 Hi, welcome back to fill 320 deductive logic. I'm Matthew Brown and today we're going to be talking about truth tables Truth tables are a method that we have for evaluating complex expressions in SL Before we get into the details first. I want to share with you a quick logic puzzle of the day Here's how it goes a boat has a metal ladder coming down the side the metal ladder has six rungs spaced one foot apart At low tide the water came up to the second rung from the bottom of the ladder Then the water level rose two feet as the tide came in Which rung did the water now hit? Think about it. Let us know What do you think the answer is you can weigh in on discord or in the comments of the video? Okay, now let's start talking about Truth tables truth tables take advantage of a feature of SL, which is that it is truth functional The truth value of the compound sentences in SL depends only on the truth values of the atomic sentences That make it up Plus the rules for the connectives and how they transform the truth values We already saw some examples of truth tables. We call them the characteristic truth tables of the connectives We'll go back over them here in just a second and when we did that last time in unit two we We used t and f to represent the values of true and false We can also use the numerical values one and zero one representing true and zero representing false and in the book Typically, they use those one and zero instead of true and false. We'll use both In different examples and homework assignments just so we learn both ways of doing it So when we use the ones and zeros for example in the truth table for the negation It looks like this in the left hand column. We have our simple atomic sentence a and then on the right hand side the expression not a right and in the In the left hand side where we have our atomic sentence We just want to make sure we go through all the permutations all the possible options and then we can give the the values for the negation on the right Here it is It just flips the truth value from one to zero or true to false and from false to true or zero to one And we can do this for all of the different expressions all the different Connectives of SL, right? So for example for the conjunction, it's true when both a and b are true for the Disjunction, it's true when either a or b are true or both the conditional is a little bit more complicated in that it is True when both a and b are true and it's true whenever a is false or But it's false when a is true and b is false, right? That's the only case in which the conditional connective gives you false and Then the biconditional is true whenever a and b have the same truth value either when both are true or both are false So what the truth table shows you is how the different connectives take truth values as input and Spit out a new truth value as the output, right? And we can do this for really complex expressions as well, right? so Here'd be a good place to pause the video look at this rather complex expression and try on your own to Come up with the truth table So have a look pause the video for a second and see what you come up with Have you done it? What did you think? Let's go through it step by step So you get a better sense of how we do this So first we want to take our truth values for our simple atomic expressions a and b and copy them over Wherever those appear so we could copy the a truth values over Under the a and the b truth values under the b You might ask Why do we have the truth values one and zero in the order that they are under a and b? And it doesn't really matter what order they're in as long as you get all of the possible combinations but this sort of pattern of 1100 1010 You'll see it again and again It's easy way to remember and I'll show you what it looks like when you had a third Truth value and so on so so first we copy over The truth values from our atomic sentences, okay, and eventually we're going to get the truth value for the whole expression Based on what we put underneath the main connective here, which in this case is the biconditional But first we have to go through all of the sub expressions all the parts of this complex expression So let's start on the left with the disjunction a or b as we know a or b is true Whenever either a or b is true, so that's how that part goes Now let's look at the right-hand expression. It's also kind of complex. It's got that negation in front of the antecedent a So we want to make sure we have got that right Okay, we know that that is I'm gonna be just the reverse of whatever a is and Then we have the conditional right and again the conditional is true Whenever the antecedent that's not a in this case is true So it's true the conditional is true in both of these These first two rows It's also true in this third row because the antecedent not a is true and the consequent b is also true But it is false in this last row where not a is true, but b is false And then for the biconditional what we need to do is we need to compare the truth values on either side Right, so the left-hand side of the biconditional It's one and on the right hand of the biconditional it's one in this row And so what we're gonna put under the biconditional is also one Right, we can go line row by row line by line and do that same comparison here It's gonna again be one because they both are one they match They match in this row. So it's one they match in this row. So it's one again And then we see once we've completed the truth table for this expression. It's one in every Row right in every condition it is one. It's true This means that this complicated