Video Transcript: Venn validity for categorical syllogisms

Hi, I'm David Feddes, and this talk is about the Venn test of validity for categorical syllogisms.  We're continuing in our logic course, based on material by Dr. Matthew Van Cleave. Now what  are categorical syllogisms? Well, a categorical syllogism is an argument with two premises,  and then a conclusion where every statement of the argument is a categorical statement.  What's a categorical statement? Just a reminder, there are four different forms that can take  all S are P a universal Affirmative, no S are P a universal negative, some S are P, a particular  affirmative and some S are not P, a particular negative those four kinds of statements are  categorical statements. And then a categorical syllogism uses only statements of one of those four kinds with two premises, and then a conclusion. Let's consider some examples. Here's an argument with all universal Affirmative. In other words, all of the form all S are P premise one  all humans are mortal. Premise two, all mortal things die. Premise three, therefore, or not  premise three conclusion all humans die. Now how do we evaluate that? We first translate  into the form all S are P, remember you have to have categories. So we say all humans are  things that are mortal. Our second premise all things that are mortal are things that die, we  don't just say all mortal things die because die is a verb, we turn it into a category, things that die. And then our conclusion. All humans are things that die. Once we've translated it into  where we have all nouns, all things, then we move on. And we make our Venn diagram. In this case, we have to have a three category Venn diagram, because we have three categories. We have humans, we have things that are mortal, and we have things that die in making the  three categories. And here's a strategy always to follow. Look at the conclusion. In the  conclusion, you see an all S are P all humans are things that die in the upper left, make your  circle to be humans. Then in the upper right, use the P the predicate category of the  conclusion, things that die and put it in the upper right. So the upper left is your subject of the conclusion. The upper right is the predicate of the conclusion. And then your third category,  you draw in the lower part of the diagram so that they intersect and look like this with your  category, humans, things that die, and things that are mortal. Now we start filling in or  shading in our Venn diagram, we take our premise, all humans are things that are mortal. And that means that there is nothing in the category of humans except what's in the intersection  with things that are mortal. So you shade out everything that's not in that intersection. Now  we move on to the second premise. And we shade out to some more. The second premise is  all things that are mortal, are things that die. So we look at the category, things that are  mortal, and the only thing that exists in that core category is in the intersection with things  that die, everything else has to be shaded out. And so we do that. And we end up with a three category Venn diagram that looks like this. And that's the three category Venn that captures  what our two premises are saying. Now we move on to map out our conclusion, Venn  diagram, it's a two category statement. So we only need two categories in that Venn diagram. All humans are things that die. So everything in the category of humans is shaded out, except what's in the intersection with the category things that die, and then there may be some  other things that die that don't intersect with humans. So that's what the Venn diagram looks  like, for all humans are things that die. And then the final step is to compare the two Venns.  And this is a valid argument, because the Venn for the conclusion doesn't contain any  information. That's not already in the premise diagram. If you look at the conclusion, then you see that the entire humans category is shaded out except what's in the category things that  die. Now, look in the premise diagram and you see, is there anything in the category humans, that's not also in the intersection with things that die? No. So it's not giving any information in the premise? Or in the conclusion that's not already in the premise. Now the premise has a  little information Should, that's not in the conclusion, but that doesn't matter. What matters is that the conclusion Venn diagram doesn't contain any info that's not already there in the  premises. And as you can see in the premises, if there's anything in the category human, it  has to be in the category, things that die because everything else is shaded out. When you  look at Venn validity for categorical syllogisms, an argument is valid. If there's no information  in the conclusion, then that's not also in the premise event. And that's true, even if the  premise Venn has some more information than the conclusion, the argument is invalid. The  conclusion then has any information that is not already there, in the premises Venn diagram.  Take another example. Premise one all pediatricians are doctors, premise two, all pediatricians

