 So we're talking about propositions. We've already seen propositions. We just finished with categoricals not too long ago, and now we're going to move on to kind of a discussion about other kinds of propositions, especially complex propositions. Now propositions, I think we discussed this when we talked about categoricals, that which is true or false, that's what propositions are, and this is contrasted to terms. Terms are meaning definitions, concepts. Concepts, definitions meaning these are not true or false. It's either accurate or inaccurate. It either defines the thing or doesn't define the thing. Propositions are different. Propositions are what are true or false. Now categoricals are one kind of proposition. We've just seen that, and from here on out, we're going to delineate two main different kinds of propositions, atomic propositions, and complex propositions. Now atomic propositions, it's just a subject and a predicate, specifically a subject described to the predicate. Now this should sound familiar. We just finished with that with categoricals. So atomic propositions are, you know, what we would think of the smallest kind of proposition. You can't break it apart anymore without just not having a proposition anymore. It's a subject and a predicate. You take away the subject, you don't have a proposition, just have a predicate. You take away the predicate, you don't have a proposition mark, just have a subject. So atomic propositions are just the subject and the predicate. It's the building block for all other kinds of propositions. Everything else beyond this, right? It's a complex proposition. It's built out of atomic propositions and some sort of logical connective. So what are our atomic propositions? I said it's a subject and a predicate. We just finished with all kinds of atomic propositions. Well, these two kinds of propositions, atomic propositions, the universal affirmative and the particular affirmative. Now you might ask, well, what about the negatives? They're not atomic. Those are not atomic. You can take them, you can take pieces away specifically from the negation. You can take it away from the negative universals and then you have an atomic proposition. So negatives are not or negations are not atomics. Only the universal affirmative and the particular affirmative. Those are atomic propositions, a subject and a predicate. Complex propositions, as I just got finished saying, are built out of at least one atomic proposition and a logical connective. At least one atomic proposition and a logical connective. Negations are complex. Negation is when you claim that the subject is not described by the predicate. It is not included in the predicate to use the language of categoricals. So to look around here, right, that tree is green. That is a universal, I'm sorry, well actually that'd be a universal affirmative because it's about one. It's a singular, right? Remember that from that discussion? That's a singular subject. But every member of that subject class, namely that tree, is also green. Or at least it has green leaves. That is an atomic proposition as the subject and the predicate quantified in a copula. That tree is not blue. That is a universal negative. Because again we're dealing with one member but all also coincidentally all members of the subject class, namely that tree. That tree is not blue, is a negation. It's a complex proposition. It's a claim that that subject is not a member of the predicate. So we've got two down right away. We've got the atomic proposition and we've got negations. And like I said, all of the complex propositions are composed of at least one atomic proposition and a logical operator. Now at least for our purposes, negation is the only complex proposition with only one atomic proposition. For what it's worth, by the way, you can negate a complex proposition but let's not get ahead of ourselves. So the negation is the only one that requires only one atomic proposition. All the other logical operators, all the other complex propositions require at least two atomic propositions and it could also be for the complex propositions. So the logical operators we have are conditionals, which is an if-then statement. If that tree, right, if that is a tree, right, if that organism is a tree, then that organism is a plant. That's a condition. If that organism is a tree, then that organism is a plant. That's an if-then statement. A disjunction is either or. Either that is a tree or that is a bush. That's one of those two. Either that's a tree or that's a bush. Okay, so that's a disjunction that says at least one of them is the case. Then you have conjunctions. A conjunction is a both and or just and. So that is a tree and that tree is green. I'm gonna put you together. You can have combinations of complex propositions and if that is not a plant, then that is not a tree. That is a conditional of negations. You can have a disjunction of negations. Either that is not a tree or that is not an animal. You can also have a conjunction of negations. That is not an animal and that does not give birth to live young. So there could be more combinations beyond that, right? You can have the component propositions. You can have either one or the other. So if that is a tree, then