hi welcome to online logic I'm mark doors me in this course overview is the basics of categorical propositional predicate logic because of a technical difficulty we're doing it the old-fashioned way today and hopefully we'll get our equipment fixed up so people can see so it looks a little bit nicer but today we're going to be using the indirect going over the indirect proof and talking about exactly what the indirect proof is now the indirect proof can be used really is sort of your a good trick in your bag when you're talking about doing deductive proof so let's sort of start off here and say maybe it's sort of brief review just to talk about some of the things we've looked at right now remember ultimately whenever we do a prove what we're trying to prove is the logical argument for the deduction of the conclusion such that we can prove that an argument is valid right now each line in a sequence right each line in a sequence right that it's not considered right except except the conditional proof right each line in the sequence counts as a valid line as a valid line of argument right and additionally let's remind ourselves we also took a book some weeks ago we are looking at propositional logic in particular we looked at the indirect truth table method and one of the things to remember is when we look to the indirect truth table method right what we did is we assumed remember if you'll recall right we assumed in validity right and then we used our rules of operation to deduce the conclusion and if we found a contradiction right if a contradiction gets discovered right then we ultimately argued that the argument must be valid okay I'm sorry I know that's sort of sideways right you can see here the indirect truth table method right we made the assumption validity work to everything out if we found a contradiction then we found out the argument if there was no contradiction the argument truly was invalid okay you're going to see that today even though what we're working on today the indirect proof method here you're going to see that it's not identical with indirect retailer method but it is very much related to this core idea here that every line you argue that you deduce with an approved sequence counts as valid so the question is what happens if you assume the opposite of something that's valid and so what I want to do here is let me show you a real quick indirect true proof let me actually change pages here right here's the sequence here and it looks a lot like the conditional approved sequence so let me just write this problem out we have a or B then C and D as our first line second line is if C then not D and our conclusion is not a you can see here there's a lot of trouble what could we do here right well you can see here we have a disjunction here a conjunction here that's not going to work for the conditional I'm sorry for the constructive dilemma you can see here we could maybe do a implication there's a number of different things to do but it looks like it's pretty difficult to finally get to the conclusion here and in fact although we won't show it you can't get to it using just the rules of replacement implication so we're going to do is we're going to do this what if because we remember we're going to if this argument truly is valid then that means that not a right if we derive it as a conclusion will eventually be considered a valid line of argument right that's assuming this is dodd and because this is logic course we're only asking you to prove actually logical sequences with the exception of the application project we'll be doing here in a couple weeks so what happens though right if not a is going to be valid then that means that a must be invalid right am s p.m. dawn so we're going to do for the interact truth that is what happens if we assume this right and then as we do our deduction we discover a contradiction if we discover a contradiction means that our assumption was wrong and that we should stick with the negation of our assumption which is this and so what the what the indirect proof of laws is to do is essentially that in fact if we want to write up the steps for the indirect proof here's what they would look like steps one assume the opposite of what you need right to write all assumptions and deductions whoops to the right just the way we did in terms of the conditional proof right and what we're going to do in terms of our steps is once we do our deductions right what we want to do is discover a contradiction and what does a contradiction look like in low logical notation a contradiction would look like this B and not B right this is a logical contradiction why because you can't both have Bob can't both be at the party and not be at the party simultaneously and that's what this statement says Bob is at the party and Bob's not at the party right that's a contradiction so what you want to do is discover a contradiction now the contradiction doesn't have to be this using the same variables as your assumption did right so that doesn't have to be the same just have to discover some sort of contradiction once a contradiction is discovered number four you discharge the proof and conclude the opposite the logical opposite or the negation of your assumption and all the same rules regarding discharging the prove and so on apply so it's really simple it's really four basic steps assume the opposite of what you need right all of your assumptions and deductions to the right you know write them out do your proof sequence look for discovering a contradiction if you can find discover contradiction right here's an example right then once you've found that you can discharge the proof of conclude with the opposite of your assumption and five all the all the basic rules for the conditional proof apply to the indirect proof okay so let's do let's do a couple of these examples using that that process right so let's start with this problem right well you can see here I need not a what's the opposite of not a right well right off to side the opposite of not a is a right so let's assume that that's going to be number three we're going to put a and this is going to be a IP notice instead of saying CP we're saying assumption for the indirect proof right we're going to write it off to the side over here right I don't know how long it's going to be so let's do that but once I have a what can I do well you can see I can add B so I'm just going to run through this quickly I don't want to take your time out just running through going over those same rules of replacement implication right so that's going to be line three addition right line five is once I have this I can unlock C and D so that's going to be C MD that's lines four and one modus ponens right