hi this is an introduction to logic I'm mark doors B and this course overviews the basics of categorical proposition or predicate logic today we're going to actually be taking a look at propositional logic and in particular this lesson focuses on truth tables for propositions in essence what we're going to talk about is what a truth table is how to build a truth table to analyze compound propositions or statements and then we'll also talk about how to classify and compare propositional arguments so okay welcome back everyone let me see if I can make this a little bit bigger okay so let me start off with sort of a brief review in our last lesson here we actually took a look at propositional I'm sorry we looked we talked about the idea that every compound statement has the property of five valence right bivalent to be the proposition isn't it true or false and what we did is we talked about the idea that if you create compound propositional statements for instance I was making this up a and B therefore C right there's an example we talked about the idea that if you build a compound statement and you simultaneously know the values of each of these things for instance we have what if a is true B is true and C is false right you could actually calculate the truth value of the entire proposition by looking about by looking at what comes under the main operator let's see here in this case this is the main operator right here so for instance if this is true and this is true and this is false right here's where you have to just memorize it hopefully by this point I'm just going to assume that you've memorized these definitions if you have it you should pull out the definitions that we talked about in earlier video I think six play two or maybe just explain one pulled up now earlier and keep it with you right because remember I call a conjunction here always has to be true and then now using this value and this value in the rule for the conditional here we can say that this state must actually be false right now here's the problem that's we looked at in our last lesson and hopefully we explained that good enough for you you understand that and it you did well in your exercises but we're gonna look at now is the question here of what about what we don't know the values right because after all most of the time when we hear people give arguments if we're gonna put in a propositional form the truth is we probably don't know whether or not each exact proposition is it a true or false it may just be beyond the scope of what we can do and so what we're gonna see today here it is true we're gonna talk about truth tables right and what truth tables allow us is they allow us really just this they allow us to calculate all possible truth values all possible truth values whether or not so we don't actually have to know whether or not a specific proposition like this a for instance is true or false we can calculate all possible truth values and then you look at all possible truth eyes we can actually classify specific types of statements some of them are always going to be bad one class offense so they're gonna be good or two tautological well good enough to right term there but some of them revita logical and then we can also talk about how to compare separate compound statements and sort of compare maybe what one person says compared to what another person says and see if they're those two arguments are those two statements ultimately ended up being consistent or inconsistent or contradictory so that's we're going to be covered in this value in this chapter now it's interesting truth values is something that's very interesting i think most people find actually that's not that difficult and you actually plan to do fairly well with them so let me see here let me create a new page you'll notice I have a little a new notebook thing that I'm using for our video instead of Microsoft Word I'm hoping they don't in this video the Microsoft where he's been doing let me sort of go over some of the real key stuff you need to know here I'm gonna change the color of my pen here so let's talk about a very important equation in terms of truth tables you're gonna see that the basic structure that we're going to do well let me start the equation that you absolutely have to just memorize and that is l equals two to the nth power and what this equation tells you it tells you how many lines how many lines a truth table needs to be in order to adequately capture all possible truth values how many lines a truth table must be okay now in order to actually explain all this what I need to do is explain to you what all this stuff means right so out here right this obviously refers to the number of lines right - there's no numerical and in here this is the important and refers to the number of propositional variables and I should really say here I could have right and refers to the number of unique propositional variables so let me see if your a fight give you a little example here let's go back to the example I gave earlier I said a and B implication C right so this is the proposition so you're gonna see here is that they first off what is the truth table going to be I need a first figure out how many number of lines maybe so I asked myself how many propositional variables so I have in this case I have one two three unique propositional variables because for instance here would have added a bracket here and then it said and B again right so this a and B therefore C and D in this case I still only have three unique variables so I would still need a pound of three up here in my things so that means that it's two to the third power right is the number of lines so what's two to the third power two to the third power eight right and that just mean side go 2 times 2 times 2 right that equals 2 times 4 2 times 2 is 4 or 4 times 2 is 8 right so that's the number of lines I would need yeah let me show you what that's gonna look like because ultimately to fully get this you're gonna have to see I just let me just erase this because this was just an example to write so so now remember in terms of 5 minutes so here's strictly speaking how you're gonna do this the very first variable you're always going to you're always gonna write because it's gonna be very long so I need 8 lines right this is gonna go down here and it's gonna