 In proofs, it's common to use words that connect two different statements together. These could be things like since. Since this holds, that holds, or because, because, you know, because I've been given much I too shall give. You know, something like that. You're connecting things together as for, so, these again are all connecting two statements together. But it's not just, it's not just a conjunction that's arbitrary. It actually suggests some type of implication. Since P holds, you know, since, let's write that one out there, since P holds, Q holds. So like, since P is true, Q is true. We could say something else like P is true, so Q is true. Okay? There's an implication going on there because, because of P, we also get Q as P holds Q holds. So we have all these examples of things like that. And these are all suggestive of the implication that P implies Q. We can, of course, reverse the direction. Sometimes we'll say things like Q holds since P holds. We could also say like Q holds because of, I guess my because is on the other box over here. Sorry about that. Q holds because of P. All right? If we want to do something like so, we could say something like, let's finish this one up, Q holds as P holds, something like that. So I mean, if you did like Q happens so, so it kind of the directions a little bit problematic, but you could switch it up with like a four. It's like Q happens for the sake of P. Again, these are, these are all examples. And I know with, with the primitive statements P and Q, it makes a little bit awkward language, but we connect these things together. This second line is you're really just going the other way around. We're saying that you put Q first, but you're saying Q is implied by P. These are all implications that we're considering here. And so notice that the meanings of these constructions is a little bit different than saying if P holds then Q. Okay? Now you might be like, what are you talking about? You kept on writing these implications. This, these statements as we have written down here are actually they, they have this implication P implies Q, but they also have the implication that, let me rewrite this. We have this implication, let me get rid of this for a moment, each and every one of these statements, like you take something like this. It's actually an argument. We're saying that yes, P implies Q, but we're also saying that P is true. Therefore, because P implies Q and because Q is, or because P is true, we have that Q is true. This is our modus ponens argument. And so when we say things like since P then Q or since P Q holds, this right here is just a compact way of making that valid argument. And so when you're writing these things, it needs to be understood that's what these words mean because since, et cetera, it means it's the entire modus ponens argument here. There's an implication that must be true and the hypothesis, the premise must also be true. P is true. And that is why we can infer that Q is true. You also want to be careful about the directions. So saying since P comma Q, that's not the same thing as saying since Q comma P, right? That would then give us the converse of the statement, that would be no good right there. You want to switch it up something like this, right? Be careful where you place the words, where you place the commas. And so be mindful of this as you're writing things. All right, so let's look at a bad example and then fix it to make it a good example here. Take the sentence n is a natural number comma so z. So one thing that's problematic of here is that when you have too much mathematical symbols, you should try to buffer it using words, which we'll see like that in just a second. But also you need to make sure that the statements involved are actually statements like z is not a statement, it's a set. So what the person's trying to say here is that because n is a natural number, n is also an integer, right? But again, they're lacking words. This isn't even a complete statement right here, it's just a set. We can improve it in the following way. Since n is a natural number, we also have that n is an integer. Notice the word also here is just a different version of so. The also conjunction there is just suggesting that there's a so, but you're saying that this thing has the same property as the previous thing because every natural number is an integer. And so be very cautious with your, maybe not cautious, you shouldn't be scared of them, but be careful, be intentional with your words since, because, as for so, also, etc. Because these words don't just suggest an implication, they actually suggest the entire modus ponens argument. Don't use them unless, of course, that argument is applicable here. It's a valid argument. We also have to make it a sound argument, right? We need to make sure these things are true. Then we can conclude q and also make sure your directions go in the right direction. Just be intentional with your words and you'll be just fine and feel free to use words like sense and because it's a good way of combining sentences to make the composition more natural instead of making everything so choppy. Sometimes a long sentence is good, something like this to break up the monotony of all these short sentences over and over and over again. All right. And so with that, that brings us to the end of lecture 27. Thanks for watching. If you learned anything in this lecture, please like the videos, subscribe to the channel, see more videos like this in the future. Feel free to share these videos with friends and colleagues who might also be interested. And as always, if you have any questions on any of the topics in any of the videos I have here, feel free to post them in the comments below and I'll be glad to answer them as soon as I can.