1.4.3. Compound Propositions

Now let us next define compound propositions. A compound proposition is a larger proposition or a bigger proposition, which is obtained by combining many propositions using what we call as logical operators.

A compound proposition is formed by taking two or more simple propositions (which can be true or false) and joining them together through logical operators. This results in a new proposition whose truth can be assessed based on the truth values of the component propositions and the logical operators used.

Think of constructing a sentence using smaller phrases. For example, if you say, 'It is raining' (proposition A) and 'I will take an umbrella' (proposition B), combining these two with a logical operator to say 'It is raining AND I will take an umbrella' creates a compound proposition that expresses a new idea.

So the simplest form of the logical operator is the ¬ operator, which is a unary operator because it operates on a single variable and the truth table or the truth assignment for this negation operator is as follows. So imagine p is a variable propositional variable and this propositional variable can take two values either true or false.

The ¬ operator, also known as the negation operator, reverses the truth value of the propositional variable it applies to. For example, if p is true (T), then ¬p is false (F), and if p is false (F), then ¬p is true (T). This is an important concept as it allows us to manipulate the truth values of propositions.

Imagine you have a light switch. If the switch is 'on' (true), flipping it to 'off' (negation) means it's now 'off' (false), and vice versa. This simple operation of switching between two states is similar to how the negation operator works on propositional variables.

Now we define another logical operator which is called as the conjunction and it is also called as AND, logical AND. We denote this operator by this notation (˄) again in some books they used a notation p dot q. This is a binary operator because it operates on two propositional variables.

The conjunction operator (AND) produces a true result only when both propositions involved are true. For example, if p is 'I will go to school' and q is 'It is sunny', the conjunction (p ˄ q) is true only if both are true; otherwise, it is false. This operator helps combine the truth values of two different conditions.

Consider making a cake. The statement 'I will bake a cake if I have eggs AND I have flour' represents a conjunction. Only if both conditions are met (you have both eggs and flour) can you successfully bake the cake.

The next logical operator is the disjunction operator which is also called as OR operator denoted by this notation (⋁). It is defined as follows: If any of the variables p or q is true then the disjunction of p and q is defined to be true. But if both p and q is false then disjunction is defined to be false.

Disjunction (OR) allows for flexibility in truth values. The resulting proposition is true if at least one of the propositions (p or q) is true. So, if p states 'It is raining' and q states 'It is Monday', then p ⋁ q would be true if either of these conditions is met.

Imagine you are deciding whether to go outside based on the weather and the day of the week. If you can go outside when 'It is raining' or 'It is the weekend', your decision is true (you can go outside) if at least one of these conditions is true.

Now let us define another logical operator, which is the conditional statement. This is also called as if then statement and we use this notation p → q. The truth table for p implies q is defined as this.

A conditional statement (if p then q) indicates that if the first proposition (p) is true, then the second proposition (q) must also be true for the entire statement to be considered true. It is only false when the first proposition is true while the second is false.

Consider the statement: 'If it rains (p), then I will take an umbrella (q)'. The statement is considered true unless it rains and you do not take an umbrella. This illustrates the nature of conditions and consequences.

--