Video Transcript: The Truth Table Test of Validity

Hi, I'm David Feddes. And this logic talk is about the truth table test of validity. That's a  mouthful, but we're going to learn about truth tables, and how to use them to show that an  argument is valid. We've already seen just a little bit about truth tables. When we were  learning about our various truth functional connectives. A conjunction has the following truth  table. And a truth table basically shows you what is true of the conjunction. If you know about the two atomic propositions, so in the first row, if P and Q are both true, then in a conjunction, it's also true P and Q is true. However, if P is true, and Q is false, and that second row, then  the conjunction is false. In the third row, if P is false, and Q is true, then their conjunction is  false. And in the fourth row, if they're both false, obviously, the conjunction is false. So the  truth table for a conjunction shows that the only time that a conjunction is true, is if both of  the conjuncts P and Q are true. The truth table for negation is quite simple. If P is true, then  the negation of P is false. If P is false, then the negation of P is true. That's all there is to it for  the truth table of a negation. For a disjunction. The truth table look like this. If P and Q are  both true first row, then P or Q is obviously true. If P is true, and Q is false second row, then P  or Q is still true, because one of them is true. And that's all it needs. Third row if P is false, and Q is true, then it's still true to say that P or Q is true. And only if P and Q are false in that  fourth row is the disjunction P or Q false. So we've learned about the truth tables for our truth  functional connectives. Now we're going to give some thought to how to use truth tables more broadly. And that brings us deeper into the realm of propositional logic. Thus far, we've talked  about translating sentences into logical symbols. And a first thing that we would do is identify  what the atomic propositions within a sentence are. And then we would choose a constant to  represent each one of those atomic propositions. And then we had to figure out what truth  functional connectives to use. If it was a conjunction, we would use the raised dot. if. It was a  disjunction, we use the wedge if it was a negation, then we use the tilde. So we did those  things to change English sentences into symbolic sentences. What was the point of all that?  Well, the point of it was so that you could determine the validity of an argument using a  formal method that will work every time, if you know how to do it, if you know the, whether  the atomic propositions are true or false, and you use the constants and you know how to use connectives. And then you use a formal method, you can show whether an argument is valid  or invalid. Earlier in the course, we thought about informal tests of validity, and basically the  informal test was try to imagine a scenario in which the premises of the argument are true.  And yet the conclusion is false. And the argument is valid if we can't come up with any  situation in which the premises of the argument would be true, and yet the conclusion be  false. But it's an invalid argument. If we can imagine a way where yeah, all the premises are  true. And yet, the conclusion is false, then we know that the argument is not valid, because  every valid argument is such that if the premises are true, the conclusion has to be true. And  the informal test again, is just Well, let's think about it. Can I imagine a scenario in which the  premises would be true, but the conclusion false if you can't think it's valid. The informal test  of validity would work on an argument kinda like this. The convict escaped either by crawling  through the sewage pipes, or by hiding in the back of the delivery truck. That's the first  premise. The second premise is the convicts did not escape by crawling through the sewage  pipes. And the conclusion is, therefore, the convict escaped by hiding in the back of the  delivery truck. Now, can you imagine a scenario in which that argument would be false? If the first premise is true, that he did it either one way or that other way? There was only Two  possibilities. That's the first premise. It was either the supine sewage pipes of the delivery  truck, it's got to be one of the two. And your second premise was, it wasn't the sewage pipes,  then you know, it had to be the other one. The convict escaped, they're hiding in the back of  the delivery truck, you can't imagine that argument not working. If you say it's one of two  things. And it wasn't the one, it's got to be the other. And that's our informal test of validity.  Now, if you're going to try to translate that into symbolic logic, here's how you go about it.  The first premise, the convicts escaped either by crawling through the sewage pipes, or by  hiding in the back of the delivery truck has two atomic propositions. The first is symbolized by the constant S escaped through the sewage pipes. The second atomic proposition is the  constant D escaped in the delivery truck. And you translate that by saying S wedge D. S or D,  one of those two atomic propositions is true. The second premise is the convict did not escape

by crawling through the sewage pipes. And so in terms of your atomic propositions, it's tilde S if crawling through the sewage pipes was the atomic proposition, then the negation of that is  saying he didn't do it through the sewage pipes. And that gives you your second premise, the  

