 So in the last video, we looked at atomic propositions and truth relations, and I mentioned but didn't really go into explaining complex propositions. The only thing I said is that complex propositions are composed of atomic propositions. And that's true. Complex propositions are composed of atomic propositions, and at least for this course, what we call logical connectives. Now what a logical connective does, it has meaning, but not in the same way that a subject has meaning or a predicate has meaning. Logical connective has meaning by telling us something in addition to the atomic proposition of the subject of the predicate. For most of these connectives, what it gives us is the truth relation between two atomic propositions. And so we had sufficiency, contrariety, subcontrariety, and excessive. So let's take a look at each of these connectives, or the list of connectives, and take a look at each one in turn. Okay, so I mentioned logical connectives. Now the logical connectives we'll be using, we'll be using a few of them. And some are equivalent, right? But the broad versions for the logical connectives, we have not, or it's false that, either or, or simply or, or unless, right, so we got not, either or, and, that's another connective and, or sometimes the but is used, or however, or although. So not, either or, and, and the last one is if then, and if then could also include only if, or simply if, or given that, right, these are different ways of expressing, or using these logical connectives. Well, let's just try these, yeah, just these, just these kinds, right? So not, and, or, and if then, let's try those four kinds. Now let's start with not. This is, this is, this kind of connective is called a negation. Now remember the last video I made a distinction between an assertion and a, and a denial. And an assertion, if you remember, is the claim that a subject is predicated, right, and described by the predicate. So, let's say the, the tree is green. That's an assertion. The tree is the subject and is green is the predicate and that assertion is true. A denial is the claim that a, an assertion, right, is, is false or that the subject is not so predicated, right? So if I say the tree is not blue, right, that's a denial. That's saying that the, that the atomic proposition, that assertion, the tree is blue, right, that, that's not so. It's not so described. And that's the difference between an assertion and a denial. Well, you know, strictly speaking, denials really don't tell us something about the subject. They tell us what it's not, right, they tell us what it's not. And a negation claims that an atomic, or a proposition is false. So I have the tree is blue. And I can use a couple of different negations for that. I can say it is false that the tree is blue. That's another one. The tree is not blue. These are equivalent in terms of negations. And then those two negations that I use are, it is false that and not, right? Those are two different ways of saying, at least for these purposes, the same thing. And a negation, as I said, negation claims that a proposition is false. So, so a negation is not an atomic proposition, even though there's just the one connective and the one proposition. Atomic is not, excuse me, a negation is not an atomic proposition, right? An atomic proposition is a subject and a predicate. Well, not being something that's not a predicate. Simply says, you know, negation is saying is that it is false that the subject is so predicated when you're dealing with an atomic proposition. You can negate complex propositions too, but let's just start there. Okay, so a negation, we have not, or it is false that, or even non, right? That says that a proposition is false. And a negation itself is true when that proposition is false. Well, I said that most of the time, connectives are used to express truth relations. Well, negation isn't a truth relation. And now I'm going to deal with conjunction. It also doesn't deal with the truth relation. I'm not a liar, just starting with some of these other ones first, all right? Now, what a conjunction does, a conjunction uses a word like ant. That's a logical, that's a logical connective used often for a conjunction. And what a conjunction says, conjunction does is it, you know, brings together two propositions and it claims that each one is true. Claims that each one is true. So here's a proposition, this tree, or this organism is a tree. This organism is a tree, okay, it's a proposition. Here's another proposition, this organism is short. This organism is short, okay. So each of those is true, right? This is a tree, I think it's true, let's assume it's true. This organism is a tree, that's true, and this organism is short. To express that they're both true, I would say this organism is a tree, and this organism is short. This organism is a tree, and this organism is short, and is the logical connective. Other logical connectives include words like but, and however, and although. Now this might throw you a little bit, because we use the word but for, and however, and although, for a variety of purposes, right? So if I say this tree, this organism is a tree, but this organism is short, that's still a conjunction. That's still the claim that each proposition is true. Usually though the word but is used to express something like surprise, right? So the other subtext is, this organism is short, right? You say this organism is a tree, but this organism is short, and the subtext is, and