(An Allowable Psychologism?, pt. 1)
In Alice’s Adventures in Wonderland, the Mad Hatter asks Alice, “Why is a raven like a writing desk?” After several pages of delay the Hatter finally admits, “I haven’t the slightest idea.” Cracked characterizes this as “one of the biggest dick moves in literature.” Apparently, Lewis Carroll was plagued with demands for an answer to the riddle. In response to the numerous accusations of a “dick move” (or the Victorian equivalent thereof), he addressed the issue, saying, “Because it can produce a few notes, tho they are very flat; and it is never put with the wrong end in front!” This is disappointing. However, apparently, “never” was originally written by Carroll as ‘nevar’. This is indeed ‘raven’ “with the wrong end in front” (i.e. “nevar” is ‘raven’ backwards). This is a little better, but still not satisfying. WiseGeek suggest that the answer is simply that neither is made of cheese, which, if this post looks TL;DR, is close enough. But if you’d like to see the specifics of my answer, and the possible resolution of some philosophical paradoxes, then please, read on.
Paradoxes, so many paradoxes
It doesn’t seem highly likely that the answer to one mathematical paradox and one logical paradox (and a few others besides) could be illustrated by, and ultimately help solve, a flippant and ultimately throwaway riddle from Alice’s Adventures in Wonderland. That being said, stranger things happened in Wonderland, and truth is stranger than fiction, so here we go.
Paradoxically, I want to start my discussion about the aforementioned mathematical and logical paradoxes with another paradox entirely. Epimenides’ Paradox is a version of what has come to be known as the Liar’s Paradox. Epimenides lived in Knossos, Crete, around 600 BC, and is attributed with the original version of this paradox. Doug Hofstadter (1979), in his book Gödel, Escher, Bach, puts the paradox like this:
Epimenides was a Cretan who made one immortal statement: “All Cretans are liars.”
This is a paradox of self-reference. Epimenides cannot be telling the truth about Cretans, and also be a Cretan; and if he is a Cretan, then he cannot be telling the truth. It is with some amusement that I note that Hofstadter’s book was originally published with the tagline “a metaphorical fugue on minds and machines in the spirit of Lewis Carroll.” Curiouser and curiouser.
This paradox of self-reference helps give us some traction on Russell’s Paradox, which Bertrand Russell uncovered whilst writing his Principia Mathematica in 1901. Russell was trying to ground mathematics in rigorous logical terms using set theory. An example of this rigorousness being that all empty sets are the same empty set, for a set with nothing in it has nothing to differentiate it from other sets with nothing in them. An empty set is zero, by definition, and there is only one empty set, so from zero, we have derived one… and so on, for several hundred pages… across three volumes.
If we concern ourselves with sets, as Russell did, we come to a paradox of self-reference when defining sets into categories. A set is normal, unless it contains itself, then it is abnormal. That being said, it seems as though the only sets that could contain themselves are sets that are descriptions of things, rather than actual things, i.e. self-reference. To borrow from Russell (1919, p. 136), “normally a class [set] is not a member of itself. Mankind, for example, is not a man.” So, now, what of the set of all normal sets (R)? Is it normal or abnormal? If R were normal it would be a member of the set of normal sets (R), thereby making it abnormal, but if it were abnormal it would no longer be a member of the set of all normal sets (R). Thus R is neither normal nor abnormal, R is both R, and not-R, and this is a paradox, Russell’s Paradox. Curious-R and curious-not-R.
Russell’s Paradox, to me at least, bears a striking resemblance to both Epimenides’ Paradox, as outlined above, and Hempel’s ‘Paradox of the Ravens.’
The Paradox of the Ravens
The Paradox of the Ravens, was put forward by Carl Gustav Hempel (not Jung), and arises from the confluence of two logical rules, Nicod’s Principle and The Equivalence Condition. The Paradox reads something like this:
- All ravens are black.
- Everything that is not black is not a raven.
