hi this is an introduction to logic I'd mark doors B in this course overviews the basics of categorical proposition or in predicate logic in this video we're looking at propositional logic and we'll be discussing how to build indirect truth tables so welcome back everyone to indirect to tables let me start off with a the first day which is after I recorded our last video which was how to build truth tables for arguments I watched it I realized that video is incredibly bored because if you go back and watch that video I'm marking down building these truth tables but if I have 16 lines or anything I did eight my truth table in the last video you can see it takes a lot of time actually and what I have for you here to look at here is I want to show you here this is a little graphic put together where this you know it's just a sort of kind of brief review here a sort of brief review remember our equation here al the number of lines for a truth table equals two to the nth power or N equals the number of unique variables so I have here is each of these are unique variables and what I've done is I've done the equation for you you can see that when we get to 10 unique variables we need well over a thousand lines at our truth table right and now think about it for a moment when we make ordinary language arguments for instance of it I don't know I'm well I'm last night when I record this was a presidential debate when you watch these presidential debates think about how many propositions the political candidates are actually putting together to make their arguments right in terms of variable size it's probably well over 10 you can see when we get to 10 the capacity to actually analyze the argument using truth tables becomes basically just impossible you would really use literally be you need an Excel spreadsheet or something you definitely need a computer program to actually analyze what people are doing or their neighbors now that's sort of interesting because one of the things it means is that it represents the tremendous capacity of the mind to actually we have an ordinary language conversation with anyone which is conversely to making arguments that our bonds are actually fabulously rich in terms of their capacity to analyze multiple things at once on the other hand you can see with as more propositions come that means there's greater and greater likelihood frankly that it arguably here invalid okay which is why some people have been incorrectly led to think that we do deductive arguments we begin with the smaller points and move forward in terms of in terms of our knowledge so and you can see why that's the case is because if you start with a lot of variables there's a high degree of likelihood most likely that it's going to be a valley right but that's just an astronomical number but it also means that what we're doing in logic here there's only really scratched the surface of what our minds are already capable of so you watched that last video it was it was it was difficult to watch because it was so long because it took me a while to build this so what are we going to Jade today we're going to introduce the short method the quick method for actually analyzing arguments we're still utilizing the basic mechanics of truth tables but we're going to see by using the indirect method we'll be able to in a much shorter time be able to assess arguments to have many many variables at once actually so in there are places in which this is a bit treated so what are the basic steps well it goes back to our concept of validity right and I know I for many of you this is sort of kind of me like a broken record here but what does a valid argument Valon target is argument where all of the premises all premises are true right and the conclusion is false that's what it I'm sorry invalidity that's what an invalid argue it is right so that means that what we've been doing in truth tables as we've been building these truth tables and then looking for specific lines in which all the premises are true but the conclusion is false if we can find one like that line like that that proves the argument invalid so for instance let me do a quick brought here I want to spend too much time on it let's say we did this example not a if not a then B or C right you have not B therefore C if C then a right so here's the arguments right if the Andrew doesn't go to the party then either Bob's commit or Charlie's going Bob's not quite there for it Charlie goes Andrew is going to go all right that's the argument it's interesting once you put it into sort of ordinary like language you can sort of assess now that doesn't sound like you're on target but let's see how it works right so if I have to build a truth table here you can see I already have three lines it's going to take me a while to do this so let's do something in Reverse this is called an indirect truth table method we might even just say it's the reverse true tale eleven indirect truth table method and what we're going to do here what are our steps let's see if we can write this over on this side here the first thing we're going to do is we're going to assume that the argument is invalid so the first thing here is assume invalidity and we're going to then we're going to enter our assumption into the truth table enter the assumption enter the assumption so how would the assumption look like if this was a truth table under all of the may operators for the premises the parenthesis would be true and under the main operator of the conclusion it would be false so let's just enter that in now right and that would be this would be true and this would be true okay since these are the made offers you can see this would be an example of which this is involved so we're going to do here is by entering our thing we're going to do and go backwards we're going to sort of work our way backwards and what we're going to try to do is see if if it works out if our associate doesn't break any of the logical definition rules we've been looking at in this that we've been using for propositional logic then this will turn out to be a case in which it's false so we'd add to our assumption and add to an ascension well religion we're going to work your way backward looking to see if you break any rules you're going to get a sense of what I mean by this in just a minute okay so let's do this now so how do we do this well then we're good let's it's often times and I'm gonna always recommend sort of hit is always start with the conclusion it doesn't always work your way work that worked in that it doesn't always work best to start the collision but typically it does so let's start with the conclusion here remember if we go back to our definitions here and I wonder if I could pull it out there but in a textbook here I think it's helpful for you guys to see those