we thought it'd be tall, that's what we thought of it. Now, so we use the word but to express like surprise, or we thought otherwise. Right. All right, but surprise and thinking otherwise, these aren't logical properties. These are emotional ones. So we're only going to consider that the logical properties with that connective. So this organism is a tree, and this organism is short, is a conjunction. Another one is, this organism is a tree, but this organism is short, is also a conjunction, doing the same thing. This organism is a tree, although this organism is short, right? Or this organism is a tree, however, this organism is short. Or even mix it up a little bit. Although this organism is a tree, this organism is short. Each one has that connective that expresses a conjunction. There might be a little more emotional or rhetorical work going on the background, that's fine, but for a logic course, we're not worried about that. The only thing we're worried about is, and, but, although, and however, are all logical connectives expressing a conjunction. And the conjunction says each of the component propositions, which are called conjuncts, by the way. The component propositions for a conjunction are called conjuncts. Each of the conjuncts is true. Okay, this brings us to either or, or just or. So either or actually does express a truth relation. We finally got there, it's a truth relation. Either or is used for a disjunction. This logical connective creates what's called disjunctions. And disjunctions, their component propositions are called disjuncts. Now, what a disjunction does is it says that the truth relationship subcontrary exists between the component propositions. Subcontrary exists between the component propositions. So we have, I don't know how well you can see this. Try and bring it up close. Here's a proposition. This mineral is chert. Here's another proposition. This mineral is flint. Now, I'm not actually a geologist, but I'm pretty sure it's one of those two. Given my limited knowledge of minerals. So I have the subcontrary relationship between these two propositions. This mineral is chert and this or this mineral is flint. So I say at least one of those is true. Remember what a subcontrary truth relationship is. At least one of the component propositions is true. At least one of the component propositions is true. So I'd use either or. Either this mineral is chert or this mineral is flint. That's one way of saying it. I could simply use or. This mineral is chert or this mineral is flint. There's another one that we've used or that can be used and it might be a little unfamiliar or you might think that that's weird is unless this mineral is chert unless this mineral is flint. Believe it or not, unless also expresses a subcontrary truth relation between those two propositions. Because think about it. If this is not chert, then this is flint. And if this is not flint, then this is chert. That follows from using the word unless. And remember what subcontrary is if one is false, the other is true. And vice versa for subcontrary. Remember that. Okay, so that's disjunction. Use either or. Either this mineral is chert or this mineral is flint. Either this mineral is chert or this mineral is flint. Or just simply say this mineral is chert or this mineral is flint. And finally, this mineral is chert unless this mineral is flint. Any one of those three expresses a subcontrary truth relationship between propositions. Okay, now we have the connective if then. Connective if then. This connective is called a conditional. It has two component propositions, the antecedent and the consequent. Now just a word of warning, the antecedent is not always the first component. And the consequent is not always the second component. A lot of times it is, but not always. So let's just start one, take this one step at a time. We'll just give it that warning right off the bat, right? So what if then expresses is that the truth relationship, a sufficiency, exists from the antecedent to the consequent. From the antecedent to the consequent. Okay, and the antecedent is sufficient for the consequent. That's what makes the antecedent the antecedent. So here we have a proposition. If this organism is a tree, then this organism is a plant. Okay, so we have two propositions. This organism is a tree. That's one component. The second component or the second proposition is this organism is a plant. And with the connective if and then we're saying if this organism is a tree, that's sufficient for then this organism is a plant. There are other ways to express this with kind of variations on the connectives. You could simply use the word if. So I'd say if this organism is a tree, then if this organism is a tree, this organism is a plant. Just skip the then from want to be really cheap with my words. We could also use the component only if or sorry, the connective only if. This organism is a tree, only if this organism is a plant. Now here's where we have to be careful. When you simply have the word if by itself, that indicates that the component proposition that immediately follows is the antecedent. That's sufficient for the other one. So when you have if then or simply if that tells you that the component proposition that immediately follows, that's the antecedent. You can kind of reverse the order of this, right? So if I have this organism is a plant, if this organism is a tree. Notice I kind of change the location of the antecedent. But I'm still using