By: Franca Driessen,
on 9/18/2016
Let us say that a sentence is periphrastic if and only if there is a single word in that sentence such that we can remove the word and the result (i) is grammatical, and (ii) has the same truth value as the original sentence.
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By: Helena Palmer,
on 7/12/2015
Here I want to present a novel version of a paradox first formulated by José Bernardete in the 1960s – one that makes its connections to the Yablo paradox explicit by building in the latter puzzle as a ‘part’. This is not the first time connections between Yablo’s and Bernardete’s puzzles have been noted (in fact, Yablo himself has discussed such links). But the version given here makes these connections particularly explicit.
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By: Alex Guyver,
on 4/5/2015
The collection of infinite Yabloesque sequences that contain both infinitely many Y-all sentence and infinitely many Y-exists sentences, however, is a much larger collection. It is what is called continuum-sized, and a collection of this size is not only infinite, but strictly larger than any countably infinite collection. Thus, although the simplest cases of Yabloesque sequence – the Yablo Paradox itself and its Dual – are paradoxical, the vast majority of mixed Yabloesque sequences are not!
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By: Alex Guyver,
on 3/7/2015
A generalization is a claim of the form: (1) All A’s are B’s. A generalization about generalizations is thus a claim of the form: (2) All generalizations are B. Some generalizations about generalizations are true. For example: (3) All generalizations are generalizations. And some generalizations about generalizations are false. For example: (4) All generalizations are false. In order to see that (4) is false, we could just note that (3) is a counterexample to (4).
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By: Alex Guyver,
on 2/8/2015
Imagine that Banksy, (or J.S.G. Boggs, or some other artist whose name starts with “B”, and who is known for making fake money) creates a perfectly accurate counterfeit dollar bill – that is, he creates a piece of paper that is indistinguishable from actual dollar bills visually, chemically, and in every other relevant physical way. Imagine, further, that our artist looks at his creation and realizes that he has succeeded in creating a perfect forgery. There doesn’t seem to be anything mysterious about such a scenario at first glance – creating a perfect forgery, and knowing one has done so, although extremely difficult (and legally controversial), seems perfectly possible. But is it?
In order for an object to be a perfect forgery, it seems like two criteria must be met. First of all, the object must be a forgery – that is, the object cannot be a genuine instance of the category in question. In this case, our object, which we shall call X, must not be an actual dollar bill:
1.) X is not a dollar bill.
Second, the object must be perfect (as a forgery) – that is, it can’t be distinguished from actual instances of the category in question. We can express this thought as follows:
2.) We cannot know that X is not a dollar bill.
Now, there is nothing that prevents both (1) and (2) from being simultaneously true of some object X (say, our imagined fake dollar bill). But there is an obstacle that seemingly prevents us from knowing that both (1) and (2) are true – that is, from knowing that X is a perfect forgery.
Imagine that we know that (1) is true, and in addition we know that (2) is true. In other words, the following claims hold:
3.) We know that X is not a dollar bill.
4.) We know that we cannot know that X is not a dollar bill.
Knowledge is factive – in other words, if we know a claim is true, then that claim must, in fact, be true. Applying this to the case at hand, this means that claim (4) entails claim (2). But claim (2) and claim (3) are incompatible with each other: (2) says we cannot know that X isn’t a dollar, while (3) says we know it isn’t. Thus, (3) and (4) can’t both be true, since if they were, then a contradiction would also be true (and contradictions can’t be true).
Thus, we have proven that, although perfect forgeries might well be possible, we can never know, of a particular object, that it is a perfect forgery. But an important question remains: If this is right, then what, exactly, is going on in the story with which we began? How is it that our imagined artist doesn’t know that he has created a perfect forgery?
In order to answer this question, it will help to flesh out the story a bit more. So, once again imagine that our artist creates the piece of paper that is visually, chemically, and in every other physical way indistinguishable from a real dollar bill. Call this Stage 1. Now, after admiring his work for a while, imagine that the artist then pulls eight genuine, mint-condition dollar bills out of his wallet, throws them on the table, and then places the forgery he created into the pile, shuffling and mixing until he can no longer identify which of the pieces of paper is the one he created, and which are the ones created by the Mint. Let’s call this Stage 2. How do Stage 1 and Stage 2 differ?
