 Previously in our lecture series, we've introduced the notion of a conditional statement. In this video, I want to introduce what it means to be a bi-conditional statement for which, as the name seems to suggest, bi here means two. There's two conditionals actually in play here. Now, in order to define what a bi-conditional statement is, I first have to talk about that. Given any conditional statement, there are actually three other related important conditional statements from which we'll construct a bi-conditional statement. Given the conditional, P implies Q, which this is our conditional we start with, three related conditional statements exist. There is the so-called converse of the conditional statement, which is Q implies P. Notice that the converse switches the roles of premise and conclusion. For our original conditional statement, the premise was P, the conclusion was Q. For the converse, the original conclusion Q is now the premise, and the original premise, which was P, is now the conclusion. So the converse just changes the direction of the implication. So if the statement was P implies Q, the converse is Q implies P. Now, related to this is also the notion of an inverse, the inverse statement. The inverse statement keeps the same direction, P implies Q, but now the premise and the conclusion are negated. So if the original statement is P implies Q, its inverse will be not P implies not Q. And then the last one on this list of four is known as the contrapositive. At this moment, this is the least intuitive why you would even consider such a thing. This one reverses the direction, this negates things, but this one does both the contrapositive. The premise and the conclusion reverse roles, and we negate both of them. So if the original statement is P implies Q, then the contrapositive is not Q implies not P. Let's use a specific example to try to illustrate what's going on here. So let's consider the statement, if marijuana is legalized, then drug abuse will increase. So this is our statement P implies Q. P is the primitive, marijuana is legalized, and then the conclusion is the primitive statement Q, drug abuse will increase. So that's what this statement is. And I'm not making the argument that this statement is true or false, although I'm not presenting any data in this video right now, but it does seem like a very reasonable situation that as drugs are more legalized, marijuana, at least with the timing of this video, is one that is being debated in the United States. Some states have it legalized, some don't. Some have it legalized for medicinal purposes, some for recreational purposes, some don't at all. So yeah, this is an open, a current debate in the United States right now. And one of the concerns is that if marijuana is legalized, then people will abuse drugs, marijuana included, but drugs in general will be abused that more than they were. And I mean, it's not too hard to see why this is probably the case. To what extent will that abuse increase? Again, I'm not making any statement about that, but this statement is probably true just from our common sense understanding of what these things mean here. Okay, now consider the converse here. The converse would swap the roles of the conclusion and the premise. So P, which was marijuana is legalized and Q, drug abuse will increase. We're now swapping the roles of this. So now drug abuse increase comes first and marijuana legalization comes second. So the converse switches the roles of this. So you get P implies Q here. And like I said, sorry, the original saying was P implies Q. The converse is Q implies P there. Now this original statement is probably true. And so for the simplicity sake, I'm going to say it's true right here. Again, I'm not trying to quibble on politics right here. It's just this is probably a true statement here. On the other hand, the statement of the converse here is probably false. Okay, so like if we see drug abuse increase in the United States, in the United States, we can't necessarily be like, aha, marijuana was the culprit to that. I mean, it could be, it could be. But I mean, there's other other concerns, right? There's the opioid epidemic, where perhaps doctors and pharmaceutical companies being too liberal with prescribing patients, painkillers of an opioid variety could lead to increased drug abuse that had maybe nothing to do with marijuana. Some people even advocate that legalizing marijuana would reduce drug abuse because while people use marijuana in a medicinal sense, often for pain relief, then we're less likely to prescribe more dangerous drugs like opioids and therefore drug abuse might actually go down. Again, I said this statement earlier is while it's probably true or reasonably true, I'm not actually making a stance on this whole drug debate right here. Just using it as an example here. But in particular, whether this statement is true or false, it's really not, you can argue with it that the converse is a false statement. That is, if drug abuse increases, then there are many factors that could contribute to it other than the legalization of marijuana. Remember, we've talked about this before, for a conditional statement to be true, it means that whenever the premise is true, then the conclusion has to also be true as well. Because there are situations for which the premise can be true, but the conclusion is false, this makes it a false statement. And the reason I include these is that the truth value of these statements, although