like children. Conclusion, therefore, all doctors like children, is that a valid argument? Well,  let's map it out. We need three categories, pediatricians, doctors, and things that like  children. And when we make a three, category Venn, for that, we remember, draw your circle  in the upper left with the with the subject of your conclusion. So the subject of your  conclusion is doctors. So you make a circle in the upper left, the predicate of your conclusion  is things that are children. So you draw that circle in the upper right. And then you take your  third category, pediatricians and you draw that circle below the two intersecting circles there.  So once you've done that, now, you take your first premise, and you say, all pediatricians are  doctors. That means that in the category, pediatricians, you have to shade out everything  except what's in the intersection with things that are doctors, your next step is to go to the  second premise. And it says, All pediatricians, like children, and so you have to take the  category, pediatricians, and you have to shade out everything in the category of pediatricians that doesn't intersect with things that like children. And that's what you end up with. For your  three category Venn diagram. Now you have to diagram your conclusion. Your conclusion is all doctors are things that like children. So in the category doctors, you have to shade out  everything that doesn't intersect with the category, things that like children, you have your  conclusion mapped out. Now you have to compare it to your premise then. And what do you  find? Well, you look at your conclusion, then. And you find that there's nothing in the category doctors that doesn't intersect with things that like children. But when you look at the premise  Venn, and you look at the category doctors, you see that there are some things, possibly in  the category doctors that don't intersect with the category things that like children, there's  information in the conclusion, that's not also in the premises. And that means that it's an  invalid argument. Again, to remind you, an argument is valid. If there's no information in that  conclusion, then that's not also in the premises, then, even if the premise Venn does have  more information than the conclusion, and argument is invalid if the conclusion Venn has  information that's not in the premises. Example number three, some mammals are bears.  Some two legged creatures are mammals. Therefore, some two legged creatures are bears.  Let's figure that one out. We need a three category Venn for the premises. And remember,  again, look at the conclusion. What's the subject of the conclusion? two legged creatures. So  that's what we put in the, that's the circle we make in the upper left corner. What's the  conclusion of the what is the predicate of the conclusion? Well, creatures that are bears. So  bears is the predicate of our conclusion. So we put that in the upper right corner. And then the remaining category is mammals. And so we make that circle intersecting the other two, but  draw it lower down. So the subject of the conclusion goes in the upper left, the predicate of  the conclusion goes in the upper right, and the third category goes lower down. We've done  that a couple of times. So hopefully, it's getting familiar to you next step. We've got a  problem. Our premise one says, some mammals or bears, and the convention for handling  that is then you put an asterisk in the intersection of the categories, mammals and bears, but  the question here is, where do you put that asterisk? Do you put it outside the category of two legged creatures? Or do you put it inside the circle of the category for two legged creatures?  Well, Do we do, because premise one doesn't tell us whether it belongs inside or outside of  the category two legged creatures, all it tells us is that it's somewhere in the intersection of  the category between mammals and bears. So we don't know where to put that asterisk.  Here's how we handle a situation like that, we put the asterisk on the line of the category, two legged creatures, because we don't know whether it's in the category two legged creatures or outside of that category. So we put the asterisk on the line of the category two legged  creatures at the intersection, you know, in that intersection area between bears, and  mammals. So you put that asterisk right on the line. And then when you move on to premise  two, we do something quite similar. You have some two legged creatures are mammals. But  we don't know whether they're bears or not. So we put that in the intersection between two  legged creatures and mammals, but we put it on the right on the line for bears because we  don't know if it's a bear. Or if it's not a bear. So we have to put it on the line. That's all the  premises tell us. And so that's how you would map out a Venn diagram for those two  statements involving these three categories. Now, we move on to the conclusion and do the  Venn diagram for those two categories. Some two legged creatures are bears. And so in the 

intersection of two legged creatures and bears, we put our asterisk that's how we do it when  we make a particular statement, a thumb, or this or that. And then we compare, we compare  the conclusion Venn with the premise Venn, and we find that it's an invalid argument.  Because the conclusion Venn gives us information that is not in the premises in the conclusion Venn tells us that there is something in the intersection between two legged creatures and  bears, and the premise of and doesn't tell us that our two asterisks are on the line, but we  don't know whether they're in or out. And when you have information in the conclusion, that's not contained in the premise, you have an invalid argument. Example number four. This one's a little more abstract, we're not going to know what the categories actually are, we'll just use  letters to represent them. Premise one, some S are M premise to all M are, P. Conclusion,  therefore, some S are P. Now, notice that in this argument, we have a mixture of kinds of  statements, we have universal statement, all M are P as a premise. But we also have a  particular statement, some asked our M. And our conclusion is also a particular statement,  some S are P. Now, when you have a mixture of universal and particular, the strategy is first  map out the universal statements before mapping the particular statements. And I'll show you why here in a moment, we'll just go ahead and map out the particular statements. First, to  show you what happens. If we map premise one first, which is a particular statement, some S  are M. Here's what you get, you've got the S category and the M category, you know, the  asterisk has to go in the intersection. But you don't know whether it's inside P or outside P,  because nothing is said about that in that first statement. So you put the asterisk on the line,  like we learned in the previous example. However, if we were to map premise two, first, here's what we get. All M are P. So that means that you shade out everything of M, except what's in  the intersection with P. And once you've done that, once you've dealt with the universal  premise first, you can go back and look at the particular premise, some S are P and you know, right away where the asterisk has to go, because we already shaded out everything in M  That's not also in P. So you know that it's not going to be in that shaded part, because shaded  means there's nothing in there. And that means you can put the asterisk right where it is, in  that Venn diagram, because your universal statement, all M are P already ruled out a certain  part of that area. And so you've got the asterisk right there and then you can move on to map out a two category Venn diagram for your conclusion, some S are P, you put the asterisk in  the intersection of S and P and now to test for validity. We compare our three category then  for the premises with our two categories, then for the conclusion, look at it carefully. The  conclusion says that there is something in the intersection of S and P. Look at the premise,  there is something in the intersection of S and P. So, there is nothing in the conclusion Now  that the premises don't already tell us, and that means that it's a valid argument. Remember,  again, if you have an argument involving both universal statements and particular  statements, take the premise that's a universal statement first, even if it's not listed first, and  map that onto your Venn diagrams first, then map your particular particular statement.  Remember that the universal can determine how to map particular statements, but particular  statements can't tell you how to map out the universal and that's why you want to map out  the universal part first, all S are P before you map out anything that says some S are P. So  remember that strategy, always map out the universal statement before you map out the  particular statement, even if they don't occur in that order in the premises, Venn validity for  categorical syllogisms just to remind you one more time, an argument is valid if there's no  information in the conclusion Venn diagram that's not already in the premises Venn diagram.  And that's so even if the premise event has more information than the conclusion that doesn't matter, because that remember, that's a three category, then it's going to include more  information than the conclusion but the conclusion cannot include more information, the  premise or you've got invalid argument. The argument is invalid. If you have a conclusion  then with information that's not in the premises. So that's how we deal with categorical  arguments and categorical syllogisms. And trying to use Venn diagrams in order to decide  whether we're dealing with a valid argument or an invalid argument.