 Okay. So we dealt with terms last chapter. This chapter, we're going to start putting those terms together into cohesive units of meaning, right? Go from terms to propositions. And then we're going to deal specifically with atomic propositions. Start with the building blocks of more complex propositions. And to really understand, you know, how even to get to complex propositions, we're going to look at how these terms are related to each other. We'll talk about the truth relations, the impact that the truth value that one proposition has on another. And then to wrap it all up, we're going to take a look at some exercises. So as I mentioned, this chapter we're dealing with atomic propositions. Now a proposition, like I said, is something that's either true or false, right? And we express these using sentences. Now last chapter, we dealt with terms. Now terms are not what are true or false. Terms are either defined or failed to define. Propositions are composed of terms, but they themselves are not the same thing as terms. So I can have a term, I mean, tree, another term, tall. And these have definitions. These have meanings. It isn't until I combine them that I get a proposition. And specifically an atomic proposition, the tree is tall. The tree is tall. So an atomic proposition is composed of a subject and a predicate. In this case, the tree, the subject, what's being described, and the predicate, tall, what's doing the describing. So again, atomic proposition. This is a statement that's either true or false. Propositions are statements that are true or false. Now, we're talking about atomic propositions. Well, they're not atomic in the sense that they're used in fission, right? In nuclear fission. That's not what I'm talking about. The word atom that we use in English is derived from a Greek word, etomos. And this in itself, it means indivisible. And this word itself, etomos, is a combination of two other words. In Greek, ah, meaning not. And I think it's pronounced tymenin or tymenin, something like that. In Greek, meaning to cut. So in Greek, literally not to cut. Indivisible, not to cut. So these propositions are atomic in this sense. You can't break them apart anymore and still have a proposition. I could simply say the tree. Right, that I haven't actually uttered a proposition. I haven't even uttered a sentence, at least not a complete one. I could say it's tall. When you separate either the subject or the predicate, you no longer have a proposition. I can still break apart the proposition, but it's atomic in the sense that it is no longer a proposition. So what a good chance. So, you know, that's one way to understand atomic propositions. That's what they are. It's a subject and a predicate. You know, I want to be careful to contrast this to complex propositions. Now complex propositions are still propositions. They're things that are true or false. So here's an atomic proposition. The tree is tall and there's another one. The sky is cloudy. Well, I can combine these two propositions to form a complex proposition. The tree is tall and the sky is cloudy. There's a complex proposition. It is composed of two atomic propositions. Now a complex proposition could be broken apart and you still have a proposition. So the two parts of a complex proposition. The tree is tall. That's one part. The other part, the sky is cloudy. I take out the tree is tall from the complex proposition. I still have a proposition. Complex propositions are composed of atomic propositions or even smaller complex propositions. But an atomic proposition, you can't break that apart even further. And this is, the atomic proposition is the basic unit of truth. This is the basic unit of, at least what we can express, how we can express the truth. When I say the tree, I haven't said anything as true or false. In the definite article in a subject, that's it. It isn't how I say the tree is tall that have uttered a proposition. So I want to offer kind of a quick word of warning. Atomic propositions need not be short. So I just used the tree is tall as an example. That's a very short sentence. It's not a long complicated sentence. But atomic propositions can be long and complicated. So say something like, the first form of government that existed immediately after the American Revolutionary War differed in important respects from our current form of government. That's a long-ish sentence. At least it's longer than the tree is tall. Now, believe it or not, that's an atomic proposition. It's got a lot of modifiers in it. It's kind of a complex description of a subject and a predicate. But that's an atomic proposition. So, just to kind of spell it out, the first form of government that existed immediately after the American Revolutionary War, that's not a complete sentence. That is not something that's either true or false. And in this case, it's the subject. The phrase, right, differed in important respects from our current system of government. That is also not a complete sentence. That's a phrase. It isn't until you put those together that you actually have an atomic proposition. Because the first warning, atomic propositions are not always short. They can be long and an atomic proposition. Second warning, not just any sentence is a