Lesson 7.2 — Direct Proof

Chain inferences from the premises to the conclusion. One step at a time.

The Shape

A direct proof walks forward. You start with the premises and apply inference rules — Modus Ponens, Modus Tollens, the others — until you reach the conclusion. Each step lands on a new line, justified by which earlier lines it came from and which rule was used.

The pattern:

1. Write down the premises.
2. Write down what you're trying to prove (the goal).
3. Apply a rule to some earlier line(s) to derive a new line.
4. Keep going until the goal appears on a line.

That's it. No tricks. Just a forward walk.

A Simple Worked Example

Suppose you have these premises and want to prove R:

Reading the premises: P is true. P implies Q. Q implies R. So R should follow.

Step 4: from lines 3 and 1, Modus Ponens gives us Q.

Step 5: from lines 4 and 2, Modus Ponens gives us R. Done.

Two steps. Each one a single MP application. Clean.

Build a Proof Yourself

Different setup. Premises and goal below. Pick the right step at each stage.

What Makes a Proof Direct

Two things distinguish direct proof from the other techniques (contradiction in 7.3, cases in 7.4):

1. It walks forward. You start at the premises and move toward the conclusion. No assumptions are added; you only use what's given.
2. Each step is a single rule application. No clever moves. Just MP, MT, De Morgan, or another rule, applied to earlier lines.

Direct proof is the workhorse. Most proofs you'll ever read or write are direct proofs.

How Long Is Too Long?

A direct proof can be 2 steps or 200. For sentential logic exercises, most are between 2 and 8 steps. For real mathematical proofs (geometry, number theory, etc.), they can be dozens of pages.

The length doesn't matter. What matters is that each step earns its place. Every line follows from the rules and earlier lines, with the justification written down.

Practice (on paper)

Prove each goal given the premises. Write out the steps with justifications.

1. Premises: P → Q, P. Goal: Q. (Single step — Modus Ponens.)
2. Premises: P → Q, ¬Q. Goal: ¬P. (Single step — Modus Tollens.)
3. Premises: P → Q, Q → R, R → S, P. Goal: S. (Three MP applications.)
4. Premises: P → Q, P, Q → ¬R. Goal: ¬R. (Two MP applications.)

Work them out. The first two are warmups. The third and fourth are real chains.

Recap

• Direct proof walks forward from premises to conclusion.
• Each step applies one inference rule to earlier lines.
• Each step gets a justification (which rule, which lines).
• The conclusion lands as the last line. Done.
• This is the workhorse technique — most proofs you'll write are direct.