hi this is an introduction to logic I'm Mark Thorsby and this course overviews the basics of categorical propositional and predicate logic today we're looking at propositional logic we'll be looking at a a method for U building proofs in propositional logic sometimes referred to here as the method of natural deduction we're going to be looking at the first four rules of implication today um so chances are since you're watching this video you're probably not familiar what whatever what natural deduction is and or you're not familiar with what the rules of implication are so we're going to get to that today um before we get going in terms of uh looking at natural deduction today let's do sort of A Brief Review um and if you've been following along in our course we in our last couple videos here we've been looking at propositional logic in particular we've been looking at how to use a method known as the true table method uh in order to determine the validity of argument ments the truth table method effectively uh operates by building a truth table for any any basic argument right and then looking to see if you can discover an invalid instance so the truth table right you basically after building a truth table you um you look for an invalid instance within the truth lines or and within the rows and that means you have all truth true premises right and a false conclusion so for instance let's do a quick argument here let's say U this is a real basic argument if Joey goes uh we'll say if P then Q right if Joey goes to become a circus then he'll end up a clown Joey does go um to join the circus therefore Joey will end up as a clown right um so this would be I'm using here our basic sort of um variables here to show if you if you have this argument if P then q p is a premise and then the conclusion was Q this would be an argument that we could use a trueth table for so for instance if we're going to use a true table it look like this true true I'll try to do it quick here uh and then of course I need to fill in this conditional that' be true true false true and true and then I have to look for any instance where all the premises which means these rows is there an instance where these are all both all going to be true and this one is going to be false because this would be an invalid valid instance and of course I can look here if I went T TT that's not a case f t f no T ft no t f f no there is no instance and so what we've been arguing in this video is that since there's no AR since there is no um instant invalid instance in any of these rows we're therefore saying that the argument must be valid now you'll notice here that what we've been doing in the truth table method is effectively proving arguments that are invalid so we've been proving invalid arguments rather than looking to see how to prove a valid argument in the positive sense so we're going to be doing in natural deduction is we're going to be looking at specific argument forms that are always work and we're going to see that we can actually use those arguments to build proofs and this is what we're going to be doing is we're going to be building propositional proofs and what you're going to see when we build a proof it very it looks like it's the same sort of proof you would give in trigonometry so I want you to think about if you've taken a trigonometry class what you did in your trig class to build proofs it's going to consist in essentially learning a number of specific argument forms and then using those forms to um to then do a deductive proof so that's essentially what going to what the lesson's going to look like today and let me S there's a number of things I sort of need to lay out for you right is that first thing I need to lay out is there's four key argument forms that I want to talk to you about four key argument forms and these argument forms are known as the first four rules forms of let's just say they're the four key argument forms of implication okay what does implication mean uh well the word here of course is implies and what we're going to see is that there there's four key argument forms that we can use to imply and ultimately deduce other sorts of conclusions so let me just lay these out for you these are very famous and the truth is you're just going to have to memorize these argument forms the first one here is called the modus ponents and you're going to see that when we're doing deductive proofs we're going to write things vertically so I'll just put this over here as the hint uh right so instead of writing the arguments P because we've been writing the argument we're going to write it like this if P then q p therefore Q right we're going to write them vertically we're not going to write them horizontally because this is the same as saying P then Q q p therefore Q you'll notice that we're just going to write it down instead okay so so this is a very classic argument form the one I just gave you U about clowns or something of that nature um so if P then q p is the second premise Q is the conclusion modus ponent the second one here you're going to have to learn is know as modus tolerance it looks like this if p e q not P I'm sorry not Q therefore not P so the modus toet says has the same conditional setup if P then q but instead of saying not Q then not P that makes sense right this would be similar to saying if it rains outside then you're going to get wet you're not wet therefore it must not be raining downside right um so we have modus tolerance let's just put the third one up here just to see if we can get them all on the same page this next one is called the hypothetical syllogism this is the hypothetical syllogism it looks like this if P then Q if Q then R therefore if P that R right that's the hypothetical syllogism it it looks something like this if um Noah drops his milk uh I'm sorry if Noah drop if Noah drops his milk then his mother's going to get mad if his mother gets mad then Noah will get in trouble therefore if Noah drops his milk