proposition. There are lots of sentences that are not propositions. So, the type of sentence that expresses a proposition is called a declarative, right? It's something that's not the true or false. Other kinds of sentences would be something like imperative, right? Look at the tree. Look at the tree, right, imperative, that's a command. That's a command. The idea is that's not something that's true or false. Look at the tree is not true or false. That's either something that you follow or don't follow. And interrogative is a question. It's a request for information. So, what color is the tree? That is also not something that's true or false, right? It's requesting something that's true or false, but it's not itself true or false. And then you have exclamations, right? These are expressions of, you know, emotional approval or disapproval or maybe pain or sadness or something like that. So, something like, oh, what a beautiful tree. Right, that's an exclamation. That's not something that's true or false. So, two warnings. First, atomic propositions need not be short. And second, not just any sentence is an atomic proposition. So, this brings us to a kind of a distinction that we want to make between assertions and denials. This is not, you know, the most revelatory distinction in the world. It's probably a pretty common sense, but when I talk about assertions, I'm going to talk about the claim that a proposition is true, be it atomic or complex. We'll get into complex propositions more in later chapters, later videos. But for now, when we deal with assertion, just think that a proposition is true. And in the case of an atomic proposition, we're saying that the subject is predicated. Right, the subject is so predicated. So, if I say, you know, if I assert that the tree is tall, I'm saying the tree is tall. With a denial, we're saying that the subject is not predicated. We're dealing with an atomic proposition, the subject is not predicated. So, it is false that the tree is a mammal. Right, there's a denial. There's a denial that the tree is a mammal. Okay, there's other ways to express it. Well, it is false that the tree is a mammal, but the tree is non mammalian. The tree is not a mammal. All three of these are predicates, pretty much expressed in the same thing. A tree is not a mammal. So, an assertion is the claim that a proposition is true. A denial is the claim that a proposition is false, which brings us to another concept that I'm going to use this phrase pretty often, the truth value of a proposition. And truth value probably sounds more complicated than it is, but all I mean by the truth value is whether the proposition is true or false. So, the proposition, the tree is tall, has a truth value of true. And the proposition, the tree is a mammal, has a truth value of false. Again, not Earth Saturn revelations, but it's helpful to explain the lingo. So, we've got assertions, denials, and truth value. Now, kind of a little bit of a word of warning. Logicians are not primarily interested with whether a particular proposition is true or false. In fact, most of the time, logicians are worried about the relationship between these propositions, which will, you know, the relationship between the propositions will determine whether further propositions are true or false, true. But most of our work is not investigating the truth value of any particular proposition. Most of our work is looking at the truth relationships between propositions and then what that can do or what it helps us to do with thought. So, I very briefly mentioned truth relation. Some propositions have an impact on other propositions. Specifically, the truth value of one proposition can impact the truth value of another proposition or a second proposition. So, we have a proposition. See, this tree is a plant. This tree is a plant. Well, that has an impact on another proposition. This tree is a living organism. If it's true that this tree is a plant, it's also true that this tree is a living organism. So, the truth value of the first has an impact on the truth value of the second proposition. It's not just true to true, right? It's true that this tree is a plant. So, it's false that this tree is a mammal. If it's true that this tree is a plant, it's false that this tree is a mammal. So, it's true to false. Now, I mentioned truth values. So, we're only dealing with two kinds of truth values, true and false. We're not dealing with mostly true or kind of true or sort of false, right? These are the only two truth values, true and false in this course. So, if we're dealing with one proposition to a second and that first proposition has two possible truth values, it's either true or false, and we do the relationship to a second and that second proposition has two possible truth values, true or false. Then we have four kinds. We have four, not four kinds, but we have four simple truth relations. Four simple truth relations. Now, to be clear, right, they're not simple in the sense that they're easy to understand, although they might be. They might be. They're simple in a different sense. So, the word simple, we get from Latin, means simplest, simplest. And this word was used to refer to certain kinds of medicines that only had one plant. Or it