# The (mathematical) logic behind the scenes

- Autore:
- Simona Riva

## 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.

*counterexample*. Discover more about counterexamples in the next section of this book.