let's see I think I've got my apologies here nevermind there was a there's an insert that if you buy the textbook there's an insert or if you have the textbook for instance you'll notice if you go to in-between the back cover it whereas the rules I'm sorry here I should have prepared they don't happen ok over then always have to add it this out here let's go back here so let's just work our way through this problem okay so so there's only if you go back to the definition for a conditional the only way that this can be false is if this is true and this is false right well if this is true this is false now what do I be by work your way back word once I know the values of would I could enter those same guys in the rest of life is where they'll all be consistent so enter your values consistently across the equation across factors of the truth the argument so that means that if a was false here anywhere there's an ant needs to be false okay anywhere there's a C the C needs to be true okay now let's work our way backwards now you can see here so okay this product.this the conclusion I'm entered in by assumptions working out so far now let's move here this premise here if the if this is true the B has to be false so now let's enter it B is false okay so now my assumption for in validity is still working right because what I'm looking to see if I break any rules because if I break a rule that means that I've actually committed a contradiction so let's see here now let's start with a face false the vacation of its true right okay so if the negation of this line is true and maybe here we can put a little all right so here is the main operator if the B Gatien of this is true that in order to maintain my assumption here because this depends upon this being true if this is false that it fails so let's go back to our subject let's just see if we break any rules if B or C it would see if they in order because if this was false then we would break the rules but this is actually true because in the disjunction rule all you get is one of them to be true in order for the whole thing to be true and since the C is true that means this is true if this is true a true a true using the conditional rule is true which means that this also works out this is in this is a invalid argument because you go to Netta is instead of looking for because this I would have to create eight lines instead of create all eight lines I said let's just start from the back and start with all the possibilities for it to be false and work our way backwards that see if we can discover an instance in which we commit a contradiction because we did commit a contradiction here I think the best way to do this is let me do another example here you can see how to do these problems over another so let's try another one here sorry I feel like my explanation skills are sort of waiting if you okay so here's the other problem if a then B or C be there for D a therefore not see if not see that D oops step to D that's nice okay so let's do this problem here now let me go back here and actually finish the steps oops so what are the last steps here number four is enter your values consistently number four if a contradiction is discover the argument is valid if not that's an R it's invalid okay so this is an argument so going back to our example here are we made the assumption of the ability we did not commit a contradiction B we did break any of our rules therefore this argument valid so let's do this problem here okay let's start off by making our inception that's the first thing we do under the bait offers that's going to be false oh and then this is going to be true true true okay so the only way for a conditional to be false is if this is true this is false now if the negation is true that is the C is false so now I have all the values for C of C da actually so let's enter them now so this is f this this is going to be half let's see all the DS all the C's are X okay and all the A's are true okay so let's take a look at this let's start with this now okay so in order for this conditional because you'd see this would checks our substance working without of this premise here if B then D the only way for this to be true D is false because remember if this was true right B was true then that would break our rule is that it would be true B false so B must be a false value so B must be false okay now I'm going to move over here I now to the value of B B is false now what's the rule for disjunction this parenthesis here is there at least one of them has to be true door richer so this is false but wait a second now I'm comparing a false and a true statement using the conditional rule but it's but it spoke but that is actually false but my assumption was there was true you can see here that that means that this conditional needs to be both true and false at the same time for my assumption that work do you see that because I need because it needs to be true for my assumption but of course my assumption required view also say it's false they can't both do the same which means that this is a contradiction because I've discovered a contradiction then navigates my assumption is false or by assumption fails and the argument is valid okay so you can see here what I would do is I would just circle this it right valid right because this is where obviously you commit the contradiction right so I get the steps soö invalidity enter the Assumption work your way backwards looking to see if you break any rules enter all your values consistently therefore if a contradiction is discovered that the arguments valid if not it's invalid so let's do another problem here because we're gonna I wanted to two more pause here just so you get a sense it but I hope you can see if this is much much faster than doing it the way we've been doing it in terms of building truth tables right so what if we have if a then B if C then B therefore give a then C now by the way we put this in our before if you like if it reads particularly birthday cake it charlie / - gate the Bob Lee birthday cake thus if Andrew eats for the cake that Charlie's gonna okay so the question is around making sense so let's begin our assumption here if you want pause this video and I'm just going to do this real fast pause this video and then see if you get the same answer than I do alright so this is going to true/false a it's true C is false so if this is fault whoa now it's interesting here so you can see here if this is false then B could be true or false so my goal is ultimately to make sure that I test all possibilities for validity here so let's enter both the values now so let's say we have if this could be either be false in order to make this true or this could be true it would also maintain the consistency you can see here whenever you recognize through the rule that