Before looking at the person-less variant of the Bernedete paradox, lets review the original: Imagine that Alice is walking towards a point – call it A – and will continue walking past A unless something prevents her from progressing further.
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.
For many months now this column has been examining logical/mathematical paradoxes. Strictly speaking, a paradox is a kind of argument. In literary theory, some sentences are also called paradoxes, but the meaning of the term is significantly different.
Imagine that we have a black and white monitor, a black and white camera, and a computer. We hook up the camera and monitor to the computer, and we write a program where, for some medium-ish shade of grey G.
One of the most famous, and most widely discussed, paradoxes is the Liar paradox. The Liar sentence is true if and only if it is false, and thus can be neither (unless it can be both). The variants of the Liar that I want to consider in this instalment arise by taking the implicit temporal aspect of the word “is” in the Liar paradox seriously.
A theory is inconsistent if we can prove a contradiction using basic logic and the principles of that theory. Consistency is a much weaker condition that truth: if a theory T is true, then T consistent, since a true theory only allows us to prove true claims, and contradictions are not true. There are, however, infinitely many different consistent theories that we can construct.
Imagine that you are an extremely talented, and extremely ambitious, shepherd, and an equally talented and equally ambitious carpenter. You decide that you want to explore what enclosures, or fences, you can build, and which groups of objects, or flocks, you can shepherd around so that they are collected together inside one of these fences.
The Liar paradox is often informally described in terms of someone uttering the sentence: I am lying right now. If we equate lying with merely uttering a falsehood, then this is (roughly speaking) equivalent to a somewhat more formal, more precise version of the paradox that arises by considering a sentence like: “This sentence is false”.
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”?
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).