convict do not escape through the sewage pipes by symbolizing it as tilde S. And then when  you come to symbolize the conclusion, therefore, the convict escaped in the back of the  delivery truck, well, D is our constant for escaping the delivery truck. So therefore D, and we  use that little three dots symbol to mean therefore, anytime you read that little symbol, it  means therefore, it means this is the result of the whole argument. This is the conclusion,  therefore. And so that's how you'd symbolize that argument that either he escaped through  the sewage pipes or the delivery truck, it wasn't the sewage pipes. Therefore, the delivery  truck. truth tables are used to represent how the truth value of a complex proposition  depends on the truth values of the propositions that compose it. So if you have propositions  that contains several smaller propositions, then the truth table represents the truth values of  the various possibilities. So if we wanted a truth table for this, convict escaping either to the  sewer or the delivery truck, we would symbolize it in the following way. When you set up a  truth table, the first two columns have to be the constants for your atomic propositions, D.  for, the delivery truck, S for the sewer, usually put put them alphabetically, so we put D first  in that top row, and then S. So those two are called the reference columns. The reference  columns are the columns, where you have the constants for the atomic propositions you work  with, then the next column will be your first premise. And remember, your first premise was S  or D, either it was through the sewer, or the delivery truck S wedge D, your second premise is tilde S, not S, he didn't do it through the sewage. And then your conclusion was D. So that's  how the argument reads across the top of the truth table, your reference columns have your  atomic propositions, and then your the columns after that will have the premises in the order  that they were stated. So S or D. you do a truth table, the column to the farthest right will  always be your conclusion. And now you want to test was this a valid argument. So to start  out with your reference columns, you have to fill in all the possibilities of what happens with  your atomic propositions, and in this case, you only have two, so D and S, you could have in  that first row, D and S could both be true. You could also have D being true or as being false.  You could have D being false and S being true, or you could have both of them being solved.  That's how you set up the truth table. When you have two constants to atomic propositions,  you have the two reference columns. True, true, true, false, false, true, false false. The next  column, you have to figure out whether S or D is true. Well, what do we get look in the  columns for D and S? And if D and S are both true and that first row then you know that  answer D is true. Look at the second row if D is true, but S is false The disjunction S or D is  still true because one of the members is true. Same is the case with the third row. If D is false, but S is true, then S or D is still a true proposition. The only time that S or D is false is if both  D and S are false. And so the final row where both of the atomic propositions are false, shows  that S or D is false as well. So now you've got that row, you've got the truth values for S or D.  Moving on to the next column, not S, that's very easy. You just look at the S column. If S is  true, they're not S as false. Second row, if S is false, they're not as true. Third row, if S is true,  then not S is false. And fourth row, you're looking again in the S column, and you see the S is  false in the fourth row. And so that means that not S is true in the fourth row, so wherever S  had one value, not S have the opposite value. And that's about the simplest thing that you  can do in logic is just negate the value and it goes the opposite. And now we have to get the  conclusion, the conclusion is D. Well, in this argument, that's really, really, really easy,  because D is just one of our constants in the first column. So you just copy the first column as your conclusion, D is true in the first true in the second row false and the third row false in the fourth row. And so that's what your D column looks like. And D is your conclusion. Now, you've got everything filled out in your truth table. Your constants, D and S have been filled out with  the possible values, that's all you put in there is what the various possibilities are. You've got  your S or D columns filled out, you have your not S columns filled out, and the D conclusions,  columns filled out how do you know whether the argument is valid? Here's how, you know the argument is valid if the premises If anytime the premises are true, then the conclusion is true. If you find a row where the premises were true, but the conclusion is false, then it's not a valid

argument. So in this case, the truth table test of validity looks at the row where both premises are true, there's only one row where both of the premises S or D, and not S, there's only one  row, the second row where both of those were true. And it turns out that if both of those were  true, then the final column your conclusion, column, D in this case is true. So if you find a row  where all the premises are true, the conclusion has to be true. If you had found a row where  you had both premises being true, but the conclusion was false, then, you know, you had an  invalid argument. An argument is valid if and only if, for every assignment of truth values to  the atomic propositions. If the premises are true, then the conclusion is true. And argument is  invalid. If there exists an assignment of truth values to the atomic propositions on which the  premises are true. And yet the conclusion is false. So in short, if the premises are true  conclusion has to be true, in order for an argument to be valid, and we saw with the sewer  and the delivery truck, that if you did a truth table for that there was only one column were  both premises, there was only one row, I mean, we're both premises were true. And so the  truth value in the final column was true as well. And that meant it was a valid argument.  Here's another example. Again, like it's a real simple with only one premise, and then a  conclusion. The first the premise, the only premise is not C. And the conclusion is therefore,  not C or A with C or A and parentheses in a truth table can show whether this arguments  valid, even if you don't know what C or A mean. So again, we construct our truth table.  