Paradoxes and Their Logic
Interview by Richard Marshall.
Margaret Cuonzois a single malt scotch from a cask wood hogshead chill filtering paradoxes. She likes Schiffer's distinction between happy and unhappy solutions, thinks paradoxes demand serious thought, thinks solutions to paradoxes re-educate our intuitions, thinks Bayesian degrees of belief useful, discusses six types of solution, thinks there's a great irony in trying to solve paradoxes and thinks the sorites one of the deep ones. As a mighty winter starts read this and warm to her task...
3:AM:What made you become a philosopher?
Margaret-Cuonzo: I have been interested in philosophical questions as far back as I remember. As a child, I would wonder about the beginning of time, its direction, and how things change. I shared a bedroom with my great-grandmother, who was a very religious Roman Catholic and prayed all the time. Our room was, in essence, a shrine, and I couldn’t help wondering about whether all the rituals, like the holy water she sprinkled on me every night, were really doing anything, and whether the God and saints to whom she was so devoted really were there and listening. There seemed to be too many inconsistencies in what I was told. Later, in college at Barnard, I attached the name “philosophy” to what I was doing. Of course, there’s a big difference between, on the one hand, wondering about such topics and, on the other, formulating positions about them and providing logical arguments, but the motivation to do so was there very early.
At Barnard, I first encountered an eccentric group of people who were devoting their lives to actually providing reasoned answers to questions about the nature of time, reality, truth, beauty, and so on. Mary Mothersill, who was thinking and writing about beauty, first introduced me to Hume’s paradox of taste, a paradox associated with how our judgments about what is beautiful are subjective and beauty is “in the mind of the perceiver,” and yet we make seemingly correct judgments about a work’s aesthetic value all the time. If beauty is in “the eye of the beholder,” then on what grounds can one rightly assert that the sunset off the coast of California, Raphael’s School of Athens, or Bach’s Violin Concerto in A Minor are beautiful?
Professor Sue Larson, a philosopher of language, introduced me to the liar paradox, which is associated with my present statement that “What I am now claiming is false.” If the statement in the last sentence is true, then it is false, and if it’s false, then it is seemingly true. I had never thought that people could spend their lives working on such problems. I’ll never forget a day in class when, after a long, awkward silence in which a few students got up and left, Professor Larson announced, “there are as many truths as there are falsehoods.” I thought she was nuts at the time, especially given the long silence that came before her pronouncement. After a while, though, I started to see what she was getting at. While it looks like the book next to me, which is black, could have been blue, orange, etc., and there are many falsehoods and only a one truth about it, it is also true that the book is not blue, true that the book is not orange, and so on. So, since truth refers to statements, and every statement or its denial is true, when you total the truths and falsehoods for claims about the color of the book (and their denials), the number should be equal. At least this is what I think she meant. Anyway, I was hooked after that.
3:AM:In your book on paradoxesyou say you had to rethink their nature. So what did you think they were before the book and what did Stephen Schifferwrite that started to change all that?
MC:Up until the point I started working on paradoxes in earnest, I thought what a lot of people think, namely, that paradoxes are there to be solved, have some kind of hidden flaw, and it is the job of philosophers and other theorists to point out that flaw. Stephen Schifferstarted to wonder about this and whether the deeper philosophical paradoxes admit of such solutions. He divided solutions to paradoxes into two camps, “happy-face” solutions, which point to some flaw in the paradox, and “unhappy-face” solutions which don’t attempt to do this, but rather show what leads to the paradox and why it can’t be given a happy-face solution. So, for example, take a version of the liar paradox I just mentioned, one about my present claim: I am a liar. An unhappy-face solution, like Alfred Tarski’s, argues that the liar shows that our ordinary natural language conception of truth is irredeemably flawed, that there is no solution to the paradox, but a new notion (which he calls “satisfaction”) can be given that doesn’t lead to paradox. A happy-face solution to a paradox, though, tries to show that there is some flaw in the paradoxical argument, and then explain why that flaw seemed so plausible.
I took up Schiffer’s distinction between happy-face and unhappy-face solutions, and based on my study of the history of solutions to paradoxes, added some categories. So, I’ve got six categories of solutions, four are what Schiffer would call “happy face solutions” (The Preemptive Strike, the Odd-Guy-Out, the It’s All Good, and the You Can’t Get There from Here) and two are “unhappy-face” solutions (The Detour and the Facing the Music).
