 bit of a rocky area around here. So far what we've seen may not look like it's real familiar or look like it's actually reasoning, but all of what we've seen so far is going to be used to justify the rules of implication. We're going to have rules of implication and these are rules that allow us to make inferences. Now we've been talking about validity. We tested the validity of premises and conclusions with these truth tables, which is fine. It's not bad. But telling us that in argument is valid is not the same thing as understanding the steps from the premises to the conclusion. And sometimes it's going to be more than a few. I wager that when you all looked at some of these arguments with the truth tables, you had these premises and conclusions, sometimes you looked at the conclusions like, I don't think so. Well, the rules of implication are valid. If we follow these rules properly, using the premises, whatever we infer from these rules, whether they're surprising or not, whatever we infer from these rules, that's going to be valid. If these premises are true, these conclusions must be true. So we'll be solving lots of formalized arguments. And when we look for a solution, we'll have a set of premises, which you're going to talk about when we looked at the sequence. We'll have a set of premises. And from these premises, we are going to try to infer that conclusion. Now to be clear, we can infer lots of conclusions, but we're not going to look for all the possible ones I would take forever, literally. Instead, we're going to look for the specified conclusion given in the sequence. So we'll be solving these formulas. We'll be solving these problems in logic. I'm going to show you these rules. Actually, here we go. Let's take a look at just kind of a group shot of all these rules of implication so far. This is going to be more later on. But so far, these are the rules of implication we'll be looking at. Now what I'm going to do is I'm going to take these rules in smaller chunks and use them to draw inferences. As for the explanation of the rules themselves, I'm going to go for that a little bit in this video, but I'm going to let you read the text more to get that fuller explanation. Now here primarily I'm going to show you how to reach the conclusion from these premises using these rules. Hopefully, it won't be as rocky as this area. So the first set of rules we're going to look at deal with conjunctions. If you remember, as we got from the text, we got conjunction introduction and conjunction elimination. This is just a really fancy way of saying we're going to, for conjunction introduction, we're going to create a conjunction using two already agreed upon truths. And conjunction elimination just says, well, from the conjunction, we're going to be able to infer either one of the conjuncts. So let's take a look at a quick problem of this. So here we have, forgive me for using notes, I'm not quite as talented to remember all the problems at once. So here we have two premises. We just had the atomic proposition p and then we had the atomic proposition q and r. And from this, we're trying to try to refer one big conjunction p can join with the other conjunction q and r. And so when you have a conjunction, right, so we look at the conclusion, we have a conjunction. When you have a conjunction, this should already tell you that, you know, there's a good chance here that the rule that you're going to use is conjunction introduction. And this is something to keep in mind when you see the conclusion, spot what kind of complex proposition it is already that can not always, but can provide a clue as to the rule that's going to be used, at least at the end, to infer that conjunction. And as you're going to see in a little bit, the first few problems are going to look at it pretty straightforward, pretty simple. But as you see a little bit later on in this video, there might be several steps, many steps to reach the conclusion. And having said that, we have this conclusion of p and the conjunction, p can join with the conjunction of q and r. Well, again, this is a really simple thing. So we just look at the premises, we have available our two assumptions, right, marked with a, when we find p, we find the conjunction of q and r, this is not a difficult thing to figure out. We just simply use the rule conjunction introduction to infer that whole conjunction that we're looking for in the conclusion p can join with the conjunction of q and r. So again, not a difficult one. Let's look at something maybe a little bit different. All right. So this time, we've got p, we've got our same two premises, right, we got p and the conjunction of q and r. But this time, instead of conjoining p to that other conjunction, we just want to conjoin p to r. So you look at that conclusion, we got p and r. Well, the conclusion is still a conjunction. But this time, right, we don't have p just simply, excuse me, we don't have r just simply by itself. So we can't just make a simple inference from p and one line and r and another to p and r. We're still probably, since it's a good junction, we're still probably going to use conjunction introduction at some point. But this first step, but it's not just a one-step or like the last one. Well, take a look at the premises that way. Take the two assumptions in lines one and two. Well, we got p and one, well, hey, there we go. We already know what one half of the conjunction is coming from. Then we got q and r in line two. Okay. Well, r, right, this