then Noah will get in trouble right that's the basic argument for him there number four here is called the disjunctive syllogism the disjunctive syllogism and you'll notice that U the whenever we say syllogisms there's always going to these are classic arguments so this if P or Q not P therefore Q so this looks like something like this um either you're going to get an A or you're not going to get an a right uh you're not going to get an A therefore well that's a confusing one here let's let give you a different example here um um if um either you're going to be happy in your life or you're not going to be happy it doesn't look like you're going to be happy therefore you're not going to be happy I just did the same argument for him there but this is known as a disjunctive Sy saying it's either P or Q if it's not P then you must just have q right and these are the basic four arguments that we're going to be focusing on today now but here's something I have to tell you is that you're going to have to be careful because whenever we do for instance a hypothetical syllogism or any of these notice that we'll see that the P here this could stand for a for this is could this could be a variable for a long for a very long u compound equation for a compound statement so let me see if I can give you an example here to show exactly what I mean by this um let's see like this okay now let's put and from here not I'm going to refer to modus opponent is just NP so modus opponents here remember just P then q p therefore Q right so let's imagine I have an argument that looks like this here's what the proofs are essentially going to look like well let go like this if I had T therefore s therefore J I have t therefore s therefore J now here's what I want you to see is that this right here is an instance of modus ponents this is a modus ponents okay but what's the difference here is see this this t therefore for S this is essentially our p and the J here is the Q right and so you can see here if the P if we say that the p is going to be equal to T therefore s then that means that this is a if this is p this is Q This is a p and this is a q so this is exactly a modus poent here I hope you can see that I know it can be sort of confusing so what this means is that any one of these variables that P the Q or in another case in for instance the hypothetical syllogism the the r right the any of these variables can stand for much longer equations let me give you another example what if we had um P there no let give you an example let's say e if an only of J and T therefore w e if it only of J and T therefore W okay this is a modus ponent in the exact same sense that we talked about earlier right because what we would say is that for instance this thing here is the P this whole thing that is and this right here is the Q so we have if p then q p therefore our Q right here so this is a modus ponents is two right so this is what I want you to see here is that sometimes the modus ponents really do just have one two variables but they can have many many longer so what's part that typically you're going to find when we start doing these natural deductive proofs is that often it's going to require you to be able to look at possible substitution instances and that's what we're going to call all of these sorts of things these are all substitution instances right where we could see that this can be a substitution for the p and this could be the Q and that sort of thing so what we're going to have to do is you're going to have to learn to do substitution instances with regard to doing these proofs so I think the best way here is for me just to actually do a couple proofs to show you what it's going to look like uh and in fact maybe actually let me just write out the basic steps that I want you to learn in order to do this U the first thing here is you basically just need to learn and when I say learn I mean memorize learn the argument forms those four argument forms you're going to see in the next couple videos in the next video we're going to add more forms but so you just need to learn and memorize these argument forms and these are classic valid arguments I guess I should put that these are all valid arguments and if we if you want you could do it true trth table or an indirect truth table to check and that's something interesting here is that any argument form that works in the truth table method works here but here's the thing is proofs you when we do these proofs we can only prove things that are valid we can't prove an argument invalid whereas the truth tables prove an argument invalid and valid um in terms of a valid argument being the non invalid argument okay so the first thing you do is just learn the argument forms you literally just have to memorize them uh but don't worry as you do these problems it won't be too difficult to do now in order to and then once you've learned the argument forms then what you're going to need to do is you're going to have to start to construct proofs and here's where I'm going to sort of give you the lowdown of what some of the key things you need to do when you construct a proof the first thing you need to do is the or the first thing you need to know rather is that the order of the premises is irrelevant the order of the premises are irrelevant it doesn't matter what order the premises are you'll remember when we did propositional I'm sorry categorical logic we seemed to argue that it mattered it doesn't matter anymore okay the second thing is that the conclusion uh gets written on the right whoops now the whoops I misspelled right there r that sort of right okay the conclusion gets written on the right you're going to see what it essentially looks like is you're going to blank the conclusion okay the conclusion gets written on the right on the other side of a Ford U Back slash of a forward slash okay now the so to constructive proof the order of the premises is irrelevant or are irrelevant the conclusion will be written on