Intermediate truth values

I have not written a blog post in a while, so I decided to write up a short observation about truth values in intuitionistic logic which sometimes seems a bit puzzling.

Let $\Omega$ be the set of truth values (in Coq this would be the setoid whose underlying type is $\mathsf{Prop}$ and equality is equivalence $\leftrightarrow$, while in HoTT it is the h-propostions). Call a truth value $p : \Omega$ intermediate if it is neither true nor false, i.e., $p \neq \bot$ and $p \neq \top$. Such a “third” truth value $p$ is proscribed by excluded middle.

The puzzle is to explain how the following two facts fit together:

1. “There is no intermediate truth value” is an intuitionistic theorem.
2. There are models of intuitionistic logic with many truth values.

Constructive gem: double exponentials

In the last constructive gem we studied the exponential $2^\mathbb{N}$ and its isomorphic copies. This time we shall compute the double exponential $2^{2^\mathbb{N}}$ and even write some Haskell code. Continue reading Constructive gem: double exponentials

Constructive gem: juggling exponentials

Constructive gems are usually not about particular results, because all constructive results can be proved classically as well, but rather about the method and the way of thinking. I demonstrate a constructive proof which can be reused in three different settings (set theory, topology, computability) because constructive mathematics has many different interpretations.

Constructive stone: minima of sets of natural numbers

I promise I will post a constructive gem soon. This constructive stone came up as a reaction to the cardinality of finite sets stone. I show that inhabited sets of natural numbers need not have minima, constructively.