Remember, the first thing to do is to make your reference columns, you have two constants, A and C. And you have to give all the possible values that those could have the possible mixture of values, so to speak. So first row, they're both true. Second row, A is true, C is false, third,  row A is false, C is true, fourth row is false, C is false. Those are the four possibilities. When  you have two constants either both are true. The first is true, the second false. The first is  false, the second true, or they're both false. So now that you have it set up with your  reference columns, the next step is to do the premise column. And your step on that one is  pretty easy. If if the premise is not C then look at the C column and do the opposite if C is  true, in First row then not C is false. If C is false in that column, then not C is true. If C is true  in that third row, then not C is false. And in the fourth row C is false. So not C is true easy  enough with a tilde, you just take the opposite truth value of what the constant was. Now we  move on to our columns for the conclusion. And here, it's we can't just right away in our head  unless you're really good at it, decide what the truth value of those are. So because we have  a quantity in parentheses, and then a truth functional operator outside the parentheses, the  first thing we have to do is decide what's going on in the parentheses, in this case, the truth  value of C or A. C, wedge A. So we have to look at A and C in those first two columns. And in  that first row, they're both true and so C or A is true. In the second row, A is true, C is false.  And so we say well, then C or A is true because a disjunction is true, even if one of the  disjunct is false, but the other true. Third row, same story, A is false, C is true. So the  disjunction is true. C or A is true, because one of the members is true. And only in that bottom row, do you see that A and C are both false, and therefore C or A is false. Now you still aren't  finished with that column, you did the portion in parentheses. That's what we've done so far.  Now we have to use the Tilde on that. And we get the opposite. So we calculated what C or A  is now we're going to do what Tilde or not C or A. So in this column, we saw earlier that we  computed that it was true, but now we change it to false because we negated it. Second row,  it was true. So we negate it. Now it's false. Third row, it was true. Now we negated it's false.  Fourth row, the disjunction was false your A was false. But therefore tilde is C, or a is true.  And now we have actually the truth values in the conclusion column. And we're ready to move on to determining whether this argument was valid. If the premises are true, then the  conclusion is true. If the argument is valid, we see one row where the premise is true. See the bottom row not see as true. And then the quantity, not C or A is also true. But the second row, you see not see as true. And yet the conclusion is false. You know what that means? It's not a  valid argument, because you just found a circumstance in which a premise was true. And the  conclusion was false. Anytime you find a situation where all premises are true, and the  conclusion is false, then you don't have a valid argument. And remember, this was an  argument with only one premise not see. Well, let's take another example. And this one  involves three constants. The first premise is A or B and parentheses, or C, the second 

premises, not A, and then the conclusion is therefore C. And again, we can use a truth table to show whether this argument is valid, even if we don't know what A, B and C mean. Now, how  do we go make a truth table, just a note on making truth tables when you have more than  two. When you have more than two atomic propositions, it gets a little more complicated,  though, not terribly, but you have to account for all the possible truth values of all the  constants. So if you have to add them, if you have one atomic proposition, then you only need two rows in your truth table. If you have two atomic propositions, then you need four rows in  your truth table as we've seen, in some truth tables we worked on. If you have three atomic  propositions as we do in this case, then you need to have eight rows in your truth table. If you had four, you'd have to have 16 rows. So it starts getting to be really big and ugly. If you have a lot of different atomic propositions. In this case, it's still kind of manageable, because we  have three atomic propositions and you need two to the n rows in a truth table for that  amount of proposition. So again, two to the zero S is is just one, but when you raise two to the third power, it's eight to the second power is four to the third is two times two times two. So  anyway, you get that I don't want to review all of earlier level math courses that you may  have taken. But when you're making a truth table, just know this. If you have three constants, you're going to need eight rows, because you need as many rows as there are constants to  the third power in this case. So here's what we do. You have A, B, and C are your reference  columns, you have eight rows down below. And then you put your premises in your  conclusion, your first premise, as you can see is A or B in parentheses, or C, your next column is not A, you find A column and see in those reference columns with your constants, here's  what you need to do. You start with the reference column, that's the farthest to the right of  the three, and you're just alternating true false, true, false, true, false, true false is enough.  