they're related, a statement in this converse are related to each other, their truth values are not identical. One can be true while the other is in fact false. Now let's compare what the inverse of this statement would look like. Going back to the original statement, P implies Q here, if marijuana is legalized, then drug abuse will increase. The inverse, which just negates the values and keeps the same implication, would be if marijuana is not legalized, then drug abuse will not increase. So that's what the inverse is. We negated the premise, we negated the conclusion. But again, when we look at this, this statement is probably not true. And it's actually not true for the exact same reasons we were considering a moment ago when we looked at the converse. Is there other reasons drug abuse could increase, not related to the legalization of marijuana? There certainly could be. And so it turns out that the converse and the inverse of a statement do not have to be true, even if the statement is true. But of course, if the statement is false, that doesn't mean these are false. They could be true as well. The truth value of the converse and inverse could be different than the truth value of the original statement itself. And in fact, what we're going to see in the future is that the converse and the inverse statement of a statement are in fact logically equivalent, that if this one is true, then so will this one. But if this one is false, then so will this one. The converse and the inverse will have the exact same truth value. But that truth value is not necessarily the same truth value as the original statement itself. With that in mind, that then leads to the very last of these statements, the so-called contrapositive. Remember, the contrapositive, it changes the implication direction, and it negates things. So it's like doing the converse and the inverse together. So if our statement was, if marijuana is legalized, then drug abuse will increase. The contrapositive will then be, if drug abuse does not increase, then marijuana was not legalized. So if drug abuse stayed the same, then that means that marijuana wasn't legalized because marijuana legalization will increase drug use, at least, again, that we're going with that current premise, that assumption. This contrapositive, which looks a little bit more mouthy, seems like it's more verbose than the original one, is actually logically equivalent to the first one. That if this is a true statement, this will be a true statement. And if this is a false statement, this is a false statement. And so these two, we have four conditionals on the screen right now. They go into two families. The original statement and the contrapositive, although phrased differently, are essentially the same thing. They have the same meaning in the end from a truth value point of view. And the converse and the inverse also have the same truth value. And so even though there's four versions of this statement, some with different truth values, there's basically only the two families. There's the converse-inverse family, for which we'll just focus on the converse. And then there's the original statement slash contrapositive family, for which we typically focus on the original statement in that one. And so there's these four versions of the statement into two families of logical equivalents. So with that in mind, we now introduce what it means to be a bi-conditional statement. A bi-conditional statement represents the idea of if and only if. We've talked about this before. As we've talked about conditionals, we often use the phrase if then. But we've also described how you can talk about if, then. You can also just use the word if without then, and that usually reverses the directions. You can use the words only if. And so in that usage that we've seen before, if and only if actually gave us different directions. And so the phrase if and only if, which we often abbreviate as just iff for short, so for if and only if, right there, iff, this gives us both implications. This symbol, the symbol we use is often a double arrow here. So we write something like p double arrow q. This means that p if and only if q. And this is the same thing as saying p implies q, the conditional statement, and q implies p. It's converse. The bi-conditional statement then infers implications in both directions. The premise p implies the conclusion q, but conversely, the conclusion q implies the original premise p as well. We don't have to include the inverse or the contrapositive because as we've already alluded to, these are logically equivalent to the notions we've already, we already have here listed the statement and its converse, hence the bi-conditional. It's a conditional with its converse, which is a different conditional statement. We have two statements there. And these, this is what we mean by something being logically equivalent. The two statements p and q are with regard to truth values, the exact same thing. They can capture different ideas, but they have the same truth values. In particular, what we'll say more about that in a second. Let's do some examples of bi-conditional statements at the moment. So an example of this, a polygon has three sides if and only if it is a triangle. Being a triangle is a different thing than a polygon necessarily having three sides, perhaps. It depends on how one defines triangle, I guess. But these two notions are in fact the same thing. So if this primitive, a polygon has three sides, we call that p, and it's