the single word as the connective if for that. Now this is different when you have the connective only if. When you have the connective only if, you're saying the component proposition that immediately follows, that's the consequent. The other component is sufficient for it. So I say, if I say this proposition, this organism is a tree, only if this organism is a plant. This organism is a plant is the consequent. That makes the other component proposition the antecedent. And it'd be really awkward to say it this way. I don't think anybody really writes this way, but you could still write only if this organism is a plant. This organism is a tree. That'd be very awkward. I don't think anybody really does that, but just in case be on the lookout. Still that only if tells you that the component proposition that immediately follows it, that's the consequent. So we have if then, this is a conditional, that kind of complex proposition is a conditional. And it's true when the antecedent is sufficient for the consequent. When the antecedent is sufficient for the consequent. And the antecedent is a part that is sufficient for the consequent. If the connective if by itself that indicates the antecedent, the word then, or the connective only if that indicates the contemplation. Well, if you've been paying attention, we run out of connectives, but we still have two more truth relations to go. You've got necessity and contrariety. Okay, well, what are we going to do? Well, at this point, we start combining connectives. You can combine connectives. You can have a disjunction with that in conditional. You can have a conjunction within a disjunction. You can have a conditional within a disjunction. There's a whole lot of combinations there. We're not going to cover all the combinations. I mean, there's probably literally infinite number of them, but let's leave that aside. In the meantime, though, if we still want to be able to express two more truth relations, necessity and contrariety, well, let's do necessity. Well, like sufficiency, we're going to use a conditional. Now, remember the last scene, I used a tree. I said, if this is a tree, then this is a plant. Now, I can't express necessity the same way. I can't say if this is a plant, then this is a tree. We do this in a lot in English, right? We're very sloppy with our language sometimes when we're expressing conditionals. And this is a big problem, right? This is a big mistake. But we're going to be much more precise in logic. We're going to be much more precise in logic. Instead of trying to use a conditional in the same way to express efficiency and necessity, we're going to use a conditional of negations. Remember what sufficiency is. If the first is true, then the second is true. For a necessity, if the first is false, the second is false. So if this is not a plant, then this is not a tree. So I've got the if then, and the connective is still connecting two propositions, but they're not connecting, the connective is not connecting atomic propositions. It's connecting complex ones, namely negations. So it's a conditional of negations. So if this is a tree, then this is a plant. If this is not a plant, then this is not a tree. So this is a little kind of secret I haven't really mentioned up until now. I might have mentioned it. I can't remember. I might have mentioned it. But these two propositions go hand in hand. If this is a tree, then this is a plant. This expresses the fact that the antecedent, if this is tree, is sufficient for the consequence. Or that the proposition, this is a tree, is sufficient for the consequence. Well, if one proposition is sufficient for a second, the second proposition is necessary for the first. This always, always, always happens. If one proposition is sufficient for a second, the second is necessary for the first. So if we say if this organism is a tree, then this organism is a plant, what's, what follows from that is if this is not a plant, then this is not a tree. This is not a tree. Okay. So, with the first conditional that we dealt with, and this is where it gets a little bit tricky. With the first conditional, if this is a tree, then this is a plant. The antecedent is this is a tree. With the second conditional, if this is not a plant, then this is not a tree. The antecedent is if this is, this is not a plant. So it gets a little, it gets a little weird. When you get the negation thrown in there, things get a little complicated. Things get a little complicated. So, just to sum up, for sufficiency, we use the connective if then, for necessity, we use two connectives, the if then, the if then and not. So sufficiency is if then, necessity is if not, then not. If not, then not. A conditional is used for sufficiency. A conditional of negations is used for necessity. Okay, last truth relation. So we already looked at a bunch of connectives. Not gives us a negation, says that the proposition is false, but or and is a conjunction, tells us the component propositions, the conjuncts are both true. Disjunction, using either or or, says at least one of these is true, and that expresses a subcontrary truth relationship. Looked at conditionals if then, and that expresses a sufficient truth relation from one proposition to the second. We had to combine negation and conditionals, have the conditional negations to give us necessity, and we're going to combine again for contrary. I remember what contrary says, at least one of these propositions is false. And so here we