>> Today's lesson's going to work on the Venn diagrams that we learned a while back. But rather than just looking at single statements we're going to use Venn diagrams to analyze entire arguments. So if we remember the old Venn diagram system we had the subject term and the predicate term. And this, if you wanted to say all S or P shade the left. If you wanted to say no S or P shade the middle space. If you wanted to say some S or P, put an x in the middle. And if you wanted to say some S or not P you put an x on the left. That's going to hold still. In this case the S is going to stand for the subject position at the conclusion or the minor term. The P is going to stand for the predicate of the conclusion or the major term. And we're going to add one more term and that's going to intersect the two circles and that's going to be, we're going to use an M for the variable for that. And that's going to stand for the middle term or the term that's repeated in both premises. So to see how this would look in an argument form, I have an argument example to the top left here that's a IAI figure 3 argument. So I is some blank are blank. A is all blank are blank. And I is some blank are blank again. The figure 3 argument tells that we need to put at the middle term in the subject position of both premises. And then whatever goes in the subject position of the conclusion needs to go on the bottom premise. And whatever goes in the predicate position of the conclusion needs to go on the top premise. Okay, so here's our completed argument. And now the question is is how do we go about evaluating this argument using three part Venn diagrams. What I'm going to do is go ahead and talk through and solve this argument using the three part Venn diagrams. And on the next slide I'll walk you through each of the steps for doing this for constructing the Venn diagrams for evaluating them. And then on the later slides we'll actually look at step by step a couple of arguments and see how that's done after we've seen the formal steps. So this is just kind of warm you up, get you ready for what we need to be doing. So step one we need to take and set up three circles for our Venn diagrams. I'm going to have circle one, circle two and then circle three intersecting the two. The bottom left circle is going to be your S term, bottom right circle is going to going to be your P term and the top circle is going to be the M term. Now the first step for constructing your Venn diagram or for evaluation your Venn diagram is you're going to plot out the premises. And the first thing you're going to do is find your universal premises and plot those first. So in this case the universal premise is all M are S. And in order to do that if we numbered these spaces so we're all sure what I'm talking about, one, two, three, four, five, six, seven spaces total. The top space is an M that is not an S. So space one is an M that's not an S. And space four is also part of the M circle but it's not part of the S circle. So if we looked at these two circles without the P circle in the way, you would see M S, if we wanted to say all M's are S, we need to shade out the part of the M circle that is not an S because we want to say all M's are S. And now we have all M's are S. Now if we took those, these two circles and put them right on top of the three part Venn diagram you'd see the shaded part overlaps that space one and four. So you'd shade in the space one and four. And don't worry as we go we're going to practice this quite a bit. Okay then the next step is to plot out your remaining premise, in this case some M are P. And here some M are P, if we look at our old Venn diagram system. So we're used to seeing the M on the left side, the P on the right side. Some M are P, you'd put an x in the middle. But now we're going to have to rotate the circles down so that P is on the bottom and M is up top. The x still stays in the middle. Now if we slid this rotated two part Venn diagram over the three part Venn diagram, you'd see that x can either go in space three or space four. Both of those are M's that are also P's and both of those say some M are P. But because space four is already shaded out, the only space the x can go in, the only space the x can go in is space three. Okay, so that's part one of this evaluation of arguments using Venn diagrams. Part two, you're going to put your pen down and ask a question. The question is can I read the conclusion in what I've plotted out here? And so the conclusion says some S are P, the old school Venn diagram. In this case the old school Venn diagram looks the exact same as the new three part one except that M circle's kind of in the way distracting us. But some S are P, what you need to see is an x somewhere in the middle. There's two spaces in the three part Venn diagram that would have an x in the middle. Either space three or space six. If you see an x in either of those spaces it would say that some S's are P's. And sure enough space three has an x in it. And so it does say that there is an S that's also a P, some S are P. And so what this tells us then, the Venn diagram tells us that if these premises are true we do know with certainty this conclusion. And so we would say this argument is valid. So again what you're doing is you're going to plot out your premises and then see if you can find your conclusion in what you've already plotted out. If you can find it, the argument's valid. If you can't find it, the argument's invalid. Okay, let's go ahead and look through step for step the way we would analyze three part Venn diagrams. Part 1, setting up the premises. So you're going to put the minor term on the bottom left, the major term on the bottom right and the middle term goes up top. Another way of saying that, you're going to put the subject of your conclusion on the bottom left, the predicate of your conclusion on the bottom right and the middle term is going to go up top. That's where you're going to put the terms for the circles. Step two, you're going to plot your universal premises first. Universal premises always take up two spaces. And there's a little hint here, you're either going to be looking for a football or like a crescent moon or a Pacman thing. And what that means is this. So if you have your three circles and you're trying to say no S's are M's. You'd be shading in two spaces. It's the spaces between the S and the M. And if you look at those two spaces that got shaded it looks kind of like a football. There is logic teacher Mark Thorsby who has on YouTube posted his entire lectures for Patrick J. Hurley's, A Concise Introduction to Logic Textbook. They're really good lectures if you want to use those as supplements, by