look like this it's gonna be true true true true false false false false right and what i'm gonna - generally speaking is because we remember since this is - today in power this equation will always and can only yield an even number right since there's an even number what you're gonna do is for the first the first variable here you're just gonna say I'm just gonna go eight divided by two right and you can see here if I just separated this I have four pure now for here that I thought but the first floor guy is true the second four values is false now for B I'm just gonna then just divide this again so this is gonna look like true true false false true true false false so notice what I did here is I basically write I basically just divided these it has and that I just repeated the same thing down here right and then what you're gonna do next is when you do see you're just gonna take this you're gonna divide these in half and your lats or it will always look like this it's just going to be true false true false true false true false so it'll look like this true false true false true false true false okay so now what I've done is I've created a table here which actually is that you raise this I'm creating a table here work now I've listed all possible truth thighs right because it and you just have to accept this at this but of course we could go in and your book does talk about why this works exactly the way it does but that means that that means that here on each line here I have every possible condition for each of these propositions because I don't know if they is true of these true C is true or false but now I can see these are all the possible all possibilities that exist right and the first word of course is where they're all true the last words were they're all false and you can see here none of these lives are ever going to look the same now what's an important thing to do now I mentioned what if for instance if you had the be back here again right let's go back to that let's recreate that example so I said and that and be now whatever falls under the B comma needs to be identical in the second time around so let's just do that here so that's gonna be true true false false true true false false now for those of you watching if you're doing these exercises by hand I actually highly highly encourage you to use graph paper right because the graph paper gives you the lines because in a complex equation what's gonna happen is the light to be very very long because remember L equals two to the nth power what happens if you have six unique propositional variables that means you're gonna end up in those to the bathroom you're gonna end up with the petruchio that needs 64 lines that's very very long in fact for the most part you're never gonna see this in you know later video walks we talked about how to do these problems easier without having to write all these these truth tables down okay so so essentially what you're gonna do with these truth tables is you're going to build up truth tables like this now what I want to do now is I'm just gonna go over I'm gonna give you a couple examples here just to show you how to do it cuz they won't you can see here how do we go ahead and do the problem now that we've done this you remember always want to keep the whatever falls under what needs to follow know that the same unique variable identical in each case he's trying to mix it up you're gonna completely screw it up by and by the way I should say this it's very easy to make that mistake I admit I've done to myself god forbid I may even do it on this video but I hope not so and now what you do is you basically just run through the problem the same way we do know at last Probab except this time you just analyze according to the definition of the rule we gave earlier so let me give you my green one here so I remember what's the rule for the condition which you always want to start just as we said with the inside parentheses and then you want to calculate the truth values and it may be helpful here just in the same way I'm using different colors it may be helpful for you use colored pencils or you've a highlighter like because you can see it's gonna get fairly complicated or it looks complicated but it's fairly simple okay so what's the rule for the conjunction they both have to be true in order for it to be true so that means that this would be true this would be true any falsehood in it's false that's gonna be the basic definition so that's false false false false false okay so those ones are all false and now what I'm going to do that means I have to compare this line with this line right because whatever falls under the main operator the parenthesis what you're gonna use and I'm gonna compare these two using the conditional roll here right so then I'm just gonna look at it and here I would could use both your fingers if you do it by hand right that's gonna be true to true rule is the rule for the conditional the rule for the conditional was that yeah it's always true unless you start with truth in the consequent is false so that means this is okay so that means this is true true false true true true true true true yeah so you're saying how did I know that those were true so quick remember it only time it was just gonna be false as if it but antecedent is true the consequence false in all of these cases they have to say that it's false which tells me that it has to be true okay now that I've done what falls under that need operator now I need to go to the I'm sorry within the brackets here there now it moved to the actual main operator which is this guy right here and usually what I do the final mate operator would well I'll just do right here so and the conditional rule again we're backing up started a conjunction rule is that they both have to be true you know let's put actually use red here just to see if I were doing the main operator so that's gonna be true false false false true true false balls now typically when you're booting what you do is then I've used it a different color so you can see what falls under the operator but you can also square it or I actually prefer to highlight it if I the highlighter and now we see that these are the possible conditions in which this statement this entire statement would be how we would analyze it right so that means that this