not been settled.

Anonymous said...

"In short, the fact that a concept is disputed in a field devoted to the study of such concepts doesn't prohibit those outside the field making use of the concept, especially when the relevant disputes have not been settled."

First, you have to explain what version of the concept you are using if you borrow it. If I read an anthropological article, I can usually tell what concept of culture (or if they even buy into the idea of culture) based on the theoretical perspective that they are using. If an anthropologist is wants to use a different conceptualization of culture than those which have previously existed, he or she will explicitly define how they are defining culture.

In the case presented above the use of "culture" is too ambiguous. Is culture synonymous with national identity in this case? Does it represent a certain national ethnic group which is taken to be dominant? From the above use of culture it is unclear how culture is conceptualized.

Second, if you borrow a concept from another field, you better be aware of the debates surrounding the concept. In some cases the version of the concept that you have borrowed has been rejected because of now obvious inadequacies. When anthropologists borrow concepts from other disciplines (even philosophy), they usually make clear which understanding of concept they are using. For example, if they are talking about a certain philosophical concept of rationality, they might say, I am using philosopher x's view of rationality. This clearly places their view in the context of wider philosophical debates.

If your going to use "culture" outside of anthropology (or sociology) you have to define it. Otherwise, you are assuming that readers share a certain understanding of culture which they might not actually share. This is especially the case if you are constructing a research methodology which is predicated on the idea of culture.

Anonymous said...

I wonder how a survey in the 2nd century BCE of whether the earth is round or flat would have turned out? I suspect the majority would have been flat-earthers. Of course, the way a question is asked often results in a given response. For example, to question whether human decision making is equivalent to the movement of billiard balls on a table will likely get no for an answer. To ask whether human decision making can be effected by the brains neuroanatomy and physiological chemistry would likely receive a yes answer. We know the movement of billiard balls is an oversimplification of causation. Anyway, the survey wasn't about whether the masses thought psychological causality was true or not. But whether they thought a person was morally responsible in a determined universe.

I too find it amazing that people can see that a brain damaged person isn't morally responsible for his (or her) behavior, but they don't allow the brain damaged person to run amok and commit horrendous acts. And they turn around and say that if moral responsibility for a so called "normal" person isn't true, the he (or she) shouldn't be "incarcerated" to protect society. Too me moral responsibility is really irrelevant to the issue in both the case of brain damaged person's behavior and the behavior of a so called criminal.

Anonymous said...

It seems misleading to suggest that anyone who answers "NO" to the second question must be committed to the thesis of 'incompatibilism.' After all, a person who thinks that, regardless of the truth or falsity of the thesis of 'determinism,' it is simply never "possible for a person to be fully morally responsible for their actions," perhaps (say) because people can only ever be partly responsible for their actions (in either a deterministic or a non-deterministic universe). That latter view would not amount to any kind of answer to the question addressed by the thesis of 'incompatibilism,' since to formulate a position on that question is to take a view about the relation compatibility or incompatibility between determinism and moral responsibility (or the sort of free will necessary to ascribe moral responsibility).

I mention this, not to quibble, but to suggest that, perhaps, many philosophical questions only arise at all against the background of a certain conceptual 'geography' -- and that without that conceptual and presuppositional background, there might not be any intuitions to discover. If so, the way we word questions would take on exaggerated importance, because we would be artificially inducing people to generate intuitions, by inviting them to think in certain ways (e.g., to think about a mechanically deterministic universe, or to think about responsibility as an 'either/or' rather than a 'more/less' contrast, or to distinguish sharply between causal and moral responsibility, or between moral and role responsibility, and so on). People not predisposed to frame any questions at all within the horizon of this highly determined system of concepts may have no intuitions about these conventional philosophical questions, until we ask certain 'leading questions' which usher them into our conceptual framework, and force them to take a 'yes or no' position, when their pre-reflective intuitions might have led them instead to take a "but-that's-not-how-I-would-frame-the-question" position.

This thought is, of course, familiar to philosophers: we can all think of many examples of competing 'thought experiments' or 'intuition pumps' which, with even subtle shifts, can generate seemingly very different intuitions.

But what conclusions can we draw? For instance, think of Judith Jarvis Thomson's "Trolley Problem" paper. (I won't recount it, because that would take too long, but one can read about it on the net, e.g., on wikipedia). It seems that these thought experiments, by their very nature, are 'leading questions,' which don't so much 'discover' intutions, but extract them, in the sense of forcing them out of people by forcing them to think about what the salient variables and relationships are.

Unknown said...

I find this discussion thread very interesting. My concern would be towards the sample of people that were surveyed. When reading your post and taking the "quiz" I agreed with your statements but I don't think that your sample is good enough in order to represent the human population as a whole. For starter you do not have the same amount of people to respond for each different country. You cannot survey 66 people for the U.S., 55 for India, 40 for Hong Kong and 70 for Columbia, and believe to have accurate answers. Even though I agree with the statements made and the responses from your surveys I don't agree with the fact that you can survey 40 people for one country, 70 for another, and hope to achieve somewhat of a fair result. First of all in order to make the process somewhat fair, the same amount of people should have been surveyed for each country. Also, I agree with one of the posts asking what were the criterions used in order to select the samples. Were they college students? Did they study philosophy? What was their social standing? All of these factors are important, and it would have been a good idea to mention them somewhere in your explanation. A person who studies philosophy would have a much different response to those questions compared to someone who is completely new to the topic.

Unknown said...

I agree with the statement most people seemed to have made. Universe B is the one that resembles our world the most yet people should not entirely be held responsible for their actions. Considering everything is determined based on what happened in the past, even though human decision-making isn’t according to the description of universe B, the fact that everything is happening because of what initially occurred must somehow have an effect on the decisions we make. People have free will but isn’t it true that we are somewhat conditioned to make the decisions we do? In this case even though human decision-making isn’t caused by what initially happened in universe B, the universe should somewhat be held responsible. What I’m saying is that even though Mary decided to get French fries for lunch, and since according to universe B her decision isn’t linked to what initially happened to the universe there must still be somewhat of a link because she could have simply decided on getting sweet potato fries instead of regular French fries. In this case even though according to the description it was her free will to decide on the French fries the universe might have somewhat of an implication.