 Well, we spent a whole lot of time looking at conditionals, and conditionals express sufficiency, which is the truth relationship. Well now, this time, we're going to look at a different truth relationship, necessity. Well, we already talked about one kind of truth relation, and that's sufficiency. And if you remember, sufficiency means that if one proposition is true, then another is true. Right? If one proposition is sufficient for another, that means if the first is true, the second one must also be true. So if my pet is a dog, then my pet is a mammal, right? My pet is a dog, it's sufficient for my pet is a mammal. Necessity is different from sufficiency. It's another truth relationship. You remember, a truth relation is where the truth or error of one proposition makes another proposition true or false. Sufficiency means true makes true, right? Necessity is a truth relationship between propositions such that the error of one proposition means that another is false, right? Or false makes false. So, you know, looking at some propositions here, my pet is a mammal is necessary for my pet as a dog. If it's false and my pet is a mammal, then it's false and my pet is a dog. It's pretty straightforward. But we don't want to confuse sufficiency and necessity. There's some other propositions that are necessary. If an organism is a plant, sorry, an organism is a plant is necessary for an organism is a tree. Organism as a plant is necessary for an organism is a tree. A figure has four sides is necessary for a figure is a square. So sufficiency and necessity are kind of in a way real similar. But let's not confuse them, right? Sufficiency means the truth of one means the other is true. Necessity means the error of one means the other is false, right? Error of one means the other is false. Now, sufficiency and necessity, they do actually go hand in hand, right? If one proposition is sufficient for a second, then the second is necessary for the first. My pet as a dog is sufficient for my pet as a mammal. Well, on the other end of that, my pet as a mammal is necessary for my pet as a dog. A figure is a square is sufficient for a figure has four sides. A figure has four sides is necessary for a figure is a square. An organism is a tree is sufficient for the organism is a plant. The organism is a plant is necessary for the organism is a tree. So sufficiency and necessity are not the same thing. They're not the same kind of truth relationship. But you can't completely separate them. If one proposition is sufficient for a second, the second is necessary for the first. Okay, so that's sufficiency and necessity. Now the question is, how are we going to express these as propositions? So we discuss necessity as a truth relation. That's where the error of one proposition means that another is false, right? Now here's the question. How do we express this as a complex proposition? So remember where we dealt with sufficiency? We had that kind of clumsy phrase is sufficient for and we said, you know, an animal, my pet as a dog is sufficient for my pet as a mammal. And yeah, I mean, you can do it that way, but it makes reasoning very, very clumsy makes reasoning and discussion pretty difficult. So we replace that is sufficient for with the connective if then we also use only if and sense and all these other versions of it for for conditionals. So we're going to do the same thing when necessity. So if we're dealing with sufficiency, let's say, you know, my pet as a dog is sufficient for my pet as a mammal, we say, if my pet is a dog, then my pet is a mammal. If a figure is a square, then the figure has four sides. If an organism is a tree, then the organism is a plant, right? The first, the antecedent sufficient for the consequence. Well, let's think about necessity. We have this phrase is necessary for. My pet is a mammal is necessary for my pet as a dog. And that's pretty clumsy and trying to reason that way gets gets complicated. And then we could create, I suppose, another connective for this, but that gets a little cumbersome, right? That gets a little cumbersome and we really don't want to go through that, especially since the connectives we already have will work to express necessity. So we'll use our conditionals and our negations to express necessity. So remember, my pet as a mammal is necessary for my pet as a dog, right? Well, we'll use the conditional, we'll put negations on each of the propositions. We'll have, you know, the one that's necessary for the other to be the antecedent. So if my pet is not a mammal, then my pet is not a dog. That expresses necessity from my pet as a mammal to my pet as a dog. If the figure does not have four sides, then the figure is not a square. If the figure does not have four sides, then the figure is not a square. If the organism is not a plant, then it's not a tree. So we'll use our conditionals and our negations together as, you know, a complex proposition, a conditional of negations to express necessity between propositions, between these propositions. So that takes care of the complex proposition. How are we going to symbolize it? Okay, so we discussed how we are going to express necessity using complex propositions. It's going to be a conditional of negations. Well, the