is again another clue when you're trying to solve these proofs. Find the predominant propositions in the premises. This is going to give you some clues to which rules you use to infer the conclusion. Well, r is one part of that conclusion. In line two, r is part of a conjunction, q and r. Okay. Well, we need to infer or pull out that r from that conjunction. Well, conjunction elimination allows us to do this. From that conjunction q and r, so we go to line three, that conjunction q and r, we can infer just r, just r. Well, and then we give our citation line two using conjunction elimination. Well, now we have p on one line and r on another. We want to infer the conjunction of p and r, so we can do that using lines one and three with conjunction introduction. Okay. Let's take a look at one last problem for conjunctions. A slight difference this time. We have p for one premise, for one assumption, q and r for another, except this time we want to infer p and q. Well, that's fine. We can, right? Using pretty much the same steps as before. The only difference here is that we're going to infer q from the conjunction of q and r. And this is fine with conjunction elimination. You can infer either conjunct, just using conjunction elimination. You can infer either one or you'll pull them out separately. That's fine. You can do that. We can infer either conjunction from the conjunction, not just one or the other, and then we just simply can join p and q together using our rule conjunction elimination. So the conjunction rules are not that complicated. And we're going to pretty much use the same strategy in solving all this. You look at the conclusion, figure out one kind of complex proposition is there, and from that it's going to give us a clue as to what kinds of rules are available to us to infer that conclusion. And then you'll look at the premises, try to find the conclusion in there, where is it located in the premises, and what rules can you use to pull out those atomic propositions or sometimes complex propositions, but can you just pull them out of those premises to infer that conclusion? A little less rocking. I'm going to go this way. Okay, well, we got a fork in the road. Since we can go either way, it seems like a good time to mention the rules about disjunctions. Now disjunctions, we got two main rules for now. Two main rules for disjunction. We got disjunctive syllogism, and we've got disjunctive introduction. Now disjunctive syllogism allows us to take a disjunction. Disjunction is true. And if we have the negation of one of the disjuncts, remember disjunctions are subcontract, at least one must be true. If we have the negation of a disjunct, then we can infer the other disjunct. So from the claim that at least one of these atomic, or one of these propositions is true, we have the negation of one, we can include the other. So that's disjunctive syllogism. Disjunctive introduction allows us to infer a disjunction. Now we can infer a disjunction pretty much from anything. I know this sounds weird, but remember, a disjunction just claims that at least one of these propositions is true. Well, if we already have something that's true, I am outside. I can infer a disjunction. I am outside, or I am inside. Now it's in fact true that I'm outside. Okay, but remember, we're just concerned about validity. If I am outside is true, the disjunction, either I am outside, or I am inside is also true. So that's disjunctive introduction. It seems a little weird. I mean, you could literally you can infer a disjunction with anything if you really want to. So again, from the proposition I am outside, we can infer either I'm outside or I had a turkey sandwich for lunch. We can infer either I am outside or there are 13 space colonies on Mars. We can infer either I am outside or get off the gray really wants me to go on a quest. We can really infer pretty much anything, any disjunction, excuse me, we can infer any disjunction from one of the disjuncts. That's what disjunction introduction allows us to do. So if you see a disjunction as a conclusion, well, that you might be so decent chance that that dis or at least a possibility should say that the rule used to infer that disjunction that could be just disjunction introduction is not the only possibility, but it's at least a clue. Okay, so let's take a look at a couple of problems then using disjunction introduction and the disjunctive syllogism and really we're going to help ourselves to conjunction introduction and conjunction elimination as well. Okay, so here's here's a problem. We got two premises P is the first premise Q is the second and from this we want to infer P or R the disjunction P or R and Q. Okay, well something to take a close look at that conclusion. We have a disjunction which seemingly appears out of nowhere. That's a clue that disjunction introduction might be used. We have a disjunction that seemingly appears out of nowhere and it's conjoined to Q. Since the entire conclusion is a conjunction, we probably use conjunction introduction to infer that conjunction. Now I want to I want to make a word of caution here. Disjunction introduction allows us to infer any disjunction we please from a single proposition. So any disjunction containing the proposition from that single proposition. So if we have P, we can infer P or Q. If we have P, we can infer P or R. If we have P, we can infer P or the conjunction Q and T. If we have P, we can infer P or the conditional S or T. We can infer any