A calculus of the absurd

Chapter 20 High-level logic

This chapter could also have been titled “the logic you really must know to do mathematics.”

20.1 Introduction

At first sight, logic can be confusing. A lot of this is because it can be quite abstract (well, at least it can feel abstract compared to things such as algebra 124124 Not abstract algebra!). The key thing (in my view) to understand is that logic is well-defined; there is a strict set of rules according to which logical expressions can be manipulated, much like there is with "normal" algebra.

For example, consider this well-known fact about any \(x \in \mathbb {R}\) and addition

\begin{equation} x + 0 = x \end{equation}

There are some analogous rules for logical expressions. However, instead of a whole series of numbers, in logic every value is in the set \(\mathbb {B} = \{T, F\}\) - i.e. it is either “true” or “false”. We can define an "operator" (i.e. something which takes two values and outputs a new one, just like the familiar addition operator). For example, let us create a new operator \(\land \). We can define the result of \(A \land B\) by considering what happens in every case

  • • If both values are true, i.e. \(T \land T\), then the output is also true.

  • • If one value is true and the other is false, i.e. \(T \land F\) or \(F \land T\) then the output is false.

  • • In the other case (i.e. \(F \land F\)) then the output is false.

Based on this definition (which corresponds to a logical operator known as “logical and”, so named because the output is true whenever both inputs are \(T\) and false in every other case) we can determine a similar relationship. Specifically,

\begin{equation} T \land A \equiv A \end{equation}

Hopefully this makes intuitive sense, but first we need to define \(A\)! It’s a variable, just like \(x\) was above - i.e. it can be anything we want, so long as it is a boolean (whereas \(x\) can be anything we want, so long as it is a real number). If \(A\) is true, then as \(T \land T = T\) \(T \land A = A\) in this case, and if \(A\) is false, then as \(T \land F = F\) for this case \(T \land A = A\).

This is logic which is generally useful in other areas of mathematics.