just really meant having one part. So, this is the sense in which these truth relations are simple, that they are their own part, right? They're not broken down any further into any smaller truth relations, right? These first four ones are the simple ones. There are other truth relations. They're complex ones, and they're built out of the simple truth relations. But these four are the simple ones. So, think of a Lego, right? A single Lego is simple. It's that one part. A Lego model, when you put it together, that's complex. But the Lego itself is simple. And it's in this sense that we have these four simple truth relations. And again, we have four possibilities, right? Either the first proposition is true or the second proposition is false. And so we have... If the first proposition is true, then either it means that the second one is true or it means the second one is false. Or if the first proposition is false, then either that means that the second proposition is true or the second proposition is false. Or in short, true makes true is one. True makes false is the second. False makes true is the third. And false makes false is the fourth. Well, look at those in a little bit more detail. Okay, let's start with those truth relations that begin with the proposition the first proposition being true So first one is true makes true, right? This is sufficiency probably familiar with sufficiency So the what this means is the truth of the first proposition Means that the second proposition is also true. Okay, so This or this organism is a tree This organism is a tree Well, if that was true that is sufficient for another proposition Namely this organism is a plant This organism is a plant If this organism is a tree is true, then it's also true that this organism is a plant Suppose we have another proposition My pet is a dog My pet is a dog If that is true It's sufficient for a second proposition. Namely my pet is a mammal So my pet is a dog is sufficient for my pet is a mammal So I have a couple of words of warning, right? Um, the first warning is this sufficiency does not necessarily run in both directions That this organism is a tree Is sufficient for this organism as a plant But that an organism is a plant is not sufficient that an organism is a tree Uh, there are sometimes sufficiency does run in both directions, but we'll you know, we'll talk about that later and in other chapters It can run in both directions, but it doesn't necessarily run in both directions. Okay, so we could have a proposition Uh today is monday And that is sufficient for tomorrow is tuesday well By the way tomorrow is tuesday is sufficient for today is monday It can run in both directions, but it doesn't necessarily run in both directions. Now. That's the first word of warning just because Uh, just because one proposition is sufficient for the second does not necessitate that the second is sufficient for the first It can be but it doesn't necessitate second word of warning All sufficiency claims that if the first proposition is true then the second is also true sufficiency does not claim that the first proposition is true So we have another uh So so far example Let's try another one. Uh my my pet is a cat If that's true It's sufficient for my pet as a mammal now My pet in fact my pet is not a cat My pet is not a cat But the truth relationship still stands between those two propositions All sufficiency claims is if the first if the first proposition is true the second proposition is also true It makes no claim as to whether the first or the second proposition is in fact true So that that's sufficiency true makes true okay, uh sufficiency doesn't necessarily run in both directions, but it can And uh sufficiency does not claim that the first first proposition is in fact true All right, let's look at another proposition. Let's look at another truth relation beginning With the first proposition being true. So we looked at true makes true That's sufficiency. Let's look at true makes false right the truth of one proposition means another is false Okay, a second is false so, um This is called contrary Or contrariety. Okay, so this Organism is a tree You know if that's true. It's contrary to another proposition. Namely This organism is a dog The first proposition is contrary To the second if it's true the second is false uh You know if it's true that that organism is a tree then it's false that the organism is a dog. Okay So a couple of words of warning about Uh contrary um as with uh sufficiency We're not necessarily claiming that the first proposition is true so, um You know, you think about Actually what's kind of related to that is by the way, it's possible that both are false Right, so let's take a proposition. My pet is a cat That is contrary to uh My uh, uh, my pet is a tree right My pet is a cat and that's contrary to my pet is a tree now As with sufficiency nobody's it isn't necessarily a case that the first proposition is in fact true because as I mentioned it's false My pet is a cat Nevertheless, it's still contrary to the second proposition. My pet is a tree And having said that you know when we're talking about contra contrary All that's being claimed is if the first proposition is true then the second is false. By the way, it's