3:AM:Paradoxes sometimes get presented as just party tricks, kind of trivial bits of philosophy that are fun but nothing else. You think this is a mistake don’t you? So, as the Joker puts it: why so serious?
MC:Paradoxes are fun, but they demand serious thought, too, since they arise from concepts we use in our lives like truth, knowledge, set, beauty, and so on, and often don’t have definitive explanations or answers. They expose ways in which the concepts that we use to organize the world around us are flawed.
Now that the book is out, I’ve been speaking with many people about paradoxes, and almost without fail, I encounter people in the audience who are dealing with paradoxical situations in their own lives. One woman, for example, told me that she is now wondering whether she is living with the same person she married twenty years ago. “He’s the same man, of course,” she admitted, “but he has changed so drastically, I’m not really sure he’s the same person.” This woman was intuitively and intimately aware that the concept of a “self” that stays the same through time leads to trouble in many situations, for example, when the “self’s” memories, habits, and beliefs are radically changed.
Also, paradoxes are sometimes what lead to developments in science, philosophy, mathematics, and the other fields in which they occur. By noticing that a concept assumed by a scientific (or other) theory leads to a paradox, these paradoxes invite those working in the field to revise the concept and perhaps the theory, too.
3:AM:Do you see solutions to paradoxes as a re-education of our intuitions then? Can you give us an example of what this might mean?
MC:If you look at any of the standard definitions of “paradox,” such as, “a set of mutually inconsistent statements, each of which seems true,” you’ll find that the definition includes the word “seems” or some other phrase that illustrates that counterintuitive nature of paradoxes. The etymology of the word indicates this, too. “Para” derives from the ancient Greek word for “beyond” or “against” and “doxa” from the ancient Greek word for “expectation” or “opinion.” Because paradoxes call into question our intuitive understanding of the world around us, solutions that attempt to show that some part of the paradox is flawed need to send our intuitions back to school, as it were, so that the paradox doesn’t look so troubling.
For example, take a paradox from statistics, Simpson’s paradox, which highlights how, when you combine different groups of data, a trend that occurs in each group can be negated or even reversed. A famous example comes from the batting averages of baseball players Derek Jeter and David Justice for the 1995 and 1996 seasons. In 1995, Jeter had an average of .250, which resulted from 12 hits on 48 times at bat, while Justice had an average of .253, which resulted from 104 hits out of 411 times at bat. Justice also had a higher average for 1996. His average was .321 (45 hits out 140 times at bat), while Jeter had only a .314 average (from 183 hits from 582 at bats). Yet, despite the fact that Justice had higher batting averages for both years, it turns out that Jeter’s average for the combined two years of 1995 and 1996 was higher. When you add up the hits and at bats for both years, Jeter had 195 hits out of 630 at bats, giving him a .310 batting average, while Justice had 149 hits out 551, giving him only a .270. (For further discussion, see Ken Ross’s A Mathematician at the Ballpark, Pi Press, 2004). When we saw that Justice had higher batting averages for 1995 and for 1996, our intuitions probably led us to conclude that his average for the combined years of 1995 and 1996 were higher, but this is not the case. So, Simpson’s paradox forces us to take our intuitive understanding of averages and sharpen it.
3:AM:How does a Bayesian idea of belief become important for your approach to paradox solution and understanding our own intuitions?
MC:Intuitions are messy, since there are so many different types, such as perceptual intuitions (see for example, discussions of the McGurk effect), mathematical intuitions (2=2), and conceptual intuitions (bachelors are unmarried). Also, definitions of “intuition” are all over the map. “Non-inferential belief,” “what we would say in a given situation,” and “seeming truth,” have all been offered.
Since our intuitive understanding figures prominently in what it means to be a paradox, I wanted to sharpen up the discussion of intuition in a way that would allow for degrees of paradoxicality, that is, that would allow us to say how paradoxical something is. This is where the Bayesian notion of degrees of belief comes in.
Since it is possible to quantify the degree to which something is believed, we can tell how strong our beliefs about parts of a paradox are. For Bayesians, complete certainty would be assigned 1, complete disbelief assigned 0, and anything in between would be given a degree in between. For example, .5 would be neither believe nor disbelief. I believe to .7 degrees that Hillary Clinton will seek and win the Democratic Party’s nomination for U.S. President in the next election. That belief can be revised up or down, depending on a number of factors. If, for example, a huge scandal involving Clinton is brought to light, my degree of belief will be lowered, perhaps even to 0.