the right hand side and C is every deriv deduction every derived deduction must be noted to the right and includes the argument form and the lines used for deduction so every um derived deduction must be noted to the right including the argument form including the argument form and um the L the the lines used for the deduction you're going to get it you're going to see here in just a second here what I'm talking about including the lines used for deduction now and then of course D is you're always going to uh look for substitution instances uh and then the last bit of advice I'm going to give you number three this is just your advice if you see something do it why because every step every step deduced is valid so you can't um so you jeez this handwriting looks horrible God I'm really embarrassed by my handwriting there uh it's hard to write on this tablet so that's no excuse so let's go back through here and look at so what are the steps first you need to learn the argument forms two is then to construct you have to then construct the proofs and here are the three four things you need to really know the key things the first thing is that the order of the premises are irrelevant it doesn't matter what order they're in B is that the conclusion will always be written on the right hand side right with a forward slash every derived deduction that you do must be noted to the the right and it must include the argument form and the number of lines used for the deduction now D look for substitution instances now the final advice I give you is if you see something do it because every step done in the process of natural deduction is valid because what we're going to see is we can always prove valid arguments we can never prove invalid arguments so let me go through here and see if I can actually just do a number of problems here so that you can see exactly what I mean um I encourage you to read your book here read it slowly but what I find is it sort of the the the natural direction is one of those things where you sort of just have to jump in and do it in order to really begin to understand it so let's say okay so let me go back here just create a new file okay so let's do a number of problems whoops sorry about that okay so let's just a number of different examples here um excuse me sorry so let's say we see this argument right here uh let's say someone gives us this argument if F then G uh F or H number three is not g then we have number four h therefore G therefore I if G then I I always make that mistake you'll notice in these videos uh okay so this is essentially an argument and we want to see if we can prove it valid so now here's what everything means right all of these These are um lines of Prem lines of premises and then this right here is the conclusion okay so we have this forward slash now well we're our goal in building the proof is to see if we can use those argument forms the modus ponents the modus tolin the hypothetical syllogism and the disjunctive syllogism these are our four basic argument forms and what we're going to be doing is we're going to be looking to see if we can figure out how to prove that this conclusion is true using these premises and then using these rules essentially as a a way in which we can find substitution instances and then find the derivation okay so this is the first sort of thing now let's go go back here and take a look at our argument forms actually I think the easiest thing here for me to go here um okay and here's two things that are really key that you should know as well that were in the book that I should mention for you is that since each rule the modus pents the most TOS the hypothetical syllogism the dis disjunctive syllogism since each of these rules is a valid argument form to themselves any conclusion derived from their correct use results a valid argument the method of natural deductions thus equal in power to the truth table method as far as proving validity is concerned however since natural deduction cannot be used with any facility to prove invalidity we still need the truth table method for that purpose now applying the rules of inference rest upon the ability to visualize more or less complex Arrangements of simple propositions as substitution instances of the rules and I just thought that'd be worth writing down for you so let's go back here and take a look at the basic for four basic rules here they are modus ponent monus tance pure hypothetical syllogism disjunctive syllogism so let's use think about those rules and here you probably need your book because I can't get it on this page as well in order to solve this problem now the first thing to do when you're looking at these problems is always start with the conclusion so these are some of my helpful hints um is I always say start with the conclusion and see what we can learn by looking at the conclusion before we begin to build the proof I need an F therefore I'm sorry if F then I that's what I need you can see I have an F here and I have an I here I have an F here how am I supposed to get out of this now wait a second I'm not I don't exactly see so start with the conclusion if you don't see what you can do with the conclusion then look to the premises and see if you can see any substitution instances so then look at the premises so let's go over here right I have if F then G not g wait a second isn't this a modus tance look back at our modus tolin rule right modus talin says let me make it just a little bit bigger for you right modus T here if P then Q not Q right that's not this then you don't have this right this says if this then this I don't have this so therefore I must not have this so number five here is not F right I hope you can see that because this s became right basically we had if P then Q not q and here we're just writing down that well can you deduce you can deduce that you if you don't have G you must not have F okay so and then so we' write number five not F and over here we'd write lines 1 and