1.4. Propositional Logic

So let us begin our discussion on propositional logic. So we first define what is a proposition. So informally, a proposition is a declarative statement which is either true or false. But it cannot take both the values simultaneously that means it has to be either true or it has to be either false. For instance consider the statement that New Delhi is the capital of India. It is a declarative statement because it is declaring something about the city called New Delhi. It is declaring that the city New Delhi is the capital of India or not.

A proposition is a statement that has a clear truth value, meaning it can either be true or false, but not both. For example, 'New Delhi is the capital of India' is a proposition because it can be verified as true. In contrast, a statement like 'X + 2 = 4' is not a proposition until the value of X is known, as it could be true or false depending on X's value.

Think of a proposition like a light switch: it can only be ON (true) or OFF (false). Just like you can’t have the same light switch being both ON and OFF at the same time, a statement can’t be both true and false.

Now let us next define what we call as propositional variables and these propositional variables are typically denoted by lower cap letters. So for instance p, q, r etc. So, what is the propositional variable? It is a variable which represents an arbitrary proposition. That means it is a placeholder to store an arbitrary proposition and the truth value of this propositional variable depends upon the exact proposition which we assign to these variables p, q, r.

Propositional variables, such as p, q, and r, act as placeholders for different propositions. The truth value of these variables will change based on the specific proposition assigned to them. For example, if we say p represents 'It is raining,' and we confirm that it is indeed raining, then p is true.

Imagine propositional variables like boxes labeled with letters. Each box can contain a true or false statement. Depending on what you put in the box, the truth of the statement changes, just like what you might find inside a gift box on your birthday.

Now let us next define compound propositions. So 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 combines multiple simpler propositions into one by using logical operators like AND, OR, and NOT. For example, if p means 'It is raining' and q means 'It is cold,' then the compound proposition 'It is raining AND it is cold' represents both conditions together.

Think of a compound proposition like a smoothie. Just as you combine different fruits to create a mix, you combine different propositions using logical operators to form a new statement that incorporates all of them.

The simplest form of the logical operator is the ¬ operator, which is an unary operator because it operates on a single variable. The truth table for this negation operator is as follows. If the variable p is true, then the ¬ of p will be false; whereas if p is false, then ¬p will be true.

Logical operators perform operations on propositions to create new logical statements. The NOT operator (¬) negates the truth value of a proposition. If we say that p is true (like 'It is sunny'), then ¬p (not p) is false (it's not sunny). Similarly, the AND and OR operators combine the truth values of two propositions in specific ways.

Think of the NOT operator as a light switch. If the switch (the proposition) is ON (true), then turning it OFF (negation) is equivalent to saying it is not ON (false). If you flip a switch that was OFF to ON, that’s like changing the truth value.

We define another logical operator which is called as the conjunction and it is also called as AND, denoted by (˄). This operator is a binary operator because it operates on two propositional variables. The conjunction p and q is true only when both p and q are true. The next logical operator is the disjunction operator, also called the OR operator, denoted by (⋁). This operator is defined to be true when at least one of p or q is true.

The AND (˄) operator requires both propositions to be true for the compound proposition to be true, while the OR (⋁) operator requires just one of the propositions to be true. For instance, if p is 'It is raining' and q is 'It is a weekend', 'It is raining AND a weekend' is true only if both are true. In contrast, 'It is raining OR a weekend' is true if at least one of them is true.

Think of playing a game with friends: you can only win (AND) if everyone is playing and cooperating. However, if you just need one person to play with you to enjoy (OR), then you can still have fun even if not everyone is present.

Let us define another logical operator, which is the conditional statement, also called as if-then statement, denoted by p → q. The truth table for p implies q shows that it is false only when p is true and q is false.

The conditional operator (→) expresses a relationship of implication between two propositions. 'If p is true, then q is true' is defined to be false only when the first proposition is true, but the second is false. For example, if 'It is raining' (p) implies 'the ground is wet' (q), this implication is false only when it is raining but the ground isn't wet.

Think of the condition ‘If it rains, then I’ll take an umbrella.’ This statement only fails (is false) if it rains, but you forget your umbrella at home. Otherwise, if it doesn't rain or you take your umbrella without rain, the statement holds true.

Now, the question is why p implies q denotes q is necessary for p and why p implies q is considered as p only if q. The statement p → q, along with ¬q → ¬p, is logically equivalent.

In logical reasoning, p implies q can also be interpreted to mean that q must be true for p to be true. This establishes a necessity relationship. Additionally, expressing 'p only if q' similarly indicates that p cannot be true without q being true. The equivalence means the two ways of stating the relationship maintain the same truth conditions.

Consider a club that requires a membership (p) and good conduct (q) to let you in. You can’t enter the club (p true) if you haven’t behaved well (q true). Hence, having good conduct is necessary for entry into the club.

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