Definition:Direct Proof

Definition

A direct proof is an argument form which establishes the truth of a conclusion by assuming that the premises are true and demonstrating that the conclusion is necessarily true as a consequence.

Examples

Greater Than

Let it be assumed that:

the greater than relation is transitive.

the two premises $12 > 10$ and $10 > 8$ both hold.

Then the following is a direct proof that $12 > 8$:

\(\ds 12\)

\(>\)

\(\ds 10\)

by hypothesis

\(\ds 10\)

\(>\)

\(\ds 8\)

by hypothesis

\(\ds \leadsto \ \ \)

\(\ds 12\)

\(>\)

\(\ds 8\)

greater than is transitive by hypothesis

Also see

• Modus Ponendo Ponens

• Results about direct proofs can be found here.

Sources

• 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.5$: Theorems and Proofs
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): direct proof
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): direct proof