Logic

What is logic?

Logic is the formal study of what makes an argument valid — that is, when a conclusion truly, necessarily follows from its starting statements, rather than just seeming to.

A valid argument: modus ponens

Consider this argument: "If it is raining, the ground is wet. It is raining. Therefore, the ground is wet." This pattern — "if P then Q; P; therefore Q" — is called modus ponens, and it is always valid: whenever both starting statements are true, the conclusion is guaranteed to be true too.

Here's why, checked completely rather than just accepted: "if P then Q" is only false in the single case where P is true and Q is false — in every other case, "if P then Q" is true. So if we're told "if P then Q" is true, and also told P is true, the only case that's ruled out is "P true, Q false" — leaving only the case "P true, Q true." So Q must be true. This covers every possible situation, not just the raining example, which is why the pattern itself — not just this one instance — is valid.

An invalid argument that looks similar: affirming the consequent

Compare that to: "If it is raining, the ground is wet. The ground is wet. Therefore, it is raining." This looks similar, but it's a different pattern — "if P then Q; Q; therefore P" — and it is not valid.

Here's a case that proves it: imagine a sprinkler was left running, with no rain at all. The ground is wet (Q is true), and "if it rains, the ground gets wet" is still true (P then Q holds). But it did not rain (P is false). This single case shows both starting statements can be true while the conclusion is false — which is exactly what "invalid" means. One clear counterexample like this is enough to prove the whole pattern is unreliable, even though many individual instances of it might happen to have a true conclusion anyway.

Why logic matters

Distinguishing a genuinely valid argument from one that merely resembles one is the basis of mathematical proof, and it matters well beyond mathematics too: every computer program's behavior is determined by exactly this kind of if-then reasoning, and recognizing invalid patterns like affirming the consequent helps catch flawed reasoning in everyday arguments, not just formal ones.