disjunction we please from a single proposition as long as that proposition is contained in the disjunction. You cannot do this with conjunction introduction. Conjunction introduction both conjuncts have to already appear in your list of premises. So I just want to make that clear. So the conclusion here is P or R can join with Q. So look for the conclusion. Look for the component propositions and the available premises. Well we just got P and we got Q. Well they're Q. That's one half of the conjunction is Q. The other half is P or R and so far we don't have it. That doesn't appear in our list of premises. But we do have P. That's one half of that disjunction there in the conclusion. We do have P. Well from P, we'll go down to line three. From P we can infer the disjunction P or R. This problem is mostly solved. We can infer the disjunction P or R from that first line P using disjunction introduction. That's fine. Now we can join P or R to Q and that's our conclusion line four using lines two and three and the rule conjunction introduction. Let's try another one. Okay so this time we have a disjunction and we have a conjunction. That's our premises. This junction is P or Q and the con junction is not P and not R. And what we want to infer is Q. Okay so now we have a single atomic proposition as the conclusion. That's not really any help to tell us necessarily which rule we can use. So you have to go hunting through the premises to find the conclusion. And we find the conclusion Q as one half of a disjunction and line one we got P or Q. Well if we want to get one half of a disjunction out we use a good candidate for this is the negation of the other disjunct. This is using disjunctive syllogism. We have P or Q. If we want Q out of there using disjunctive syllogism we need not P. Well where's not P? Not P is in line two in a conjunction. Now we simply just can't you know say oh well lines one and two disjunctive syllogism. No right because the half of that disjunction is not a conjunction. Instead we need to pull not P out of that conjunction to use for the disjunctive syllogism. So let's do that we already have a rule that lets us do that conjunction elimination. So we pull not P out of the conjunction in line three using conjunction elimination then we can infer Q using disjunctive syllogism and that's how we get Q finally out of that disjunction. So far so good. So you notice got two steps there at that time a little bit more to it. So these are the rules we got disjunction elimination excuse me we got disjunction introduction we got disjunction introduction and we have disjunctive syllogism. We have one more rule about disjunctions we're not going to cover that to the end because it actually combines a couple of different things. Well that's going to get us started with disjunctions. So so far we've covered conjunctions and disjunctions. This area is safe on the condition you don't go and jump off any rocks. That's my really clumsy way of trying to introduce conditionals. So you remember conditionals if P then Q and this expresses the truth relation of sufficiency. P is sufficient for Q it also expresses necessity Q is necessary for P one step at a time right. So we got we're going to start with two rules with conditionals. Got modus ponens and modus tolens. Now what modus ponens allows us to infer is the consequent of a conditional. It allows us to for the consequent of a conditional when we can assert the antecedent. So if I have if P then Q and I also have P well then I can infer the consequence modus tolens allows us to infer the negation of the antecedent not the antecedent the negation of the antecedent from the negation of the consequence not the consequence. So you can't take if P then Q assert Q therefore we can include P no that that's a fallacious inference that's called affirming the consequence is a classic deductive fallacy. Similarly a classical deductive fallacy is denying the antecedent that's where you take a conditional you'd say that the antecedent is false therefore the consequence is false no that is a classic deductive fallacy called denying the antecedent. But we do have modus ponens and we do have modus tolens you might even think of modus ponens as affirming the consequence that's excuse me affirming affirming the antecedents right we affirm the antecedent therefore we can affirm the consequence you can think of modus tolens as denying the consequence we deny the consequence therefore we can deny the antecedent. So you know conditional works one way all right most of the time there you can just can't run both ways but it doesn't get that automatically all right so modus ponens we have if P then Q we also have P therefore we can infer Q modus tolens we have if P then Q we have the denial of Q therefore we have the denial of P all right but let's see how this works let's try a couple of problems here okay so we have this we have two lines for our argument we have the assertion of P then we have the disjunction of P or Q and that's a the antecedent of a conditional we have the disjunction of P or Q then R and what we're trying to do is we're trying to infer R okay so let's take this step by step we look at our uh uh conclusion R and we try to find it in our premises well lo and behold it's the consequent of a conditional cool right we have R as the consequent of the conditional chances are we're going to use modus ponens to get that consequent out but then so we look at the antecedent P or Q well P or Q does not appear as a