statement would be true if these conditions existed right it would be false or under these conditions it would be true under these conditions it'll be false for these conditions so that means that the hold it this statement is true here here here there's three cases in which the state can be true one at which a B and C are all counted it's true another one in which a is true I'm sorry a is false but B and C are true and then it's also possible everything is false but B happens to be true then actually the statement is still true of sort of interesting because and we're not starting with an ordinarily much example here we'll look at that homework but you can see here I now have now that if say for instance I read an argument I did the truth table now in order to know that the statement was true I actually just need to investigate whether or not really B is true because if B is true then it may be has to be true in each of these cases and in this case only be asked to be true in order for the whole statement to be true right that means I just really I can actually just really live it what I have to go research and find out whether or not something is true given that statement so I hope that makes sense in terms the way I did this I feel like what I feel like sort of when you teach you talk about these pot of truth tables you sort of just jumped in in the deep water and learn to swim and we'll just do example after example here so you get a sense of it okay so let's do another example york or let's actually write down what the basic steps the steps are for doing a truth table the first step again was up to is to determine the number of lines right you need to make sure to determine the number of lives you have to put in your truth table to make sure you consistently you could with tomorrow say you consistently record the truth table right and when I say consistently record it when I beat this make sure that whatever it falls under one variable always is the same as what falls under the next variable and they can utilize that system the very first variable you're just could have divided half the first half is getting true the second how is false the second burger is gonna divide again and you'll do that on and on until eventually your Lots registers be true false true false true false and that's essentially the process for crius so make sure you consistently write that down because I do not consistent you'll get it wrong and that's really critical actually and then once you've done that then you're gonna compute the table using the truth functional rules so let me do it example another example here well let's see here let's say we have this and up pull these from your block actually see we have this example C and not T therefore I'm sorry and I don't know this could be for is Charlie's going to the party and Derrick's not going to the party if that are safe if Charlie and if Charlie was the party - Derrick does not go to the party then Edward will go to the party right so the question here is under what conditions with this statement actually be true or what way would we say whoever said that's him it would be a liar for instance so the first thing I want to do is to use the steps here is that three variables three unique variables so that means how long does my truth table have to be right don't be afraid just to write it again l equals two and 3/8 so I need eight lines long and so I'm going to take that first variable here and I was correct true-false true-false by the way I apologize I've got a cold right now so this video presents out - good so that's gonna true true true true false false false false right now the first thing and then the next thing it's wanna make sure it's consistent so I look over there any more C's no so let's see so then the D here I'm just going to divide that house so that's gonna be true true false false true true false false and then e is just giving true false true false now it's important here that to actually be really careful in terms of how you write this out and that's why I encourage you to use graph paper because if you want to make sure that when you compare them it's not like crooked so you don't wanna compare this this best you want to make sure that everything's on the same okay so where do I start from here what's the first operation should start with I hope you see it I should start with the occasion here I always start with the negation because they may occasionally depends on what they're in aviation in the parentheses I start with what's in the parentheses start with a vacation once the role for the Geisha what this is true this is gonna be false false true true false false true true and then next I'm gonna use this one here right which is going to be now remember I'm gonna compare this of this now once I fill in the negation value I don't have to worry about this I can ignore it it may even help you just to sort of exit out to make sure you don't miss compare them right and really the conjunction rule says they both have to be true otherwise it's false so that means that this is false false true true false false false false right and now this is the main operator the parentheses which means I'm going to be comparing this line this line using these conditional rule here okay and now I'm going to change the color here you may not be able to do that just so we can know that this is the final line that actually helps us determine under what conditions this compound stable would be true so the conditional right in each eye it has to be true that false or that's gonna fall so that means that this is true right false and false is true true and true true true false and right that's false and the rest of these since all of these are false is gonna be all true and then I usually box it so that way you know what to look at okay now so this is essentially how we do each and as you can see that this statement is actually always true there's only one case in which this this statement would be considered a lot and that would be a case in which C was true D was false and he was false those things were all happening that this day would be false otherwise it tends to be truth so you can imagine this statement is has a high probability of being true obviously of course in order to actually that's true not to go to the world it's right and check it