 Hello everyone, welcome to this chapter rules of implication. Let's take a look at some of these practice exercises So ah for some of these you're going to fill in the citation and for some of these you're going to fill in the formula So with the citation one quick tip the first premises listed You know, excuse me for the first citations are almost always going to be the premises that are given for the Conclusion so those are pretty straightforward, right? So we got the two premises listed in this sequence Well, they each receive an a for you on the assumptions or the premises within the argument the only time when this isn't going to happen is when we have a Formula that's you know, impossible to be false, right? It must be true We might get to those later on but for right now don't normally need to worry about it so we have Our two premises here if p then not q and you know, not not q and so the the question is well What rule now do we use to get to the conclusion not p? using these two Premises so you can think about the different rules that we have as a disjuncted syllogism. Well, we'll know right? There's no disjunction here. Is it dilemma? No for the same reason. There's no disjunction here so you think about the rules that are available and How the at what inference is used so the two premises? The first premise there should say has a conditional And if you notice the second premise is the negation of the consequent of the conditional Well, then that lets us infer not p using the rule modus tonens or it's like most all it's not Tony's most all So, you know take a look at the premises to see what rule is available to draw that inference Let's take a look at another one. So we've got P or q and not p as the premises and notice they're always going to be in order You always list the premises in order as they appear in the sequent and again They both receive a right and that's our citation for an assumption So the question is well, what rule allows us to infer q from this disjunction? Well, you know, that's just junked of syllogism, right? So we have that disjunction and the negation of one of the disjuncts so we get to infer the other disjunct So we have our citation one comma two comma DS, right? make sure you have that space between the last comma and the The rule and make sure there is no space between the numbers, right? I just want those numbers listed one right after the other Okay, one more now. We just got two lines here are two entries So we have a single premise That's p and q and so we get to infer p. Well, what do we well? We got two rules for conjunction conjunction introduction and conjunction elimination or not creating a conjunction here We're pulling out one conjunct from a conjunction. Well, that's conjunction elimination. Let's try another one now This looks a little different doesn't it? So we have a single premise p. So we got that citation there and then we have the sequel. Well How do we how do we do this? Well, think about the rules that allow us to infer a conditional right so we're not good. So the the Conclusion is not a conjunction. So we're not going to use the conjunction rules The conclusion is not a disjunction. So we're not going to use the disjunction rules And so what's what's the conditionals? We got modus ponens. Well, we don't have a conditional by which we can infer, you know This long conditional on the conclusion. It's not modus ponens We're neither getting the affirmation of the consequent or the demilion to seed him or not And we're not going to be able to conclude either one of those It's not a hypothetical syllogism allows us to conclude a conditional. True But we have to use two conditionals to do that. Well, what's left? What is conditional proof? Conditional proof is the last rule available. So look at that Conclusion and the conclusion is a conditional if p then q That's the intercedent and the conclusion for that. Excuse me the consequence for that is then q Well, what do we? How do we you know have you know if p then q what you get to assume that for the sake of conditional proof? And that's why you had that citation there assume for conditional proof. That's what that means Okay, well remember for an assumption for a conditional proof We assume the intercedent and we infer the consequence. So that's q That's the consequent there of that whole conditional. How do we get to do that? We get to use one and two Lines one and two using modus pollens, right? That's how we get q And from all of this we get to infer the conditional The using the first you know the first number cited there is the assumption of the intercedent The second number is the is where we infer the consequent and from that we get to infer the whole conditional Let's look at another one where we're typing in the formula So here's another little tip right when you're typing in the formulas the first First lines listed should always be the premises the last line listed should be the conclusion. So you already know All right, we got one premise here You already know that that's going to be p and q that's going to be listed first and somehow that line four Has got to be q and p Well, then how do we do that? How do we get q and p? Can we just switch it around? Well, you can but that's an equivalence rule which we're going to learn later So right now we're not going to do that right when I know I'm going to do it the hard way Well, you know we get to pull out q You know look at the citations on the side there the citations on the side give you clues to what you're going to infer and Using what rule so from line one using conjunction elimination. We get to pull out q Well line three also uses line one using conjunction elimination with we only have one conjugal left Well, that's p and then we get to put them together in the order that we wanted q and p Okay, using lines two and three with conjunction introduction All right, let's take a look at another one. It's a little bit longer and this time we're doing the citation again All right, we got the citation again So we have two premises if p then q and if q then r right those are two premises And the conclusion is if p then r So think about the rules that we can use now we can use you know if we were doing this differently Right, we could use hypothetical syllogism and just simply conclude if p then r But I thought I might challenge you a little bit give a little practice using conditional proof So here I have p. Well, how do I have p there? Well, think about it. What rule do you use? right To infer a conditional And look at that because we're inferring conditional right the last line is there if p then r inferring a conditional What rule do we use to infer a conditional where we just start out by assuming the intercedent? Well that again, that's a assumption for conditional proof So by using assumption conditional proof we can get q we use you get q from lines one and line three See that q that's up there line one and line three using modus ponens And now we're looking for r. Well, how do we get r? We'll look at line two and line four Right again using