premise what are we supposed to do well does any part of that antecedent appear in the premise yes it does right we got P in line one well we need the disjunction P or Q if only we had a rule that allowed us to infer a disjunction from a premise that already was given well we do right it's disjunction introduction so we take that first line P and we infer the disjunction P or Q and from that inference I mean from that disjunction we have P or Q well that's the antecedent of a conditional so we take lines two and three together and we can now infer R using modus ponens modus ponens allows us to infer the consequent of a conditional when we have the antecedent let's try another problem we got P or Q we have P wow so far so good easy peasy well the conclusion is Q or R huh where does Q or R exist where is it present in the premises not anywhere right we don't have the disjunction Q or R and you might yeah but still you know we have that disjunction so maybe maybe maybe disjunction since the conclusion is a disjunction we already talked about how you know that can be a nice clue as to what rule we're going to have to use it well if the conclusion is the disjunction we might have to use disjunction introduction okay well let's look through the premises do we have anything in there from where we can include you use conjunction a disjunction introduction well we've got Q and line one P is not going to be any help to us but we got Q and line one so maybe if we get Q out of there we can use disjunction introduction on it and for Q or R well as luck has it Q is the consequent of a conditional and we have the we have the antecedent in line two so in line three we can infer the consequent we can infer the consequent that's Q and then line four we can infer the disjunction from that consequent right so now we got Q or R from uh uh using modus ponens to get Q out of there now we got Q or R okay let's try another problem because this this stuff can get interesting real fast let's look at this one whoa what's this Q or R and T then S business what the heck is that and R and the the conditional T or S is not anywhere in our two premises we've only got two premises well remember that's a disjunction Q or and all that mess that's behind it okay do we have R and if T then S no we don't but we got Q Q is the consequent of a conditional so we've got and we got the antecedent there with P so now we can infer Q using modus ponens again all right so far so good what do we get well we can just simply infer the disjunction of Q or all that mess that's what disjunction introduction allows us to do we can infer any disjunction from a proposition provided that proposition that that disjunction contains that proposition no problem I know that seems weird but you can do that with disjunction introduction it is completely valid go ahead and construct a truth table if you don't believe me take those two premises in that conclusion you'll still get you'll you'll you'll still get a completely valid a truth table there completely valid argument okay let's try another problem this one's a little different we've got the conditional if P then Q we have the negation of Q then we have P or R and the conclusion we're looking for is R okay well let's take this step by step the conclusion isn't any particular complex proposition so it's not necessarily clue there and what rule we might use to find it so we just have R well where is R well R is in line three as one half of a disjunction well if we want to get that that half out right we have to have the negation of the other one okay so we have if P sorry we have the disjunction P or R we want to get R out of there so we need not P well where's not P well not P is anywhere in there right we don't have a not P but we do have P right P is the antecedent of a conditional in line one all right well we can't infer antecedents using any any rule but we can infer the negations of antecedents all right we use the negate we can infer the negations of antecedents using modus tolens if we have the negation of the consequent lo and behold line two we have the negation of the consequent so from that negation of the consequent right we have the negation of the antecedent that's where we can infer in line four using lines one and two of modus tolens and then we can infer R in line five right it's uh using that that disjunction P or R well since we have not P in line four we can just infer R right well that's kind of nice we're able to pull that out all right let's see let's try another problem okay we've got if P then Q we've got if Q then R and not R all right and what we want to infer is not P and not Q okay so we've got a conclusion that's a conjunction well it's probably since it's a conjunction there's a chance we'll use conjunction introduction to get that conjunction all right well let's look at our uh premises do we have not P and not Q anywhere in there well no we don't right we don't have not P and not Q um do we have the parts of it well okay well we got P and we got Q in lines one and two but they're antecedents of a conditional we can infer antecedents but we can infer the negations of antecedents if we have the negation of the consequent using modus tolens all right well we have the negation of the antecedent of line of two excuse me we have the negation of the consequent of the conditional in line two that's not R well then using lines two and three line two says if Q then R and in line three you have not R so we can infer not Q using lines one