 So let's take a look at this distinction that humans give us between relations of ideas and matters of fact. Now, humans still basically an empiricist. He thinks that platonic form should be cast to the side, right? If it's not empirical, right? It should be cast to the fire. So here are two ways that he tells us that we can justify our beliefs. Either our beliefs are justified as a relation of ideas or our beliefs are justified as a matter of fact. Now, matter of fact, these beliefs are true, right? So here's a matter of fact. I am wearing a white shirt. I am six feet tall. There is no present king of France, right? These are all matters of fact. They are true, but it's possible they could be false, right? France did have a monarchy for a long time. It's possible in some sense of the word that they could have, you know, either the monarchy survived or they started a new one. I am six feet tall, but it's possible I could have been at different height. I am wearing a white shirt, but I have a wide variety of shirts. I could have worn something other than white today. So these are all true, but they could have been false. They're in fact true, but they could have been false. And we verify that these beliefs, these propositions are true mostly through experience, right? Through our everyday experiences, okay? So I observe my shirt as white. Hey, I'm wearing a white shirt. That's justified. I look at the French former government. It's not a monarchy. Well, then there I know this. There's not a present king of France. I measure myself. I'm, you know, go to the doctor's office. I measure myself. Hey, look at that. I'm six feet tall, right? These are all verified, proven through experience. The other way that beliefs are justified the way that they're proven is as a relation of ideas. Now, relations of ideas are true and they must be true. They cannot be false. They are true and they must be true. They cannot be false. And they're true and virtue the meanings of the terms, right? The meanings of the terms necessitate that these propositions are true, okay? So it's something like all squares have four sides. That must be true. How do I know this? Well, through the meanings of the terms, right? What's the definition of square? It's an equilateral, equi-angular, quadrilateral. Quadrilateral means four-sided. Therefore, all squares, which are equilateral, equilateral, quadrilaterals have four sides. So it must be true in virtue of the meanings of the terms. That's one way we could test it. We could test it another way. Let's assume it's false. Assume the proposition is false. And if we derive a contradiction from that assumption, then it can't be false. Okay. So all squares have four sides. If I say that that is false, I say that some square does not have four sides. And again, the definition of square is equilateral, equi-angular, quadrilateral. So when I say some square does not have four sides, I am saying some four-sided figure does not have four sides. That's a contradiction. It's inconsistent in and of itself. So it can't be false. That means it must be true. It must be true. Okay. Now, that's the difference between relations of ideas and matters of fact. Let's try one more relation of ideas. There is a largest number. There is a largest number. Well, I'm sorry. There is no largest number. Excuse me. There is no largest number. There is no largest number. Well, how do I know that? Well, by the definition of number. And so we take something like, you know, these three basic axioms of a number, zero is a number, successor of a number is a number. These are numbers and those three axioms, I think called pianos axioms. Okay. Well, we take that together and from this we can infer there is no largest number. Why? Well, because any number that we say is the largest number, I can simply add one to it and boom. I've got another, I've got a larger number. Okay. Well, that's one simply virtue of the meanings of the term. Try just boom, I've showed that that there is no largest number because any contender that you pick, it doesn't matter which one, I can add one to it and it's larger. Well, let's, let's assume it's false. There is a largest number. There is a largest number. Well, okay. So what happens if I did that? Well, I take that number and by the definition number, right, I add one to it and I've got a largest number, a larger number. So then that number is both the largest number and it's not the largest number. So now the contradiction. Okay. So there's two ways to prove something is true as a relation of ideas. Uh, you know, it's true simply in virtue of the meanings of the terms, right? It can't, it can't be false. So it's true simply in virtue of the meanings of the terms. The other way is to assume it's false. And if you assume it's false and derive a contradiction, then it can't be false. It must be true. So what we're going to do is, we've got a collection of propositions here. All right. I want you to go through and try to figure out which ones are true as a relation of ideas and which ones are true as a matter of fact.