3 Conditional Proposition

A conditional proposition is a proposition, but it is special because it involves a condition and a consequent. Let us look at some casual conditional propositions.

``If Jimmy breaks a window, he is punished.''

This is a conditional proposition because it consists a condition and a consequent. The condition is ``Jimmy breaks a window'', which can be true or false. The consequent is ``Jimmy get punished'', which can be true or false, as well.

As a conditional proposition, when is it true, and when is it false?

In our particular example, the conditional proposition is true if and only if the following are true:

• It is always the case that Jimmy breaks a window, and he gets punished.
• It is never the case that Jimmy breaks a window, and he does not get punished.

But what if Jimmy does not break a window? As it turns out, it does not matter!

Functionally, to say that ``if Jimmy breaks a window, he gets punished'' is the same as saying ``(Jimmy does not break a window) or (he gets punished).'' The parantheses are only there to help group the components of the disjunction (``or'').

In the context of mathematics, we can now have some fun. Let's consider the following proposition:

``if $x < y$ , $x \le y$.''

Your first reaction is that this is not a proposition, because its truth value depends on the values of $x$ and $y$. However, a closer examination suggests that the values of $x$ and $y$ do not matter!

Let us consider the alternate form of this proposition:

``not $x < y$ or $x \le y$.''

Or even better,

``$x \ge y$ or $x \le y$.''

Is it true (for real numbers) that at least one of $x \ge y$ or $x \le y$ has to be true? In different module, we'll explore how to say more about $x$ and $y$. For the time being, we'll just say that we know that $x$ and $y$ are real numbers.