While most of you probably don’t believe in Santa Claus (and some of you of course never did!), you might not be aware that Santa Claus isn’t just imaginary, he is impossible! In order to show that the very concept of Santa Claus is riddled with incoherence, we first need to consult the canonical sources to determine what properties and powers this mystical man in red is supposed to have.
According to philosophical lore many sentences are self-evident. A self-evident sentence wears its semantic status on its sleeve: a self-evident truth is a true sentence whose truth strikes us immediately, without the need for any argument or evidence, once we understand what the sentence means.
In a 1929 lecture, Martin Heidegger argued that the following claim is true: Nothing nothings. In German: “Das Nichts nichtet”. Years later Rudolph Carnap ridiculed this statement as the worst sort of meaningless metaphysical nonsense in an essay titled “Overcoming of Metaphysics Through Logical Analysis of Language”. But is this positivistic attitude reasonable?
Imagine that, on a Tuesday night, shortly before going to bed one night, your roommate says “I promise to only utter truths tomorrow.” The next day, your roommate spends the entire day uttering unproblematic truths like: 1 + 1 = 2.
A ‘Liar cycle’ is a finite sequence of sentences where each sentence in the sequence except the last says that the next sentence is false, and where the final sentence in the sequence says that the first sentence is false.
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.
The Liar paradox arises when we consider the following declarative sentence: This sentence is false. Given some initially intuitive platitudes about truth, the Liar sentence is true if and only if it is false. Thus, the Liar sentence can’t be true, and can’t be false, violating out intuition that all declarative sentences are either true or false (and not both). There are many variants of the Liar paradox. For example, we can formulate relatively straightforward examples of interrogative Liar paradoxes, such as the following Liar question: Is the answer to this question “no”?
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!
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).
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.
One of the central tasks when reading a mystery novel (or sitting on a jury, etc.) 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.
Early twentieth century packaging for Quaker Oats depicted the eponymous Quaker holding a package of the oats, which, in turn, depicted the Quaker holding a package of the oats, which itself depicted the Quaker holding a package of the oats, ad infinitum. It inspired a generation of philosophers.
Why 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 spoke with Raymond Smullyan; a mathematical logician and renowned author of ‘Knights and Knaves’ (K&K) puzzles.