possible that both are false This is this Another word of warning, right? So I guess we're on third word of warning. That's the first word of warning um Or second word. Anyway, first you know first word of warning It need not be that it's not necessarily the case. The first proposition is true second word of warning So your second word of warning is possible that both are false, right? uh My pet is a cat Is contrary to my pet as a tree and both are false because my pet is neither a tree nor a cat Here's the third word of warning a third word of warning um sufficiency Runs only art. No, no, sorry Can run excuse me sufficiency runs in one direction can run both but not necessarily Contrary Necessity it runs in both directions if one proposition is contrary to a second the second is also contrary to the first all right, so This organism is a tree is contrary to this organism is a dog And this organism is a dog is contrary to this organism is a tree okay So with sufficiency It can run both directions, but that's necessarily run both directions contrary It runs both directions all the time So the last two truth relations we looked at started with the first proposition being true the second two Truth truth relations we're going to look at start with the first proposition as false So, uh, let's try this thing do with this one false makes true All right, so the error of the first proposition means that the second proposition is true This is called sub contrary sub contrary Some really good examples of subcontrary relationships are things like explanations So we have a certain phenomena and you know given that phenomenon. There's a couple different things are possible, right? This is the way talking about it So maybe a really writing example for something like this would be uh when you're logging in to say your account at school And you get that error message and the error message says either you Either the username is incorrect or The password is incorrect Okay so Well, this is these are subcontrary relationships or subcontrary propositions, right? Your username is incorrect It you know, I suppose you like you double check right you look at the little field You see your username and lo and behold, it's all typed incorrectly Uh Well, then it's false that your username is incorrect. So It must be it must be the case that your password is incorrect I mean So these two propositions are subcontrary the error of the first means that the second is true So a couple of words of warning Uh as before with sufficiency and contra contra riety Yeah, this is just the truth relation And this is just the truth relation The truth relation says if the first is Falls that the second is true, but it doesn't necessitate that the first is in fact false I mean, it doesn't doesn't mean that the first is in fact false Uh, you know, it could be that you entered your username incorrectly, right? Um, but there still would be a truth relationship between your username is is incorrect and your uh password is incorrect So that's the first word of warning doesn't necessitate that the first proposition is in fact false second word of warning It's possible that both propositions are true right So with subcontrary relationships, you know with contrary it was possible that both are false right contra means They're the first is to the second is false But both could be false This one for subcontrary says it they both can be true okay, so Suppose you you type it in that first time you get the error message and you look at Your your name in the in the fields like oh, I did give my username incorrect Well, it's also still possible that you typed in the wrong password too Right that this can't happen So that's the second word of warning. It's possible that both are in fact false Uh, sorry, both are in fact true. It's possible both are in fact true uh third word of warning You know contrary relationships run both directions always and everywhere So do subcontrary relationships if the first proposition is some contrary to the second The second is subcontrary to the first Okay So that's false makes true that's subcontrary Last one is false makes false This is necessary necessity false makes false. So uh Let's try a tree again right say that organism Ism is a mammal That organism is a mammal. Well if that's false In fact is right if it's false then it's also false that that organism is a dog It's false that the organism is a mammal. So it's false that the organism is the dog um The first proposition If the first proposition is false the second proposition is also false That means that the first is necessary for the second I take a look at that you know over here again, right? We say this organism Is a plant if that's false it's also false that that organism is a tree Okay, so that's false makes false that necessity false makes false So again some words of warning here As we saw This is just the truth relationship If one proposition is necessary for a second All it's saying is if the first is false and the second is also false That doesn't mean the first is in fact false We're just looking at the truth relationship between the two propositions That's the first word of warning second word of warning like sufficiency Necessity can run in two directions, but