three modus tolins right because look we used this line and this line and the rule we used was the modus tolins rule right here so we just write that over here and what we're going to do is we're going to list this out until we find the conclusion in order to get the proof so let's go here now that and you typically here's another thing is you can use the same premises again and again so there's no rule that says that once you use number one for modus T you can't use it again that's not true you can use the premes multiple times and now I can use number five here as a premise to look for substitution inance of another one of the rules maybe I should just write the rules up here this is if P then Q you have P therefore you have q modus tance is if P then Q not Q therefore not P hypothetical Sol is if P then Q if Q then R therefore if P then R and the disjunctive syllogism was uh if it's either P or Q if it's not P then it must be Q right that was the disjunctive syllogism and you can see it's right here as well if P or Q not P therefore you have q okay so let's take a look at it here what can I do now this is not F what about this line line two right it can't I use the disjunctive syllogism here where this becomes my P this becomes my Q and then I have not P therefore line six I must have not um if I don't think about this if I don't have F then I must have H then I must um hold on wait a second no no I'm sorry six here is just H because if I don't have F then I must have H so that would be lines two and five disjunctive syllogism DS okay now that I have H wait a second can you see this here is now we're in much better shape let me erase this because now look here now I have an H line seven here is going to be if G that I I write this down yeah how did I get if G then I because look here look at line four line four is say say that if I have H then I have G then I I do have H which means I can sort of plug it in here the way what I find it's helpful here is to imagine it and I know this may sound silly for a lot of you out there who are watching this video but I think one of the ways that's helpful to understand is to think about it is a series of keys right this proposition here basically says that if you can get H then you can unlock G then I I do have the key h I just sort of plug it in and then I get G there for I and I can unlock it so that means I've used lines four and six modus ponents because that was the if P then Q I had the P so I could unlock the Q right so means I have G there for I now remember my goal here is if is going to be if F then I how am I going to get there it seems like it's taking me quite a long time to get there um let's see let's take a look is there another way wait to second notice line one here look at line one and look at line seven this says if F then G and this says if you have G then you have I this says if Frank goes to the part then Gene is going to go this says if G is going to go then Isaac's going to go and the conclusion we want is if F then I this is a hypothetical syllogism isn't it because it says if P then q q is the same one so it links back up like a chain and then r so that means you can just simplify the chain to say if P then R so can we do the same thing here and that's the answer line eight here is going to be if F then I and we use lines one and seven hypothetical syllogism okay and now since we our last line of our proof is where our conclusion is we've proven the argument valid we've actually built the proof we' built the proof now some of the things that I needed tell you that a really key here is it's incredibly important you always write down the problem correctly as it's in your book because remember this method can only prove validity if an argument's invalid I'll literally just keep trying to do things until I could do nothing but this method only proves validity so let's try it again here U that's a good example here let's do another one here take a look at this problem U let me change the color this says line one says John is not going to the party line two either John or Ken is going to go to the party if Ken goes to the party then uh Lena will go to the party therefore Lena is going to go to the party so imagine if I had this argument can I use the deductive argument method to prove it valid now just here you again you just got to remember our we what are our tools well we have four rules of implication modus P modus ponents I'm sorry modus tolet the hypothetical syllogism and the disjunctive syllogism you're going to see in the next couple videos we're just going to keep adding to this tool set but the basic method Remains the Same I write the premises vertically with the conclusion off to the side and then I'm going to try to deduce steps using these argument forms you can see these patterns of reasoning I want to see if I can find the same patterns of reasoning here okay so let's take a look at line four what's the first thing you can see here if I have I have either I forgot what how I I said either Jack is g to go or Ken's goingon to go to the party Jack's not going to go if both of these things are the case then that means I can use the disjunctive syllogism right remember the disjunctive syllogism is it is either P or Q not P therefore Q so line four simple it's going to be K and I used lines one uh I'm sorry one and two disjunctive syllogism okay now number five once I get the K can't I get the L here using modus ponents because see doesn't the K here effectively unlock my L here right and that's what the modus ponent rule is right it's if P then Q if you have P then you have q right so that means I can actually conclude L right now right where the K here sort of goes in here and then unlocks the L so that means that's my L and I use line three and four modus ponents okay now since i' since I've now reached the where the conclusion is I'm done and I can say this argument is valid okay now here's the trick though this this method this natural deductive deduction