In science we standardly use a logically non-valid inference — the fallacy of affirming the consequent — of the following form:
(1) p => q
or, in instantiated form
(1) ∀x (Gx => Px)
Although logically invalid, it is nonetheless a kind of inference — abduction — that may be factually strongly warranted and truth-producing.
Following the general pattern ‘Evidence => Explanation => Inference’ we infer something based on what would be the best explanation given the law-like rule (premise 1) and an observation (premise 2). The truth of the conclusion (explanation) is nothing that is logically given, but something we have to justify, argue for, and test in different ways to possibly establish with any certainty or degree. And as always when we deal with explanations, what is considered best is relative to what we know of the world. In the real world all evidence has an irreducible holistic aspect. We never conclude that evidence follows from a hypothesis simpliciter, but always given some more or less explicitly stated contextual background assumptions. All non-deductive inferences and explanations are necessarily context-dependent.
If we extend the abductive scheme to incorporate the demand that the explanation has to be the best among a set of plausible competing/rival/contrasting potential and satisfactory explanations, we have what is nowadays usually referred to as inference to the best explanation.
In inference to the best explanation we start with a body of (purported) data/facts/evidence and search for explanations that can account for these data/facts/evidence. Having the best explanation means that you, given the context-dependent background assumptions, have a satisfactory explanation that can explain the fact/evidence better than any other competing explanation — and so it is reasonable to consider/believe the hypothesis to be true. Even if we (inevitably) do not have deductive certainty, our reasoning gives us a license to consider our belief in the hypothesis as reasonable.
Accepting a hypothesis means that you believe it does explain the available evidence better than any other competing hypothesis. Knowing that we — after having earnestly considered and analysed the other available potential explanations — have been able to eliminate the competing potential explanations, warrants and enhances the confidence we have that our preferred explanation is the best explanation, i. e., the explanation that provides us (given it is true) with the greatest understanding.
This, of course, does not in any way mean that we cannot be wrong. Of course we can. Inferences to the best explanation are fallible inferences — since the premises do not logically entail the conclusion — so from a logical point of view, inference to the best explanation is a weak mode of inference. But if the arguments put forward are strong enough, they can be warranted and give us justified true belief, and hence, knowledge, even though they are fallible inferences. As scientists we sometimes — much like Sherlock Holmes and other detectives that use inference to the best explanation reasoning — experience disillusion. We thought that we had reached a strong conclusion by ruling out the alternatives in the set of contrasting explanations. But — what we thought was true turned out to be false.
That does not necessarily mean that we had no good reasons for believing what we believed. If we cannot live with that contingency and uncertainty, well, then we are in the wrong business. If it is deductive certainty you are after, rather than the ampliative and defeasible reasoning in inference to the best explanation — well, then get in to math or logic, not science.
The kind of fundamental assumption about the character of material laws, on which scientists appear commonly to act, seems to me to be much less simple than the bare principle of uniformity. They appear to assume something much more like what mathematicians call the principle of the superposition of small effects, or, as I prefer to call it, in this connection, the atomic character of natural law. The system of the material universe must consist, if this kind of assumption is warranted, of bodies which we may term (without any implication as to their size being conveyed thereby) legal atoms, such that each of them exercises its own separate, independent, and invariable effect, a change of the total state being compounded of a number of separate changes each of which is solely due to a separate portion of the preceding state. We do not have an invariable relation between particular bodies, but nevertheless each has on the others its own separate and invariable effect, which does not change with changing circumstances, although, of course, the total effect may be changed to almost any extent if all the other accompanying causes are different. Each atom can, according to this theory, be treated as a separate cause and does not enter into different organic combinations in each of which it is regulated by different laws …
The scientist wishes, in fact, to assume that the occurrence of a phenomenon which has appeared as part of a more complex phenomenon, may be some reason for expecting it to be associated on another occasion with part of the same complex. Yet if different wholes were subject to laws qua wholes and not simply on account of and in proportion to the differences of their parts, knowledge of a part could not lead, it would seem, even to presumptive or probable knowledge as to its association with other parts. Given, on the other hand, a number of legally atomic units and the laws connecting them, it would be possible to deduce their effects pro tanto without an exhaustive knowledge of all the coexisting circumstances.
