Apply Bubble Logic to Two-Column Proofs

By Mark Ryan

You can add bubbles and arrows to a proof solution to show the connections between the statements and the reasons. Although you won’t be asked to do this when you solve a proof, it’s a great way to help you understand how proofs work.

Here’s an example of a bubble proof.

geometry-bubble-proof
Follow the arrows from bubble to bubble.

Note how the bubbles and arrows show how the logic flows through the proof.

In a two-column proof,

  • The idea in the if clause of each reason must come from the statement column somewhere above the reason.
  • The idea in the then clause of each reason must match the idea in the statement on the same line as the reason.

The arrows and bubbles in the figure show how this incredibly important logical structure works.

Here’s another example to show you how a deductive argument all hangs together. The following proof establishes that Clyde the Clydesdale won’t be giving an address at your high school commencement. Here’s the basic argument:

  1. Clyde is a Clydesdale.
  2. Therefore, Clyde is a horse. (Because all Clydesdales are horses.)
  3. Therefore, Clyde can’t talk. (Because horses can’t talk.)
  4. Therefore, Clyde can’t give a commencement address. (Because something that doesn’t talk can’t give a commencement address.)
  5. Therefore, Clyde won’t be giving an address at your high school commencement. (Because something that can’t give a commencement address won’t be giving one at your high school commencement.)

Now take a look at what this argument or proof would look like in the standard two-column geometry proof format with the reasons written in if-then form. When reasons are written this way, you can see how the chain of logic flows.

Given: Clyde is a Clydesdale.

Prove: Clyde won’t be giving an address at your high school commencement.

geometry-logic-chain

Follow the arrows from bubble to bubble. Note again that the idea in the if clause of each reason connects to the same idea in the statement column above the line of the reason; the idea in the then clause of each reason connects to the same idea in the statement column on the same line as the reason.