out to know if the propositions themselves are but this is essentially how you build the truth tables and I'm not going to do too many more examples like this because I think that you can see how to do it and you have to take a look your book and also have other videos that are posted on the YouTube channel you can take a look at so what I want to focus on now is how do we classify statements once I get all of these together what kinds of possibilities exist in terms of classifying types of statements so let's move here to the next page and take a look at that okay so what we're going to be looking here is classifying statements okay classified statements so there are really three types of statements the first is what we call it to taller G and that's when all of the values are true and when I say all values all values to fall under that main operator not all of the other guys but all of the main operator values number two another example here if you don't besides a tautology you can have a self-contradictory save it contradiction essentially and that's when you have all false values the fault or the main operator so for instance that I mean this of course would be the worst sort of saving because dominated all possible statements I'm sorry in all possible conditions a confident that whatever state is self-contradictory is always bad it's always false so you always want to obviously avoid those and at the two of course the majority or what we call contingent and basically that means do you have mixed values you have at least one true and one false value that falls under the main operator okay so let me instead of you can see here if we go back to our page we were looking at earlier if you go back here what kind of statement was this well it almost was a technology right because they're all true but it's not had one false which means this was if we're gonna classify this day right we would say that this statement is contingent it's contingent upon well really these can be this line okay similarly if you go back to our I think one of our other page right this was true false true false this is also contingent the grand majority of all these are actually going to be contingent statements so let's see you me give you an example here let me actually show you the book here and because this is easier for me just to show instead of me doing all true tales one of the things you'll find in this job six point three is two it actually takes a while to do all this stuff to run these retail especially if they have more than three especially if that four or five-year-old it takes a long time generally speaking in your book is never gonna ask you to do more than sixteen lines at any one time but here's the basic definition again a truth table gives the truth value of a common proposition for every possible true truth value of it's simple components each line of the truth table represents one such possible array of values you can see here six lines 64 I'm sorry number of different simple propositions I think it's easier to say unique variables because that's easier for me to remember the way they sort of Miss the wording sort of odd so let's say let's scroll down here take a look at some of the examples that Hurley gives in terms of this you can see one of the things that they they're Hurley is gonna do I don't think it's necessary to do but it may be helpful to do for some of you just to get used to it is to write the a and the B separately and then to fill them out cuz you can see here here's a B the B comes up multiple times so it may be helpful to do it to write it out here so then you can just sort of write them in real quick without having to think it through I don't usually do that because it's sort of just extra writing it doesn't seem necessary but it is probably helpful for people who are just and learning how to do this okay you can see he does the same thing since this is the main operator there's why Paul's this is the truth buzzer look at it if you classify this statement right it's contingent okay so let's take a look at here again all true all false the contingent you can see here here's a tautology here right let's see how to say if George goes through the party that Harry is going to go and George is gonna go then Harry is gonna come right if I make that claim that's true under every possible conditions ironically even if it's false that George's where the party in areas that statement is still true right this is what we call it - Tala ji one of the things that's interested - philosophically about technologies is tautologies never really tell you anything because if you make a statement it in every possible condition is always true that doesn't really tell you anything new about anything right so you have to be really careful about tautologies because even just because we're saying that college is always true under all possible conditions it's actually pointless thing because if it's always true that doesn't really tell you anything right especially if George and Harry are quite a party that statement sort of meaningless and similarly self contradictory statements are obviously bad in another way right this is an example here either George school party or Harry's going if and only if it's not the case that George goes and it's not the case that here he goes right now you can imagine we don't usually talk like that but if someone actually gave you an argument like that it would be very difficult to know if what they're saying makes any sense I mean here herein lies the value of propositional logic because you can see that when you actually rub through the truth table and you figure out what falls under made operator you see the under every condition the statement is false if it's false under every condition that it's totally contradictory totally meaningless this at least has meaning this has no meaning it doesn't even make sense it smells seven circle so this is necessarily true and this is necessarily false right okay now let's talk about comparing statements comparing statements what if we're gonna compare now we should mention here that when we're comparing statements we're only talking about comparing two statements to compound statements plus about two compounds we're not talking about comparing three and four five statements we're mainly only talking about