that is not an animal, right? That's a conditional of one negation. You can have a disjunction with one negation. You can have a conjunction with one negation. You can have a negation of conjunctions. You can have a negation of disjunctions. You can have the negation of a conditional. It goes, and there's a lot, a lot at that point. There's literally an infinite number of combinations, but we're not going to try and list them all because, hey, we don't have the time. One other logical operator, if I got to mention, is if and only if. And that's, you know, that is a plant. If and only if that organism has roots and uses sunlight to produce food. I actually don't know the definition of plant, so I might have to fake that later on. I don't know that at the top of my head. If and only if is quite often used for definitions that we talked about. We might have touched on that when we looked at definitions earlier. Okay, so that is, you know, we got the atomic propositions. We got the negations, which is a complex proposition. So we have the other complex propositions, which begins with a conditional, a disjunction, a conjunction, and then the bike additional, which is really a conjunction of conditionals, but one thing at a time. Now, you know, a question you might have real quick is, well, why do we use one logical operator to form a complex proposition versus another? What's the difference between these logical operators? I'm glad you asked. So I'm going to introduce a concept to you called a truth relation. A truth relationship is a relationship. And we're just going to deal with truth relations between two propositions. We can probably talk about truth relations with three propositions or more, but let's just deal with two and only two propositions. So a truth relationship is a relationship between propositions such that the truth value of one affects the truth value of another or has an impact on the truth value of another. And there's four basic kinds. True makes true, true makes false, false makes true, false makes false. I said they're very fast on panic. The first one is called sufficiency. That's where the truth of one proposition means that another proposition is true. I had one example of this already, right? If that is a tree, then that is a plant. If that organism is a tree, then that organism is a plant. Okay, so it's composed of two atomic propositions. That organism is a tree is one proposition. That organism is a plant is another. And there's a relationship from the first to the second such that the truth of the first means that the second is also true. That's sufficiency. True makes true. The truth of the first means that the second is always true. Now this truth relation pretty much just goes one way. You should just presume that it just goes one way. There are cases when it goes both ways, but let's ignore that for right now, right? Can be the case, but it's not necessary the case that sufficiency runs from the second to the first. So this is the truth relation, right? True makes true. If the first is true, then the second is always true. I'm sorry, it's also true, right? The truth of the first necessitates the truth of the second. If that organism is a tree, then that organism is a plant. So that's one truth relation. The second truth relation strongly related to that, right? Is called necessity, right? Necessity. It's where the error of the first means that the second is also false, right? So false makes false. So if what if it's false that that is a man that that if it's false that that's a mammal. Right? Excuse me. If it's false that that's an animal. If it's false that that's an animal, then it's also false that it's in that it's a mammal, right? If it's false that that's an animal, it's false that it is a mammal, right? So that's false makes false, right? We usually express sufficiency using the conditional, right? If that is a tree, then that is a plant. We also use conditionals for necessity, but we use a conditional of negations, right? So if that is not an animal, then that is not a mammal. If that is not an animal, then that is not a mammal. Sure makes true, false makes false, sufficiency and necessity. And like sufficiency, we're in deal with necessity, right? Necessity pretty much just you should just presume it goes one way, right? It just goes one way. It can go both ways. It's getting ahead of ourselves here, but if one proposition is sufficient for the second and the second is also sufficient for the first, right? Sufficiency goes both ways and in your rare cases, necessity also goes both ways between those two. So sufficiency, true makes true, necessity, false makes false. Then you have contrariety or contrary and subcontrariety or subcontrary. So contrary might be a little bit more familiar, right? What contrary means is that the truth of one means that the second is false, or another way of saying this, at least one is false, right? So that is a tree is contrary to that is an animal. There's nothing that's both an animal and a tree. Somewhere out there, there's a biologist that's starting to fiercely type away in the comment section, but I'm pretty sure that's right, right? There's no animals that are also trees, right? So that is a tree is contrary to that is a