The (mathematical) logic behind the scenes

Material implication: ⇒

Let's consider the statement , and the statement . We say that implies , and we write to mean that if is true, then also is true. (If is true, is necessarily true). The symbol connects a premise and a conclusion and is very used in proofs, because it's a symbolic way to show deductive reasoning. The statement " implies " is also written "if then " or sometimes " if ". Does this sound complicated? No... let's see a few examples of implications.
  • If you score 68% or more in this problem, then you will pass the exam.
  • Your head will hurt if you bang it against a wall.

Exploring implications in geometry

Given a quadrilateral Q, use the applet below to find out the reciprocal implications between the following statements: a: Q has an obtuse angle. b: Q has three acute angles. c: Q has no right angles. (drag the red points to explore different quadrilaterals)

Which are the implications between statements a, b and c?

Implication is confused by fake guys

Consider this example: We started with a false premise and implied a true conclusion. Now consider this: We started - again - with a false premise, and implied a wrong conclusion. Implication doesn't like false premises. If we start with a false premise, the conclusion obtained by implication can be anything.

Showing why things go wrong

In the example above, we had the following three statements about a quadrilateral Q: a: Q has an obtuse angle. b: Q has three acute angles. c: Q has no right angles. We can say that:
  • a doesn't imply b because a rhombus (that is not a square) has an obtuse angle, but not 3 acute ones.
  • a doesn't imply c because a right trapezoid (that is not a rectangle) contains an obtuse angle, and two right angles.
  • c doesn't imply b because a rhombus (that is not a square) has no right angles, but doesn't have three acute angles.
We explained that implication doesn't hold using a "tool" that in mathematics is named counterexample. Discover more about counterexamples in the next section of this book.