At Stage 1 we do not, strictly speaking, have a case of a perfect forgery. Although the piece of paper the artist created is physically indistinguishable from a dollar bill, the artist can nevertheless know it is not a dollar bill because he knows that he created this particular object. In other words, at Stage 1 he can tell that the forgery is a forgery because he knows the history, and in particular the origin, of the object in question.
Stage 2 is different, however. Now the fake is a perfect forgery, since it still isn’t a dollar, but we can’t know that it isn’t a dollar, since we can no longer distinguish it from the genuine dollars in the pile. So in some sense we know that the fake dollar in the pile is a perfect forgery. But we can’t point to any particular piece of paper and know that it, rather than one of the other eight pieces of paper, is the perfect forgery. In other words, in Stage 2 the following is true:
But the following, initially similar looking claim, is false:
We can sum all this up as follows: We can know that perfect forgeries exist – that is, we can know claims of the form “One of those is a perfect forgery”. But we can’t know, of a particular object, that it is a perfect forgery – that is, we can never know claims of the form “That is a perfect forgery”. And it is this latter sort of claim – that we know, of a particular object, that it is a perfect forgery – that leads to the contradiction.
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By: Alex Guyver,
on 1/11/2015
One of the central tasks when reading a mystery novel (or sitting on a jury) is figuring out which of the characters are trustworthy. Someone guilty will of course say they aren’t guilty, just like the innocent – the real question in these situations is whether we believe them.
The guilty party – let’s call her Annette – can try to convince us of her trustworthiness by only saying things that are true, insofar as such truthfulness doesn’t incriminate her (the old adage of making one’s lies as close to the truth as possible applies here). But this is not the only strategy available. In addition, Annette can attempt to deflect suspicion away from herself by questioning the trustworthiness of others – in short, she can say something like:
“I’m not a liar, Betty is!”
However, accusations of untrustworthiness of this sort are peculiar. The point of Annette’s pronouncement is to affirm her innocence, but such protestations rarely increase our overall level of trust. Either we don’t believe Annette, in which case our trust in Annette is likely to drop (without affecting how much we trust Betty), or we do believe Annette, in which case our trust in Betty is likely to decrease (without necessarily increasing our overall trust in Annette).
Thus, accusations of untrustworthiness tend to decrease the overall level of trust we place in those involved. But is this reflective of an actual increase in the number of lies told? In other words, does the logic of such accusations makes it the case that, the higher the number of accusations, the higher the number of characters that must be lying?
Consider a group of people G, and imagine that, simultaneously, each person in the group accuses one, some, or all of the other people in the group of lying right at this minute. For example, if our group consists of three people:
G = {Annette, Betty, Charlotte}
then Betty can make one of three distinct accusations:
“Annette is lying.”
“Charlotte is lying.”
“Both Annette and Charlotte are lying.”
Likewise, Annette and Charlotte each have three choices regarding their accusations. We can then ask which members of the group could be, or which must be, telling the truth, and which could be, or which must be, lying by examining the logical relations between the accusations made by each member of the group. For example, if Annette accuses both Betty and Charlotte of lying, then either (i) Annette is telling the truth, in which case both Betty and Charlotte’s accusations must be false, or (ii) Annette is lying, in which case either Betty is telling the truth or Charlotte is telling the truth (or both).
This set-up allows for cases that are paradoxical. If:
Annette says “Betty is lying.”
Betty says “Charlotte is lying.”
Charlotte says “Annette is lying.”
then there is no coherent way to assign the labels “liar” and “truth-teller” to the three in such a way as to make sense. Since we are here interested in investigating results regarding how many lies are told (rather than scenarios in which the notion of lying versus telling the truth breaks down), we shall restrict our attention to those groups, and their accusations, that are not paradoxical.