method method will never prove an argument invalid so that's why the indirect Tru table method is still very important to know because the indirect truth table method will give you the capacity to determine whether or not it's invalid and then if you want to construct proofs you could do you could do construct a positive proof using this method so let's try a different problem here you can see here uh the the there's there's no foolproof method it's much more like a game like chess and I think the way you'll do well in this section of the course is if you can start to have fun with these problems and really begin to see these as logic logical games and logical puzzles so really see these as puzzles uh instead of problems and I think you might have fun with these puzzles instead of problems when you view these as problems you ultimately can get frustrated with them when you see them as puzzles you have fun with them and you'll get better and better it's like playing chess the more and more you do these right uh so let's go here let's do another problem here I'm just picking these out of the book at random this says A or B line two not C therefore I'm sorry I always say it if not C then not a if C then d ever four here is not D therefore B so can we figure out this problem now what did I say there some good strategies here always begin with the conclusion because the conclusion will tell you where to go number two make sure you've written the problem correctly because if you don't and it's invalid you'll have a lot of trouble um and then you're just going to start with the conclusion if necessary work your way backwards right okay I see that I have the B here so up here in line one is where my B is located is there any you can see how could I get this B well if I had not a I could use a disjunctive syllogism here I do have a not a right here okay but in order to get that I need to get the KN C first do I have a not C no I don't have a not C but then I do have this not D and I have a d here wait a second maybe I can use LINE five and maybe let me change the color here line five here isn't line five not C using lines three and four modus tallin remind ourselves what does modus toallin look like right modus toin is right here it says if you have P then you get Q is the first premise if you don't have q then you don't have P right so what I have here is I have this one is my P and this is my Q I don't have the Q thus I must not have the the P thus I have not c as my line number five okay now that I have line five you can see I actually by beginning with the conclusion I was able to sort of get how this is all going to work since I don't have C I can unlock the not a so six here is not a using lines two and five modus ponents if you don't have a then you must have b line seven is B that's using lines six and one all one and six here uh disjunctive syllogism right and now since I've ended up with the B I'm done this problem is done I've proven this argument valid okay now one of the things you guys are going to have to do is you're just going to have to do these problems a lot over and over and over but I'm hoping that just by you watching me do these problems you're getting a sense of how they work so let's here let's take a look at what your homework actually looks like yeah that is if you did your homework on the book in the book um what what you're straightforward they have to do is they're going to ask you to U demonstrate your understanding of each of the argument forms demonstrating your ability to recognize substitution instances and then number three is finally build the proofs actually I should go over these with you here's some strategies that since there's no foolproof method here there is sort of things you can do you can think about and I guess it's worth mention these in the video strategy one here is always begin by attempting to find the conclusion of the premises that's the way I showed you always start with the conclusion strategy two if the conclusion contains a letter that appears in the consequent of a conditional statement in the premises consider obtaining that letter via modus ponents right and that was the problem I just did for you um since I had a I could get B using modus ponents strategy three right if the conclusion contains a negated letter and that negated letter in the antecedent of a conditional statement in the premises consider it the negated letter via Lotus tolin right they're basically saying if you have a negated letter right and you have what a conditional here then that means you can get not a by using modus tolerance line for if the conclusion is a conditional statement consider obtained via pure hypothetical syllogism right remember if you need to get a conditional a hypothetical syllogism right you can use if a then B if B then C therefore if a then C so that's Lin one and three hypothetical syis and the last strategy here which is if the conclusion contains a letter that appears in a disjunctive statement then think about the disjunctive syllogism right so we have a not a you can see here as a destructive syllogism here so what do your homework exercises look like well if you're doing it in the book here's what they're going to look like U they're basically going to say here are the premises what is should the next line be using what rule right they want to basically um these homework exercise will give you practice recognizing substitution instances and and in the case of natural deduction I actually encourage all of you to do all of your homework it's pretty important stuff actually when we get when we're looking at these sorts of problems so let me just give you this example let's just do a couple of these problems um you can see here if J then F if I have not F then that would be not G right lines one and two modus tolerance now it's easy for me because I've memorized all of these forms modus TS modus opponents and where it's gonna be difficult for you is you haven't memorize them