and two excuse me using lines two and three and modus tolens cool well that's one half of the conjunction what about the other half the other half is in line one all right well guess what we just infer the negation of the consequent of the conditional in line one okay well now we can use that negation not Q to uh and we use that negation not Q in line four and the conditional in line one to infer not P really so now we got both half of our of our conjuncts we got not Q in line four we got not P in line five well then using conjunction introduction we can infer the conjunction not P and not Q that's our conclusion all right so modus ponens and modus tolens allow us to pull out in some way shape or form part to the conditional right modus ponens allows us to pull out the consequent of a conditional when we have the antecedent modus tolens allows us to pull out the negation of the antecedent when we have the negation of the consequent but we want to be clear we can't infer antecedents from consequence we can't do that and we can't infer the negations of consequence using the negations of antecedents we can't do that either that is not modus tolens that is not modus ponens all we can do is infer consequence from the affirmations of antecedents and we refer the negations of antecedents from the negations of consequence. Well, that was cool, but let's look at two more inference rules using conditions. I like you guys, but I'm not walking down that path to get an interesting shot. That looks a little hazardous. If I were to go down that path, then, you know, maybe bad things would happen. It's not to say that bad things have happened, or even that I'm going to go down that path. I'm just saying if. Well, that if, that's a hypothetical. If I were to walk down that way, you know, bad things might happen. Well, we can make inferences that are conditions. And we got two rules that allow us to do this, right? One rule is called hypothetical syllogism. And another rule is called conditional proof. You know, now here's a little tip, right? So we've looked at different rules. If your conclusion is a conjunction, chances are there's a good chance you're going to use conjunction introduction for that rule. If your conclusion is contained in a conjunction, you're probably going to have to use conjunction elimination for as a rule, right? If your conclusion is a disjunction, chances are there's at least a possibility that you're going to use disjunction introduction for that, for that inference, right? If your conclusion is part of it, one half of a disjunction, chances are you're going to have to use disjunctive syllogism to get to that conclusion. Not the, not just like a one-stepper, but you know, as part of the way. Well, we got two rules that allow us to infer conditionals. We already dealt with conditionals before. We got modus ponens and modus tolens, but that doesn't allow us to infer a conditional, just to allow us to allow us to infer either the consequent of a condition provider we have the antecedent or the negation in the antecedent provider we have the negation of the consequent. Hypothetical syllogism and conditional proof allow us to infer conditionals. So chances are, there's at least a chance that if your conclusion is a conditional, you might have to use hypothetical syllogism or conditional proof for that. Okay, so let's take a look at a problem here. All right, so this should kind of look familiar. We already saw this earlier with using modus ponens and modus tolens, but you know, here we have ifp then q and we have ifq then r and then not r, and the conclusion is not p. Well, it shouldn't be hard to figure out, since we kind of already did it, that since the conclusion is the negation of an antecedent of one of the conditionals, we're probably going to use modus tolens to get that out. Okay, so in order to do that, we need not q. Well, q is not anywhere in there, but it's contained as the antecedent of a conditional. So if it's a negation of an antecedent, we're probably going to use modus tolens to get that out. Okay, and wouldn't you know we have the negation, we have not r, and that's the negation of that consequent. Okay, so far so good. But it'd be nice if we could just have that conditional ifp then r, and then use not r to infer not p. Well, we can do that. That's what hypothetical syllogism allows us to do. So look at lines one and two. We have ifp then q. That says that p is sufficient for q. That's line one. We have line two, q if q then r. That says q is sufficient for r. Okay, that says q is sufficient for r. Well, if p is sufficient for q, and q is sufficient for r, well that means p is sufficient for r. So using lines one and two, using lines one and two, we can infer ifp then r using hypothetical syllogism. Right. Well then now we have not r. We have not r as one of our assumptions there in line three. We have not r as our assumption there in line three. Well then using not r and the conditional ifp then r, well then we can infer not p using modus tolens, using lines three and four modus tolens. It's pretty straightforward. Okay, let's try something a little more different. Whoa, look at that. That is a complicated formula. p is one half of a disjunction. The other half is a conjunction of two negations, not s and not t. That's our only premise. And what we want to infer is if not p, then not t. Well, our singular premise is a disjunction. So we're not going to be able to use hypothetical syllogism, at least not immediately, to