The following are two simple results that constraint the number of liars, and the number of truth-tellers, in any such group (I’ll provide proofs of these results in the comments after a few days).
“Accusations of untrustworthiness tend to decrease the overall level of trust we place in those involved”
Result 1: If, for some number m, each person in the group accuses at least m other people in the group of lying (and there is no paradox) then there are at least m liars in the group.
Result 2: If, for any two people in the group p1 and p2, either p1 accuses p2 of lying, or p2 accuses p1 of lying (and there is no paradox), then exactly one person in the group is telling the truth, and everyone else is lying.
These results support an affirmative answer to our question: Given a group of people, the more accusations of untrustworthiness (i.e., of lying) are made, the higher the minimum number of people in the group that must be lying. If there are enough accusations to guarantee that each person accuses at least n people, then there are at least n liars, and if there are enough to guarantee that there is an accusation between each pair of people, then all but one person is lying. (Exercise for the reader: show that there is no situation of this sort where everyone is lying).
Of course, the set-up just examined is extremely simple, and rather artificial. Conversations (or mystery novels, or court cases, etc.) in real life develop over time, involve all sorts of claims other than accusations, and can involve accusations of many different forms not included above, including:
“Everything Annette says is a lie!”
“Betty said something false yesterday!”
“What Charlotte is about to say is a lie!”
Nevertheless, with a bit more work (which I won’t do here) we can show that, the more accusations of untrustworthiness are made in a particular situation, the more of the claims made in that situation must be lies (of course, the details will depend both on the number of accusations and the kind of accusations). Thus, it’s as the title says: accusation breeds guilt!
Note: The inspiration for this blog post, as well as the phrase “Accusation breeds guilt” comes from a brief discussion of this phenomenon – in particular, of ‘Result 2′ above – in ‘Propositional Discourse Logic’, by S. Dyrkolbotn & M. Walicki, Synthese 191: 863 – 899.
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By: Alex Guyver,
on 11/30/2014
Supposedly, early 20th century packaging for Quaker Oats depicted the eponymous Quaker holding a package of the oats, where the art on this package depicted the Quaker holding a package of the oats, which itself depicted the Quaker holding a package of the oats, ad infinitum. I have not been able to locate an photograph of the packaging, but more than one philosopher and mathematician has attributed an early interest in the nature of the infinite to childhood contemplation of this image. Here, however, I want to examine a different phenomenon: whether artwork that depicts itself in this way can lead to paradoxes.
Let’s begin with two well-known puzzles. The older of the two– the Liar paradox – was known to ancient Greek philosophers, and challenges the following platitudes about truth:
(T1) A sentence is true if and only if what it says is the case.
(T2) Every sentence is exactly one of true and false.
Consider the Liar sentence:
This sentence is false.
Is the Liar sentence true or false? If it is true, then what it says must be the case. It says it is false, so this means it is false. If it’s false, then, since it says it is false, what it says is indeed the case. But this would make it true. So the Liar sentence is true if and only if it is false, violating the platitudes.
The second puzzle is the Russell paradox, discovered by Bertrand Russell at the beginning of the 20th Century. This paradox involves collections, or sets, of objects, and two central theses:
(S1) Given any property P, there is a set of objects containing all and only the objects that have P.
(S2) Sets are themselves objects, and can be contained in sets.
Given (S2), we can divide objects into two types: Those that contain themselves (such as the set containing all sets whatsoever) and those that do not contain themselves (such as the set of all kittens). Thus, “is a set that does not contain itself” picks out a perfectly good property, and so by (S1) there should be a set – let’s call it R – containing exactly those things that have this property. So:
A set is a member of R if and only if it is not a member of itself.
Now, is R a member of itself? Either it is or it isn’t. If R is a member of itself then R isn’t a member of itself. And if R isn’t a member of itself then R is a member of itself. Either way, R both is and isn’t a member of itself. Again, a contradiction.
There is another puzzle that seems intimately connected to these two paradoxes, however, that has not (as far as I know) been noticed or studied – the paradox of the impossible painting. This paradox stems from two principles governing the notion of depiction (or representation) rather than truth or set-theoretic membership.