- Nevermore, my pet raven, is black.
- This green (and thus not black) thing is an apple (and thus not a raven).
Both 1 and 2 are equivalent. 3 is evidence for 1, and 4 is evidence for 2, but because 2 is equivalent to 1, 4 is also evidence for 1.
To expand on this a bit, under Nicod’s Principle, if we state that ‘All ravens are black’ (All Rs are B), and we encounter an instance of a raven that is black, we have, to some extent supported the hypothesis. Famously, of course, it was thought that all Swans were white, until they discovered black swans in Australia. So, as with the scientific method, inductive logic, and Bayesian reasoning, each instance of a case that supports the hypothesis lends incrementally more credence to the hypothesis, but certainty is seldom absolute. Indeed, courtesy of Karl Popper, the scientific method now requires that a hypothesis be able to be falsified (shown to be untrue) in order to have any use as a hypothesis. As such, the hypothesis ‘All ravens are black’ can be falsified by an instance of a raven that is not black, such as an albino raven. This being said, the definition of ‘swan’ could include the fact that they are white, and that thus black swans are in fact not swans at all. Conversely, an albino raven is still a raven, albeit not black, and not ultimately a case which falsifies the hypothesis, because we know what albinism is.
With the equivalence condition, whatever can be confirmed by a statement can also be confirmed by an equivalent statement. For example, if we were to say that ‘All Ravens are not Writing Desks (All Rs are not-WD), we could logically say that ‘No Writing Desks are Ravens’ (No WDs are R). From this, each instance of a writing desk not being a raven also supports the idea that no raven is a writing desk. But we expected that.
In combination, then, Nicod’s Principle and the Equivalence Condition should allow us to say ‘All Ravens are Black’ (All Rs are B), and that its equivalent statement is ‘All non-ravens are non-black’ (All not-R are not-B). Which seems fair enough on the face of it. The problem arises when you realize that the hypothesis ‘All Ravens are Black’ would now be, at least to some extent, supported by any instance of a non-black thing that also happened to not be a raven. Curious-R and Curious-B.
This idea is nicely explained, here:
It’s amusing to note that the Raven in this paradox is called ‘Nevermore’. Another solution to the riddle ‘Why is a Raven like a writing desk?’ courtesy of WiseGeek, that actually does seem to answer the question in a satisfying way, is also a reference to Edgar Allen Poe: Poe wrote on both. Cracked credits yet another solution to American Chess puzzle composer, Sam Loyd: They both have inky quills.
These aren’t bad, but not where I’m going with this.
The Paradox of the Ravens is supposed to illustrate a conflict between inductive logic (as exemplified by Nicod’s Principle and the Equivalence Principle) and intuition (as illustrated by our reaction to the paradoxical conclusion that the greenness of an apple in anyway supports the hypothesis that all ravens are black). So it seems that the paradox is somehow a product of some flaw in human thought. But does that flaw extend to logic itself, or is that an unallowable psychologism?
Mentally processing a negative
There is a truism in self-help literature that when trying to stop doing something, like smoking or drinking, we do better by mentally representing what we will do, rather than what we will not do. Saying that we will not do something forces us to represent all of the alternatives. Saying that we will do something that is not the thing we are trying to not do, is a successful way of not engaging in the behaviour we are trying to avoid.
Likewise, if we are trying to remember things, or make logical inferences about things, it helps to represent these things in positive, and thus concrete, terms. We will more quickly get to the correct state of affairs if we remember ‘The door was closed’ rather than ‘The door was not open’ (Kaup, Lüdtke & Zwaan, 2006). The delay is greater, and means of representation different, if the negation opens up an even wider scope of possibilities than the binary represented by the door (open or closed). “Not wearing a pink dress” (Kaup & Zwaan, 2003), for example, gives rise to everything from an amber dress to a yellow dress.