now that I have C and D what look here if I can get C by itself I can get not D so in fact I actually already see what's going to happen so let's go ahead see that's line five simplification seven since I have C I can use modus ponens to unlock my negation D so that's going to be line six and to modus ponens but take a look I have a D here so line 8 let's calm you take these remember that's when you switch them so that's going to be in line five commutation and line nine here is going to be D right line eight simplification right and then finally line 10 is going to be D and not T because notice right if you can join these two things together you're going to end up with a contradiction so that's exactly what we're going to do D and not D that's line seven and nine right conjunction you can see Ivers that means that and with this assumption this problem can only result in a contradiction if the results and contradicts with that assumption then the opposite of the Assumption must necessarily be true so that means we can discharge the opposite of a here is not a right and what I've done here is I've discharged I've done outlines three through ten in direct proof okay now again right since I got a contradiction here that means a right I have to put it through its reverse the opposite of this right the opposite of this becomes not a okay so you can see how this problems works I've discharged it and so that makes its true so ironically I using the indirect proof I can sort of say well if this the opposite of this if what I'm looking for is obviously this let's assume this if it results in a contradiction then the opposite must have been true or the negation of it must be true and this is essentially how the indirect proof works it's not really too difficult all the same rules apply in terms of you can embed it you can do multiples and in some cases you may not be able to find the conclusion directly off right so let's do a kind of different problems here and you could also employ conditional proofs in the line of sequence as well so let's try this this is a sort of funny problem on s and then the conclusion here is T or not TV right this is crazy how am I supposed to figure this out right we can see that this is my conclusion right so if I want to use the indirect proven here this always think of that the indirect proof is the nuclear option which isn't funny in terms of actually think about nuclear war anything like this but I think it's the nuclear option in the sense that if you really don't know what to do just do the indirect proven see if it works out for you right so that means that that if I wanted the negation of this if I was in a direct proof need the negation of this sequence here so that means line 2 here let's write it off to the side is going to be not T or not T right that's going to be line 1 I know I'm sorry not like one let's just give you our a IP right and now that I've done that let's do it - Morgan's because whenever I see an engaged in outside of parentheses I always always think - Morgan's rule remember the de Morgan's rule is if you have not P or Q that's the same as saying you have not P and not Q okay let's do that now so that means we're going to get not T and not not T right if you get not T and none ought to write that's giving line to - Morgan's right line for here's let's get rid of this this is a double negation so that's going to be not T and T that's line three double negation wait a second this is a contradiction isn't it to say that Tony is going to the party and Tony's not going to the party that doesn't make any sense which means that this has to be invalid so that means I can simply discharge it here they write the opposite of this was this so that's T or not t that's line two through four indirect proof okay and that's how that sequence does it that looked like a really difficult problem because this s and the T that was sort of crazy but you can see the enjoy proof a lot of students it was actually very short in terms of actually building a proof okay let me do one more problem here well you're all do two more problems let me do it one more as an example and then we'll do a problem using from the overt that it's okay let's take a look at this problem here this problem is a little bit uglier all right it's if L then not M then N and OH line two is not ad and P and the conclusion we're looking for is its L then M and P right now the reason I've chosen to do this problem is because you can see or if you're stuck with this problem you'd be like you would start off by saying I need this but how can I get that from here you can see what we probably need here because notice what the main operator of the conclusion is it is a conditional since it's a conditional it makes sense that probably the first thing we should start with is a conditional proof so remember the rules for the conditional proof an indirect proof apply sit in the same way the goal is different though right so you just have to take care of that when you choose well your assumption is for the conditional proof you're thinking that it's going to result in a conditional sequence with these two things so the indirect proofs do the opposite so here we have we're going to assume L so that's going to be our a CP the assumption for the conditional proof slide four is once we have L I have not M then N and O right and that's going to be line three and one modus ponens okay now that I have one two three modus ponens the question is what I need is I need to get the N and O by itself right because then right I would seem that I need to get the N in the bottom selves at least to begin to unlock this problem what can I do here well I'm not sure because what I really need is I need the opposite of this because if I don't have this well let's try this let's do an indirect proof ragnar instead let's assume that we don't have em right so our indirect proof is here is going to be not add and you're going to see why I've done this in a minute right so we're going to assume not m and this is our a IP right now remember my goal here is to discover a contradiction right so if I don't have M then what happens well then obviously I have edit out stuff line six here is going to be N and now remember because this wasn't discharged I can use these lines within my indirect proof sequence so this is going to be lines five and four modus ponens now obviously as soon as I see these you can imagine well I should be able to combine these right or simplify these so let's get n here is going to be line six simplification right and then notice I'm going to simplify line two here to get the not n right so it's going to eight it's going to not n from line two simplification now of course I'm