infer the conclusion. What are we going to do? Well, let's take a look at that disjunction. Right. Let's take a look at that conclusion. We want to get if not p, then not t. All right. Well, where is not t? Not t is one half of that conjunction. And it's that conjunction inside a disjunction. Well, it would be great if we had that one half of the, if we can get that part of the disjunction out. If only we could assume not p, well we can for the purposes of demonstrating a conditional. So we can assume not p. That's what that ACP means. We can assume not p. And frankly, in logic, you can assume whatever the heck you like. You can't infer whatever the heck you like, but you can assume whatever the heck you like. Right. So we're going to assume not p for the purposes of a conditional proof, for the purposes of inferring a condition. So just like on the assumption, I were to walk down that path, something bad would happen. I'm not walking down the path, something bad hasn't happened. But if, right, on the assumption, I walked down that path, something bad would happen. Same thing here. If we're going to assume not p. Well, if we assume not p, then we can infer not s and not t using lines one and two and disjunctive syllogism. Great. So we've got that not s and not t. We need to pull out that not t. Since it's a conjunction, we can do that quite easily using conjunction elimination. So we, so line four, we pull out not t from the conjunction in line two, line three. Right. So we've assumed not p. From that assumption, we've inferred not t. Great. Magnificent. Since we inferred not t, using that assumption, we can infer a conditional. If not p, then not t. We haven't said not p is true. We just assumed it for the sake of proving a conditional. We haven't said not t is true. We just assumed, we just inferred it from the assumption. Right. I haven't gone down that path, but if we assume we, I did, if we assume I did, what would follow is bad things would happen. That's how you infer that that's, that's how you can infer a conditional using conditional proof. Okay. So to be clear, we haven't said not p is true. We've only assumed it for the sake of proving the conditional. We haven't inferred not t. We've only inferred it on the assumption. So we get to infer if not p, then not t. Let's try something else. Wow, this looks weird, doesn't it? If p then q. And if r then s, those are our two premises. Right. And we want to infer if q then r, then if p then s. Boy, golly. Take it easy on us, Dr. Haugen. It's actually not as complicated as it looks. Look at the complex proposition, right? The conclusion is a complex proposition. It's a conditional. That means, I mean, here's two good clues, right? You're either going to use hypothetical syllogism or conditional proof for that. Well, if it's hypothetical syllogism, do we have what? A conditional where q then r is the need to seed it, followed by a consequent. And then, you know, that consequent appears another conditional with if p then s. No, right? We don't have that nice handy chain as it stands. So we'll probably have to use conditional proof. So we've got our two premises, if p then q. And we've got if r then s. Right. We want to infer if p then s. How could we infer if p then s? Well, we can infer if p then s, just if p then s, if we had a linking conditional, namely, if q then r. Right? If we had a linking conditional, if p then q, and if q then r, and if r then s, then we can infer if p then r, and then we can infer if p then s using hypothetical syllogism. But we don't have that condition. But we could assume it. Right? So let's assume for the purpose of conditional proof. If you assume for the purpose of conditional proof, you better infer a conditional from that. Right? So let's assume for the purpose of conditional proof if q then r. Well, great. Now we got our linking chain. So then we can infer if p then r using if p then q, and if q then r, you know, that q then r is our assumption that's lines one and three using hypothetical syllogism. Right? Then we can, so we got if p then r, and we got also r then s, so using lines two and four, using lines two and four, we can now infer if p then s using hypothetical syllogism. Cool. We've reached the consequent of that conditional. Now we assumed the antecedent in the conclusion. Right? In the conclusion, we assume the antecedent of q then r. We infer if p then s, so we can infer a conditional. Well, that so for the conclusion, if q then r, then if p then s. That's our conclusion. Conditional proof looks a little weird. We don't like making assumptions. We think making assumptions is bad for logic. Oh, it's not true. You can make assumptions all the time. You just can't make assumptions as a conclusion. I can make assumptions for the purpose of proving something in the conclusion. Okay. You know, that assumption has to be noted and is noted as a conditional. I can't just assume the conclusion to walk away. If I'm going to make an assumption, I better be inferring a conditional or later on, as we'll see, I better be inferring a disjuncture with dilemma. So I can't just assume the conclusion to walk away, but I can't assume the antecedent for the purpose of inferring the consequent and then conclude a conditional where the assumption is the antecedent of what I inferred as a consequent. Right? All right, let's try another one. We got a conditional if p then r. And now we've got an even more complicated conditional. If q, then the conditional is the consequent