there's multiple ways in which this could be true you have to list them how to build truth tables for all of them and that's one of the difficulties of using the direct route table method is you have to work through all possible contingencies to make sure that your assumption for immunity doesn't work because it may not work on one line but it may work on another line so this is all an inherent difficulty because if we had this would means that if we had a thousand unique variables it would also be very difficult to solve the problem for the same reason because you died you have less things to rather than a truth table but still it'd be quite a few so that means that B could be false or true for something to work so let's go look over here in order for a if a would be to be true right if this is true B has to be what B has to be true right you can see if I put false here it would look like I committed a contradiction right so that's not the case but let's go back down here so this line looks like it could it's a cluttered action right you can see right there because true/false can't be true but if I assume that it's true because this was a possibility do that it a is true that means this would be true so that means that I found this this is the assumption works now and because it works it's invalid because I assumed it was it not argument I've proven it right now let's see the reason this is a good example here is because you can see that if you just did this first line then would on your very where you think it's a valid argument when it's not actually a valid argument it is possible for the premises to be true and calaf all conclusion simultaneously okay now let me quickly talk about one more thing before I do any more problems is we can also use the indirect treadmill to test for consistency of statements in here just to give you example let's imagine that we have even though we are detectives right and then we've actually we have were detectives and we're trying to solve a murder case right so the example here is think about detectives we're trying to solve a murder case I mean we have three different witnesses right we have three different witnesses we have witness a and we have witness B we have witness see in each of these witnesses has given us specific statements about what they saw in terms of the murder right let's say that it's the witness a here the first witness says I don't know what their proposition is but it can be it can be symbolized as see if see that B right and let's imagine our our second witness here says no it was not B or C and then our third witness here says it was see if it only of B right let's imagine that each of our witnesses gives us these three statements as a detective we would want to know whether or not these statements are consistent right we wanted to what about they're consistent because if they're consistent then that means everything they're saying checks out or at least that you know it's all its sake if you will if there's standard Turk it consistent right that that would be that solo is lying right someone is lying it would seem so especially if to favor consistent for that it was done so we're going to see here is that we can use you can notice here there's no conclusion these problems have there's no conclusion so these are not arguments we're just trying to test whether or not arguments could be consistent here's what we're going to do is we can use the truth table depth into the same way right but if a contradiction is discovered huge party in contradictions mean inconsistency and then note no contradiction means that the arguments are consistent okay so a contradiction means inconsistency and if there is no contradiction discover it means the sales are consistent so let's test for them now so what we do is we assume that they're all true put enter T for the main operator and then let's see let's work our way backwards and see if we can discover a contradiction of that now let's start I always say start with the conclusion but you can really start anywhere on these problems as long as you're consistent consistent in terms of your approach and you test all possibilities so you can see here what are the ways in which see if it only B could be true well there's two ways it could be true if C is true the B's routes true but also if C is false and B is false that would be another example in which this conclusion could be true here so you can see C or B can be true or false so let's run through this personal and see if we can find a contradiction so that would mean that C is true right and B is true if B is true then that is the negation it's false right if B you can see here you have it of false or true right that doesn't break the rule for disjunction because disjunction is that you just have to have one of them be true in this case the C is true so our associate here seems to be working so let's I join our values our B value is true and our C value is true true if C then B if true 2 therefore true that's true so that means it works our social works out here but we still need to check this second line very if we find a contradiction we'll know that there's a need that these sales could be inconsistent and since we're detectives we want to do a good job with it so this is going to be false B what if B is false how does the how do things work out it would have C is false right okay so if B is false the negation of B would be true and so if you know true using the disjunction rule here true to false is still true right as are true or false is still true and so that means okay that means it seems to be working out here oh that's probably better for us so if we said C was false let's take a look at it here if C is false right if she'd be the rule for the conditional is that just that would be true okay so now we've checked all possibilities right in which that could be the case yeah so that seems to be I think I'm trying to make sure that we were totally consistent here is there any other possibilities could this is true this is false what happens if this was true no because the yeah this is dumb I'm sorry I'm not doing girl this video but that's right so this proves that we did not there's no contradiction so I mean these statements are consistent they're consistent statements the reason I was sort of sitting there trying to double-check where I was double checking is whether or not I had to move the disjunction but you can see here the biconditional requires this so it determines that these are the lines that these have to be but both of them are some short doubts not if this was in a consistent argument okay let me do this go here I want to show you one last thing here if from the textbook okay so in terms of indirect retails this is the the chapter website but I want to draw your attention to a