Oscar on the island of two truths
Imagine that one day you woke up in a world having another logic underling all our thinking and all our reasoning. That is exactly what happened to Oscar on a murky winter day when he was playing with his magic set, picturing himself as Harry Potter fighting with Voldemort. Suddenly he found himself on an island inhabited by people who had a way of thinking completely different from anything he had ever encountered before. Instead of the two truth values 0 and 1 as in the world he used to live in, there were three: 0, Q and K. For the people of the island the proposition "Oscar is a ten-year-old." wasn't either TRUE or FALSE, but rather Q-TRUE, K-TRUE or FALSE. How exactly are we to understand that, and how did it effect the interactions among the inhabitants of the island? And how was Oscar to explain to the people of the island that he really was ten years old?
The island was ruled by a queen and a king. It is important to stress that the queen was neither inferior nor superior to the king. Rather than as a married couple one should think of the queen and the king as two parallel powers, somewhat like the Queen of the Night and the King Sarastro in Mozart's famous opera The Magic Flute. The queen and the king had their own castle each, each of them had their own court, their own advisers and servants, and most importantly each of them even had their own truth value.
On the island, a proposition p is either FALSE, Q-TRUE or K-TRUE; in each of the cases we say that p has value 0, Q or K, respectively. The queen finds the truth value Q to be superior, while the king values the most the value K. The queen and the king have their opinions on all issues, while other residents typically have their opinions on some issues but not all.
A native A can either have an opinion on a matter (or proposition) p or not. If A has an opinion on p, then exactly one of the following holds:
- A considers p to be FALSE, in which case we say that A assigns value 0 to p;
- A considers p to be Q-TRUE, in which case we say that A assigns value Q to p;
- A considers p to be K-TRUE, in which case we say that A assigns value K to p.
Having an opinion on p, A can be either right or wrong about p. We say that A is right about p if the truth value that A assigns to p equals the value of p. Two residents of the island are said to agree on a sentence p when they both assign the same truth value to it. Moreover, if the queen assigns the value 0 to p, then so does the king, and vice versa. One could argue that the two royalties share the same FALSE but have different TRUEs.
The queen and the king share equal power, and every resident of the island is loyal to one of them. A native A is loyal to the queen if and only if A agrees with the queen on all those issues p on which A has an opinion. Similarly, A is loyal to the king if and only if A agrees with the king on all those issues p on which A has an opinion. Oscar discovered all that soon after his arrival. However, it remained unclear to him for quite some time whether a person could be loyal to both the queen and the king at the same time.
A native that is loyal to the queen is said to believe a sentence p when he assigns the truth value Q to p, and a native that is loyal to the king is said to believe p when she assigns the value K to it. When a native A believes p we also say that A considers p to be true. Finally, we say that A knows p when A believes p and is right about p.
See if you can solve the pair of exercises below. You are most welcome to post your solutions in the comments below.
Exercise 1. Show that the queen is loyal to herself and the king is loyal to himself.
Exercise 2. Show that the queen believes a sentence p when she assigns the truth value Q to p, and that the king believes p when he assigns the value K to it.