animal. Now, I'll be curious to see if somebody finds an example that expands my horizons, right? Usually, we express contrary propositions, or at least in this course, so where we're going to express contrary propositions is using a disjunction of negations, either that is not a tree or that is not an animal. This might sound a little weird to you first. That's because we use disjunctions to express contrary, subcontrary and contradictory propositions, so we don't really differentiate between them in natural language. So that's contrary. That is a tree is contrary to that is an animal, right? At least one of those is false. Okay. Then there's subcontrary. Subcontrary means that the error of one means that the other is true, or another way of saying this, at least one of the component propositions is true. At least one of the component propositions is true. That is either a tree or that's a bush. So if it's false, if that's a tree, then it's a bush. If it's true, if it's false, that it's a bush, then it's a tree. Now, we express subcontrary propositions with just a disjunction. Either that's a tree or that's a bush. Now, again, you're probably thinking, whoa, whoa, whoa, we use disjunction. That could mean that that's, you know, one or the other, but not both, and it could mean that at least one is true in this class. I know natural language gets it all messed up. Natural language is sloppy. It loves to use the disjunction for contrary, subcontrary, and contradictory. But we're using the disjunction for subcontrary. And the reason why is because we don't have to add on a negation to it. Subcontrary means that at least one of the propositions is true. So we don't need to put an negation on it. Contrary means at least one is false. So either that is not a tree or that is not an animal. Those are contrary to each other. So and then contradictory, well, that's when two propositions are both contrary and subcontrary. We're going to leave that alone for a second. We're going to leave that alone for a second. So just for now, we got contrary, we got subcontrary. So that's four truth relations. True makes true, sufficiency. False makes false, necessity. True makes false, contrary. False makes true, subcontrary. Now there's one other thing to keep in mind. With contrary and subcontrary, if two propositions, if one proposition is contrary to a second, the second is also contrary to the first. Contrary runs both directions. It runs both directions. Same thing with subcontrary. Subcontrary, if one proposition is subcontrary to a second, the second is also subcontrary to the first. Subcontrary runs both directions. Sufficiency runs one way. Necessity runs one way. Most of the time. Most of the time. Sometimes it doesn't. But contrary and subcontrary, they always, always, always run both directions. They always, always, always run both directions. So four truth relations, four simple truth relations. There are complex truth relations that we build out of the simple ones. So sufficient and necessary. We put those together, we get equivalent. Contrary and subcontrary together, we get contradictory. If one proposition is contrary and sufficient for a second, then it's impossible. If a proposition is necessary and subcontrary to a second, then it's absolutely the case. That's a lot to lay on. You said just pay attention to the four simple truth relations, at least for now, sufficient, necessary, contrary, and subcontrary. And these truth relations are going to be what determines which logical operator we use. All right, that's what determines which logical operator we use. And I kind of already touched on that, but we'll do it a little bit more in a bit. All right, four truth relations. Let's look at the logical operators a little bit more. Okay, so we have conditions. I mentioned conditionals already. We use conditionals to express sufficiency. A conditional is composed of at least two propositions. And for now, let's just deal with atomic propositions. But it can be like further complex, right? We can have conditionals composed of disjunctions or conjunctions. It gets complex real fast. But let's just deal with atomic propositions for now. So it's composed of these two propositions. The antecedent is one, the consequent is the other. The antecedent is sufficient for the consequent. The antecedent is sufficient for the consequent. So we, there's lots of ways to express a conditional, probably two many ways, frankly. We typically use if then. So if that's a tree, if that's a tree, right, that's a plant. The if indicates the antecedent. Then indicates the consequent. If indicates the antecedent, then indicates the consequent. The antecedent is sufficient for the consequent. We could use only if. That is a tree. That is a tree. Only if that is a plant. Now it's easily confused to say, well the if is there, so that's the antecedent. No, it's only if. Only if. So if indicates the antecedent, only if indicates a consequent. We could have just if, right, and have no indicator word for the consequent at all. If that's a tree, that is a plant. If that's a tree, that is a plant. We're completely living out the then. The if indicates the antecedent. It's kind of implicit that the other component proposition is the consequent. Okay, we