When you take the degrees to which we believe parts of a paradox and plug them into something called a paradoxicality rating, you get a number that tells you how paradoxical something is. That’s useful, because it will also give you a sense of how likely the paradox is to have a standard solution-type (types 1-4).
3:AM:You have six types of solution, or non-solution in the ‘facing the music’ and 'detour' responses? Why isn’t there just one way?
MC:Due to the variety of paradoxes, no one type of solution would be adequate. There are hundreds of known paradoxes with varying degrees of strength, probably over a thousand if you define “paradox” broadly enough. Some are not very paradoxical at all and can be given a quick solution that points to some flaw in the paradox. An example might be the “Buttered Cat Paradox,” which is constructed from putting together two common adages, “Cats always land on their feet,” and “Buttered toast always lands buttered side down.” If you strap a piece of buttered toast onto a cat’s back, with the buttered side facing out, and assume both are true, then it seems to follow that, when thrown out the window, the cat/toast combination will keep turning and never land. Such a weak paradox can be solved using an “Odd Guy Out” approach by rejecting at least one part of the paradox, namely, the adage about buttered toast. While cats have biological mechanism that, usually, gets them to right themselves before landing, the adage about the toast is metaphorical and not literally true. Other paradoxes, like the liar paradox I mentioned earlier, have been around for millennia without non-controversial solutions, and are most likely, capable of being solved using the more restricted “Detour” solution, which creates an alternative notion than the one that leads to the paradox, or the “Facing the Music” solution, which shows why there can be no solution to the paradox. So, different types of paradoxes engender different solutions.
3:AM:Are the ‘pre-emptive strike’ and ‘you can’t get there from here’ solutions showing us that the paradox was pretty weak and our intuitions weren’t under too much strain?
MC:I think that the “Preemptive Strike,” where the whole notion that leads to paradox is shown suspect, the “Odd Guy Out,” where a flaw in the paradox is identified, the “You Can’t Get There From Here,” where the reasoning involved is questioned, and “It’s All Good,” where the conclusion of the paradoxical argument is shown true, are all successful only for the weaker paradoxes. They sometimes get offered for the stronger ones, too, but trying to solve the deeper paradoxes with one of these strategies generally fails.
3:AM:'Odd guy Out' is the solution that brings down our belief in certain parts of the puzzle so the paradox kind of withers. Are you convinced that Zeno, Quine/Duhem and the sorites fade away via this approach?
MC:The “Odd Guy Out” generally works for the weaker paradoxes and not the Quine/Duhem, sorites, or most of Zeno’s paradoxes, which are all deep paradoxes. Of Zeno’s paradoxes, the stadium is a weaker one, and invites a kind of “It’s All Good” solution where you acknowledge the claims of the paradox, but hold that there is no troubling consequence to be derived from it.
3:AM:The last two approaches accept the paradox and so in a sense show that it might seem counter-intuitive but nevertheless there are no inconsistencies. The difference between 'It’s all good' and 'the detour' is about lowering or not lowering the subjective probability of the paradox isn’t it? Can you explain this?
MC:The “It’s All Good,” which shows that there a supposedly problematic result generated by a paradox is not very problematic at all, while the Detour creates an alternative notion to the one that leads to paradox. Take the sorites, for example, which can be put in the following way:
1. A person with 1 hairs on his head is bald.
2. For any number n, if a person with n hairs on his head is bald, then a person with (n + 1) hairs on his head is bald.
3. Therefore, a person with 1,000,000 hairs is bald.
Here we have an argument with two premises (1) and (2) and a conclusion (3). An “Odd Guy Out” solution to this paradox would say that the conclusion (3) is true. Of course, arguing for the claim that a person with a million hairs on his head is bald is going to be a challenge, and probably not get off the ground very easily, but it can be done. A Detour would claim that while the paradox cannot be solved for our ordinary language conception of baldness, we can create another concept that mirrors this concept, but doesn’t lead to paradox. Let’s call this concept baldness*. Baldness* might be defined, for example, as having 1000 hairs or less. While the sorites paradox arises for baldness, it doesn’t for baldness*, because the second premise will turn out false. A person with 1000 hairs is bald* and 1001 hairs not bald*. The problem with this Detour is that baldness* as we have defined it doesn’t resemble the way we normally talk about baldness. Both these strategies can be used on the same paradox. The difference is whether you accept the conclusion as true, or whether create an alternate concept that mirrors the original one.