it doesn't but it doesn't always run in two directions It can but it doesn't always so because a proposition is a first proposition is necessary for a second It doesn't mean the second is also necessary for the first It can But it doesn't mean that So, you know thinking about our monday and tuesday example, right? um You know Today is monday Is necessary for tomorrow is tuesday if it's false that today is monday. It's also false that tomorrow is tuesday By the way, if it's false that tomorrow is tuesday, it's also false That today is monday Necessity can run in both directions, but it doesn't necessarily run in both directions, right? So, um You know it said this organism is a plant Is necessary for this organism as a tree If it's false this organism plant it's false this organism tree, but The reverse doesn't run the the other direction, right? If it's false and an organism is a plant that doesn't mean it's false that the organism Excuse me if it's false the organism tree that doesn't mean that's false the organism as a plant, right? It's because there's plenty of plants out here that are not trees So first word of warning Sufficient just says if the first is false and the second is false, but it doesn't mean that the first is false Second word of warning necessity can run in both directions But it doesn't always run in both directions Okay, so just quick summary of the truth relations, right? You got true makes true true makes false false makes true false makes false True makes true that's efficiency true makes false. That's contrary False makes true that's subcontrary false makes false. That's necessary With all of them, right with efficiency and contrary It's just the relationship but Just because one proposition first proposition is sufficient for a second doesn't mean That the first the first proposition is true. Just the relationship stands same thing contrary And with subcontrary and necessity necessity, you know They both start with the assumption that the first proposition is false, but it doesn't necessarily mean the first proposition is false, right? Uh, the the relationship's just here with the relationship not whether the first proposition is in fact true or false Uh With sufficiency and necessity It can run in the relationship can run in both directions, but doesn't that doesn't always run in both directions All right, you can but it doesn't always with Contrary and subcontrary that relationship always runs both directions that the first is contrary to the second the second's contrary to the first If the first is subcontrary to the second the second subcontrary to the first Okay And then finally, right Contrary it's possible both are false It's possible both are false all contrary says is really saying is at least one of these is false and maybe both Not always but maybe and subcontrary at least one of these is true And maybe both not always but maybe So that's a quick summary right quick sum up of the four simple truth relations Now I want to mention one other thing and that's irrelevant right irrelevancy Irrelevant isn't really a relationship. It's the denial that there is a relationship But we're fond of saying well that first proposition is irrelevant to the second one It's not really a relationship just the denial that there is one all right, so you know this happens right this organism is a tree This organism is tree That you know as soon as true that has no truth that has no impact on the truth value of the sky is blue it doesn't Um, it could be that the sky is cloudy Right, it could be that uh, it's dark outside. That's the sky wouldn't be blue at all right be pitch black Uh, there's all kinds of way the sky could be there's no but that's not impacted at all by whether this organism is uh a tree Uh, so there are irrelevancies between propositions And the way you discover that is like okay assume right assume the first proposition is true And does it have an impact on the truth value of the second? If it doesn't move on to the false it's soon the proposition is false Does that have an impact on the truth value of the second? If it doesn't have an impact or out of the way the first is irrelevant to the second First is relevant to the second okay now this is important because um Yeah, it's important not only to be able to understand which truth relation exists between propositions But whether a truth relation exists between propositions if the truth relationship doesn't exist Probably can't make an inference from the first to the second Okay So we've got atomic propositions We're done with that. We've got truth relations between tonic propositions Now let's look at some exercises dealing with these kinds of propositions So let's take a look at some different kinds of exercises for atomic propositions of truth relations The first set of problems that you're going to attempt Uh is to attest your ability to spot an atomic proposition versus some of the kind of sentence so, uh You know this is going to have basically two tests right trying to figure out whether you can spot an atomic proposition Uh versus uh say an interrogative or an imperative or an exclamation All right, these are different kinds of sentences or even just sentence fragments See if you even have a sentence or you