The raven paradox: logic and psychology

I. Logic.

Numerous attempts to solve the paradox have been made over the years, which vary in the extent to which they rely on logical versus psychological explanations. Hempel concluded the paradox is a "psychological illusion", and suggested two possible causes.

(A) We assume that the hypothesis "All ravens are black" applies only to ravens -so we think white shoes are irrelevant to the hypothesis. (B) We know beforehand what the non-black non-raven object is (e.g. a shoe) and we have tacit knowledge already as to the properties of those objects. 4 In those cases there is no information gained by observing the non-black non-raven. However, if we do not know the object's identity beforehand, then the observation does support the hypothesis, as logic dictates.

The most obvious explanation is that the number of non-black non-ravens far outnumbers the black ravens in this world (Hosiasson-Lindenbaum 1940). Therefore we are not surprised at seeing a non-black non-raven. So even if this observation does confirm the hypothesis, it provides such a negligible boost to our confidence that we are not impressed and we dismiss the evidence as totally unconvincing. 5 Some have even attempted to quantify the absolute, relative or Bayesian probabilities of the hypothesis given the various possible pieces of evidence (e.g. Glymour 1980;Horwich 1982;Achinstein 1983;Howson and Urbach 1989). The relative numbers of ravens and nonblack things are predicates for logical argumentation -but whether we know the relative numbers, and whether we use that knowledge, are matters of psychology. For example Mackie (1963) suggested that we confuse 'negligible' information with no information at all, hence putting the final brick in this psychological account of the paradox.

At the opposite extreme are attempts to rubbish the whole of confirmatory logic by pointing out that an observation of a white shoe 'confirms' many hypotheses (e.g.

Things exist; There is only one thing; Nothing is either a raven or black; All things are white shoes; etc. : Hempel 1965;Goodman 1955;Horwich 1982;Popper 1983). As part of his attack on confirmationism, Popper (1983, p. 257, footnote) answered Hempel's paradox by calling observations of white shoes "too cheap to be accepted as corroborations, because they are not in general the results of genuine attempts to refute the theory".

Others have attempted to rescue logic from such reductios by creating logical systems where evidence always bears on the relative credibility of two hypotheses, rather than on the absolute probability of one hypothesis (e.g. Goodman 1955;Glymour 1980). Several further schemes realize that we always have background information in addition to the focal hypothesis and evidence, and it is a methodological fiction to assume otherwise. Lipton (1991) for example analysed the raven paradox as a causal explanation: while reasons can be conceived why ravens might be black 6 these would have no bearing on what colour shoes might be, or what objects should not be black, because ravens and shoes (or whatever) are already known to have no shared causal history which might create any similarity or difference between them. Kitcher (1993) argues that background knowledge is used to set up contests between rival hypotheses which can then be eliminated one at a time by collecting evidence in a controlled fashion. For example 'All ravens are black' might be compared with 'All ravens are yellow' because we know ravens are birds and some birds are yellow, or with 'All male ravens are black and females brown' because we know some species of bird have 4 different colours for the two sexes. 7 Lipton's and Kitcher's visions accord with other attempts to circumvent the raven paradox by delimiting the choice of objects that it is relevant to observe. Thus Hempel (1965) recommended only objects whose identity was not known in advance (and Horwich 1982 even calculated the changes in Bayesian probability that would result from various degrees of foreknowledge). Also, Achinstein (1983) claimed the raven paradox arises because we have background knowledge that tells us how we could collect ravens without prejudice to see if they are black, but we also know we would be unable to inspect non-black things without showing an inevitable bias to chose nonravens. We therefore know the latter type of observation would be invalid. He then discussed at tedious length the practical problems of how one might go about selecting ravens, or non-black things, without any bias (e.g. select ravens from different places at various times of year). 8

Responses to the raven paradox, which originated in studies of symbolic logic, thus differentiate into those which blame psychology, those based on logic, and those which involve considerations of both logic and psychology (e.g. those involving background knowledge and its use or mis-use). I will now turn to approach the problem from the psychological perspective.