specific passage the Patrick Hurley includes it's worth reading here's the three funnel points need to be made about indirect retail art its first if a contradiction is obtained in the asylum choose rise every step leading to it must be logically applied by some prior step in other words the contradiction must be unavoidable if a contradiction is obtained after 2000 haphazardly or by guessing then nothing's been proved the objective is not to produce the contradiction but to avoid what if possible okay you can see here it's a great little thing if you sue the are going to invalid there's a cluttering so that's valid without contradiction ensued and valid okay and the same thing here goes in terms of consistency right if you card it shows it's discovered inconsistent there's no coloration of the arguments consistent you can see here's a number of different examples it's not very difficult to do let me just do two problems here from the homework side just so you can get a sense of how we're going to do them let's do number 17 here let's do 17 they'll let us a get better check the instructions here it structures when possible compute truth values of the simple components of the following compound propositions if no truth value computed write a question mark so let's take a look at every 17 here it's not a full truth table problem let's take a look at it right so so here's what we got okay so if this is true if we want to maintain the truth functionality of this David right this is going to be true this is true then the negation notice we're starting to be gauged about the P the negation cannot be false because if it was false remember we said that right we said A to B in order for the studio is true true false false let's get it true false true false right this is true false true true here's the defamation here right so since this is true and we want to maintain our assumption of truth here then that means these are the two possibilities we're going to look at in terms of the definition right and now you can see that if it turns out to be false then that would make our main operator false we don't want that we want to keep our truth condition so that means that this thing has to be true in litigation so that's going to be a tea right there if that's a tea then P is false right to see how we did that we're just using the definitions in order to figure it out that's some fairly easy problem to do let's take a look at the section to use the indirect truth table method to determine whether the follow targets are valid or invalid I'm gonna scroll down to do one here that's maybe a little longer so you can see how to do it what do one of the things you'll find in terms of doing these problems is that when you use the whole universe sakes you're one of the things you'll find when you do these problems is that the conditionals you always like the conditionals because there's only one way for the conditionals to be true the same thing goes with the conjunctions there's only one way for those to be true the ones that are tricky is when you have the disjunctions the disjunctions mean you can do multiple things that's why would I see a problem what I generally try to do is do my disjunctions and bye-bye conditionals last I don't rush into doing those because I want to let the conditionals do the work for me if that makes sense if you have no idea if you're confused by what I'm talking about if you just start doing these problems over and over it's sort of you'll see exactly what I'm talking about naturally it's kind of like when you learn to play poker first you have no idea what you're doing I still know but how is my buddy but but after a while you start to get a sense of the game it's the same way and that's by the way what I recommend is you'll do really well of these problems and logic yacht out if you could really just sort of see what we're doing here is a game and the goal and what we're Jews were is changing the rules with each session with each video and then what we're going to do and then you'll enjoy it ultimately I think that's the trick is to learn to sort of enjoy what you're doing the logic we'll see how if we can do that to see it as a gate as sort of a puzzle is how you ultimately enjoy the bat so I'm going to always start with the conditionals you can see I want to end over here with the disjunction because the disjunction there's three to five so if I start here I've got to lay out three different possible lines I don't want to do that that takes more time little player the indirect YouthBuild is do it quick so this is false then this has to be true this has to be false right because in which would be Big E true so now did I know a now I know the a values let's put it true and now let's put in the e value which is true okay now be good so that was done let's look here at this one if this is true the B has to be false let's input that falls okay now let's jump over here you can see this is where it starts to suddenly get a little tricky which one should i do first I have more Val I have the same thought of do I have more values in put it here so let's take let's stick with this one actually a little dolphin do this third one you could you can start here it's just going to take you a little more time to do it but we're going to see why we're going to do this because in order for these since I know that a is true and I have a conditional the only way that this could be true is if this is also true right so in order for this to be true right because B or C if they're both false it turns out to be false which means that it needs to be true so C's value is true that helps me tremendously now because now I can take a look at this one if C is true right at I would have maintained one condition then that is the conjunction cannot be false it must be true right what must be the case in order to be this would be true if you have dat the only way that compound conjunction could be true as if both values are true which means this must be true so you can see here what I what I've discovered is that this argument is invalid because my assumption works out I never discovered a contradiction it also means simultaneously that my parentheses are consistent so you might say that this argument is an invalid argument but that my original assumptions by premises here we're actually kids logically consistent right even though it's a dot okay but ever through it's also done okay so that's our that's our video for indirect truth tables in our next video we'll be moving on to natural deduction we'll be talking about how to actually make positive routes instead of trying to prove invalidity in our next series of videos we're going to try to prove how to we're going to look and see how to prove that something is valid okay that's enough for me I'll see you guys online bye