yet so but you will if you just do all these problems okay if you have ifs you can see here I could get M because my S unlocks my m so that must mean like one and two modus ponents remember the order of the premises doesn't matter notice how twos the back end it doesn't really matter um this would be if R the D if e then R well you could see here it would be if e then D cuz e to R and the r here operates as sort of the middle term which gets me to the D right so that's lines one and two um hypothetical syis let's see over here number six you can see number eight Let's cross that out this would be this was not J or P not J well you could see here what could I conclude I could say p using lines one and two to Str of syllogism what about this s in the J I don't see that as really doing anything for me I think this is a p line one and two disjunctive syllogism and then you can see here H what about this problem what can I deduce from here whenever I see this F right if I have F therefore H H therefore D you can see this H here gives me that middle term so I can might as well say deduce if F then d lines one and three hypothetical syllogism right so you can see that these problems really aren't too difficult to do u a lot of what you're going to see in these first things is since you can only do one step some of these prives are just going to be totally irrelevant and they know and despite how long they get I don't think it's really that difficult right let's do maybe problem 20 so you can see what that looks like um okay okay so line 20 so this one looks like this you can see here there what you need to be thinking here is when you see lots of parentheses and things stuff like this is you need to think in terms of that look for a substitution instance look for substitution so let's see here I have a Q then G do I have another G well look I have a Q then G right here which means if I have this I can conclude a or n right because this whole thing here becomes my q and this whole thing becomes my P so I really have a modus ponents right here where remember modus ponents is if P then q p therefore Q right take a look at it if P then Q I have this is my P then Q I have the P thus I must have a or n lines one and four modus ponents right so it may look messy but you can see that the logic behind it isn't that difficult it's actually pretty simple so that's part one what's part two going to be part two gives you a little bit different exercises they're going to give you premises and they're going to give you a conclusion or they're going to give you a deduction and you need to figure out what the missing premise is right um so think about this one you have b or K and in K what needs to go here that what is the intermediate step well it's probably not B right so actually let me just do this problem for you so you can see we'll do all these problems for you all right so let's take a look this must be not B that's lines one and two destructive syllogism what cued me off there is a disjunctive syllogism is the disjunction right here when you see a disjunction ask yourself is this going to be a disjunctive syllogism when you see a conditional ask yourself is this going to be a hypothetical syllogism and things like this so in fact let's move over here I already know there's a hypothetical syllogism because look at this this is uh if C then H well they C if R then H so maybe this is um if R oops if R then C if R then C if C then H thus this creates that middle term so you can just say including if R then H so this must be are there for C and that's lines one and two hypothetical Solis let's look to this next one this one's not f it looks like what I need here is just a modus tolerance right I'd say Not N right because if this is not n then that would be so this would be lines one and three modus to right over here these problems this would be an ed that lines one and two modus ponents if Ed then s and therefore s here we have another one I have a not K you could see a modus to a not t would give me that that must be a one and two modus ta right so you can see I could I'm doing these problems very quickly it's going to take you a little bit longer but don't feel bad about that it's because I've memorized these argument forms and I've done problems like these thous thousands of problems like these uh but as you do these you're going to see that they're not really that difficult now finally where it's this is what's most important in your homework here is to actually build these proofs so here are some of the proofs they're going to give you let's see if we can do a couple of these problems um just to give you a sense here of what they're going to look like let's do both of these problems let's do them separately okay so let's start with here if I have this not C right the first I'm sorry start with the conclusion here which is not a where's my a it's here well in order to get that I'd have to get rid of this thing so not see but wait I can't interact with these and that's another important thing is that when you re when you use rules of implication right you have to use the entire line you must use the whole line so I can't just take it out I can't just for instance take this out right if it's in the premises it's locked in so I need to unlock it so number three here line three here is going to be if a then C right since. C this is my P is p then Q so that I can write down the Q so that's lines one and two modus pents okay if I don't have C then I must not have a that's line four not a lines two and three modus toet right now that I have the conclusion I'm done that's all I need to do so let's do the proof for this one right uh let's take a look here if I don't have this and here these problems you could see start with the conclusion and here you can see it's locked in this thing again so what can I do and if you see something you could always just do it because remember you're only deducing according to valid rules which means or Val valid