Proof by Deduction Examples

Is the following situation an example of deductive reasoning? Why or why not?

The area of any circle is given by the formula A = πr². Therefore, the area of a circle with a radius of 5 cm is 25π cm².

Deductive reasoning always starts with a general principle, then applies that principle to a specific example. Start broad, then get detailed. Is that what's going on here?

Right out of the gate, we're given the area formula for any circle. That sounds pretty general to us. We could apply that formula to any specific circle in the known universe, and it'll still work.

But we're applying it to one particular circle with a radius of 5 cm. If we plug that into our general formula, we'll get A = π(5)² = 25π cm².

Hmm…we put the general area formula to work on an actual circle. Yeah, that's definitely deduction.

Is the following situation an example of deductive reasoning? Why or why not?

Marceline carefully measures the internal angles of 500 different rectangles, and she finds that every time, the angles add up to 360°. Therefore, she concludes that all rectangles must have internal angles that add up to 360°.

Our pal Marceline started out with a bunch of very specific examples: she measured the angles of 500 different rectangles. Hopefully she took a lunch break in there somewhere.

After that, she used those specific examples to come up with a general theory: all rectangles must have angles that add up to 360°. Is that deductive reasoning?

Nope, it isn't. She did things backwards: specific to general. That doesn't mean she's wrong—in fact, she's totally right—but this isn't an example of deductive reasoning. It's actually inductive reasoning, which is its own special thing.

 What general theory or idea can we use to deductively prove Statement #3 in the following formal proof?

Given: A = B and X = Y
Prove: AX = BY

Since deductive proofs go top-down, our general theory should get us to the specific statement AX = BX.

We want the multiplication property here, which says that if A = B, we can multiply the same number by both sides of the equation without changing its truth. In this specific case, we're multiplying by X.

Solid deduction, Sherlock.