 Well, we just got finished studying our categoricals, our types of categoricals, and already we're going to get to jump into some inferences. These are called immediate inferences, because we don't need any other proposition other than the one we're dealing with. So with these categoricals, the universal affirmative, the universal negative, the particular affirmative, the particular negative, we're going to be able to make some inferences from them like that. Let's get started. These are some big rocks. Okay, so to understand some of these inferences, we're going to start with a couple of concepts. The first is deductive validity. Deductive validity for an argument means that if the premises are true, the conclusion must be true. It's impossible for the premises to be true and the conclusion false. A premise, as you might have figured out, are the propositions from which we draw an inference. And specifically from which we draw the conclusion. The conclusion is that which is inferred from the premises. So deductive validity means if the premises are true, the conclusion must be true. By the way, this doesn't necessitate that the premises are true. So, okay, so for instance, let's look at this. So if my pet is a dog, then my pet is a mammal. That's a true proposition. If my pet's a dog, then my pet's a mammal. My pet, Penny, is a dog. These are two true propositions. And from these, when inferred the conclusion, my pet is a mammal. It's impossible for the premises to be true and the conclusion false. Now, let's try another argument, which is so that that argument is deductively valid. And it happens to have true premises, and from that we have that true conclusion. Here's another argument, if my pet is a cat, then my pet is a mammal. My pet is a cat, therefore my pet is a mammal. Now, this is still deductively valid. If those premises were true, the conclusion must be true. But, by the way, the second premise is false. The first premise is true, if my pet's a cat, then my pet's a mammal. That's true, because cats are males. But my, about the second premise is false because I don't have a pet cat. So, just because an argument is deductively valid, doesn't mean that the premises are true. By the way, the conclusion is still true, right? My pet's a mammal, but that's just kind of accidentally true. Even though, it's deductively valid, okay, but deductively validity doesn't necessarily mean that any conclusion is true. The conclusion that follows is true. It just happens to be, in this case. You know, even if the premises are false and the conclusion is false, the argument can still be deductively valid. The deductive validity is just, an argument is deductively valid just because of the relationship between the premises to the conclusion. So, here's an argument. If I teach a Senate attorney college, then I have wings on my back. I teach a Senate attorney college, therefore I have wings on my back. Now, the conclusion's false, really false, but the argument's still deductively valid, and it's deductively valid because of the relationship from the premises to the conclusion. So, that's the first thing, deductive validity. Now, you might ask, well, why are some arguments deductively valid and some arguments aren't? Because just because you have true premises and a true conclusion that doesn't make a deductively valid argument. So, for instance, if my pet's a dog, then my pet's a mammal, my pet's a mammal, therefore my pet's a dog, all true premises, true conclusion, not deductively valid. So, what makes an argument deductively valid versus not? And what makes an argument deductively valid are the truth relations from the premises to the conclusion. Okay, so I promised you an explanation for why some arguments are deductively valid and some are not. It has to deal with what's called truth relations. A truth relationship is a relationship between two or more propositions, and we're just going to actually, we're just going to deal with two, right? Two propositions, such that the truth or error of one proposition affects the truth or error of another, right? So, the truth or error of one proposition affects the truth or error of another, okay? Now, we're going to do with a handful here of truth relations just with, or at least let's start here, just with our categoricals, right? Just with our categoricals. We're going to have subaltern, subcontrary, contrary, and contradictory. So, we're going to have four truth relationships with our categoricals, and then these relationships are going to form this square, traditionally called the square of opposition. We're going to chart them out according to the, you know, we're going to put the right categorical in the right corner, right? We could chart out those relationships, and we'll be able to infer the truth or error of one proposition, one of the categoricals, at least some of the time, from the truth or error of others. Subaltern, the particular affirmative is a subaltern to the universal affirmative, okay? And what that means is, if the universal affirmative is true, then the particular affirmative is true. If the universal affirmative is true, the particular affirmative is true. So, all trees are plants. All trees are plants. This says everything in the category tree is also in the category of plant. All right. Now, since that's true, what's also true is some trees are plants. Now, immediately this sounds weird, because you're used to thinking of some, when we say some, we mean, well, some are and some aren't. Some are and some aren't. That's probably what happens in conversational English at least some of the time, not all the time, but some of the time. But remember that some just means that at least one, right? Some means, just means at least one. So, since all trees are plants, it's also true that at least some of them are, at least some of them. Now, a word of warning, subaltern, the subaltern relation does not go from the particular affirmative up to the universal affirmative. Here's something that's also true. Some plants are trees, right? That's true. Some plants are trees, but it's false that all plants are trees. So, the inference from the universal affirmative to the particular affirmative is deductively valid, but not the inference from the particular affirmative to the universal affirmative. That is not deductively valid. So, the inference from the universal affirmative to the particular affirmative is