a triangle, the polygon's a triangle, we call that q, then what we've written here is that p is equivalent to q. So that's how you read this. p is bi-conditional to q, p is equivalent to q, p if and only if, q, all of these are the same thing. The two notions are the same. Another example, the software can be returned if and only if the seal is not broken. Again, if we call this one statement s for software, and this one over here, s is broken. So we might say that like b equals the seal is broken. Then what this statement right here looks like is that s, that is the software can be returned, is equivalent to not b. And so we get another type of bi-conditional statement. The two statements mean exactly the same thing with regard to truth value. So let me elaborate on this a little bit more here. So we've seen these types of statements a lot as we've been trying to develop Boolean algebra. So if you have two statements, p and q, and you suppose a bi-conditional on them, what you're saying is that this thing will be, this is equal to true if and only if, I know it's weird, I'm sort of talking about bi-conditionals with a bi-conditional, but whatever, p is equivalent to q. That's a true statement exactly when p and q are equal to each other. That is, they're both true or they're both false. The bi-conditional is true when they have the same truth value. So if p is true and q is true, then the bi-conditional is true. Similarly, if p is false and q is false, then the bi-conditional statement is true. Remember that we have these primitive statements, but then we put them together to form a compound statement. The compound statement is itself a statement with a truth value. If the two primitives have the same truth value, then the bi-conditional statement is true. But conversely, if they disagree, if one is true and the other is false and it doesn't matter which one, then the bi-conditional statement is in fact false. And so throughout these lectures we've had about compound statements, we've seen operations, we have disjunctions, conjunctions, negations, conditionals and now bi-conditionals. We can evaluate these things. So much like one in arithmetic can do something like 2 plus 3 is equal to 5 or 3 times 4 is equal to 12. You can do these calculations and thus simplify the expression. 2 plus 3 simplifies to 5. 3 times 4 simplifies to 12. We can do the same thing in Boolean logic using the rules that we have seen previously. So consider this one right here, not true and false. Now whenever you see a negation symbol, we are going to assume it applies only to the truth value immediately following it unless there's parentheses. So be aware that not true and false is not the same thing as not true and false. For which without the parenthesis, this is how we interpret the statement. And so with that negations will always go first. So not true becomes false. Negation just switches the truth value from true to false in this case, but it goes the other way as well. Now in this situation you have false and false. Remember a and statement, a conjunction is only true when both statements are true. And so this would be false in this situation. I confess we could have jumped here immediately because who cares what this is. If you have something and false, that's going to be a false statement. Now when you look at this one, same basic idea here, this not true becomes a false. You end up with false or false. Like we saw before, when it comes to an or statement, a disjunction, the disjunction will be true if any part of it's true and it's false only if both portions are false. So we can simplify this likewise to be false in that situation. Let's do one with some conditionals here. Let's take the statement true implies false or false implies true. We will simplify what's in the parentheses first. So with the first one, true implies false. This is the only time a conditional is actually a false statement. If the premise is true, but the conclusion is false. So that first conditional is a false statement. But then when you look at the second statement, false implies true. This is what we saw before as a vacuously true statement. A conditional is true if the hypothesis is false. The only time a conditional is false is in this situation. We're not in that situation. So it actually is a true statement. So then we have false or true with an or statement. If any part is true, the whole thing is true. So this statement is a true statement as well. All right then, let's look at this last one. Notice now the parentheses. The not and the negation applies to the entire conjunction here. There's this and statement in the middle. So we're then going to do what's inside these parentheses first. So false implies true. Like we mentioned a moment ago, that actually is a true statement. And we have this by conditional right here. By conditional means that it'll be true if the two values are the same. False is different from true. So that by conditional is a false statement. Now if you have true and false, that makes it false. And if you have not false in the end, this statement evaluates to be true. So we're now putting together all of these types of statements of the following form that we've been seeing here in our lecture series. We've talked about conjunctions, disjunctions, negations, conditionals and by conditionals. And so now we've reached a point where we have to start evaluating these Boolean expressions. And that's what we'll be talking about in, of course, the next lecture.