could have the if like out of order, right? That is a plant. If that is a tree. That is a plant. If that is a tree. Now it might be confusing because you think, well I got the second part there and you have the if, so that's a consequent. No, consequent is only if. Only if. If you have just if, it's an antecedent. If you have just if, it's an antecedent. Only if is the consequent. Let's see. There's other indicator words. I listed them in the in the reading. So you take a look, study the list. But you know, the really the sure far way to determine this is to figure out, well, which one's sufficient for the other? Which one's sufficient for the other? Okay, so that's conditions. And we use a conditional to express sufficiency from the first to the second. Now here's the thing. The antecedent is sufficient for the consequent. The consequent is necessary for the antecedent as sufficiency and necessity are kind of you can go hand in hand. Now I said earlier, you know, if one proposition is sufficient for a second, it does not not necessitated. The second is sufficient for the first. And that's true. But if one proposition is sufficient for a second, the second is necessary for the first, right? Necessary for the first. So that is a tree is sufficient for that as a plant. That is a plant is necessary for that as a tree. Now we express necessity. And we could just have the conditional and maybe that that does the trick. But one great handy way to express necessity is to use a conditional of negations and a conditional of negations. And that's where you have a thing I've seen the consequent, but you use a negation. Now to do this, so you have to be careful, right? We have that is a tree is sufficient for that as a plant. Remember, necessity is the error of the first necessity means that the second is possible. If that is not a plant, then that is not a tree. If that is not a plant, then that is not a tree. And that tells us that that is a plant is necessary for that as a tree. Another sounds complicated now, but after some practice, we get a hold of it. You'll be all over it. Okay. So we have a conditional and we have a conditional of negations. Now, here's something else. You can also have the negation of a conditional. That's not the same thing as a conditional of negations. A conditional of negations is when you have a conditional and the component parts, the incident, the consequence are also negations. The negation of a conditional is when you have a conditional and the whole thing is negated. So here's a conditional that's, you know, by the way, false, right? If that is a tree, excuse me, if that is a bush, no, that's not, if that, if that's a tree, then that's a bush. That's a false conditional. There's plenty of trees that are not bushes. So we would say it is false that that it is false that if that is a tree, then that is a bush. If that is a tree, then that's an animal. That's also false. Plenty of trees are not animals. Again, the biologist, some biologists out there is fiercely typing what. So you have a conditional if then. You have a conditional of negations, if not, then not. And now we have the negation of a conditional. It is false that if then. Those are, that's going to get you started. You can also have, there's one of the others negated, but we'll get to that a little while, probably another couple of chapters or so. Okay. So that's conditions. Let's look at disjunctions next. In addition to conditionals, we have another very prominent kind of complex proposition, the disjunction. The disjunction is an either or proposition. Either it will rain, which is increasingly look like it's going to be the case, or it will be dry. Either it will rain or it will be dry. That's a disjunction and it expresses what is expressing there is subcontrarity. At least one of those is going to be the case. Either it will rain or it will stay dry. That's a disjunction. At least one of those is true. Okay. So that expresses subcontrarity. The error of one means that the other is true. So if it's false, that it stays dry, then it will rain. And if it falls, that it will rain, then it stays dry. That's disjunction, either or. You can leave off the either. It will rain or it will stay dry. There you go. It will rain or it will stay dry. You can even use unless. This is a weird one, but unless gives us a disjunction. It will rain unless it stays dry. That's an unusual one, but it's an unusual thing to think about, but that's a disjunction. It will rain unless it stays dry. Okay. So that's subcontrarity. That's subcontrarity. Disjunction. We can use a disjunction of negation for contrary. Disjunction of negation is for contrary. So either that is not a plant or that is not an animal. It's true that that is a plant. So it's not an animal. It's true that that's a plant. So it's not an animal. So again, we use the negations with a disjunction to express contrary. To express contrary. You all mentioned contradictory before. That's when two propositions are both contrary and subcontrary. We would actually use, it's getting ahead of ourselves, we would use a bike additional where one of the component propositions is a negation. So that's my way of saying ignore, ignore contradictory for now. But contradictory would, you