have an organism either. So this organism is a tree is contrary to this organism is a dog, right? It can't something can't be both a tree and a dog. Now again, in English, you might be really tempted to say, well, either this is a tree or this is a dog. Well, okay, I mean, maybe the maybe that subcontrary relationship holds, and it sure seems false that if it's false that it's a tree, then that's not going to work. But it's definitely contrary. Now to express that it's contrary, right? To express this contrary, we'll use the negations. Either this is not a tree or this is not a dog. That's claiming at least one of those is false. Subcontraries claiming at least one is true. With contrary, we combine the negations with the disjunction. We got at least one of these is false. Either this is not a tree or this is not a dog. Okay, so that's the last truth relation. We use either or to express subcontrary, if then to express sufficiency, if not, then not to express necessity, and either not or not to express contrariety. Now if you've been paying attention, there's still different combinations of connectives that we can use, right? We can have negations of conjunctions. We can have a conjunction of negations. We have a disjunction of negations. Well, we have a negation of disjunctions. We can have a negation of conditionals. We can have a disjunction with only one negated disjunct. We have a conjunction with only one negated conjunct. We can have a condition with only one where they can negate it into sedentary. We can have a condition with a negated consequent, right? There's all kinds of combinations. Like I said, you start combining this. You could literally have an infinite set of combinations. We're not going to get to all the infinite set. We're going to get to other ones later. For now, we just want to worry about which truth relation, our simple truth relations, and how to express them using the connectives. So, again, just to sum up, we got negation, which says the component proposition falls. We got conjunction, which says each of the component propositions is true. We got disjunction, which says each of the disjuncts is subcontrary to each other. We've got conditional, which claims that the antecedent is sufficient for the second. Expressing sufficiency, we got negation. We're the negated antecedent. This gets confusing. But a conditional negations, which expresses necessity, and a disjunction of negations, which expresses subcontrary. It's a mouthful, all at once. Don't worry, with some practice, you're going to understand what's going on. You'll be able to spot it pretty easy. Having said that, let's look at some sample problems. So the first kind of problem you're going to deal with, you're going to be given a proposition, a complex proposition, and you're going to identify the truth relationship that's expressed between the component propositions. So you're given a proposition, you identify the truth relation. So suppose we have this, suppose you have a dog in front of you. And it looks somewhat familiar. You say, well, either that dog is a husky or that dog is a malamute. Either that dog is a husky or that dog is a malamute. So here's the problem. What truth relationship is expressed between the component propositions? Okay. So first step, identify the component propositions. So taking a quick look at it, you'll try to identify the connectives. So you probably identify the connectives real quick, but you're looking for the subject and the predicate. Well, that dog is a husky, is one component proposition, and that dog is a malamute, is the other component proposition. Now, this is just going to save you in the long run. Ask yourself, okay, I'm looking at that component proposition. Is it a negation? No. There's no negation here. A negation would be that dog is not a husky, or that dog is not a malamute. That would be a negation, but you don't have that here. So since there's no negations, you can set that aside. There's no negations. You know it's not going to be a necessity, and you know it's not going to be contrary. So the second step, well, what is the connective? Well, you've identified the component propositions. The connective is not if then. That gives us sufficiency. Now we have an either or, and since we have an either or without any negations, the truth relationship expressed there is subcontrary. It's probably going to help to keep a, you know, running list in your head, the different truth relationships, and how you express them using connectives. So subcontrary either or sufficiency, if then, necessity, if not, then not. And contrary, either not or not. To keep that in mind, it's going to help you zip through these problems really fast. And you can take those steps. So I look at the identify the component propositions, figure out whether the negations, if they're not, look for the connective. If there is a negation, look for the connective, and that'll tell you the rest of the problem. So if the problem had been either that is not a husky, or that is not a malamute, that would express contrary. If it had been some, well, I gotta go on further examples, you practice with that problem. So this is the first kind of problem. You give it a proposition, and you identify the truth relationship. Okay, the first kind of problem gave you a proposition, and you are supposed to identify the truth relationship expressed by that complex proposition. The second kind of problem gives you a truth relationship, and you identify