Justifying Proof Lines

Before you build proofs, practice reading them. Given a finished proof with the justifications removed, you can supply each one by matching the line against the rules of Matching a Rule to Whole Lines.

Key idea

To justify a line, find the earlier lines it came from, then name the rule whose pattern takes those lines to this one. Write the rule and the line numbers, like "DS 2, 4". The cited lines must come before the line you are justifying.

Why it works

A justification is a claim that can be checked: "this line follows from those lines by that rule". Writing it forces you to look at the main connectives, which is the habit every proof needs. It also lets someone else check your proof one line at a time without redoing your thinking.

Fill in the three justifications, then Check.

Worked examples

Worked example: Find the source

In the proof above, where does line 4, $p$, come from?

$p$ appears in line 1 (inside the antecedent) and line 2 (a disjunct).
Line 2 is a disjunction, and line 3 denies its other disjunct.

Answer: DS 2, 3

Worked example: The next line

Line 5 is $q \land r$. Justify it.

It is the consequent of line 1.
Line 4 is the antecedent of line 1.

Answer: MP 1, 4

Worked example: One line cited

Line 6 is $r$. Justify it.

$r$ is a conjunct of line 5.

Answer: Simp 5

Common mistake

Citing the right rule with the wrong lines, or citing a line that comes after the one being justified. A line can only use what is already written above it. Before writing the numbers, put your finger on each cited line and check that it really has the shape the rule needs.

When the cited lines are in a different order from the rule's pattern, the justification is still fine: "MP 4, 1" and "MP 1, 4" are the same.

Look at the line's main connective.
Search earlier lines for it as a whole line or a part.
Ask which rule connects them, and check the pattern.
Write the rule and the line numbers.

Sometimes two different justifications are both correct. Line 4 above could not come from anything but DS, but in other proofs a conjunction may appear in two places, or a line may follow by MP from two different pairs. Any justification that really fits the rule is fine; the checker only asks that the one you wrote is correct.

Good proof readers move fast because they ask the same question every time: "where did this come from?" The answer is almost always visible within a line or two above.

Justifying Proof Lines

Worked examples

Practice

Builds on

Leads to

Sources and further reading