q, then the conjunction q and r. Oh, you fake. Well, look, this can get odd, right? If you try to backtrack it all the way from the conclusion backwards, you might get a little bit lost. But looking at the conclusion, we can see that it's a conditional. And we have a single premise. If we have a single premise, we're not using hypothetical syllogists to prove this. So if we have a conditional as the conclusion, we don't have anything else for hypothetical syllogists, we're probably going to use conditional proof. So without even trying to backtrack and figure it out ahead of time, just assume p. See what happens. So we're just going to assume p. Okay. But you notice we've got another conditional as the consequent of the conclusion. So we have if p, then if q, then q and r. Okay. Well, you know, we really don't have anything where q implies r or anything like that, right? In fact, q is just nowhere to be found in our assumptions. Well, here's the thing. You can assume more than one premise, right? You can assume more than one inch of seed. So we've got the assumption that p and from that only assumption that q, well, okay, let's go ahead and assume q, right? So assume p and line two, we'll assume q and line three. That's fine. You can do that. Then what we're going to do is take q and make an inference. We'll take, actually, we already have p. We can already use p. We have q. We can make an inference and from that we'll infer our conditions. So we've got q right there on line three. Well, we also have p and line two. Well, from p, line two, from that assumption and line one, we can infer r, right? Okay. So now we got r and line four using lines one and two modus ponens. That's great. Well, now we can infer our conjunction. We got q and r, lines three and four, q and r. Well, since we inferred q and r from our assumptions, well, we can infer a conditional. Right? We got q. We assume q and line three and from that we infer q and r. So now we have a conditional. If q, then q and r. Great. Right? Great. Lines three and five. You know, in conditional proof, the first number you cite is the assumption you made for the purpose of conditional proof. Five is the line where you draw the inference, right? And then you wrote the rule is conditional proof. So lines three, because that's where we assumed q, line five, that's where we infer q and r. So then we can infer if q, then q and r. Okay. Well, now if q, then if q, then, I'm sorry, if q, then q and r, that's an inference we made on the assumption of p. So now we can infer another conditional. Right? We have if p, then if q, then q and r. That's, you know, it's a mouthful, right? But this is fine. We've got our, we've got this conditional. So line two, that's we, that's where we assumed p to begin with. Line six, that's where we inferred the consequent. So that's our citation. And then our rule for justifying this is conditional proof. Wow. That was a lot. But it's really not as complicated as it seems. When you see a conditional like this, right, you got a conditional. That has a conditional as this consequent. That has a conditional as its consequent, right? This can go on for a long ways. Just make these assumptions, right? Just make these assumptions. If you got an ant, and you're convinced it's going to be conditional proof, just make these assumptions. Here's my warning though, work from left to right. So if you've got a conclusion, you know, if p, then q, if p, then if we assume q, then if we assume r, then if we assume s, right, then assume p, q, r, and s in order, right? Work from left to right in the conditional. Okay. Let's try another one. All right. Oh, never mind. That was the last one we tried for that. So that's hypothetical syllogism and conditional proof. And that allows us to infer a conditional. I said on the assumption that I go down there, bad things will happen. Oh, I'm not going to go down there. That doesn't necessitate bad things aren't going to happen, but let's hope so. It's a pretty cool path down there. Looks like some neat sites. Well, I can either go down the path or I can go back home. Either go down the path or I go back home. If I go down the path, you know, it's getting pretty late. I'll probably be out here a while, right? The path goes down a ways. It's a long path. If I go down the path, I'll be out here a while. If I go back home, I can get my dinner on time, more or less. And, you know, you've been watching a while and you probably figured out that I don't like to have dinner late. I don't like to skip dinner at all, frankly. But okay, so either I go down the path or I go home. That's the disjunction. Either I go down the path or I go home. If I go down the path, I'll stay out here late. If I go down the path, I'll stay out here late. If I go back home, I'll get my dinner on time. So either I'm going to go down the path or I'm going to go home. And from that disjunction, we can infer either I'll stay out late or I'll get my dinner on time. Okay, this is what's called a dilemma. This inference is called a dilemma. I said earlier, if you have a disjunction as the conclusion, there's a good chance that you use disjunction introduction as the rule. Well, dilemma is the other one. Dilemma is the other one. Dilemma says, I know in common speak, we say dilemma and we mean like a choice that we should make but we really don't want to. That's not what dilemma means. Dilemma means you have a disjunction, at least one of these is true. Each disjunct infer something else. So we can