The next column, you double that you go True, true, false, false, true, true, false false. And  then the next column, you raise it to another power. True, true, true, true, false, false, false  false. So that column over to the right, the C column, you go every other one, the next one,  you go to two to two, and then the third column, you go four, and four. And if you were to  have a fourth column, you'd be going, you'd have to have a 16 column row, we're not going to go there right now. But just remember that rule of going with as many as many rows as two to the power of the number of constants, we have three constants, you start in the column  farthest to the right, and just alternate, the next one, you go two at a time, the next one, you  go four at a time. And if you do it that way, you will have covered all possible combinations of  the truth values for A, B, and C. Now, in the next column, we have a premise that involves  several components. So the first thing we need to do is just get the truth value of A or B, and  in the in the row where A and B are both true, then the then the disjunction is true. Next row,  they're still both true A and B, so it's still true third row, A is true, B is false. So then the  disjunction is still true, because A or B is true. Next one, same story for throw. A is true, B is  false. So the can the disjunctions still true. The fifth row, you've got false true, so the disjunct  is still true. The sixth row, same story, it's still true because you've got a false but also a true,  and only those bottom two rows are A and B both false, and therefore the disjunction is false.  So now we're done with that computation of that column. Now, we're still not done with the  column, because all we did was A or B. Now we've got to say, Now A or B, and then our C. So  here's how you go, you look at the A or B value and look at C while they're both true. Second,  row, C is false, but A or B is true. So then the disjunction is still true, because it's or  remember, third row, true and true. So you're still true, the disjunction fourth row C is false,  but the disjunction of A or B was true. So then the distance, the total disjunction is still true.  fifth row, you still got two trues, sixth row, you've got C was false, the A or B was true. So still, the total disjunction is true. In the seventh row, C is true, and A or B is false. But because C  was true in the disjunction, you still got true because the only way that a disjunction is false  as if both of its components are false. And then in the eighth row, you see C is false. And the  disjunction A or B is false. And therefore, since they're both false, then the total overall  disjunction A or B, or C is false, it's false only let eighth row, the next column is gonna be a lot easier to look over at the A column. If it's true, then not as false. If it's true again in the next  row and the next row in the next row, then it's going to be false. In all four of the top and the  four top rows are not a, you'll look over in the A column again, and you see four falses in a 

row. And you say, Oh, that's easy for trues in a row for not A. So you've got that column  wrapped up. And then finally your C column, all you got to do is copy C from where it was  over there, because C was an atomic proposition, but it's also your conclusion. So you just  copy what was in the C column and copy it over again, in the conclusion C column. Now, your  next step in doing this truth table is look at the rows where all the premises are true. Look at  the fifth row. The premises are both true. Remember, your reference columns are just A, B and C those aren't premises anymore. Those are just the atomic power. positions the premises  where A or B, or C and then not A, and in the fifth row, those two are true. And the conclusion  is true. So far, so good, it might be valid. But the sixth row, you see that A or B is true, A or B,  or C is true. And you'll see that not as true, but C is false. And if you have your premises both  being true, but your conclusion false, you know, the argument isn't valid. You look at the next  row, and the premises are both true. And the conclusion is true, but it doesn't matter. You  could go through a truth table and find several rows where the premises are true. And the  conclusion is true. But if you find just one row, just one row, where the premises are true, but  the conclusion is false. It's not a valid argument. Because an argument is valid only if the  conclusion is true. Every time the premises are true. It's a lockdown failsafe argument if these premises are true, the conclusion has to be true. And if not, it's not a valid logical argument.  So that's the truth table test of validity, you're using it on this argument, you showed that it  wasn't valid because you found a row where the premises were true, yet the conclusion was  false. Let's try one more valid or invalid A or B that disjunction in parentheses. And then  you've got A or C, that disjunction in parentheses. And you have the raised dot in between so  this is a conjunction of two disjunctions, A or B, and A or C, your second premise not A, your  conclusion, therefore, B or C is that a valid or an invalid argument? Well, let's set up our truth  table. Again, remember that you need a certain number of rows to cover all the possible truth  value combinations of your atomic propositions. And once again, we have three of those  three, three constants that we're going to be using. And so because we have three, it's two to  the third power two times two times two, we're going to need eight rows for those values. So  we have our reference columns, A, B, and C for the three atomic propositions, and then we  have eight rows below those. And then we have our columns for A or B and A or C, we have a  column for not a the second premise. And then we have our column for the conclusion, B, or  C. Remember that in order to get our atomic propositions, all the possible truth values when  you have three of them, we start in that column for this to the right, and just go true false,  true, false, true, false, true false. Then the next column over we go True, true, false, false,  true, true false false, where you going in bunches of two at a time down the truth table. And  then as you move over to the last most reference column, you go with four at a time, true,  true, true, true, false, false, false