Keynes’ incisive critique is of course of interest in general for all sciences, but I think it is also of special interest in economics as a background to much of Keynes’ doubts about inferential statistics and econometrics.
Since econometrics doesn’t content itself with only making ‘optimal predictions’ but also aspires to explain things in terms of causes and effects, econometricians need loads of assumptions. Most important of these are the ‘atomistic’ assumptions of additivity and linearity.
Limiting model assumptions in economic science always have to be closely examined since if we are going to be able to show that the mechanisms or causes that we isolate and handle in our models are stable in the sense that they do not change when we ‘export’ them to our ‘target systems,’ we have to be able to show that they do not only hold under ceteris paribus conditions and a fortiori only are of limited value to our understanding, explanations or predictions of real economic systems.
Econometrics may be an informative tool for research. But if its practitioners do not investigate and make an effort of providing a justification for the credibility of the assumptions on which they erect their building, it will not fulfill its tasks. There is a gap between its aspirations and its accomplishments, and without more supportive evidence to substantiate its claims, critics like Keynes — and yours truly — will continue to consider its ultimate argument as a mixture of rather unhelpful metaphors and metaphysics.
The marginal return on its ever higher technical sophistication in no way makes up for the lack of serious under-labouring of its deeper philosophical and methodological foundations that already Keynes complained about. Firmly stuck in an empiricist tradition, econometrics is only concerned with the measurable aspects of reality, and a rigorous application of econometric methods in economics really presupposes that the phenomena of our real world economies are ruled by stable causal relations.
But — real world social systems are not governed by stable causal mechanisms or capacities. The kinds of ‘laws’ and relations that econometrics has established, are laws and relations about entities in models that presuppose causal mechanisms being atomistic and additive. As Keynes argued, when causal mechanisms operate in the real world they only do it in ever-changing and unstable combinations where the whole is more than a mechanical sum of parts. If economic regularities obtain they do it as a rule only because we engineered them for that purpose. Outside man-made ‘nomological machines’ they are rare, or even non-existant.
In my view, scientific theories are not to be considered ‘true’ or ‘false.’ In constructing such a theory, we are not trying to get at the truth, or even to approximate to it: rather, we are trying to organize our thoughts and observations in a useful manner.
What a handy view of science.
How reassuring for all of you who have always thought that believing in the tooth fairy make you understand what happens to kids’ teeth. Now a ‘Nobel prize’ winning economist tells you that if there are such things as tooth fairies or not doesn’t really matter. Scientific theories are not about what is true or false, but whether ‘they enable us to organize and understand our observations’ …
What Aumann and other defenders of scientific storytelling ‘forgets’ is that potential explanatory power achieved in thought experimental models is not enough for attaining real explanations. Model explanations are at best conjectures, and whether they do or do not explain things in the real world is something we have to test. To just believe that you understand or explain things better with thought experiments is not enough. Without a warranted export certificate to the real world, model explanations are pretty worthless. Proving things in models is not enough. Truth is an important concept in real science.
If we identify degree of corroboration or confirmation with probability, we should be forced to adopt a number of highly paradoxical views, among them the following clearly self-contradictory assertion:
“There are cases in which x is strongly supported by z and y is strongly undermined by z while, at the same time, x is confirmed by z to a lesser degree than is y.”
Consider the next throw with a homogeneous die. Let x be the statement ‘six will turn up’; let y be its negation, that is to say, let y = x; and let z be the information ‘an even number will turn up’.
We have the following absolute probabilities:
p(x)=l/6; p(y) = 5/6; p(z) = 1/2.
Moreover, we have the following relative probabilities:
p(x, z) = 1/3; p(y, z) = 2/3.
We see that x is supported by the information z, for z raises the probability of x from 1/6 to 2/6 = 1/3. We also see that y is undermined by z, for z lowers the probability of y by the same amount from 5/6 to 4/6 = 2/3. Nevertheless, we have p(x, z) < p(y, z) …
A report of the result of testing a theory can be summed up by an appraisal. This can take the form of assigning some degree of corroboration to the theory. But it can never take the form of assigning to it a degree of probability; for the probability of a statement (given some test statements) simply does not express an appraisal of the severity of the tests a theory has passed, or of the manner in which it has passed these tests. The main reason for this is that the content of a theory — which is the same as its improbability — determines its testability and its corroborability.
