Exercise 2.36

Exercise 2.36#


Let’s build a list of example statements (including those that were already made) that may be made in this language. This is part of specifying “interpretation” as defined in forall x: 30.5 Interpretations:

  • \(Q\): n is prime

  • \(R\): n is even

  • \(S\): \(0 \leq n\)

  • \(T\): \(n = 15\)

Clearly \(S \leq P\) for all \(P\) in \(n \in ℕ\), but \(T\) is only true for a single \(P\). If we use “and” as our monoidal product, we must pick something that is always true (like \(S\)) as our monoidal unit. If we had chosen “or” then we’d have to come up with a statement that is always false.

An “and” operation is clearly already associative and commutative. It’s also order-preserving, which we can use in the following proof (in a style suggested by Example 1.123):