First, it seems, at least at first glance, that we can paint anything that we can describe – if I tell you to paint a forest with exactly 28 trees, then you can produce a painting fitting that description. Thus:
(D1) Given any description D, we can create a painting that depicts things exactly as described in D.
Second, there is nothing to prevent a painting from being depicted within another painting – for example, Velazquez’s Las Meninas depicts the painter working on another painting. Thus:
(D2) Paintings can be depicted in paintings.
If some paintings can depict other paintings, then it seems like we can divide paintings into two types: those that depict themselves (such as the artwork on old Quaker Oats packaging) and those that do not. Thus, “a scene depicting all and only the paintings that do not depict themselves” is a perfectly good description, and so by (D1) it should be possible to produce a painting – let’s call it I – that depicts things as described. So:
A painting is depicted in I if and only if it does not depict itself.
Should I depict itself ? In other words, if you are creating this painting, should you include a depiction of I itself within the scene? If you include I in the painting, then I is a painting that depicts itself, so it should not be depicted in I after all. But if you don’t include I in the painting, then I is a painting that does not depict itself, so it should have been included. Either way, you can’t create a painting that depicts things exactly as described.
The paradox of the impossible painting is distinct from both the Liar paradox and the Russell paradox, since it involves depiction rather than truth or set-membership. But it has features in common with each. Most obviously, circularity plays a central role in all three paradoxes: the Liar paradox involves sentences that says something about themselves, the Russell paradox involves sets that are members of themselves, and the paradox of the impossible painting involves paintings that depict themselves.
“Who knew oats could be so deep?”
Nevertheless, the paradox of the impossible painting has features not shared by the Liar paradox, and other features not shared by the Russell paradox. First, the Liar paradox involves a sentence that clearly exists (and is grammatical, etc.) that must be accounted for, while the Russell paradox can be seen in different terms, as a sort of proof that the Russell set R just doesn’t exist, and that we need to revise (S1) accordingly. The proper response regarding the paradox of the impossible painting is more like the latter – we are not tempted to think that the paradoxical painting does or could exist, but instead conclude that there is something wrong with (D1).
There is another sense, however, in which the paradox of the impossible painting is more like the Liar paradox than the Russell paradox. The Liar paradox arguably arises because of circularity of reference: the Liar sentence refers to, or ‘picks out’, itself. And the paradox of the impossible painting arises because of circularity of depiction – that is, paintings that depict, or ‘pick out’, themselves. Reference and depiction are different, but, insofar as they are both ways of ‘picking out’, while set-theoretic membership is not, suggests that, in this respect at least, the paradox of the impossible painting has more in common with the Liar paradox than with the Russell paradox.
Thus, the paradox of the impossible painting ‘lies between’, or is a sort of hybrid of, the Liar paradox and the Russell paradox, with some features in common with the former and others in common with the latter. As a result, studying this puzzle further seems likely to reward us with deeper insights into these two much older and more well-known conundra. Who knew oats could be so deep?
The post The impossible painting appeared first on OUPblog.
By: Sara Pinotti,
on 9/7/2014
Why should you study paradoxes? The easiest way to answer this question is with a story:
In 2002 I was attending a conference on self-reference in Copenhagen, Denmark. During one of the breaks I got a chance to chat with Raymond Smullyan, who is amongst other things an accomplished magician, a distinguished mathematical logician, and perhaps the most well-known popularizer of `Knight and Knave’ (K&K) puzzles.
K&K puzzles involve an imaginary island populated by two tribes: the Knights and the Knaves. Knights always tell the truth, and Knaves always lie (further, members of both tribes are forbidden to engage in activities that might lead to paradoxes or situations that break these rules). Other than their linguistic behavior, there is nothing that distinguishes Knights from Knaves.