Although Bayesians think otherwise, to me there’s nothing magical about Bayes’ theorem. The important thing in science is for you to have strong evidence. If your evidence is strong, then applying Bayesian probability calculus is rather unproblematic. Otherwise — garbage in, garbage out. Applying Bayesian probability calculus to subjective beliefs founded on weak evidence is not a recipe for scientific akribi and progress.
Neoclassical economics nowadays usually assumes that agents that have to make choices under conditions of uncertainty behave according to Bayesian rules — that is, they maximize expected utility with respect to some subjective probability measure that is continually updated according to Bayes’ theorem.
Bayesianism reduces questions of rationality to questions of internal consistency (coherence) of beliefs, but – even granted this questionable reductionism – do rational agents really have to be Bayesian? As I have been arguing repeatedly over the years, there is no strong warrant for believing so.
In many of the situations that are relevant to economics one could argue that there is simply not enough of adequate and relevant information to ground beliefs of a probabilistic kind, and that in those situations it is not really possible, in any relevant way, to represent an individual’s beliefs in a single probability measure.
Bayesianism cannot distinguish between symmetry-based probabilities from information and symmetry-based probabilities from an absence of information. In these kinds of situations most of us would rather say that it is simply irrational to be a Bayesian and better instead to admit that we “simply do not know” or that we feel ambiguous and undecided. Arbitrary an ungrounded probability claims are more irrational than being undecided in face of genuine uncertainty, so if there is not sufficient information to ground a probability distribution it is better to acknowledge that simpliciter, rather than pretending to possess a certitude that we simply do not possess.
So, why then are so many scientists nowadays so fond of Bayesianism? I guess one strong reason is that Bayes’ theorem gives them a seemingly fast, simple and rigorous answer to their problems and hypotheses. But, as already Popper showed back in the 1950’s, the Bayesian probability (likelihood) version of confirmation theory is “absurd on both formal and intuitive grounds: it leads to self-contradiction.”
The other day yours truly had a post up on the Heckscher-Ohlin theorem, arguing that since the assumptions on which the theorem build are empirically false, one might, from a methodological point of view, wonder
how we are supposed to evaluate tests of a theorem building on known to be false assumptions. What is the point of such tests? What can those tests possibly teach us? From falsehoods anything logically follows.
Some people have had troubles with the last sentence — from falsehoods anything whatsoever follows.
But that’s really nothing very deep or controversial. What I’m referring to — without going into the intricacies of distinguishing between ‘false,’ ‘inconsistent’ and ‘self-contradictory’ statements — is the well-known ‘principle of explosion,’ according to which if both a statement and its negation are considered true, any statement whatsoever can be inferred.
Whilst tautologies, purely existential statements and other nonfalsiﬁable statements assert, as it were, too little about the class of possible basic statements, self-contradictory statements assert too much. From a self-contradictory statement, any statement whatsoever can be validly deduced. Consequently, the class of its potential falsiﬁers is identical with that of all possible basic statements: it is falsiﬁed by any statement whatsoever.
When applying deductivist thinking to economics, mainstream economists usually set up ‘as if’ models based on a set of tight axiomatic assumptions from which consistent and precise inferences are made. The beauty of this procedure is of course that if the axiomatic premises are true, the conclusions necessarily follow. The snag is that if the models are to be relevant, we also have to argue that their precision and rigour still holds when they are applied to real-world situations. They often don’t. When addressing real economies, the idealizations necessary for the deductivist machinery to work — as e. g. IS-LM and DSGE models — simply don’t hold.
Defending his IS-LMism from the critique put forward by e. g. Hyman Minsky and yours truly, Paul Krugman writes:
When people like me use something like IS-LM, we’re not imagining that the IS curve is fixed in position for ever after. It’s a ceteris paribus thing, just like supply and demand.
But that is actually just another major problem with the Hicksian construction!
As Hans Albert noticed more than fifty years ago:
The law of demand … is usually tagged with a clause that entails numerous interpretation problems: the ceteris paribus clause … The ceteris paribus clause is not a relatively insignificant addition, which might be ignored … The clause produces something of an absolute alibi, since, for every apparently deviating behavior, some altered factors can be made responsible. This makes the statement untestable, and its informational content decreases to zero.