II. Psychology.

i) The four-card selection task

In 1966, Wason introduced a new format for studying human ability to solve problems in logic. Subjects (university students) were presented with four cards placed on a table.

They were told that every card had a letter written on one side and a number on the other side. Here is what they saw:

The students were told that a Rule determined the pattern of letters and numbers on the cards, that Rule being: "Whenever there is a vowel on one side of a card, there is an even number on the other". The students were then asked to name those cards, and only those cards, which needed to be turned over in order to determine whether the Rule was true or false.

The researchers reasoned that the E card must be turned over, because if there is an even number on the reverse side, that would be consistent with the Rule's being true, but if there is odd number on the other side, it would prove the rule to be false. The K does not need to be turned over, because the rule says nothing about what is on the other side of a consonant. The 4 should not be turned over, since the Rule does not say whether or not both vowels and consonants may have an even number on the obverse.

Finally, the 7 should also be turned over, since if there is a vowel on the back, it would prove the Rule to be false.

The results of the experiments were however surprising. In four early studies, with 128 students, 42 picked only the E, 59 chose E and 4, and only 5 selected E and 7. 9 Numerous experiments since have found the same pattern: E, or E and 4, are the commonest choices, with <10% choosing E and 7 (Stein 1996). 10 Thus almost all students selected the E, which is consistent with their either seeking confirming or disconfirming instances of the Rule. However very few selected the 7. If the 7 had been revealed to be backed with a consonant, it would not have given any information either way about the Rule. Only if the 7 had been backed by a vowel would useful information have been obtained -but that would have been disconfirming information. The 4 was chosen by more than half of the students; if backed by a vowel, it would have provided another confirmation, but if backed by a consonant, it would have proved inconclusive.

The researchers concluded that people do not naturally seek disconfirming information;

instead people are biassed to seeking confirming instances of a hypothesis.

However, this was soon disproved in their next experiments. The researchers wondered whether the poor performance of the subjects was due to the abstract nature of the material. They therefore compared performance on the above task with one containing concrete, meaningful material. For example, the following cards were presented and subjects (who were at university in London) were asked to test the Rule: "Whenever I go to Manchester I travel by train". This time, the subjects were more accurate: 84% chose Manchester and Car only.

Subsequent research has confirmed that with certain kinds of stimulus materials and Rules, most subjects will decide to pick the two cards that can potentially falsify the Rule. For example asking them to look for violations of the Rule, rather than just "whether it is true or false" will often increase the probability of their picking the potentially falsifying instances. Researchers are however still debating the exact nature of the card and Rule content that causes the particular patterns of behaviour observed.

A virtual taxonomy of putative causes has been developed over the years. For example Liberman and Klar (1996) checking cars at random would be unlikely to catch him violating the Rule. Indeed some workers have developed quantitative models of how knowing the relative frequencies of the expected data (e.g. the relative numbers of car and train journeys), and the a priori expected probabilities of the various hypotheses (e.g. that I always travel to Manchester by train, versus there is no correlation between destination and mode of transport), could influence behaviour; but the verdicts are not yet in on whether the predictions are accurate (Kirby 1994;Love and Kessler 1995;Oaksford and Chater 1995;Oaksford 1997).

