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.