This is a critique rehearsed by John Earman et al. in a special issue of Erkenntnis on the ceteris paribus assumption, concluding that it is bad to admit ceteris paribus laws at all in science, since they are untestable:
In order for a hypothesis to be testable, it must lead us to some prediction. The prediction may be statistical in character, and in general it will depend on a set of auxiliary hypotheses. Even when these important qualifications have been added, CP law statements still fail to make any testable predictions. Consider the putative law that CP, all Fs are Gs. The information that x is an F, together with any auxiliary hypotheses you like, fails to entail that x is a G, or even to entail that with probability p, x is a G. For, even given this information, other things could fail to be equal, and we are not even given a way of estimating the probability that they so fail. Two qualifications have to be made. First, our claim is true only if the auxiliary hypotheses don’t entail the prediction all by themselves, in which case the CP law is inessential to the prediction and doesn’t get tested by a check of that prediction. Second, our claim is true only if none of the auxiliary hypotheses is the hypothesis that “other things are equal”, or “there are no interferences”. What if the auxiliaries do include the claim that other things are equal? Then either this auxiliary can be stated in a form that allows us to check whether it is true, or it can’t. If it can, then the original CP law can be turned into a strict law by substituting the testable auxiliary for the CP clause. If it can’t, then the prediction relies on an auxiliary hypothesis that cannot be tested itself. But it is generally, and rightly, presumed that auxiliary hypotheses must be testable in principle if they are to be used in an honest test. Hence, we can’t rely on a putative CP law to make any predictions about what will be observed, or about the probability that something will be observed. If we can’t do that, then it seems that we can’t subject the putative CP law to any kind of empirical test.
The conclusion from the massive critique should be obvious.
Ditch the ceteris paribus fairy!
• Archer, Margaret (1995). Realist social theory: the morphogenetic approach. Cambridge: Cambridge University Press
• Bhaskar, Roy (1978). A realist theory of science. Hassocks: Harvester
• Cartwright, Nancy (2007). Hunting causes and using them. Cambridge: Cambridge University Press
• Chalmers, Alan (2013). What is this thing called science?. 4th. ed. Buckingham: Open University Press
• Garfinkel, Alan (1981). Forms of explanation: rethinking the questions in social theory. New Haven: Yale U.P.
• Harré, Rom (1960). An introduction to the logic of the sciences. London: Macmillan
• Lawson, Tony (1997). Economics and reality. London: Routledge
• Lieberson, Stanley (1987). Making it count: the improvement of social research and theory. Berkeley: Univ. of California Press
• Lipton, Peter (2004). Inference to the best explanation. 2. ed. London: Routledge
• Miller, Richard (1987). Fact and method: explanation, confirmation and reality in the natural and the social sciences. Princeton, N.J.: Princeton Univ. Press
‘Critical realism’ is very similar to the jargon-dense, literary ‘critical theory’ taught in literature departments.
One of the most important tasks of social sciences is to explain the events, processes, and structures that take place and act in society. But the researcher cannot stop at this. As a consequence of the relations and connections that the researcher finds, a will and demand arise for critical reflection on the findings. To show that unemployment depends on rigid social institutions or adaptations to European economic aspirations to integration, for instance, constitutes at the same time a critique of these conditions. It also entails an implicit critique of other explanations that one can show to be built on false beliefs. The researcher can never be satisfied with establishing that false beliefs exist but must go on to seek an explanation for why they exist. What is it that maintains and reproduces them? To show that something causes false beliefs – and to explain why – constitutes at the same time a critique of that thing.
This I think is something particular to the humanities and social sciences. There is no full equivalent in the natural sciences, since the objects of their study are not fundamentally created by human beings in the same sense as the objects of study in social sciences. We do not criticize apples for falling to earth in accordance with the law of gravitation.
The explanatory critique that constitutes all good social science thus has repercussions on the reflective person in society. To digest the explanations and understandings that social sciences can provide means a simultaneous questioning and critique of one’s self-understanding and the actions and attitudes it gives rise to. Science can play an important emancipating role in this way. Human beings can fulfill and develop themselves only if they do not base their thoughts and actions on false beliefs about reality. Fulfillment may also require changing fundamental structures of society. Understanding of the need for this change may issue from various sources like everyday praxis and reflection as well as from science.