comparing two types of students in particular now there's four possibilities else right over here the first possibility here is that the two statements are logically equivalent that's one of the reasons I'm using this little fun character thank you variance on these things logically equivalent and what this means is this but what don't the values that fall under the main operators in each statement are identical they have identical values so I was hoping that this new boat they would help my handwriting unfortunate that so let's give you an example here logically equivalent statements for each line they have the same value so take a look at this one you have K then L and then you have not L therefore not K right so the first line here and see you can see why it's so important to label everything identically I mean consistently so see how the K here is true true false false and see how this gets true true false false make sure to label everything correct because I guarantee you someone who's watching this video when you take your test you're gonna make that mistake and then you're gonna get the wrong answer I want to warn you not to do that so you can see our boat if you label everything correctly when you compare them and you can also see you can only compare students to have the same number of values and the same values too so we're talking about two statements with the same values so on the first line this is true and this is true of a secular this is false this is false and these TEDx lives is true and true and then over here it's true and true which means that these statements are logically equivalent and that's actually we're not going to talk too much about it in this video but later well that's gonna come fairly important because that means that you can actually replace this statement with this statement these statements effectively have the same truth functional meaning not the same meaning per se but in terms of truth functionality they're the same okay and that's really key so the first one here is logical equivalency something is logically equivalent both stems to your compare have identical values so I'm gonna add this over here we're talking about two compound statements each with the same number of variables and the same variables - okay so I just forgot that in terms of what kind of stamens you can compare here so the first one is things that are logically coolant the second possibility of students they're contradictory in a statements contradictory if on each line the statements have opposite truth values right it's contradictory if it has all % true thoughts on each of those lines so let's take a look your earliest book here to see what he gives you an example of you see that you're making a little bit bigger for you okay so notice here so we have two statements each with the same variables and the same number of variables 2kl alright let's take a look at it wherever when in this tin it K the same is true but even this David is false and this one is false and this was true and this one is true and this one is false and then this one is true this was false see how they're opposites now if they're opposites that means that wherever this statement exists this statement will be false and under the conditions were this state is true this day always be false so which means that these two statements are contradictory now here we should have maybe identify the really principal law and logic it's actually called the log-dog fire danger I think how do you ever mentions it in earlier video the law of non-contradiction says that something cannot both be and not be at the same time in the same way and say respect and this comes from Aristotle now what that means is that it we should never ever argue in contradictions right because if we make contradictions that means we're saying something it is and it's not at the same time because thing you know imagine someone imagine you're talking to soda they say if you kill bats then the lights problem will be solved right and then they go on to say and you know I also believe that if you if you kill dots and the lice problem will not be solved right you can't believe both things simultaneously so contradiction is sort of the moral stated philosophy and when you compare statements that are contradictory you should automatically know that whoever's making those assertions it does probably know that they're making a contradiction but then whatever they're saying just doesn't make any sense so contradictions are really important to understand it spot so you can you can qu you compare statements you may discover that they're contradictory okay so that's the second type of possibility in terms of second category formerly compare statements what's the third possibility here the third possibility is that they're consistent a consistent means there's at least one line where both statements are true there's at least one line where both statements are true right all you need is one and that's what it means to say that statements are consistent fear here for this example imagine for instance that you're a police investigator for a dog maybe if Lisa been scares watching this video what does it mean to say that two witnesses have consistent statements right to two witnesses will have consistent statements if there's at least one line in each of their statements the same line at which both of their statements ends up being true in the main operator so let's take a look at the example and you know I hope just giving me showing you these examples of the book help rather than me writing all this out otherwise the video would be very long right okay so consistent here so the first line here is true and true right and notice these are all false that these are all false right so this is consistent because of this first lon right so true true that automatically means that these students are consistent this is true this is false this is true this is false and then we have false and false something that's very important here is do not think that simply because the last line has false and false that they're consistent right it has to be consistent in terms of truth not in terms of truth functionally not in terms of the non folks are now on the others right so you need truth here just because you have a line with our false doesn't mean anything all