modus ponens. We get to get to infer r. Well, now we have the inference of the consequent We've inferred our consequence So we get to infer the conditional we assume p at line three we inferred q That's good. We inferred r at line five. So by conditional proof. We now get to infer if p then r Uh, well now this what looks interesting We have a conditional in for another conditional Okay, so we get so remember our little tip here right the first Line we enter is always going to be the premise and the last entry should always be the conclusion That we're looking for in this case It's a conditional a really long conditional conditional composed of conditionals When the antecedent is if q then r the consequence of p then r How do we do this? And we got that acp there that tells us we're using a conditional proof Well, we get to assume if q then r for conditional proof, right? That's the antecedent. We want to get there So when you assume an antecedent it doesn't need to be an atomic preposition atomic proposition. It could be a complex one It could be a very long complex one if we really wanted to But you know for now, I'm just going to assume the antecedent q then r then using Hypothetical syllogism right look at look at the citation there's using the lines one and two of hypothetical syllogism Well, what would you infer if p then q q not? Well, then we get to infer if p then r Well, now there's our consequence for the conditional and the conclusion. Okay, so we conclude with our concept with our Conditional and the first number is the citation where we assume me to see the second number is what we infer the consequent We get to infer that we Excuse me site it using cp for conditional proof Well, here's a long one What are we gonna do here near now? We're typing in citations, right? We're not putting in formula But we're typing in citations. Well, what do you think we're gonna use right? We start out with the disjunction We end with the disjunction that should be a big clue if you start with the disjunction and with the disjunction You're probably you know think about the rules for disjunction Disjunctive syllogism does that finish with the disjunction? Not really I mean it can as possible if one of the disjuncts is itself a disjunction, but you know, that's not the case here We also have Disjunction and two conditions. Well, that should be a clue that should tell you that we're going to use overall is dilemma We'll use dilemma overall So we have our a I mean we've got three premises here So we have a after each premise And that tells us that's our assumption for the argument And the way dilemma works is you you start with a disjunction You assume one disjunct and you infer Something further than you assume the other disjunct and you infer and you know an even more thing, right? You can infer a disjunction with those two inferences so that that's a lot to take in We'll just look at this right, right? So we got not p. That's our assumption for disjunction. That's why we have that Citation they're assuming for disjunction. Well using p. How do we infer not r? Well, where is r if we look up in our premises? Well r is up there on line two Okay And since we have not p we'll infer not r using modus tolin. So line two and four using modus tolin Now we assume so we got one side of the disjunction, right? Look at the conclusion. We got a disjunction And we got not r from from not p So let's try and get not s. Well, we got to get not s by assuming not q for the sake of just using a assuming for disjunction And we can easily infer not s using modus tolins, right? We found same thing as the uh, as the other side We did that earlier But now we're using the other conditional Now since we've made this inference using each Part of the the disjunction we get to infer a disjunction We get not r And you say we have so look at the citation there we have line one. That's the initial disjunction The second number that number four that's where we assume one of the disjuncts Five is where we infer Not r using that assumption not p line six is where we assume the other disjunct And line seven is where we make the the inference for the other side of the disjunction in the conclusion Okay, so that's how the limo works Okay, well this one looks puzzling we start with p And we say okay. Well now I want to get this disjunction here. Oh, how do I get the disjunction? Well, if you're going to start with a single proposition you're going to infer a disjunction. That's disjunction introduction And yeah, it's a complex disjunct That you're informed but you get to do that. It's disjunction introduction. You can you can make a complex inference All right Well, wow now we got another one. Well, this looks familiar doesn't it? we got p Right as one of as is that one of our premises. We've got a conditional if p then s That's another assumption The assumption we get to listening and we got in the third conditional if q if q then t and we're supposed to infer s disjunction t Well, okay, so if you take a look you can see how we get s If you look at lines one and two we can get s using modus ponens And if we if we were just You know, there'd be a simple way we could do this right if we just have those two lines and we can infer s and then using disjunction introduction Have s or t but I thought I'd challenge you a little bit and you know, see what's going on here Uh, I want you to use we could solve this using disjunction introduction. That's fine All right, so you look at that line four and you might think oh, well, you know, that's what I'm gonna do But uh, we're not going to be able to do that yet right because look at that citation It's using line one with disjunction introduction Well with line one, we don't get to infer s or t Because that's just that's just p is right there in line one. So what do I have in mind here? Well, we've got two conditionals If p then s and if q then t So how are we going to get that? Well, we can use the lemma again You say okay great, but I don't have a first disjunction Well, you can get that disjunction using p with disjunction introduction We got p or q and now we have our disjunction that we can use for the lemma So we start by assuming p for the sake of uh disjunction Then we can even using modus ponens line two and five we get our s All right, then we assume q for the other side of the disjunction and using uh lines three And seven right we can use modus ponens to infer t And now we get our disjunction Now the first number is the initial disjunction p or q second numbers where we assume the left disjunct This third number is where we conclude Make new broad inference from that left disjunct the Fourth number is where we assume the right disjunct And the fifth number is where we draw the conclusion And then finally right we have our srt and that that's where we get that's where that's our disjunction that we can to infer uh using dilemma All right, so this is some practice good luck on the uh on the exercises You're gonna be filling in citations and you'll be filling in formulas we alternate between those two keep thinking