Along the same lines, matching a picture to a sentence describing that picture takes longer, and is more error-prone, when using negatives (Carpenter & Just, 1975; Clark & Chase, 1972; Trabasso, Rollins & Shaughnessy, 1971). Interestingly, negated items are slower to be recalled (Kaup, 2001; MacDonald & Just, 1989), which means that you will be slower to think of a white bear when you’re told not to, but think of it you will (Winerman, 2011).
So, with respect to the Paradox of the Ravens, is the fact of representing not one negative, but two – not a raven, and not black – just something our brains balk at? Are we actually trying to mentally represent all of the alternatives, or even just all of the plausible (seemingly relevant) alternatives? Or, is the problem maybe that our brains are trying to deal with the description, as given, negatives and all, and the result does not seem like it can logically be related to black ravens?
The Conjunction Fallacy
If a negative is a shorthand way of describing absolutely all possible counter-examples of a given situation or thing, what problems, aside from delayed access to relevant information, could that give rise to? The conjunction fallacy may give us some clues to the possible answer. The classic depiction of the conjunction fallacy is due to Tversky and Kahneman (as cited in Kahneman, 2012):
Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.
Which is more probable?
- Linda is a bank teller.
- Linda is a bank teller and is active in the feminist movement.
This is not a paradox, this is an outright fallacy. The number of feminist bank tellers must be less than both the number of active feminists and the number of bank tellers. So it is more probable that Linda is merely a bank teller than being both a bank teller and an active feminist. The reason that the idea of Linda being both is so attractive is that we know she is definitely a bank teller (given the two choices) and, courtesy of the representativeness heuristic, we feel that a bright, female, philosophy major, with concerns about discrimination and social justice simply must be a feminist. As Stephen J. Gould said,
I am particularly fond of this example [the Linda problem] because I know that the [conjoint] statement is least probable, yet a little homunculus in my head continues to jump up and down, shouting at me— “but she can’t just be a bank teller; read the description.”
More on the conjunction fallacy here.
Is a negative an inherent source of a conjunction fallacy, or even multiple conjunction fallacies? Is the source of the problem with the Paradox of the Ravens our inability to represent millions of alternatives to blackness and ravenness being compounded by our inability to account for the overlaps between these? Or is it just, as the conjunction fallacy illustrates, that our brains are really unintuitive when it comes to probabilities? Or is the use of the word “not” an obtuse kind of self-reference that leads to paradoxes of self-reference, just as Epimenides’ “lie” and “normal” sets do?
Wason and Confirmation Bias
Also discussed in Kahneman’s ‘Thinking, Fast and Slow’ is Peter Wason’s experiments from 1960. Wason asked people to identify a rule as represented by a number sequence, (2, 4, 6) and to then ask him if a number sequence they generated also fit the rule he had in mind. They were to do this as many times as they felt necessary to confirm that they knew the rule. Most people would then proceed to provide sequences that were consecutive even numbers (e.g. 6, 8, 10), some might provide a sequence of even numbers with gaps (e.g. 4, 8, 12). Few, if any would present odd-numbered sequences, or even numbers in reverse numerical order, or anything that deviated too far from the most obvious. Wason’s pattern was simply numbers in numerical order, which would include such examples as ‘3, 7, 9’ and ‘10, 100, 1000’.
So we should be looking for disconfirmation. We should be looking to falsify our hypothesis. Non-black ravens falsify our hypothesis. Black non-ravens neither confirm nor deny our hypothesis, and non-black non-ravens support our hypothesis.
Why IS a Raven like a Writing Desk?
A raven is more obviously like other corvids, such as blackbirds, rooks, and so on, and a writing desk is more obviously like other man-made, wooden, objects, such as bureaus and dressers. This having been said, is a wooden chess piece, which happens to be called a Rook, more like a raven, or a writing desk? What about a piece of cheese? The moon? And what about the word ‘Raven’ written on a piece of paper? Is that more like the raven it names, or the writing desk upon which it was written?