Typically, K&K puzzles involve trying to answer questions based on assertions made by, or questions answered by, an inhabitant of the island. For example, a classic K&K puzzle involves meeting an islander at a fork in the road, where one path leads to riches and success and the other leads to pain and ruin. You are allowed to ask the islander one question, after which you must pick a path. Not knowing to which tribe the islander belongs, and hence whether she will lie or tell the truth, what question should you ask?
(Answer: You should ask “Which path would someone from the other tribe say was the one leading to riches and success?”, and then take the path not indicated by the islander).
Back to Copenhagen in 2002: Seizing my chance, I challenged Smullyan with the following K&K puzzle, of my own devising:
There is a nightclub on the island of Knights and Knaves, known as the Prime Club. The Prime Club has one strict rule: the number of occupants in the club must be a prime number at all times.
The Prime Club also has strict bouncers (who stand outside the doors and do not count as occupants) enforcing this rule. In addition, a strange tradition has become customary at the Prime Club: Every so often the occupants form a conga line, and sing a song. The first lyric of the song is:
“At least one of us in the club is a Knave.”
and is sung by the first person in the line. The second lyric of the song is:
“At least two of us in the club are Knaves.”
and is sung by the second person in the line. The third person (if there is one) sings:
“At least three of us in the club are Knaves.”
And so on down the line, until everyone has sung a verse.
One day you walk by the club, and hear the song being sung. How many people are in the club?
Smullyan’s immediate response to this puzzle was something like “That can’t be solved – there isn’t enough information”. But he then stood alone in the corner of the reception area for about five minutes, thinking, before returning to confidently (and correctly, of course) answer “Two!”
I won’t spoil things by giving away the solution – I’ll leave that mystery for interested readers to solve on their own. (Hint: if the song is sung with any other prime number of islanders in the club, a paradox results!) I will note that the song is equivalent to a more formal construction involving a list of sentences of the form:
At least one of sentences S1 – Sn is false.
At least two of sentences S1 – Sn is false.
————————————————
At least n of sentences S1 – Sn is false.
The point of this story isn’t to brag about having stumped a famous logician (even for a mere five minutes), although I admit that this episode (not only stumping Smullyan, but meeting him in the first place) is still one of the highlights of my academic career.
Instead, the story, and the puzzle at the center of it, illustrates the reasons why I find paradoxes so fascinating and worthy of serious intellectual effort. The standard story regarding why paradoxes are so important is that, although they are sometimes silly in-and-of-themselves, paradoxes indicate that there is something deeply flawed in our understanding of some basic philosophical notion (truth, in the case of the semantic paradoxes linked to K&K puzzles).
Another reason for their popularity is that they are a lot of fun. Both of these are really good reasons for thinking deeply about paradoxes. But neither is the real reason why I find them so fascinating. The real reason I find paradoxes so captivating is that they are much more mathematically complicated, and as a result much more mathematically interesting, than standard accounts (which typically equate paradoxes with the presence of some sort of circularity) might have you believe.
The Prime Club puzzle demonstrates that whether a particular collection of sentences is or is not paradoxical can depend on all sorts of surprising mathematical properties, such as whether there is an even or odd number of sentences in the collection, or whether the number of sentences in the collection is prime or composite, or all sorts of even weirder and more surprising conditions.
Other examples demonstrate that whether a construction (or, equivalently, a K&K story) is paradoxical can depend on whether the referential relation involved in the construction (i.e. the relation that holds between two sentences if one refers to the other) is symmetric, or is transitive.
The paradoxicality of still another type of construction, involving infinitely many sentences, depends on whether cofinitely many of the sentences each refer to cofinitely many of the other sentences in the construction (a set is cofinite if its complement is finite). And this only scratches the surface!
The more I think about and work on paradoxes, the more I marvel at how complicated the mathematical conditions for generating paradoxes are: it takes a lot more than the mere presence of circularity to generate a mathematical or semantic paradox, and stating exactly what is minimally required is still too difficult a question to answer precisely. And that’s why I work on paradoxes: their surprising mathematical complexity and mathematical beauty. Fortunately for me, there is still a lot of work remains to be done, and a lot of complexity and beauty remaining to be discovered.
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