infer disjunction from that inference. That's what a dilemma is. Now I have the example in the text. I have the example in the text. I'm going to take you through it just a little bit step by step so you can see what's going on. So we have a disjunction of p than q. We have an additional if p than r. We have another disjunctional if q than s. And we want to infer is if r than s. Now we look at that conclusion, r than s. Well, is it anywhere in the premises? Well, both r and s are consequence of conditionals. Okay, cool. So you might think, well, I'll just infer one and then infer a disjunction from it with the other one. And then I'm done. Well, you can try that. But as you can see, so if we just take if p than r, p is the antecedent. p is not available as a premise anywhere. It's in a disjunction, p or q. If we're going to get p out of there, we have to have not q, but we don't have not q anywhere. Okay, so we're probably going to need to use the dilemma. Since p or q is available as the first premise, and both p and q are antecedent to those conditionals, well, then we can infer a disjunction. Okay, so how do we do this? Well, remember we did assuming for the sake of conditional proof. Now we're going to assume for the sake of dilemma. All right, now we're going to assume for the sake of dilemma. And just to kind of talk you through it, you assume one half of the disjunction, draw the inference, assume the other half of the disjunction, draw the inference, then we can infer the disjunction. So we're going to assume p in line four. We assume p and that ad there means assuming for the sake of disjunction. So we assume for the sake of disjunction. Well, it's very quickly we can infer r. So we infer r using line four and line two and modus ponens. So now we got r out of there. Okay, we got one half of the disjunction that's in the conclusion r or s. Now we got to get the other half. Well, we'll get the other half from q. So we've already got r. We can move on to the next disjunction. Let's assume q. We assume q. And again, we can, using, again we cited for assuming for the sake of disjunction, again using modus ponens, lines three and six. We can infer s. So we made the inference from one half of the disjunction. We've made another inference from the other half of the disjunction. Therefore, we can conclude a disjunction of those inferences r and s. So look at the citation there. Okay, the first number in the citation, that's the disjunction we're working with. The second number in the citation is the assumption of the first disjunction. The third number. So that's p, right? So the first number is p or q. The second number is where we assume p. The third number, that's where we draw that inference from that assumption. The fourth number is where we assume the other disjunct. And the fifth number is the inference from that second assumption. So it's a long citation, but all it's saying is look, first we got the dis, dis, disjunction, then so first we got the disjunction. The second number is here's the assumption of one half of the disjunct. The third number is the inference we got from that assumption. The fourth number is where we made the other assumption, right, the other half of the disjunct. And the fifth number is where we drew that inference from that second assumption. That's what's going on. So, right, the first number is either I go down the path or I go back home. The second number is, well, let's assume I go down the path. The third number is, on the assumption I go down the path, I will be out here late. The fourth number is, let's assume I go home, right? The fifth number is I'll get my dinner on time. So we draw the inference, either I'll stay late or I get my dinner on time. That's how that works. Now, delimit doesn't always need conditionals. You can make inferences with whatever rules that we have, right? You can make inferences with whatever rules that we have. So, let's try this, right? Let's try another problem. We have a disjunction p or q and we have another premise r. Okay. Well, what we want to infer is another disjunction, p and r or q and r. Now, this should be too hard to figure out what we're going to do at this point. So, we have the disjunction and we have r. Well, let's assume one half of the disjunction p. That's our line three. Well, using r and p, we can infer the conjunction p and r using conjunction and introduction. That's one half of the disjunction. Let's assume the other half of the disjunction q. From that, we can assume q and r. We can assume q and r. That's the other half of the disjunction. So, we can infer that whole disjunction p and r or q and r. Now, just keep in mind that's pretty straightforward, but you could do things like assume for the sake of conditional proof within the assumption for dilemma. These are possible. We could do modus ponens modus ponens. All right. All that's possible. We don't have to just use conditionals. We don't just have to use conjunction and introduction. Any inference we draw from that assumption, that's going to be that half of the disjunction. Okay. Well, that's a dilemma and that's all the rules we're going to use right now. We're going to have more rules later on. We're going to have equivalence rules and we're going to have rules about complex, further complex propositions using complex truth relations one step at a time. This set of rules will get us started and with it, we're going to start solving our problems. All right. Now, I'm going to go back and get dinner.