Psychologists have developed a dual-process theory of thinking to explain the pattern of results obtained by using these tasks. There are two stages of mental processing: a preconscious heuristic associative process that sets up a mental model or scenario depicting what seem to be relevant aspects of the task, and a later analytic stage where symbolic reasoning and judgments take place. Failures to achieve a correct solution are usually attributed to incomplete processing in the first stage: the words in the Rule quickly arouse associated background information and beliefs from memory, which bias the interpretation of the situation depicted on the cards -perhaps by focussing attention onto some of the stimuli at the expense of others, making some of them appear more relevant than others, and/or leading to the construction of incomplete mental models of the possible evidence that might exist on the hidden backs of each of the cards. 11 In this vein, Evans et al. (1993, p. 135) conclude that of all the mechanisms proposed, those based on heuristics, mental models and domain-sensitive rules are viable explanations of human reasoning; but the evidence clearly excludes the idea that we reason by using formal logical rules of inference. More recently however Evans & Over (1996, pp. 153-155) have allowed that conscious symbolic processing can only take place as a second stage of thinking; these processes are flexible but slow and of limited capacity, they are tiring to use and prone to lapses of attention, and are thus not greatly used in everyday tasks.

ii) The paradox of the ravens.

So what does this tell us about the raven paradox? First, consider the following fourcard task:

What cards need to be turned over to test the proposition: "Whenever it says raven on one side of a card it says black on the other"? The considerations described above

indicate that the answer is: raven and brown. In other words, you need to look at the raven to see if it is black, but also at the non-black colour to see whether it is (or is not) a raven. Omitting the 'brown' card would be irrational. The parallel with testing the proposition "All ravens are black" is clear. 12 It is necessary to examine not only the raven (to check it isn't non-black) but also the non-black thing (to check it isn't a raven).

In other words, seeing a brown sparrow is relevant (if you turn over the 'brown' card, but not the 'sparrow' card), in that it counts as a failure to disconfirm that "All ravens are black".

So what causes the paradox of the ravens? I have shown it is not a fault in the logic; therefore it must be one of psychology: it (only) seems to be a paradox. So why does it seem paradoxical?

The first answer is that the paradox is often presented in a way that maximizes dramatic effect, but is misleading. A typical line is: "Seeing a green leaf confirms the hypothesis that all ravens are black". This is misleading because readers will automatically build a mnental model or scenario in which they imagine themselves strolling down the road, wandering into a park and noticing a leaf growing on a tree as they walk underneath it -whereupon suddenly, like Newton and the apple, in a flash of mental light, they cry out: "Eureka! All ravens are black!" But this is unfair! The thesis is not that if you see a leaf and (either simultaneously or afterwards) find it is green then you have supported the proposition; the thesis is that you inspect a green thing (or any other non-black thing) and see it is not a raven. 13 The small print in the philosophy books makes this clear, and so does the above exposition of the paradox as a four-card task -but it is not always the immediate picture created in the reader's mind by the statement of the paradox. Even readers familiar with the paradox are likely to form an unavoidable mental association with the 'seeing-a-leaf' interpretation, via the first of the two stages of thinking just described (section IIi).

A second reason the situation seems paradoxical is one of numbers. First, to make the four-card task more like the real situation envisaged by Hempel, we need to imagine there are multiple copies of each of the four types of card. There will be numerous raven cards, one for each raven that exists -perhaps a million? However, the number of nonblack things in the universe is much greater -for all intents and purposes we might as well call it infinity -so the number of brown (and red, green, etc.) cards far outnumbers the raven cards. Therefore turning over any single non-black card seems to make such a tiny dent on the task of turning them all over to check there is not one single non-black raven (and we would have to turn them all over to be sure the proposition is not false) that it seems a waste of time, in that it provides such a trivial amount of information we have not really proved anything by gaining that information.

This naturally brings us full circle, back to the Hosiasson-Lindenbaum (1940) explanation of the raven paradox that I described in Section I, and to attempts to quantify the amount of information gained from each act of observation. These divide into two types: those following the positivist approach that observations of black ravens and of green leaves confirm or verify and thus increase the probability that the thesis is true (e.g. Glymour 1980;Horwich 1982); or alternatively that these observations fail to disconfirm or falsify, thus adding to the degree of corroboration (Popper 1983

III. Conclusion.

My contention has been that the raven paradox arises because psychological factors intervene when we try to exemplify abstract logic in real-world terms. It is impossible to ignore the particular scenario when we apply the logic to everyday situations. 15 The raven paradox arises because logic and psychology give us different answers as to the relevance of white shoes. Logic is correct in asserting that white shoes are relevant;

probabilistic calculations show that the information gained from white shoes is minuscule. But whether we know this is a matter of psychology. Logical arguments have to be filtered through the constraints of our ability to think. 16 Our intuitive faculties first construct a mental model in which it is clear that white shoes are irrelevant to the question of black ravens; only slow effortful analytic reasoning enables us to see the logical explanation for the relevance of white shoes.

