Lesson 7.1 — What's a Proof?

A proof shows the path from premises to conclusion, step by step, so anyone can walk it.

"I Think" vs. "I Can Show"

There's a big difference between believing something is true and being able to demonstrate it. A proof is the demonstration.

"I think the streets are wet because it rained."

That's belief. Maybe a good guess. But it isn't a proof.

"Premise 1: If it rained, the streets are wet. Premise 2: It rained. Therefore (by Modus Ponens): the streets are wet."

That's a proof — small, but real. Two premises, one inference rule, one conclusion. Anyone who accepts the premises has to accept the conclusion.

What a Proof Needs

1. Premises — the starting sentences. You're allowed to assume these.
2. A conclusion — the sentence you're trying to reach.
3. Steps — each one follows from things already established (premises, or earlier steps).
4. A justification for each step — which earlier line(s) does it come from, and which rule (MP, MT, etc.) was applied?

If any step doesn't follow, the proof breaks. Every step has to earn its place.

A Tiny Proof

Read it from top to bottom. The first three lines are given (premises). Line 4 follows from lines 3 and 1 by Modus Ponens. Line 5 follows from lines 4 and 2, also by Modus Ponens. The chain reaches R. The proof is complete.

The little justification on each line is what makes it a proof and not just a list of sentences. Anyone can check it.

The Reader's Job

When someone presents you with a proof, your job is to check each step. Does line 4 really follow from the rules and the earlier lines? Does line 5? If yes, you have to accept the conclusion — even if you didn't see it coming.

This is the social contract of logic: if the premises hold and if each step is valid, the conclusion is forced.

What Counts as Obvious

Some steps are too small to write down. "P, therefore P" — you wouldn't write a justification for that. The judgment of how small is too small is a craft skill, not a rule.

For learning, write every step. Even the small ones. As you get fluent, you can skip the trivial moves and write only the interesting ones.

Is It a Proof?

Quick recognition drill. For each item, pick whether it qualifies as a proof.

What's Coming

The next three lessons cover the three main proof techniques:

1. Direct proof (7.2) — chain inferences from the premises to the conclusion.
2. Proof by contradiction (7.3) — assume the opposite of what you want to show, derive a contradiction, then conclude what you wanted.
3. Proof by cases (7.4) — when an OR appears in the premises, prove your conclusion holds in each branch.

All three use the inference rules you already know (MP, MT, De Morgan, etc.). The technique is the framing.

Recap

• A proof is the path from premises to conclusion, written so anyone can walk it.
• Every step must follow from earlier lines (premises or prior steps).
• Each step earns its place; each gets a justification (which rule, which lines).
• "I think" becomes "I can show." That's what a proof gives you.