know, another way to do this, have a, yes, kind of long-handed way of saying this, but it's, you know, or at least one is true and the other is false. It would be a conjunction of disjunctions with identical component propositions. And one of the disjunctions is negations. That's long and complicated. Don't, don't, don't, don't do that. Don't do that to yourself just yet. But I do just kind of want to forecast it. All right. Just kind of forecast it a little bit. Okay. So disjunctions, we have either or, either it will rain or it'll stay dry. We have for, that's for subcontrary for contrary, either that is a plant, excuse me, either that is not a plant or that is not an animal for contrary. All right. So now we, we can also have, by the way, just like we had a disjunction and we've had a disjunction of negations, we can also have the negation of a disjunction. So it's false. Here's one, here's one for you. It's false that either that is a fish or that is a mammal. Right. It's just not the case of at least one of those is true. Right. It's false that at least one is true. That's the negation of a disjunction. And you can even have the negation of a disjunction of negations. Okay. An example escapes me right now. You can. And that, and what that would just be to say is that, you know, the negation of a disjunction negation is just saying that they are in fact not contrary. Okay. That's a lot to take in at once. That's a lot to take in at once. But we got four kinds of truth relations, sufficient, necessary, contrary, subcontrary, and we, and which logical operator reuse is determined by the truth relationship between the propositions, sufficiency is a conditional necessity as a conditional of negations, disjunction is, sorry, subcontrary is a disjunction, contrary is a disjunction of negation. That's a lot to take on at once. That should keep you busy. Have at it. So one last one, that's conjunctions. Conjunctions are, you know, use the phrase both and, right? Both that is a tree and that is green. Both that's a tree and that's green. Actually, I guess it's probably brown. So usually we don't use the both very much, but it comes in handy when we have the negation of a conjunction. So just keep that in mind. Both and both that's a tree and that is covered in bark. How about that? Both that's a tree and that's covered in bark. All right. So conjunction doesn't express the truth relation. Truth relation is when the truth of one affects the truth of the other. Remember a conditional expresses sufficiency, conditional negations expresses necessity, disjunction expresses subcontrarity, a disjunction of negations expresses contrariety. Well a conjunction doesn't express the truth relation at all. If you add just the plain conjunction of two atomic propositions, all this saying is both the atomic propositions are true. Both that is a tree and or that is a plant and it has bark, right? That's what that's doing. You can have a conjunction of negations and that is expressing that both component and homic propositions that are negated are false, right? Both that is not blue and that is not an animal. Both that is not blue and that's not an animal. Here we go. We have a conjunction of negation, just saying that the component atomic propositions are false. That's all it's saying. And then finally we have, you can have the negation of a conjunction, right? It's false that that is a tree and it is blue. It's false that both that is a tree and that and it is blue. And really when you have the negation of a conjunction, it's just saying that at least one of the component propositions is false. Sound familiar? So negation of a conjunction, a conjunction in of itself doesn't express a truth relation, but the negation of a conjunction does in fact express contrariety. Another way to express contrariety. Okay, so you can use both and, you know, just and, right? That is a tree and that has bark. You can have but, right? But, so that is a tree, but it's alive, but still expresses a conjunction. It's not a truth relation. It's just rhetorically speaking not in terms of truth value, but just rhetorically speaking, but is saying this is also true and it's a surprise. It's kind of the idea that is a tree, but it's alive. Let's see. I wonder if we got a good bud around here. That is an ivy, but it has bark. The thing about one right there, I don't know if you can see it's like lacing around the tree. I think that's some kind of, I'm pretty sure that's some kind of ivy and it looks like it's bark on it. There's another one. I could be wrong about that, but maybe that's a good example. That is a tree, but excuse me, that is ivy, but it has bark. That would give us a, that would also be a conjunction. And yet, that's another nice connective and yet. Let's see, what is it? That is palm and yet it's in Texas. It's in San Antonio. This is a swamp and yet we're in Texas. That would be a conjunction. Again, just kind of expressing surprise. Okay. So there's all kinds of ways, there's a variety ways to express a conjunction as well. You have to pay attention to the connective. Again, the text, I give lots of good examples, pay attention to those. So a lot of what you'll be doing is identifying what kind of complex proposition you're dealing with.