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Academic Insights for the Thinking World

• # Author: Roy T Cook

The Yablo Paradox (due to Stephen Yablo and Albert Visser) consists of an infinite sequence of sentences of the following form: S1: For all m > 1, Sm is false. S2: For all m > 2, Sm is false. S3: For all m > 3, Sm is false. : :
: Sn: For all m > n, Sm is false. Sn+1: For all m > n+1, Sm is false. Hence, the nth sentence in the list ‘says’ that all of the sentences below it are false.

The Liar paradox arises via considering the Liar sentence: L: L is not true. and then reasoning in accordance with the: T-schema: Φ is true if and only if what Φ says is the case. Along similar lines, we obtain the Montague paradox (or the paradox of the knower) by considering the following sentence: M: M is not knowable. and then reasoning in accordance with the following two claims: Factivity: If Φ is knowable then what Φ says is the case.

## The illegitimate open-mindedness of arithmetic

We are often told that we should be open-minded. In other words, we should be open to the idea that even our most cherished, most certain, most secure, most well-justified beliefs might be wrong. But this is, in one sense, puzzling.

As regular readers know, I understand paradoxes to be a particular type of argument.

A directed graph is a pair where N is any collection or set of objects (the nodes of the graph) and E is a relation on N (the edges). Intuitively speaking, we can think of a directed graph in terms of a dot-and-arrow diagram, where the nodes are represented as dots, and the edges are represented as arrows.

## Really big numbers

What is the biggest whole number that you can write down or describe uniquely? Well, there isn’t one, if we allow ourselves to idealize a bit. Just write down “1”, then “2”, then… you’ll never find a last one.

## The logic of unreliable narrators

In fiction, an unreliable narrator is a narrator whose credibility is in doubt – in other words, a proper reading of a narrative with an unreliable narrator requires that the audience question the accuracy of the narrator’s representation of the story, and take seriously the idea that what actually happens in the story – what is fictionally true in the narrative – is different from what is being said or shown to them.

The idea that many, if not most, people exhibit physical signs – tells – when they lie is an old idea – one that has been extensively studied by psychologists, and is of obvious practical interest to fields as otherwise disparate as gambling and law enforcement. Some of the tells that indicate someone is lying include:

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.

## Periphrastic puzzles

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.

## Temporal liars

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.

## The consistency of inconsistency claims

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.