patterns and reasoning so it's not going to hurt you at all so line four here is going to be if D then T because if I don't have this right then I must have this this is line one and two just junctive syllogism okay now you have D then T So L five here must be T because the D here unlocks the T using modus ponents so that's line three and four modus ponents and for some of you I'm worried maybe watching this and still not seeing how it's working here right the D here this says if you have D then you have T I do have D thus I must have t that's basically the argument but again I'm using the whole lines so those are two proofs there let me do one last proof here let me do a long one because what you'll see is they get harder and harder as you go through so let me do one really long long difficult problem so that way you at least can see that and then finally they're going to ask you to translate them and then apply it but I'm not going to do one of those problems right now let me do problem 25 right usually the ones at the end are the long ones so let's take a look here at this problem how can we do this problem see can I make it bigger no okay that's all right hopefully you can see it well enough um let me go here to block Orum blue all right so line five now here let me start by the conclusion not D so there's a d there there's a d there you can see I have parentheses here and I have brackets all this stuff has to be unlocked there's a not D I'm not sure how to get there how can I do this you can see here I had difficulty seeing what to do if I had a not n i could unlock this whole thing right cuz this whole thing here could be Rewritten as if P then Q so but I don't have the P so that's not going to help me so what am I going to do here wait a second doesn't this look like a hypothetical syllogism right here where this becomes the middle term the D then D so couldn't I just simplify to if B then e lines three and four hypothetical syllogism okay if I have if be the wait wait a second there's my same equation in the parentheses so now I have a modus poent going on here this is line six or my if B then E I can basically plant it in here turn the key I unlock my not in so my conclusion here is not in using lines two and five modus ponents uh look now I have since I have my not in I can go back to line one up here so if I if I use this to unlock this then that means I'm going to have to write out this big long ugly thing so it looked like this if b f d f n or not e okay so I have if B then D then it or not e i just unlock that now remember I still need to get that D how the heck am I ever going to get down that far right well do I have a B then D wait a second I do line three you can see here it's important this problem you're using multiple lines multiple times so that's line8 must be n or not e utilizing lines three and seven um no opponents oh I forgot to write my reasons here didn't I so line seven here I made a little mistake here we did did if B the D so we use lines one and six no opponents right you for your proof you want to make sure it's all clear number nine here so I have n or not e I need to get that D how am I ever going to get that far remember I did have a not n right here because I could use my own premises too and once I've made them and inferred them they're valid to I can use those as much so if I don't have an N then I must have a not e so that's not e right where now this whole thing here is my Q right this is my P and this is my Q whoever the disjunctive syllogism is if P then Q if you have P oops let me rewrite that for the distructive SYM if it's either P or Q if you have if you don't have P then you must have q right so I don't have this so I must have this so that's why I'm writing not e right right here and that would be lines 8 and six disjunctive syllogism so I have a not e I still need to get this not d That's the goal right is to get not D so if I don't have e wait a second then I must not have D line four because remember if I so that's so number 10 here must be not d right using lines four and nine modus tance because remember modus to is uh if P then Q if you don't have q then you must not have P so this says if D then e but I don't have V therefore I must not have D and that was the conclusion and that's the proof so the hardest I just did the hardest proof for you so the hardest proof is only 10 lines I'll warn you sometimes they can be well over 20 or 30 lines but that's you're going to see those sorts of problems for a while so that's an example of the proof now what mistakes or problems do I have with this proof well number one is you could see this looks fairly crammed right I really should I really should have written all of these things over here I should have written all these things over here so that there was a nice clear line so I could see the proof this proof is a little bit of a mess um I was just doing it for you but you can see when you actually do your own proof groups you need to try to make it as clean as possible oops I don't want to sa that um so that is our um section here on using the rules of implication to build proofs and it really comes down to memorizing those key forms the modus pents the modus tallin the uh the hypothetical syllogism and the disjunctive syllogism it's very important that you memorize those things so just to sort of show you what once again you need to memorize these argument forms and then hopefully I've demonstrated for you today how to use the rules of implication ultimately in order to um find the conclusions U and make valid argument proofs you're going to see in our next video we're just going to give I'm going to give you more rules of implication and the proofs are going to get longer and longer in fact for the next couple of videos I'm just going to we're going to be doing the same thing proofs except we're going to introduce more and more rules so those are the rules of implication thank you for watching online logic we'll see you online bye