right what they need to be here is you need one line that's true okay now what's the last one the last possibility if they're not consistent is that the two statements are inconsistent so that's the the fourth possibility it's the third inconsistent in inconsistent is there's no line now remember the same line there's no line where both statements are true my chicken scratcher nothing orrible okay I hope you can read it so but that's why you have a book right so let's give an example here it's something that's being consistent notice that in this damn it we have both truth and false falsity right you know here we both have falsely in truth so both of them have bivalent properties right it's possible for each of these statements to be true but we'll see it is not possible for them to be both be true at the same time hence they're inconsistent so for instance where this is true this is false but this is false this is true where this is false this is false and where this is true this is false right notice that it's not they're not logical contradictions because there is a lot of which both of them could be false simultaneously so not contradictory they're not opposites right but there's no line of where it's truth all is the same for both of them which means it's inconsistent right whereas well Hurley says there's no light of the calls in the Opera or truth dyes are both true right and this is important because one of the things I've noticed is that often times students accidentally make the mistake both in terms of consistency or inconsistency of thinking that if there's a lot of where they're both false is consistent that's not the case that has to be truth and simultaneously people can get it this makes that because right here this was true and this was false this was true this is false and there's as false of this true you bag tape that they're logically contradictory they're not because of this third line okay so that is essentially how you classify and compare statements let's do a sort of quick review here or a minor road right what are the possibilities when you classify statements you can either get a tautology a contradiction is self-contradictory stave it or you can get a contingent statement right and then when you compare statements right you could get a logical equivalence contradiction consistency or inconsistency right now I like that example with the detective because you can see your white consistent and inconsistent it's important and now maybe we next time you watch one of these like CSI or what are these like shows detective shows and you hear someone say the witness's students are inconsistent now you know exactly what new consistency means it consistency means that both of them could be lying right but there's no case in which they're both telling the truth all right that's it consistency consistency means they could both be line but there is a possibility that the both sides with contradictory means one of those is true one of those false logically equivalent would mean they both have the same story if you will and that might hopefully help you in terms of thinking about these things so what should we do now maybe let me just take a look here whoever gets show you what your homework is gonna ask you to do so let's go over here to the Hurley textbook we make a little smaller for you okay so here's the exercises right so first thing that says use truth tales to determine whether the following Cymbeline statements or tautologies self-contradictions or contingent stay loads now I can tell you this way and therefore in therefore and therefore in what do you think that's gonna be well maybe we should do the problem real fast and you can we can actually see what it is all right so let's do that problem see if I can make it big here okay so how do we do this now this one's easy the first thing that we do here is ask well how many lines at our church I remember here's our equation l equals two to the nth power right in this case I have one two times itself it's two so that means if this is really easy right this mean true-false true-false true-false okay and I always start with the me I'm sorry that the internal operators first if this is true this is true that's true this is true and this true that's true and then go and now I'm gonna compare this line and this line using this rule because this is the main operator you can see here why we spent time talking about it both of the last two videos why it's important to note the main operators if this is true this is true well it's true this is false or this is true it's true right and you exit out here right this is a tautology this is the technology and I hope you saw there was a tautology actually because think about it imagine if I said this if Nick goes to the party that it's the case that Nick goes to the party Nick's gonna go to the party right all of that could be just sort of simplified sale I guess mix going to the party right so that's a tautology and then they're gonna add don't think necessarily this is a digit ology just because it looks the same so we could do that later we could show you what it is but you can see it is how to make it work this first you are easy because these are two lines you're going to need this one you're gonna need four lines because you've got an estimate are right so that's great for lines E&F four lines H&K four lines MP for our lines zx4 line do you can see me and they're gonna give you frets Oh number nine is where it kicks in it gets a little harder because here you have three variables you can see now you're gonna need eight lines and then you can skip all the way down now what about this last one you've got f EG h you got four lot four unique variables wait am i missing one no yeah you got four unique variables so that means your tables gonna be 16 lines long you can see that they're not gonna usually ICJ to more than 16 because then it just becomes crazy a wieldy right okay so now what do you guys do at sexta to use the true tales that you're following a pair of simple items are logically equivalent contradictory consistent consistent furniture in whether the pairs are logically clear contradictory did if these relations don't apply the true out there Christmas very consistent so let me just do one of these problems too so you can see how to do it here maybe a little