going to conjoin these two together to create my contradiction so that means I get N and not n right that's line seven and eight conjunction you can see here this is resulted by using the rules in a contradiction which means let's go ahead and discharge it the opposite of this well what is the opposite of not end well it must be not not M right and you can see here the lines continuing down and that's going to be lines five through eight inter indirect proof now once I've discharged it remember all of these statements no longer count as a valid element within the proof sequence okay so this stuff doesn't count any longer so now I have not not M what can I do with this well not not M line ten is what well let's just simplify it that's em fly nine double negation right double negation now that I have M what can I do here take a look up here remember my goal is to get I remember I already have my else where it gets difficult remember let me use a different color I'll use orange right hopefully you can see it on the screen right so here's the L and notice here's my L here so that means my goal is still to get this thing within the conditional proof sequence all I need is M and P we'll look I have a P right here so line 11 here is going to be P and nut and what I've always done is just switch this around and commentated it so that's line 3 commutation now that I've commutated let's simplify it to PD so that's line 11 simplification and now that I've combined let's just add these two things together in this order so that's line 13 is going to be M and P right lines 10 and 12 conjunction now that I've conjoined them and I have this that I can discharge my conditional proof and get this whole sequence here so on 14 is going to be if you have L then you can in fact get 2 you're having both m and P simultaneously I've discharged the conditional proof which means line that's a lines 3 through 13 conditional proof but you can see there is the results that we were able to actually conduct the proof okay so you can see that this one was a little bit tricky ultimately the reason we chose does not M is because we got to this spot here and we didn't have any I am point new we need to get these things unlocked so I said well let's just what if we just add that in it resulted in a contradiction which eventually gave us M we already had P by itself so now that we had M and P we were able to conjoin them here and then discharge our proof to get if L then M and P and so that means that that is the proof that this truly is a valid argument using the indirect proof method okay let me do one more problem from the homework here and again I'm using the old textbook here because I'm doing this sort of archaically online with my web camera but let's just do let's do problem number three the least problem three in this book if you have C and D then e line two is if you have D and E then you get F and the conclusion we're looking for is C and D then f okay now this problem of this problem isn't too difficult I don't think right so but you can see here if I look at the conclusion immediately since I see a conditional I'm thinking in my head conditional proof will conditional proof worker and it seems like it might right because my goal if I did if I got C and D I could simplify D I could get Yi and then I could simplify do to get Yi and I can get yeah I think all I need here is a conditional proof you'll notice in the homework that they don't automatically just give you an indirect proof problems they want you to see that both of these are really very very similar are there tricks of the same trade so since this is the antecedent conclusion and there's a conditional let's assume that it's gonna be C and D and this is my acp all right once I've assumed that right obviously I can get e cut to east-west lines 3 & 1 modus ponens right slide 5 here is going to be D and C that's line 3 commutation I'm just going through this quake here you'll get quicker with this the more you do it that's going in line 5 simplification right I'm putting the D basically just reverse these and dropped out the C you get my D then line 7 I'm going to conjoin lines 4 & 6 together to get D and E let's line 4 and 6 conjunction okay once I've conjoined them I can use modus ponens to get the F so 8 here is going to be F right that's line 7 and 2 or 2 & 7 modus ponens you can see now that I've got my F and my C and D I can down discharge the problem line 9 and conclude what well I conclude if you got C and D then well you're going to get F right that's what that sequence demonstrates and that's lines 3 through 8 conditional proof so you can see actually not to do any direct proof on this problem because what they're going to do is they're going to switch in your homework switching it out the conditional noon drug crews but at least you can see it basically how to do these problems the last thing I'll put up here on the board for you again are the steps because that's really what's true what's important here to do the indirect proof and even though I just did a conditional proof sequence writers remember if you do an indirect proof you're going to assume the opposite of what you need right because the goal is to find that contradiction right make your deductions until you can find a contradiction and again the contradiction doesn't have to be you be with the same variables as your assumption right we saw that with one of these we saw that with this problem in for instance right we made this assumption we ended up with the different type of contradiction different variables that's not a problem but once we get that contradiction then we're going to discharge the proofs and write the opposite or the negation of whatever our assumption was right that's what we end up our assumption proves and then we use all the same basic rules that apply to the conditional proof they also apply to the indirect proof which means you can do them sequentially you can embed them horizontally there's a lot of different things you do but the key thing to remember is once again once you discharge a sequence right you're no longer allowed to use this so this is off the table and now that I discharges this is off the table right so literally the only valid lines it's argument are this this and the conclusion they're right so that's how you do the indirect truth table methods again thank you for your patience I apologize for posting this video and I apologize if this videos all about paper though if you think you actually like if it's easier for you to follow using pen and paper let me know and we can always post videos like that okay thank you very much for for listening I appreciate your comments and look forward to seeing you online okay bye