know something that is even a proposition And also if it is a proposition Is it a complex proposition or an atomic proposition? So this kind of goes through two tests, right? So the first test is uh Whether you're even dealing of uh with a proposition So look at the sentence, you know, is it an imperative? Does it make a command? Is it a Interrogative? Does it ask a question? Is it an exclamation? Is it just, you know, emotional response? Or is it something that's true or false, right? Is it something that's true or false? So that's the first test second test if it's that if it is a sentence that Gives you something that's true or false. Is it atomic? Is it composed of a single subject and a single predicate? Or is it somehow two subjects and two predicates or at least two such two predicates? Okay So let's take a look at an example here first Okay, so we have So the question right, which of the following is an atomic proposition? Well, look at the first one here Cats are mammals or cats are not mammals Okay, uh, is this an imperative? All right, is it commanding cats to be mammals? No, it's not right Um, is it an interrogative? Is it asking whether The cats are mammals cats? Well, it doesn't have a question mark, right? So it's not a question mark So it's not an interrogative. Is it an exclamation like yay cats are mammals or cats? No, it's not doing that either Yeah, this is a deed a proposition. This is a sentence That tells us something that's either true or false Okay, now the question is is it a complex sentence or a complex proposition? I should say is it a complex proposition? Um, well, how many subjects do we have? Well, we have cats, right? I mean cats has mentioned twice But it's you know, that's a subject and you know our mammals and are not mammals. Well, those are also two different Uh subjects. So, you know what I'm saying here is we can break apart this proposition We got cats are mammals the first part of the cats are mammals. Well, that's that's a proposition and cats are not mammals That's also a proposition So this is complex. This is composed of two atomic propositions And the two atomic propositions by the way are cats are cats are mammals the second one is cats are not mammals If I were looking for an atomic eight, you know, which sentence is the atomic proposition? Well, this one's complex So it's not going to do it second one salty ocean You know, this is at best Either a subject or a predicate, but it's not both a subject and a predicate. I mean salty sort of modifies ocean So, you know, it's maybe an adverb, but this in itself doesn't say some of this true or false And it's not imperative. It's not anything else like that, right? This isn't even a proposition. And frankly It's probably a sentence fragment. So it's something, you know, not even a complete thought Third option none atomic. So what's getting out of here is none of the options are atomic Well, we haven't gone through all of them. So selecting this one would be a little You know a little quick at this point So let's look at the last one the Atlantic Ocean is east of the continental United States Okay, now that's a long sentence. Yeah, but that doesn't mean it's not an atomic proposition Right all that we need for an atomic proposition. All that means is that there's a single subject and a single predicate Well, um The Atlantic Ocean, that's the subject And you have the predicate, you know is or is east of the continental United States of America That's the predicate. So this sentence even though it's long even though it's long. This is an atomic proposition That's the one All right, let's try Another kind of problem. So this one, right? We're given Given two propositions first and the second and the question is what is the truth relation from the first to the second? So, um, it probably you know the way to think through this problem Is to start with the first proposition and then assume it's true And if it's true Is the second proposition Impacted is the truth value the second proposition impacted. So if we assume it's true is the second proposition too But if so then it's sufficient or if it's the first is true is the second false, you know, so it's contrary If you assume it's true And you don't find an impact Then try assuming it's false And assuming it's false I mean, uh, if the first one's false is the second one true in which case it's sub contrary and or You know, if you assume the first one's false and you find out the the second one is false. Well, that is necessary Uh You know if you assume it's true and there's no impact you assume it's false and there's no impact Then you finally can conclude that it's irrelevant, but I wouldn't conclude it's relevant until you go through all that Okay, so let's let's take a look at this right some dogs are mammals Let's assume it's true. If it's true that some dogs are mammals Is it true that all dogs are mammals? Well, I mean No, it's in fact true that all dogs are mammals but not in virtue of it being true the some dogs are mammals After all it's true that some dogs are brown Right But it doesn't follow from that that all dogs are brown right dogs come in a variety of colors So the truth of the first does not mean that the second is also true So this it's not