false. And that'll give you all the possible combinations of  truth values for three different atomic propositions. Then we move over to this and again, we  can't do it all in one mouthful, we have to start out with one disjunction, A or B, and you move down from the first row A and B are both true so A or B is true second row. True again,  because both are true third row the disjunction is true because A is true even though B is  false for throw these false but A is true, so the disjunction is still true. fifth row, disjunction is  still true because B is true and A is false. So as long as one of them is true, this junction  remains true. Same story on the sixth row, one of them's false as false with the B is true, so  the disjunction is true. seventh row finally you see two falses A and B. And so the disjunction  A and B is false. And the same is true, the eighth throw both are false, the disjunction is false.  So we're partway there now we have to do the other disjunction in that column, A or C. And so we, we work our way across and we look at A and C they're both true so the disjunction is  true. A or C in the second row is true C S false but the disjunction still true. Third row they're  both true, A and C, so disjunction is true for throw, A is true, C is false. So because one of  them is true, then the disjunction still true. fifth row A is false, C is true. So the disjunction is  still true sixth row a is false and C is false. So the disjunction is false. So you see in the sixth  row, why it is junction An A or C is false, seventh row is false, but C is true. So the disjunction  is true and that eighth row A is false, C is false. So A or C is false. We're still not quite done.  We've done both of the disjunctions in that premise. Now we have to conjoin them with the  race di, A or B, and A or C and figure out what the story is on that. And remember, in order for

a conjunction to be true, both of the statements that it can joins have to be true. And so in  that first row, you see a or b was true, and AMC was true. So the conjunction is true. In the  second row, both of the disjunctions were true. So the conjunction is true. In the third row,  you have two trues you those two conjunctions and so again, the conjunction of two trues is a 

true fourth row, same story. fifth row, same story to trues always makes it true, but you get to the sixth row, and you see that one of it is junctions was true, but the other was false. And if  you have two statements combined, remember, a half truth is a whole lie. So if you can join  two statements in one is true, the other false then the conjunction is false. So six rows false  because one of those items was false. In the set, seventh row, the first item was false, the  first disjunction was false. The second disjunction was true, but when you can join them, a  false and a true you get a false when it's a conjunction, same story on the eighth or they're  both false in this case, so definitely, the conjunction is false. So that's the most challenging  row to work through here. Now let's move on to the not a row a nice easy one, just look at a  and go the opposite. So where A is true, not a is false, where A is false, not as true. So the  first floor had a being true, so not easily false. In those first four cases, the second floor in  your truth column reference column had a being false, so not a is true four times in a row. And now we need the truth value for B or C. And again, you know the rule by now for disjunctions  in the first row it's true because both B and C are true. Second row, these true C's false so the disjunction is true third row, B is false, but C is true. So the disjunction is true for throw they're both false. So they're the disjunction is false. That's the only situation which disjunctions are  false. B or C is false only both components are false. And that's the case in that row. fifth row,  they're both true, B and C. So obviously the disjunction is true six row, B is true, C is false, so  the disjunction is true, seventh row these false but C is true. So once again, if the disjunction  if one items true there, but the the total disjunction is true. And in the eighth row, A and B are I mean b and c are both false. And that means the disjunction is false because both of its  members are false. So in the B or C column, you have the fourth row and the eighth row  showing a false value. So now we've got it all set up, and we're ready to find out whether this  argument is valid. How do you do that? Remember how to do that, again, you look where the  premises are true. And then in any rows where all the premises are true. Is the conclusion  true in all those rows, look at the first row. Well, you don't have those premises being true  second row, third row, no fourth row, no fifth row, both premises are true. Sixth row No.  Seventh row No. eighth row, no, there's only one row, the sixth row in which both premises  are true. And so that is a valid argument. Because we're both premises are true. The  conclusion is true B or C is true in that column. And therefore, the argument is valid because  you do not have any row that shows the premises being true, and yet the conclusion being  false. That's how truth tables work you. In a sense, they look complicated and may seem  complicated your way through. But if you do it just one bit at a time, really that's what logic  does just take one step at a time. Think it through. If you have to watch this video again, or  read Dr. Van Cleves article carefully work through it again, till you kind of have the hang of it,  and then do the exercises and more and more of that will sink in. But remember the aim of it  of doing a truth table is to find the rows in which the premises are true, and then find out  whether the conclusion was true in those rows. Yes, the conclusion is true. In all rows where  the premises were true, then you have a valid argument. But if you find even one row where  all premises are true, but the conclusion was False the argument is not valid