Explanations of societal phenomena must be subject to criticism, and this criticism must be an essential part of the task of social science. Social science has to be an explanatory critique. The researcher’s explanations have to constitute a critical attitude toward the very object of research, society. Hopefully, the critique may result in proposals for how the institutions and structures of society can be constructed. The social scientist has a responsibility to try to elucidate possible alternatives to existing institutions and structures.
In a time when scientific relativism is expanding, it is important to keep up the claim for not reducing science to a pure discursive level. We have to maintain the Enlightenment tradition of thinking of reality as principally independent of our views of it and of the main task of science as studying the structure of this reality. Perhaps the most important contribution a researcher can make is reveal what this reality that is the object of science actually looks like.
Science is made possible by the fact that there are structures that are durable and are independent of our knowledge or beliefs about them. There exists a reality beyond our theories and concepts of it. It is this independent reality that our theories in some way deal with. Contrary to positivism, I cannot see that the main task of science is to detect event-regularities between observed facts. Rather, that task must be conceived as identifying the underlying structure and forces that produce the observed events.
The problem with positivist social science is not that it gives the wrong answers, but rather that in a strict sense it does not give answers at all. Its explanatory models presuppose that the social reality is “closed,” and since social reality is fundamentally “open,” models of that kind cannot explain anything of what happens in such a universe. Positivist social science has to postulate closed conditions to make its models operational and then – totally unrealistically – impute these closed conditions to society’s real structure.
In the face of the kind of methodological individualism and rational choice theory that dominate positivist social science we have to admit that even if knowing the aspirations and intentions of individuals are necessary prerequisites for giving explanations of social events, they are far from sufficient. Even the most elementary “rational” actions in society presuppose the existence of social forms that it is not possible to reduce to the intentions of individuals.
The overarching flaw with methodological individualism and rational choice theory is basically that they reduce social explanations to purportedly individual characteristics. But many of the characteristics and actions of the individual originate in and are made possible only through society and its relations. Society is not reducible to individuals, since the social characteristics, forces, and actions of the individual are determined by pre-existing social structures and positions. Even though society is not a volitional individual, and the individual is not an entity given outside of society, the individual (actor) and the society (structure) have to be kept analytically distinct. They are tied together through the individual’s reproduction and transformation of already given social structures.
With a non-reductionist approach we avoid both determinism and voluntarism. For although the individual in society is formed and influenced by social structures that he does not construct himself, he can as an individual influence and change the given structures in another direction through his own actions. In society the individual is situated in roles or social positions that give limited freedom of action (through conventions, norms, material restrictions, etc.), but at the same time there is no principal necessity that we must blindly follow or accept these limitations. However, as long as social structures and positions are reproduced (rather than transformed), the actions of the individual will have a tendency to go in a certain direction.
A major, and notorious, problem with this approach, at least in the domain of science, concerns how to ascribe objective prior probabilities to hypotheses. What seems to be necessary is that we list all the possible hypotheses in some domain and distribute probabilities among them, perhaps ascribing the same probability to each employing the principal of indifference. But where is such a list to come from? It might well be thought that the number of possible hypotheses in any domain is infinite, which would yield zero for the probability of each and the Bayesian game cannot get started. All theories have zero
probability and Popper wins the day. How is some finite list of hypotheses enabling some objective distribution of nonzero prior probabilities to be arrived at? My own view is that this problem is insuperable, and I also get the impression from the current literature that most Bayesians are themselves
coming around to this point of view.
Chalmers is absolutely right here in his critique of ‘objective’ Bayesianism, but I think it could actually be extended to also encompass its ‘subjective’ variety.