which complex proposition expresses that truth relationship. And you're gonna give you four of them. Okay, so let's look at the first one. There will not be a party unless there are no dishes in the sink. So your mom tells you, there won't be a party unless there's no dishes in the sink. Okay, you're trying to throw a party, there will not be a party unless there's no dishes. All right, so let's look at these component propositions. That's the first thing, identify the component propositions. There will be no party, as one, and there are no dishes in the sink, as another. Now, right away, ask yourself, are these negations? And they are, right? There will not be a party as a negation. It's a negation for the atomic proposition. There will be a party. There are no dishes in the sink as a negation. It's a negation for the atomic proposition. There are dishes in the sink. So there are the component propositions, there will not be a party, and the other one is there are no dishes in the sink. Okay, so we've got the negations, and since we're asking for subcontrary, we already know this one isn't gonna do it. Well, let's identify the truth relationship for that proposition anyway. So what's the connective? The connective is unless. Unless. So we got that disjunction. The unless is used for disjunction, and since we got negations here, this is a disjunction of negations. So the truth relationship expressed for that complex proposition is contrary. It's contrary, or contrariety, right? It's contrary. There will not be a party as contrary to there are no dishes in the sink. Okay. Well, let's try another one. If you are enrolled at San Antonio College, then you are a student. So the first step, identify the component propositions. And when we start looking for the subject and the predicate, the component propositions are you are enrolled at San Antonio College, that's one component. The second component proposition is you are a student. Okay. Are these negations? No. These are not negations, right? You are enrolled at San Antonio College is a subject and a predicate. You is the subject are enrolled in San Antonio College is the predicate. Same thing with you and student. So we don't have negations here. Do we have a subcontrary truth relationship? Well, what's the connective? Well, the connective is if then. Well, that's not subcontrary. That's sufficiency. All right. So that connective says you are enrolled at San Antonio College is sufficient for you are a student. If it's true, you're enrolled at San Antonio College is true that you are a student. Okay. So that's the second proposition. That's sufficient, but we're looking for subcontrary. Let's try a third proposition. If, sorry, a belief is not a belief is not proven only if the belief is not a fact. A belief is not proven only if a belief is not a fact. All right. Well, let's identify the component propositions. One component proposition is a belief is not proven. The other component proposition is a belief is not a fact. Do we have negations here? Yes, we do. A belief is not proven is the negation for the atomic proposition or belief is proven. A belief is not a fact is the negation of the atomic proposition or belief is a fact. So the component propositions are negations and since it's subcontrary relationship doesn't involve negations, we already know this isn't ours. Well, let's identify it anyway. So what's the other connective? We have the negations. What's the other connective? Well, it's only if. All right. So we have a conditional of negations here and a conditional of negations and only if is the is the consequent there. Okay. So this expresses necessity. This expresses necessity. And we're looking for a subcontrary so this isn't it. All right. So we've eliminated those three. There's only one more to go. It's a pretty sure about this last one's gonna be contrary. Well, let's take a close look at it anyway, just so, you know, we could talk our way through it. All right. So either it is going to rain or it is going to hail, right? And given the cloudy conditions of today, that proposition is becoming ever more relevant to my situation. Okay. So let's look at the component propositions. What are they? It is going to rain. That is one. It is going to hail. That's the other. All right. Are these negations? No, they're not, right? It is going to rain is an atomic proposition. It is going to hail is another atomic proposition. We have two component propositions here. They're not the one's negation. So we're in the running. Well, what's the connective? Well, it's not and. And it's not if then, right? It's either or. Well, this expresses the subcontrary truth relationship between these two component propositions. So it's this last answer. This last option, that's the answer. So we've got two kinds of problems for this chapter. We've given a proposition. You have to identify the truth relationship. Then you're given a truth relationship. You have to identify which complex proposition expresses that truth relationship. Okay. Well, that's the end of this video. We've taken a look at complex propositions and truth relationships and how to express these truth relationships using these complex propositions, using the logical connectives, and even a couple of other complex propositions that aren't that don't express truth relationships, but they're useful. Nevertheless, we've even looked at a couple of sample problems. So good luck on the quizzes. I'll talk with you next time. Keep thinking.