sufficient Well, is it false that some dogs are mammals? Well, no No, right if some dogs are mammals It's it's you know, can't be false that you know all dogs first all dogs are mammals. Excuse me can't be false It's not necessarily false that all dogs are mammals. Um Yeah, so it doesn't make doesn't make the second proposition false, right again Uh You know anyway, leave leave me that aside. Um So we didn't have any luck going from you know starting that from the assumption that the first one's true Well, let's assume it's false. Let's say it's false to some dogs are mammals Is it then true that all dogs are mammals? Well, I mean again not from The first one, right? I don't know from the assumption the first one's false So for instance, we can assume that some that is false that some dogs are What bright yellow? All right, we assume some dogs are bright yellow and we say that's false What does it follow them that all dogs? That is true that all dogs are bright yellow. Well, no, right. It doesn't follow from that. So, um I find the assumption that it's false that some dogs are mammals All right, it doesn't follow that it's true that all dogs are mammals Well, what about if we assume it's it's false That all dogs are males. Is it then uh false that all dogs are mammals? Well, yeah If it's false that some dogs are mammals Well, then it is not the case that you know, at least some of the dogs are not going to be mammals Well, that's not the case that that all dogs are mammals So, you know again, right? If we say it's false that all dogs are that that some dogs are bright yellow Well, that's going to be false that all dogs are bright yellow So, uh, the first one is false if we assume the first one's false. Well, then the second one All right is also false That makes this necessary All right, that makes this relationship necessary. The first one is necessary for the second Okay Last kind of problem whereas, you know, the problem we just looked at we are given two propositions And we're trying to find the truth relationship from the first to the second this one We're given a single proposition And we're asked to find Uh, which uh, you know, we say we we have this proposition. Well, which one is in this case is sufficient, right? Which one is sufficient? as this You know, we have this proposition Which probably for which proposition is this one sufficient? So we're supposed to go for the first With the truth relation to find another proposition That could be sufficient could be contrary. So contrary necessary, right? Whatever But in this case, right, we have no dogs are brown. I mean, that's, you know, the given proposition below is sufficient for which of the following Okay, so, uh, only is so the way to do this, you kind of have to test them, right? We can't just sit here and start conceiving of all propositions that are true because it's true that no dogs are brown Um, there's literally an infinite number. So I kind of to test the the propositions So if it's true that no dogs are brown Is it true that some dogs are brown? Well, no right That that can't be true, right? So, uh, in fact, if it's true that no dogs are brown, it's false. That's some dogs are brown Uh, so in this case, actually the first is contrary to the uh, the second Uh, if no dogs are brown, is it true that all dogs are brown again? No, right again? No In fact, these, you know, these are contrary to each other Some brown animals are dogs, right? Well, if no, if it's false If it's true that no dogs are brown, if it's true that no, you know, we have all the dogs and then we're brown Well, this one says well, some brown animals are actually dogs. No That's also false. The first three are actually each contrary Uh, you know, the three options are each contrary to the proposition Well, you know, by process of elimination, we figured out the last one Is actually sufficient other than, you know, the last proposition while we're looking for Um, you know, the first the given proposition is sufficient for This one here, but it's still helpful to go through the reasoning So on the assumption that no dogs are brown, let's say that's true, uh, is it true that some dogs Uh, are not brown Well, yeah That's the idea. You know, we have, you know, these dogs, none are more brown Uh, so at least some of those dogs are not brown Right now, you know, don't be fooled by the conversational, you know, convention that we have when we say well, some things are not this Therefore, some things are well, that's not true, right? So, you know, uh, some dogs are, um, are not bright yellow right Or uh, uh, well, that doesn't mean that some dogs are bright yellow That inference doesn't work. That's just a convention that we have is maybe we've got to drop that convention Okay, so there's three kinds of problems that we have for this, uh, chapter That's it for these, you know, sample problems. Um, good luck Okay, well, that's all I got for now We've taken a look at atomic propositions and the four simple truth relations that exist Between them that can't exist between them. We've also got a relevancy, but that's not really a relation Okay So, uh, we've even looked at some exercises Uh, good luck. I'll work on those exercises. Uh, I'll see you in the next chapter. Keep thinking