deductively valid. The inference from the particular affirmative to the universal affirmative is not, sometimes it's true, sometimes it's true that both the universal and the particular, namely when the universal affirmative is true, both are true. right both both are true but the relationship does not necessarily hold right from the particular ferment to the universal therefore it's not deductible valid. The universal negative right we have the universal negative the particular negative is also subaltern to the universal negative right so if the universal negative is true then the particular negative is also true so uh no tree is a mammal right no trees are mammals that's true and since no trees are mammals the particular negative some trees are not mammals it's also true right so the so the inference from the universal negative down the particular negative is deductively valid the inference from the particular negative up to the universal is not right the inference from the particular negative up to universal is not so um some reptiles are not lizards right some reptiles are not lizards that's true right there's lots of reptiles that are not lizards you know snakes for example but the inference to the universal negative therefore all reptiles are not lizards that's false right that's false uh there are some reptiles that are not lizards and there are some that are right um so the subaltern relationship goes from the universal down to the particular maybe that's the way I say that right in subaltern relationship is from the universal down the particular regardless if it's the universal negative to the particular excuse me universal negative to the particular negative or the universal affirmative down to the particular affirmative. All right, in case you're wondering, I'm in Los Maple State Park in Texas. All right, so the next one is contrary. What contrary means is that as one is true, the other's false. If one is true, the other's false. All right, so for example, well, not for example, yeah, the two universals, the universal affirmative and the universal negative are contrary to each other. So all dogs are mammals. All right, that's true. And since it's true, it's false that no dogs are mammals. All dogs are mammals is contrary to no dogs are mammals. Okay, here's universal negative. That's true. No reptiles are dogs. No reptiles are dogs. Since that's true, it's false that all reptiles are dogs. So the two universals, universal affirmative and universal negative are contrary to each other. If one is true, the other's false. Now, before you make a mistake here, right, I want to warn you, all contrary means that if one is true, the other's false. That doesn't mean that at least one of them is true. They can both be false. They can both be false. So here's one that's false. All mammals are dogs. Right, that's false. There's lots of mammals that are not dogs. So for example, you know, cats. So all mammals are dogs is false. But what's also false is that no mammals are dogs. Right, that's false and no mammals are dogs. So they can both be false. All contrary means that if one is true, the other is false. And it's the two universals, the universal affirmative, the universal negative. I'm going to keep going. You know, they say 20 minutes spent in nature, as little as 20 minutes spent in nature a week, does a lot to prevent depression. And being out here, I can see why. Okay. Subcontrary. Subcontrary means that at least one of the propositions is true. Okay. If one is false, the other is true. And vice versa. If one is false, the other is true. So let's take a look at this. Some dogs are not mammals. That's false. Right. It's false that some dogs are not mammals. Well, since that's false, it's true that some dogs are mammals. Right. And that's the duck will be valid. Some dogs are not mammals is false. Therefore, it's true that some dogs are mammals. Right. Now, as with contrary, just because one, just, you know, I'm sorry, as with contrary, remember contrary said, at least one is false and possibly both. Right. It doesn't necessitate that, that any one of them is true. Subcontrary is the same way. At least one is true and possibly both. All subcontrary says is that if one is, if one is false, the other is true. That doesn't necessitate that other one is actually false. So some dogs, some mammals are dog, our dogs, that's true. Some mammals are not dogs is also true. Now the particular affirmative is always, always, always subcontrary to the particular negative. A particular affirmative is subcontrary to the particular negative. All right. Well, let's look at contradictory. Now contradictory means that if one proposition is true, the other one is false and vice versa. If the first is true, then the second is false. If the first is false, then the second is true. Right. These two propositions have opposite truth values. Right. So here we go. Let's take a look at this. Some creek beds do not have water. Right. For example, some creek beds do not have water. Well, since it's true that some creek beds do not have water, namely this one, it's false that all creek beds have water. Right. It's false that all creek beds have water. The universal affirmative is always, always, always contradictory to the particular negative. If the universal affirmative is true, the particular negative is false. If the universal affirmative is false, the particular negative is true and vice versa. So that was an example of a true particular negative and a false universal affirmative. Here's a true universal affirmative, our very famous familiar example by now. All dogs are mammals. Since that's true, the contradictory, some dogs are not mammals, it's false. So contradictory, universal affirmative to the particular negative. Also, the universal negative is contradictory to the particular affirmative. Right. Universal negative is contradictory to the particular affirmative. Here's the universal negative. That's true. No dogs are reptiles. Right. Since that's true, it's false that some dogs are reptiles. Here's a particular affirmative that's also true. Some creeks have water, not this one, but some of them out there, right? Some of the creeks out there have water. Now, since it's true that some creeks have water, it's false that no creeks have water. So the universal affirmative is contradictory to the particular negative, and the universal negative is contradictory to the particular affirmative.