Q.Two statements are given below followed by two conclusions numbered

Creating New Statements — The Core Idea

When you first learn logic, it feels like you're just memorising rules. But the real power is this: from a handful of true statements, you can build new true statements without ever checking the real world again. That's what "creating new statements" means — you take what you already know and combine it to produce knowledge you didn't have before.

Think of it like Lego. You have a few basic bricks (simple true statements). By snapping them together in allowed ways, you can build something new — a tower, a bridge — that wasn't there when you started. The rules for snapping are the logical connectives: and, or, not, if-then, if and only if.

The Intuition First

Suppose you know two facts:

It is raining.

The ground is wet.

These are separate statements. But you can create a new statement by joining them:

It is raining and the ground is wet.

That's a new statement. It's true only if both original statements are true. You didn't need to go outside to check — you already knew the parts, so you can deduce the truth of the whole.

Similarly, you can create:

If it is raining, then the ground is wet.

That's also a new statement. Its truth depends on the logical relationship between the two parts, not on whether it's actually raining right now.

The Precise Statement

In formal logic, a statement (or proposition) is a sentence that is either true or false — not both, not neither. "Creating new statements" means applying logical connectives to one or more existing statements to produce a new statement whose truth value is completely determined by the truth values of the original statements.

The five standard connectives are:

Why This Matters

This is the engine of mathematics. You start with axioms (statements you accept as true without proof). Then you create new statements using these rules. If the rules are followed correctly, the new statements are guaranteed true — no experiment needed. That's how a whole subject like geometry or number theory is built from just a few starting points.

A common mistake is to think "creating new statements" means making up random sentences. It doesn't. The new statement must be built only from the given statements and the allowed connectives. You cannot introduce new facts from outside.

A Quick Example

Let $P$ = "The number is even" and $Q$ = "The number is divisible by 3".

From these, you can create:

• $\neg P$: "The number is not even"
• $P \land Q$: "The number is even and divisible by 3" (i.e., divisible by 6)
• $P \lor Q$: "The number is even or divisible by 3"
• $P \to Q$: "If the number is even, then it is divisible by 3" (this is false for 2, true for 6)
• $P \leftrightarrow Q$: "The number is even if and only if it is divisible by 3" (this is false for most numbers)

Each of these is a new statement. You can now reason about them, combine them further, and build more complex truths — all without leaving the page.

The Takeaway

Creating new statements is the process of using logical connectives to build compound statements from simpler ones. The truth of the new statement depends only on the truth of its parts and the connective used. This is the foundation of all deductive reasoning — in mathematics, computer science, and clear thinking.