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# 2: Two-Column Proofs

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No way of thinking or doing, however ancient, can be trusted without proof.
What everybody echoes or in silence passes by as true to-day
may turn out to be falsehood to-morrow, mere smoke of opinion. . .

Henry David Thoreau (1817–1862)), American author Walden

The aim of a proof is to show that a deduction is valid, and it does this by putting together a number of simpler deductions that are already known to be valid. Ultimately, our goal is to teach you to write clear and correct proofs, in English, of claims stated in English. But we will start with the simpler situation of proofs written in the language of Propositional Logic. This has several advantages:

• it allows assertions to be written more concisely, because entire English phrases are abbreviated to a single letter,
• it avoids the difficulties caused by the fact that sentences written in English can be ambiguous, and
• it displays the logical structure of a proof in a way that makes it easier to decide whether or not each step in a proof is valid.

After you are familiar with proofs in this simpler setting, you will employ the same principles to write proofs in English.