expression is a tautology Here's another expression I'm gonna give it away a little bit and tell you that the statement is contingent And I'm gonna ask you to show that it's contingent by filling out the truth table so here's another place where I'd like you to pause the video and Try to do the truth table on your own try to fill out all the all the rows and all the columns And and evaluate your main expression which in this case your main connective is the conditional So fill out the complete truth table. All right. Did you get it? Let's start with our just filling out for our atomic sentences here a b and c again we're just trying to get all of the permutations down and the typical way that I'm gonna do it is for the Right-hand most Expression which is c in this case. It's just alternating 101010 The next one over it's going to alternate half as often. So it's 1100 1100 And then the next one it's going to alternate again half as often. So here's 1111 0000 that gives you all the possible permutations and you can pause and check it out if you want to confirm that that's true Okay, now we're going to again copy over the atomic Sentence truth values, right and then we have to go again Column by column under the different parts of this expression moving from the atomic expressions To the more and more complex parts, right? From the inside out you might say, right? So the by conditional looks like this again It's where the truth values match. It's one where they don't match. It's zero The negation is just the opposite The conditional is true whenever the antecedent is false Or whether whenever both the antecedent and the consequence are true and again at any point You want to confirm what I'm what I'm showing you here. Just pause the video and check it over But that's right in this case and then if we go through a little bit more slowly the main connective here The conditional right we see it's one in this first row Because both sides are one. It's one in this row because both sides are one It's one in the next four rows because the left-hand side the antecedent is zero or false It's one in this row because again the by conditional is one and the conditional on the right is one So if they match again And it's only in this last row where we put a zero Because the truth value of the by conditional at a if and only if b is one and the truth that but the truth value of the But the truth value of the conditional not c if not c then a is zero right And so it's these You know these last two rows is where we see actually the statement is contingent What it means for the statement to be contingent is that it could be true or false both are possible What we show that with a truth table is we show that there are rows Where we have both truth values We can also fill out truth tables to evaluate More than one expression at a time. For example, I have two expressions here If not a then b and if not b then a right pause the video for a second and Go over This truth table try to fill it out on your own Okay, let's check and see if you got it right. So again, I'm copying over the atomic sentences I've got my negation over here that gives Just the opposite of a And I'll go ahead and do the negation over here as well gives me the opposite of b And then I fill out for The two conditionals And I get these values right And again, you can just confirm that based on The characteristic truth table for the conditional If I compare our two main connectives here the truth values for the two expressions You see that they're the same on every row. It's one one one zero All right, this proves that these two expressions are logically equivalent So we can use a truth table in this way to prove logical equivalents We can also use a truth table To evaluate an argument. So here's a very simple argument if a then b If b then c therefore if a then c it seems pretty intuitive Here's the truth table to start out with And you can go ahead and pause the video and try filling it out on your own Hopefully that was pretty easy because these expressions are not are not very complex They're just a single connective in each case, right Again, we copy over our atomic sentences and then we can see For the conditional it's true. It's true Here it's false because you've got a one in the antecedent and a zero in the consequent And then because these last four Have a zero in the antecedent. We know they're all true Similarly here, we're going to get true false true true true false true true And for our last conditional we're going to get true False because again one and zero is false true false And then we've got zeros in the antecedent the rest of the way. So it's true in all of those cases So that's now we filled out the truth table. How do we use this to think about validity? Remember the definition of validity whenever all the premises are true The conclusion also must be true So we only concerned about rows where our premises are true Let's look at this first premise. It's true here in the first row And the second row and the last four rows Look at our second premise. It's true here in the first row the and uh, these other rows, right? But we only have to worry about those rows where both premises are true, right? So that's here here Here and here, okay Now that we know Which rows matter for evaluating validity? We have to look over at the conclusion And see what the truth values of the conclusion are. Well in this row, it's true And in all of them, it's true, right? That means that this is a valid argument, right? So we've used a truth table here To show that this is a valid argument validity just means in a sense for