Lesson 7.4 — Proof by Cases

When an OR appears in the premises, prove the conclusion in each branch. If both succeed, the conclusion holds overall.

The Setup

What do you do when one of your premises is an OR? You don't know which side of the OR is true — only that at least one of them is. You can't apply Modus Ponens to an OR directly.

The trick: split into cases. Assume one side and derive the conclusion. Assume the other side and derive the conclusion. If you get the same conclusion in both branches, the conclusion holds regardless of which side of the OR was true.

The Shape

Given:

• P ∨ Q (we know at least one of these is true)
• P → R (if P, then R)
• Q → R (if Q, then R)

Goal: prove R.

The proof has two branches.

• Case 1: Assume P. From P → R and P, by Modus Ponens, R.
• Case 2: Assume Q. From Q → R and Q, by Modus Ponens, R.

Both cases reach R. The OR doesn't matter — whichever side was true, R follows. So R.

A Kid-Sized Example, Step by Step

Saturday morning logic:

• Premise 1: It's Saturday OR Sunday. (S ∨ U)
• Premise 2: If Saturday, no school. (S → N)
• Premise 3: If Sunday, no school. (U → N)
• Goal: no school. (N)

Click through the proof.

The key idea: an OR in the premises doesn't immediately let you derive anything — you can't apply MP to an OR. The case-split is the move that lets you USE the OR. You handle each branch, and if both branches give you the same conclusion, the OR's uncertainty doesn't matter.

Why It Works

An OR statement says: at least one of these is true. So in reality, ONE of the cases is the actual case (or both). You don't know which. But you don't need to — if your conclusion follows in EVERY case, it follows in the actual case (whichever that is).

The logic: if P → R and Q → R, then (P ∨ Q) → R. This pattern is called case analysis or disjunction elimination.

When to Use It

1. Whenever an OR is a load-bearing premise — it's the only way to use it.
2. When proving "X holds in all situations" and the situations split into a few possibilities.
3. In math: "the number is even or odd; here's what follows in each case."

Real proofs often have three or four cases, not just two. Same technique scales — split into every possible branch, prove the conclusion in each, conclude overall.

Tricky Bits

1. Both cases need the same conclusion. If Case 1 gives you R but Case 2 gives you S, you can only conclude R ∨ S — not R, not S. Make sure both branches reach the same target.

2. Exhaust the possibilities. If your OR has three branches (P ∨ Q ∨ S), you need three cases. If you skip one, the proof has a hole.

3. Cases can be combined with other techniques. A case might itself involve a proof by contradiction or another inference chain. Cases are a framing, not a single rule.

A Bigger Example

The classical math proof "for every integer n, n² + n is even" uses cases:

• Case 1: n is even. Then n² is even, and even + even = even.
• Case 2: n is odd. Then n² is odd, and odd + odd = even.

Both cases give "even." Therefore n² + n is always even.

Notice the OR is implicit: "every integer is either even or odd." That premise is so familiar mathematicians don't usually write it down — but it's the load-bearing OR that justifies the case split.

Recap

• Proof by cases: when an OR appears in the premises, prove the conclusion separately in each branch.
• If you reach the same conclusion in every case, it holds regardless of which side of the OR was true.
• Both branches must reach the SAME target.
• Exhaust the possibilities — every branch of the OR needs its own case.
• Same building blocks (MP, MT, contradiction, etc.) — different framing.

That's the Sentential Logic Curriculum

You now have everything in sentential logic: the four words, compounds, parens, arguments, fallacies, symbols, and three proof techniques. You can read a logic textbook, follow a math proof, spot a fallacy in everyday speech, and write your own arguments precisely.

If you want to go deeper, optional tracks branch out from here: predicate logic (objects, properties, "all" and "some"), modal logic (necessity, possibility), and informal reasoning (bad arguments in the wild). Those build on this foundation but aren't strictly necessary — the sentential half is the workhorse.