symbolizing it should be pretty straightforward. We've already got our symbolization for the conditionals and negations. All right, so suppose we have our complex proposition expressed in necessity. If the animal, if my pet is not a mammal, then my pet is not a dog. Well, following the rules, my pet is a mammal, receives the atomic proposition p, my pet is a dog, receives the atomic proposition q, and to express this, to symbolize this, we will have not p greater than not q, right? Not p, then not q using our symbol for conditions. So this is really not, like I said, we're not going to invent a whole new connective for this. We're not going to invent a whole new symbol. We'll just have not p, then not q, and this expresses necessity from p to q. Well, you might wonder if we have any new rules of implication for necessity. The answer is no. Necessity, you know, we have the conditional. So all the rules of implication that we have with the conditionals is going to work the same for necessity. You can have modus ponens using necessity, if you have not p, then not q. Well, you can assert not p and therefore conclude not q. You've got modus tolens. If you, by the way, since modus tolens, you're probably going to use double negation in there. So if I have not p, then not q, and I have q, right, then I'll have to double negate q to get not not q to infer not not p, and then probably use double negation again for p. So for modus ponens, modus tolens will work exactly the same way. If I have not p, then not q, and not q, then not r, then I can include not p, then not r using hypothetical syllogism. I can even have conditional proof, right? Suppose I have not p, then not q, and then if I assume, for the sake of argument, not q, then not r, then I can infer from that using conditional proof and use hypothetical syllogism, then not p, then not r, then finish with our conditional, right? Giving our assumption not q, then not r, then not p, then not r, right? So all of that works the same way using the rules of implication. We don't have a new rule of implication, but we do have a new equivalence rule. So the last equivalence rule we had was double negation. You remember, it's where we could take a proposition and swap it out for the same proposition, but with two negations in front of it. Now, the truth table has exactly the same truth values, right? And we could just swap it out, right? We could swap out a single proposition with that same proposition or two negations. We could swap out a double negated proposition for just the proposition. No problem, easy peasy. Well, we've got a new equivalence rule dealing with necessity. It's called contraposition. Contraposition, okay. Now remember what I said earlier? You know, the reason why we could do this. Remember what I said earlier? Sufficiency necessity go hand in hand. If I have my proposition expressing necessity from P to Q, if not P, then not Q, that is logically equivalent to a conditional expressing sufficiency from Q to P. All right, so if P is necessary for Q, then it follows necessarily that Q is sufficient for P. Similarly, if Q is sufficient for P, it follows necessarily is that P is necessary for Q. So suppose we have our conditional here of negations, not P than not Q. Well, I can swap that out whenever I want with a conditional, you know, conditional is still containing P and Q, but we take away the negations. We swap the position of P and Q, right? So not P than not Q is equivalent to Q than P. By the way, any proposition, now I got Q and P, right, if Q than P, I can swap that out for another conditional where I've switched positions for P and Q and put a negation in front of them, not P than not Q. And we could do this wherever this formula appears, like anywhere, if it's embedded in there, right, we can swap it out, no problem. So contraposition is an equivalence rule that allows us to replace any proposition expressed in necessity, not P than not Q, with a proposition that expresses efficiency, P than Q. So Q than P, Q than P. Make sure you get your, make sure you switch your antecedent and your consequent. So we could take any proposition where the antecedent and the consequent are negated, replace it with a conditional, right, where we switch positions of the antecedent and the consequent and we take away the negations. Okay. Now we could do this. It's a rule of equivalence. You can prove it using the truth table. Here's the truth table. It's really straightforward. You know, P than Q has exactly this. If P than Q has exactly the same truth values as not Q than not P, right? That's exactly the same truth assignment that's true in the exact same situations. And we could even, if we really want to do what we could even do with rules of implication, I'll let you go ahead and do that. You'll use, pretty much like I said described, right? You'll start with a conditional if P than Q and prove not Q than not P. Similarly, you'll take the condition of not Q than not P and prove the conditional P than Q. You're probably going to use conditional proof, double negation, but as long as most put them along the way, right? And that'll be fun. That'll be fun. Okay. So that's our rule. That's our equivalence rule. We could swap out those conditions, whatever we want.