Okay, now discrete gets a little more fun. Conditional statements are “conditional” in the sense that they contain an element whose truth value depends on the truth value of something else. For example, “if Zach is happy, he cackles” is a conditional statement because the cackling depends on the happiness. Now, let’s get discrete:
p: Zach is happy
q: Zach cackles
The formal statement of the above conditional would go:
The book denotes a crapload of ways to say this verbally. I’m gonna go with “if p then q.”
You can readily see how this new symbol is going to mix with the old one. For example, let’s make this statement: If Zach is sad AND there’s no cookie dough THEN Zach cries alone in his room. So,
p: Zach is sad
q: There’s no cookie dough
r: Zach cries alone.
The formal statement goes:
You can think of lots of variations. For example, flip that AND sign to an OR, and you’ve got a situation where I cry if I’m sad OR if there’s no cookie dough OR both.
[table id=5 /]
Mull that one over a bit, as it’s less obvious than the previous stuff.
It may be throwing you off that p can be false and q can be true, and the statement p→q is true. If you’re confused, it’s purely a matter of understanding terms better.
Most importantly, in this framework, calling something “TRUE” is the same as calling it ”NOT FALSE.” In fact, sometimes when I get confused, I prefer to say “FALSE” or “NOT FALSE” as opposed to “FALSE” or “TRUE.”
So, now let’s go back to the table. We have 4 cases:
1) p and q are both true. In this case, p→q is a non-false statement because if I go outside, the sky will be blue. It doesn’t matter that YOU KNOW there’s not a real relation there. We’re only talking about the statement.
2) p is true and q is false. This is the lone case where p→q is false. If I go outside and the sky is not blue, we know (if nothing else) that the statement “If Zach goes outside, then the sky will be blue” is not true.
3) p is false and q is true. This is the one that might screw with you. Think of it like this: If Zach doesn’t go outside and the sky is still blue, it doesn’t render p→q false. How could it? You don’t have a relation for what to do if p is false. So, judging solely from the information available, p→q has not been shown false. Therefore it is not false, a.k.a. true.
I wanna linger on case 3 a little longer. The easy way to remember is that in any conditional, if the premise has a FALSE truth value, then the whole conditional is NOT FALSE. Think of it this way: You have a conditional statement that tells you that the second thing depends on the first thing. If the first thing is false, you don’t have any way to know if the conditional overall is true. For example, if I said “If you meet Zach you will be turned on,” but then you never met me, you don’t know if the overall statement is true or false. Therefore, based on the available information, you must call it not false.
4) p and q are both false. This is similar to case 3. Since the premise is false, and it’s a conditional, you know immediately that the statement overall is not false. But, let’s make sure you understand why. Start with “if Zach goes outside the sky will be blue.” Now, say I don’t go outside AND the sky is red (that is, both p and q are false). That gives you no information about whether the initial statement is true. I didn’t go outside, so it doesn’t matter what color the sky is – the statement only relates to times when I do go outside. So, based on the information available, you must say that p→q is not false.
Got it? I went into all the “WHY?!” stuff because it can be a bit confusing. But, the important thing to remember is that all statements are either true or false. There’s no room for “I dunno…” because “I dunno…” is considered part of “true.” It’s analogous to how in court we say that a defendant is either guilty or not guilty. If he’s guilty, he’s guilty. If he’s not guilty, he’s either actually innocent, or he’s at least not guilty. Those are two different ideas contained in one word, but they’re lumped together because they have the same meaning in regards to whether you go to jail or not.
I’m going to try to stick to the word “true” from now on. But, whenever it’s a little confusing, I’ll switch to “not false” or “non-false,” as I think it’s a little easier on your human brain.
Next section: Converse, Inverse, Contrapositive!