We’ve established that we’re not very good at probabilities, even with something as simple as Linda’s job in combination with her political orientation.
We’ve established that negating a statement may give rise to an infinity of alternatives, but even where it only gives rise to one alternative it takes longer for our brains to represent than the same statement expressed in the positive.
And we’ve established that the greenness of apples adds to our certainty that ‘All Ravens are Black’.
Except that it still “feels” wrong, doesn’t it?
What if I were to suggest that the problem is with the way the equivalence principle makes you think of equivalency in the wrong way? Encountering a black raven does directly lend extra weight to the hypothesis that all ravens are black. But then the hypothesis specifically mentions black ravens. The greenness of an apple also lends credence to the hypothesis that all ravens are black, but to a very, very, very much smaller degree. A green apple means that you have one less thing that is both not black and not a raven, but that is only one less thing from a very long list of things. Indeed, all of the things!
The statements ‘All ravens are black’ IS logically equivalent to ‘All non-ravens are not-black’, but the weight they lend to their respective hypotheses, the degree to which they add to your certainty as to the truth of the statement, are not equivalent. Indeed 1/∞ is nowhere near 1/10,000,000 (no actual stats as to likely world raven population were discoverable, by me at least, at the time of writing). Now, 1/10,000,000 is not a massive incremental increase in our certainty on the hypothesis that all ravens are black, so it’s odd that our brains, which have trouble with numbers of that size, are still so certain that green apples do not help us to confirm that hypothesis, but that black ravens do. Then again, if ravens are black, by definition, we don’t need an incremental increase in our certainty on that point, so the logical relationship between black ravens and green apples is irrelevant. Which really does seem like self-reference by the back door.
So, the solution to the paradox of the ravens can be illustrated by using it to solve Lewis Carroll’s riddle, “How is a raven like a writing desk?” A raven is like a writing desk in that they are both unlike far more things than they are unlike each other. They also both have names that seem to unequivocally denote what they are, and many more that denote what they are not. The fact that ravens seem unlike writing desks to us is, statistically speaking, merely a rounding error, they are, in fact, virtually identical, and notably unlike a piece of cheese.
 Hofstadter, D. (1979). Gödel, Escher, Bach: An Eternal Golden Braid (a meta-phorical fugue on minds and machines in the spirit of Lewis Carroll). New York, NY: Basic Books.
 Russell, B. (1919). Introduction to Mathematical Philosophy, London: George Allen and Unwin Ltd, and New York: The Macmillan Co.
 Nicod, J. (1930). Foundations of Geometry and Induction, P. P. Wiener (trans.), London: Harcourt Brace.
 Kaup, B., Lüdtke, J. & Zwaan, R. A. (2006). Processing negated sentences with contradictory predicates: Is a door that is not open mentally closed? Journal of Pragmatics, 38(7), 1033-1050.
 Kaup, B., & Zwaan, R. A. (2003). Effects of negation and situational presence on the accessibility of text information. Journal of Experimental Psychology: Learning, Memory, and Cognition, 29(3), 439.
 Carpenter, P. A. & Just, M. A. (1975). Sentence comprehension: A psycholinguistic processing model of verification. Psychological Review, 82(1), 45-73.
 Clark, H. H., & Chase, W. G. (1972). On the process of comparing sentences against pictures. Cognitive Psychology, 3(3), 472-517.
 Trabasso, T., Rollins, H., & Shaughnessy, E. (1971). Storage and verification stages in processing concepts. Cognitive Psychology, 2(3), 239-289.
 Kaup, B. (2001). Negation and its impact on the accessibility of text information. Memory & Cognition, 29(7), 960-967.
 MacDonald, M. C., & Just, M. A. (1989). Changes in activation levels with negation. Journal of Experimental Psychology: Learning, Memory, and Cognition, 15(4), 633-642.
 Kahneman, D. (2012). Thinking, Fast and Slow. London: Penguin.