A classic example — borrowed from Bertrand Russell — may perhaps be allowed to illustrate the main point of the critique:
Assume you’re a Bayesian turkey and hold a nonzero probability belief in the hypothesis H that “people are nice vegetarians that do not eat turkeys and that every day I see the sun rise confirms my belief.” For every day you survive, you update your belief according to Bayes’ theorem
P(H|e) = [P(e|H)P(H)]/P(e),
where evidence e stands for “not being eaten” and P(e|H) = 1. Given that there do exist other hypotheses than H, P(e) is less than 1 and a fortiori P(H|e) is greater than P(H). Every day you survive increases your probability belief that you will not be eaten. This is totally rational according to the Bayesian definition of rationality. Unfortunately, for every day that goes by, the traditional Christmas dinner also gets closer and closer …
The nodal point here is — of course — that although Bayes’ theorem is mathematically unquestionable, that doesn’t qualify it as indisputably applicable to scientific questions.
Bayesian probability calculus is far from the automatic inference engine that its protagonists maintain it is. Where do the priors come from? Wouldn’t it be better in science if we did some scientific experimentation and observation if we are uncertain, rather than starting to make calculations based on people’s often vague and subjective personal beliefs? Is it, from an epistemological point of view, really credible to think that the Bayesian probability calculus makes it possible to somehow fully assess people’s subjective beliefs? And are — as Bayesians maintain — all scientific controversies and disagreements really possible to explain in terms of differences in prior probabilities? I’ll be dipped!
The advantage of randomised experiments in describing populations creates an illusion of knowledge … This happens because of the propensity of scientific journals to value so-called causal findings and not to value findings where no (so-called) causality is found. In brief, it is arguable that we know less than we think we do.
To see this, suppose—as is indeed the case in reality—that thousands of researchers in thousands of places are conducting experiments to reveal some causal link. Let us in particular suppose that there are numerous researchers in numerous villages carrying out randomised experiments to see whether M causes P. Words being more transparent than symbols, let us assume they want to see whether medicine (M) improves the school participation (P) of school-going children. In each village, 10 randomly selected children are administered M and the school participation rates of those children and also children who were not given M are monitored. Suppose children without M go to school half the time and are out of school the other half. The question is: is there a systematic difference of behaviour among children given M?
I shall now deliberately construct an underlying model whereby there will be no causal link between M and P. Suppose Nature does the following. For each child, whether or not the child has had M, Nature tosses a coin. If it comes out tails the child does not go to school and if it comes out heads, the child goes to school regularly.
Consider a village and an RCT researcher in the village. What is the probability, p, that she will find that all 10 children given M will go to school regularly? The answer is clearly
p = (1/2)^10
because we have to get heads for each of the 10 tosses for the 10 children.
Now consider n researchers in n villages. What is the probability that in none of these villages will a researcher find that all the 10 children given M go to school regularly? Clearly, the answer is (1–p)^n.
Hence, if w(n) is used to denote the probability that among the n villages where the experiment is done, there is at least one village where all 10 tosses come out heads, we have:
w(n) = 1 – (1-p)^n.
Check now that if n = 1, that is, there is only one village where this experiment is done, the probability that all 10 children administered M will participate in school regularly is w(1) = 0.001. In other words, the likelihood is negligible.
It is easy to check the following are true:
w(100) = 0.0931,
w(1000) = 0.6236,
w(10 000) = 0.9999.
Therein lies the catch. If the experiment is done in 100 villages, the probability that there exists at least one village in which all tosses result in heads is still very small, less than 0.1. But if there are 1000 experimenters in 1000 villages doing this, the probability that there will exist one village where it will be found that all 10 children administered M will participate regularly in school is 0.6236. That is, it is more likely that such a village will exist than not. If the experiment is done in 10 000 villages, the probability of there being one village where M always leads to P is a virtual certainty (0.9999).
This is, of course, a specific example. But that this problem will invariably arise follows from the fact that
lim(n => infinity)w(n) = 1 – (1 -p)^n = 1.
Given that those who find such a compelling link between M and P will be able to publish their paper and others will not, we will get the impression that a true causal link has been found, though in this case (since we know the underlying process) we know that that is not the case. With 10 000 experiments, it is close to certainty that someone will find a firm link between M and P. Hence, the finding of such a link shows nothing but the laws of probability being intact. Yet, thanks to the propensity of journals to publish the presence rather than the absence of “causal” links, we get an illusion of knowledge and discovery where there are none.
One practical implication of this observation is that it spells out the urgent need for a Journal of Failed Experiments … Such a journal, there can be little doubt, will have a sobering effect on economics, making evident where the presence of a result is likely to be a pure statistical artefact.