1.4.1. Definition of Proposition

Informally, 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.

A proposition is a statement that asserts something and can be classified strictly as true or false. For instance, the statement 'New Delhi is the capital of India' is a declarative statement asserting a specific fact, making it a proposition. The key characteristic of a proposition is that it cannot simultaneously be both true and false.

Think of a light switch. It can either be on (true) or off (false) but cannot be in both positions at the same time. Similar to how a proposition operates, a statement about the light switch must clearly define its state as either true or false.

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 capital of India or not. Indeed, this statement can be either true or false; it cannot be both. In the same way, if I make a statement like 'Bahubali was killed by Katappa,' this is also a declarative statement which can be either true or false and hence is a proposition.

Let's analyze two statements to illustrate propositions. The first is 'New Delhi is the capital of India,' which is a factual statement. Because this is a verifiable fact, it is a valid proposition. The second statement, 'Bahubali was killed by Katappa,' is also a proposition because it can be confirmed as true or false depending on the context of the narrative or movie. Both statements represent clear truths without ambiguity.

Imagine you're playing a true or false game. If I ask, 'Is the Earth flat?' You can respond with true or false clearly. Similarly, the propositions presented here all allow for definitive true or false responses, just like your game questions.

Whereas the statement 'X + 2 is equal to 4' is not a proposition. Well, it is a declarative statement because it is declaring something about X, 2, and 4. But we do not know what is the value of X. Depending upon the value of X this statement can be either true or it can be either false.

This statement lacks the definitive nature of a proposition because it relies on the unknown value of X. Since X could take on numerous values, making the overall truth of the statement dependent on that variable means it cannot be classified as a proposition. Propositions must be clearly true or false without conditions.

Consider a mystery box. You tell your friend, 'The item inside the box is a surprise.' Until they open the box, the truth of the statement remains uncertain. Just like the condition with X where we can't determine truth without additional information, the mystery is not a definitive statement.

Now let us next define what we call as propositional variables. These propositional variables are typically denoted by lower cap letters. For instance, p, q, r etc. A propositional variable is a variable which represents an arbitrary proposition.

Propositional variables are symbols that substitute for specific propositions. They serve as placeholders that represent statements which can be either true or false. The choice of letters such as p or q helps in the mathematical representation of logical arguments and propositions. These variables allow us to abstractly handle logical expressions.

Think of propositional variables like letters in algebra. When you see 'x + 2 = 5,' the letter 'x' stands in for a specific number. Similarly, in logic, 'p' or 'q' can stand in for statements like 'It is raining' or 'I will go out today.' The actual truth of those statements is what the variables represent.

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