bit of a longer one let's see you're gonna snip lets you problem this problem here okay so let's do this problem so the first thing of course to recognize is how many ask yourself how many lives does this thing need to be right and X and X got four variables l equals two to the sorry - to the second right studies this we need four lines long so I've got true true false false right and then the X well you know once I do the egg just go ahead and fill in the eggs for the other one this is a giddy important especially because if you start having to do these signs of problems on your tests if you're gonna take a test yeah this is where you're liable to make a mistake just go ahead and fill it all in and get it in there so that way you know that you you're comparing this table is correct because if because notice this one sort of treating because if you did this first you might start over here and say this is true true false false and just mix them up so the X who's gonna true false true false true false true false true false okay so now let's do let's start with this one and then we'll sort of work our way through it so if maybe I'll change the colors here so that means this is gonna be false false true true now that I compare this in this using this rule the biconditional rule and we haven't used the buy conditionals too much of the videos what's the rule for the bank conditional they both have to have the same value in order to be true otherwise they're false so if there's different thighs that follows this false visitors none of this and must be false false and false means it's true ironically true and true means it's true and then true and false means that it's what its false okay and so I'm not gonna exit because I we make that a little clearer okay so lure cuz I'm this is what I'm gonna be using this light since instamate offers what I'll be used to compare to this table so let's go and then fill in this statement and see what we get now you can see how I have two parentheses so I need to do both of these parentheses before I can get to the main operator here so that means this is gonna be false false true true and then I need that conjunction rule that's that's real strict rule everything's got to be true that's false false true or false okay and then over here let's do this that's getting false true false true and then I get that conditional again that's false true false false when I switch to red now that I do this main operator remember what's the rule for the disjunction or that means or so this says X and not a or a and not X the rule is that one of them has to be true that's all otherwise it's false so if they're both Falls it's false so what am i comparing here I'm gonna compare the main operator each of these princes which is with this line I'm sorry this column and this column so so that means that false and false is false false and true it's true true and false is true that false and false is false okay so now now that I've done this you can see here I could classify these statements both of them are contingent because you can see they both have true have false values so now it's comparable on each line where this was false this was false where this was true this was true where this is true this is true and where this is false this is false if they have identical values right go back to our notes here identical values means what right identical values means they're logically so I would just say this is logically equivalent right and that's how you do the problems you can see you you have to make sure your head rays meet what I encourage you to do is get a highlighter because when you compare this especially with their long and highlight the man what about the columns that fall out other made offers so that what you need to compare it's easy for you to see and you don't get too confused you can see using different colors is also helpful I've had students use colored pencils and things like that generally speaking I'd say a highlighter is all you probably really need and maybe graph paper but that's how you do these problems okay it's really not too difficult if you understand the basic mechanics and you just work slowly right okay now here you can see that they had the same number of unique oops Purvi they have the same number of unique variables but they have more variables here and then fighting it's going to ask you to use two tails to the answers to the following exercises so here's what you got renowned economist harold carlson mesa following predation the balance of payments will will decrease if and only if interest rates remain steady however it is not the case that either interest rates will not remain steady or the balance of payments will decrease what can we say about Carlson's prediction you can see here like when you first hear that you're like ah I'm not sure it sounds fairly confusing but actually if you go through and you take that what's in the parentheses here and you put it into propositional form right and you can see here this is just one statements without comparing statements we just want to figure out if it's tight ology if it's a contradiction or it's contingent so you just basically to do this problem you basically write out the translation perform the truth table and then once you perform the truth Channel you just need to classify the state of it and then later on you know there's other ones where you have to compare what dr. Frank says to dr. Harris so that you're going to compare in the stands and so forth but I really like how in these in these problems you actually get a sense for how here it's fairly helpful here in analyzing all this stuff so that is six point three truth tables and you're gonna see some of your putti oh wait a second what happens if I have ten variables what do I do obviously you're gonna you need seems like you need a computer to calculate that many early it's a huge piece of paper and that's we're gonna work on our next videos we're gonna look at true uh dog sorry their next video we're gonna go get truth tables for arguments not just statements and they will look at the six point five here what's the shortcut if you have lots and lots of variables so that's what we got coming up in our next videos so that's true tales for propositions thanks for watching and I look forward to seeing you guys online