sl anyway That you can show For every line in the truth table where the premises are true. The conclusion is also true That's our method for demonstrating validity So here's some more examples where we can look at comparing multiple Expressions in sl, right? And we can not only ask about equivalence We can also use truth tables to evaluate consistency or inconsistency, right? So why don't you take a second and fill out this truth table? Okay, let's see So those are the truth values you should have gotten Here I've used t's and f's instead of ones and zeros just to mix it up and show that both notation systems are valid As we did before to evaluate logical equivalence We compare the columns and in this case we see again These two expressions are logically equivalent If there was any line where they differed in their truth value We would know they were inequivalent or not equivalent What if we want to know are these expressions consistent? Well, what we have to do here is look at Whether there is any row Where both expressions are true and there are several rows like that, right? So we only need one to know that they're consistent. So this tells us that they are consistent Let's look at another example pause the video and try to fill out this truth table Okay, so again here are the values you should have gotten And again for equivalence we compare the main connectives and we see they are equivalent They're false in every row Right because both of these expressions are contradictions, right? So we've shown that each expression is a contradiction All contradictions are logically equivalent. So the two expressions are logically equivalent So that's great. We also know that the two expressions are inconsistent. How do we know that? We know that because all pairs of contradictions are Inconsistent because there's no row where they're both true, right? Because the contradictions are never true So this is how we use truth tables to evaluate complex expressions in sl Using their truth functional nature Next I want to show you a little bit about how to use Carnap to do truth tables So here I've loaded up a problem in Carnap created by PD Magnus And I'm going to show you how to Use it to fill out the truth table So here we go in this problem The truth values for the atomic sentences have already been copied over In some of the problems in your practice exercises, that'll be the case in others You'll have to do it yourself by hand Just to practice and what we do here is we go through and we We select the appropriate truth value According to the connectives, right? So we'll start here with this expression And we'll go through this is conjunction. So we know it's true Wherever they are both true And it's false everywhere else like so Okay, now that we've done this expression We can do the expression it's included in which is this conditional And we see we have a one and a one here. So that's true. We have a one and a zero here So that's zero. We have a one and a zero again zero one zero Zero zero one Is going to be one. We actually don't have to look over here anytime. There's a zero in the antecedent We know it is true. It's a one Okay Now let's move over here to this complex expression Um, let's start with a negation. That's an easy one because it just flips the Uh truth value to its opposite Okay, can't do the conditional yet because we have to evaluate this expression First we got to do the negations So that's again just flipping to the opposite Now we can do the conjunction and that's only going to be true Wherever both of the conjuncts are true. So It's false here false here false here True here. It's one and one And then false here false here false here and it's true here one and one Okay, so this expression is done now. We can come back to this conditional So first off in these first four rows the antecedent not e is Zero it's false. So we know the conditional is true in all four of those rows And then these last four rows not e is one Right, it's true. So we just have we have to look over here to the conjunction one and zero We know that's zero one and zero zero one and zero zero one and one That's one. That's true. Okay Finally now we can do the main connective for our expression, which is the Conjunction, so we're comparing this conditional here And this conditional here and it's going to be true wherever they're both true So here we've got a one and a one. That's one here. We've got a zero. Okay, so now this is false zero False zero false here. It's a one, right? Okay, and here it's a zero. Okay, so that's zero And it's it's also zero zero here. So it's zero and then in this last row. It's one here And it's one here. So it's one. Let's check our answers. See if we got it right Success. So that means we got it right That's how we're going to use the truth tables in carnapp And then sometimes there will be problems like this, which asks you to look at your truth tables And figure out what kind of sentence we have we see there both ones and zeros in our In our column for our main connective. So we know that's a contingent sentence Right. All right. So That is how we use carnapp to do truth tables Why don't you go ahead and try it out with the first three sets of practice exercises However, many of those you need to do to feel comfortable The best way to to learn truth tables is not to really watch someone else do it But to do it yourself and do lots of practice problems. Okay So give it a go and I will see you in the next lecture for unit three where we talk in A little bit more detail about truth tables and look at some additional examples Best of luck and see you soon. Bye