Notes.

evidence supporting a statement also supports any logically equivalent statement. So seeing a black raven confirms that all non-black things are non-ravens, and conversely, seeing a non-black non-raven confirms that all ravens are black.

The situation can be presented more simply with a Venn diagram. The rectangle below represents the entire set of all things. The set of ravens is circumscribed by the inner circle (which assumes: Some things are ravens, and: Not everything is a raven). The set of black things is indicated by the grey patch; this patch has an irregular shape, to signify that it is not specified in the hypothesis whether non-ravens are black or nonblack. They might all be non-black, in which case the grey patch should be coincident with the circle. However it is necessary for the paradox that non-ravens are not all black, so the grey patch does not fill the rectangle. I have drawn the intermediate case,

where some non-ravens are black and some are not. However, the grey patch must cover the circle entirely, to represent the assertion that all ravens are black.

Observation of a black raven is indicated by the x in the circle, and observation of a nonblack non-raven is marked by the X in the outer white area. Clearly, the white area does not overlap the circle, in accordance with principle (ii) above: non-black things cannot exist in the space occupied by ravens. The existence of an X is thus consistent with the hypothesis (plus the assumption that not everything is black). So Hempel's conclusion, that white shoes confirm that "All ravens are black", does make sense logically, in that the existence of non-black (e.g. white) non-ravens (e.g. shoes) is a necessary consequence of the conjunctive hypothesis: "All ravens are black and not everything is black". 5 Hempel (1965, p. 21, footnote) points out this might not work in all cases; he considers it "an empirical question" whether a given case of "All P's are Q" has the non-Q's outnumbering the P's, and that this will often be difficult to decide in practice. Lipton (personal communication) has suggested the example: "All massless particles travel at the speed of light" -is the number of things moving at other speeds greater or smaller than the number of massless particles? 6 Lipton (1991, p. 106) suggested we can convert the raven hypothesis into a causal model of explanation by rephrasing it as: "All ravens contain a gene that causes them to be black". He recognized that this implies the gene is part of the essence of ravens.

Lipton's argument touches on the first of the two problems that arise with the use of "All ravens are black" to illustrate the logical consequences of confirmationism. The first problem is "raven" and the second is "black".

(i) First, Lipton asks how we know whether the property of being black is "essential to ravens" (in which case the hypothesis must necessarily be true) or whether it is accidental (in which case the hypothesis could be false). The issue is one of definitionhow do you define a raven? There are three methods in common use for the classification of species (Hull, 1988). The first (and oldest) is essentialism. Aristotle considered that a raven is a raven because God breathed 'essence of raven' into ravens when He created them. The difference between a raven and a crow, a horse, a human, etc., is that these species all have different essences within them. In modern terms, these essences may be physical things: the existence of a certain gene or set of genes, for example (e.g. Lipton's 'gene for blackness'). However all organisms are actually multidimensional: they have many characteristics. And further, each such phenotypic characteristic depends on non-linear interactions between many genes (e.g. Kauffman, 1993). Hence there is a problem in identifying one essential property or thing that an organism possesses, that defines it as a member of a particular species and that is held by no organism of any other species. (There may be a set of such properties rather than just one, but as long as this set does not overlap with that of any other species, the principle remains similar.) This approach to classification is called cladistics. It has the problem of identifying and justifying the choice of 'essential' characteristic.