Propositional Calculus and Boolean Algebra

Arithmetic expressions and equations

Since we’re familiar with real numbers, we will start with the following arithmetic equality:

$$\begin{equation} 3 \times (4 + 5) = 3 \times 4 + 3 \times 5. \end{equation}$$

The expression on the left-hand side has three numbers: $3, 4$, and $5$. Furthermore, we have used two arithmetic operations: addition($+$) and multiplication($\times$). These operations allow us to combine two numbers to get another number as a result. The famous BODMASBODMAS stands for brackets, order, division, multiplication, addition, and subtraction. It specifies the precedence of the arithmetic operations. rule tells us that to evaluate the left-hand side, we must first add $4$ with $5$ and then multiply $3$ by the result. The final calculation yields another number, $27$, as the answer.

The calculation on the right-hand side is also dictated by rules of arithmetic (including BODMAS). It also yields the same result, therefore justifying the equality.

Equation (1) is just an example of the distributive law for real numbers. In general, it states that:

$$\begin{equation} a\times (b + c) = a \times b + a \times c. \end{equation}$$

Equation (2) is true for any real numbers $a, b,$ and $c$.

The distributive law (and other laws) allows us to manipulate algebraic equations (over real numbers).

Logical expressions and equivalences

Just as arithmetic operations like addition, subtraction, and multiplication allow us to combine numbers to create other numbers, logical operations allow us to combine propositions to create new propositions called compound propositions. Let’s start with an illustrative example. Consider the following statement:

$E_1$: If Eddie gets a bonus in January, she will buy a pair of sunglasses and a new cellphone.

In the above statement $E_1$, we can readily identify three atomic statements. Let’s make a list of them:

1. $J:$ Eddie gets a bonus in January.
2. $S:$ Eddie buys a pair of sunglasses (in January).
3. $C:$ Eddie buys a new cellphone (in January).

The truth value of $E_1$ depends on the truth values of $J, S$, and $C$.