3:AM:What’s the pessimistic meta-induction argument and why does it apply to the approaches to paradoxes you’ve discussed?
MC:The pessimistic meta-induction argument is an argument that is given by some philosophers of science regarding scientific progress. If you look at the long history of scientific theories, it becomes apparent that the vast majority of scientific theories that have been offered throughout the ages have now been shown to be false and the entities that the theories posited where shown not to exist. Based on this evidence from the past, it is rational to conclude that all present theories will suffer a same fate, as well as many future theories. According to this line of argument, the past failures provide some grounds for concluding that present theories are inadequate, as well.
If you look at the long history of attempted solutions to the deeper paradoxes, you see an even worse situation for solutions to paradoxes. Not only have paradoxes like the liar, sorites, skeptic, and so on resisted solutions for millennia, but there has never been a consensus on solutions to these paradoxes. This leads me to conclude that the proposed solutions that fill contemporary philosophy and other journals will fail, as well.
3:AM:You don’t think the deep paradoxes have a final solution do you? So how can false solutions be useful? How can they be linked to intellectual progress? Is it born out by history?
MC:This is the great irony of trying to solve paradoxes. History provides many examples of paradoxes leading to advances in thought, despite the fact that the paradoxes remain without uncontroversial solutions. Russell’s paradox was motivation for developing later set theories. The EPR paradox in physics inspired Bell’s theorem. The prisoner’s dilemma gave rise to advances in game theory. The list goes on. Failed solutions are often very useful. The utility, though, doesn’t lie in solving paradoxes, but in doing other things.
3:AM:The sorites is I take it one of these deep paradoxes and it seems to be one of your favorites. It runs through the book. Tim Williamsonand Roy Sorensenthink it’s got an odd guy out solution, but you don’t buy that do you? Michael Dummett used a ‘facing the music’ approach – is that better for you? How do you do the sorites, what’s its appeal for you, why does it still have legs and what does that tell us?
MC:The sorites is, to my mind, one of the deepest paradoxes out there and it provides a good example for understanding different ways to solve paradoxes. It is very deep because the premises of the paradox are close to certainly true, the reasoning is straightforward, and yet the conclusion is obviously false. In addition, the phenomenon that gives rise to the paradox, namely the vagueness of language, is so ubiquitous that the paradox has implications for much of our language. It tells us a lot about the concepts that we use everyday. The vast majority of our concepts, those like, baldness, wealth, strength, nearness, and so on, are vague. These admit of borderline cases and this can lead to outright contradiction. One fascinating thing about the sorites, then, is that it exposes how perfectly useful concepts lead to contradiction. I think Dummett’s solution is the best among those you list, and this is because Dummett admits that the paradox lacks a standard solution.
3:AM:If in the future we handed over our cognitive work to machines, would paradoxes disappear?
MC:That’s an interesting question! I’ll have to think more about it! Here’s a very tentative conjecture: If we programmed the machines using completely consistent rules with no room for ambiguity, vagueness and so on, then perhaps paradoxes would not arise. I’m not sure that the world modeled by those machines would really resemble, or be hospitable to creatures with our limited cognitive capacities. Another issue, too, is that since we are doing the programming there will be flaws in the system, feedback loops and so on.
3:AM:And for the readers here at 3:AM, are there five books we should be reading to get further into your philosophical world?
MC:For a general audience, I’d say Jeffrey’s book, Subjective Probability, is a good way to learn about degrees of belief. Also, for a nice catalogue (but not theory about) paradoxes, Michael Clark’s Paradoxes from A to Zgives a number of paradoxes. To go a bit deeper into the sorites paradox and also get a sense of how different systems try to solve a paradox, I’d recommend Timothy Williamson’s book Vagueness, though I’d encourage a bit of wariness regarding his own solution. Stephen Schiffer’s The Things We Meanis dense and technical, but definitely worth the effort. Also, it will give the reader a glimpse at the starting point for my own view. And lastly, the classic The Development of Logic, by William and Martha Kneale is a massive book on the history of logic that will give the reader a sense of how logical systems develop and how they are used to solve paradoxes.
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