all means use them. And his example is looking for a football, that's where I got this idea from. It's pretty clever. And then the other thing to look at, or I think it's clever at least, the other thing to look at is the Pacman or the crescent moon. This is [inaudible] ideas well. So if you want to say all M's are P's, you're going to have two spaces that you'd have to get rid of to leave us with all M's being P's. And it's going to be like that. Now the reason this is like a Pacman or a crescent moon, you have the circle for the M term. You have the circle for the P term. Then if you look right here, that part that's shaded, that part looks kind of like a crescent moon or a Pacman. If you had a Pacman it would be, the ghost would be here running away from Pacman. Because Pacman's trying to gobble him up. So you're either going to look for a crescent moon or a Pacman. The important part is you're always going to be shading two spaces when you're doing your universal premises. Okay, the next thing is is if you have a x on the line of an argument that's going to indicate uncertainty as to where the particular premise goes. Particular premises always go in one of two possible spaces or on the line. So if you can't put it in one of two spaces and it goes in the other space, if you can put it in both spaces then the x would have to go on the line. For example. [ Pause ] If you have some S's are M's, space one, two, three, four, five, six, seven. If you look at space two that's an S that's an M. If you look at space three that's also an S that's an M. And so that could go in either of those spaces. And so this is what that number three says on the left. An x on the line indicates uncertainty as to where the particular premise goes. And so in this case because it could go in either space two or three and we're not sure where it goes, we'd put the x on the line between two and three. Now what if we had already plotted the premise no M's or P's before that and shaded out space three and four? Now if you say some S's are M's that could either go in space two or three but three doesn't exist anymore. So the only space left is for the x to go in space two. And this is actually rule number two applies, we plot our universal premises first. Because a lot of times what will happen is we'll plot our universal premises and then we move on to our particulars that plotting of the universals kind of kicks the particulars out of those awkward positions of being on the line. And it says okay, you can only be in one space. So an x on the line indicates uncertainty is where the particular premises go. The particular premises are either going to go in one of two possible spaces. If one of the spaces is shaded out, it goes in the other space. If no space is shaded out then it goes on the line between two spaces. And so that's setting up the premises. Once you've set up your premises the next phase you're going to move onto is evaluating your argument. So what you do is one, stop writing. You're not going to write anything down for your conclusion. Once you've written everything out for your premises you're done. Then you ask yourself a question. Can you read the conclusion in what you've plotted? If you can, the argument's valid. If you can't, the argument's invalid. As we go through this we'll talk about some tricks for helping you see if you can identify the conclusion and what you've plotted. But for now let me give you an example. So M S P, let's say the first premise was no S's are M. And the second premise was all P's are M's. [ Pause ] And the conclusion was no S's are P's. So let's focus on our conclusion. No S are P. With that conclusion, no S are P, the question is can you read that conclusion what you've plotted? Well the old school Venn diagram system you have S on the left, P on the right, shaded in the middle. On this three part Venn diagram you do have the middle between S and P is shaded. And so this example you can read the conclusion in what you've plotted. And so this argument is valid. There's a couple hints. The first hint is more information is okay, less is not. Before I erase this so you can read everything, let me explain what that means. In the example drawn here the space between S and the M is shaded in two spots. And then you have the space where the P is is shaded out over where the P is. And this looks like a little bit more is done than the final conclusion Venn diagram. So for instance what it'd really look like is if you have the top part of the S shaded and then you had a big chunk of this P shaded as well. Now that's kind of what the old school Venn diagram would look like. This is more information than is needed and that's okay. So in this case the argument's still valid because it still does say if you're an S you are not a P, no S's are P's. Where you get into a problem is if you had less information there. So for instance if you had. [ Pause ] The space between P and M, it said no P are M. And then you had the space between S and M. It said no S or M. And your conclusion was still no S are P. Here you have less than your conclusion written. Because the bottom section in the middle between the S and the P, space one, two, three, four, five, six, seven, space six, needs to be shaded in as well and it's not. So that's less information. If that was your, if this was your argument, this would be an invalid argument. So this is what we mean by more information is okay, less is not. Again as we go through the specific examples you'll see this worked out in much more detail. And I have some practice exercises that really spend some time dealing with this. So if you're kind of unsure as to what was going on right there, don't worry we're going to go through it again. Here's the hint, more information is okay, less is not. Second hint, an x on the line means the argument is invalid. So if you're ever unsure as to where the x should go, it could go equally between two spaces, the shading doesn't force you into one space or the other and you put that x on the line, you know right away the argument is invalid. And a final hint is a space that has been shaded twice, that also means the argument's invalid. It just happens to work out that way. Same with the x on the line, it just happens to work out that way. So those are some extra tips that'll help you out. Everything before the hint is what you really, really need to know. But if you need some extra help, the hint helps as well. Okay, let's go ahead and practice with some specific arguments. This first argument is no felines are dogs. All cats are felines therefore no cats are dogs. So we'll take this one step at a time. The subject of the conclusion or the minor term goes on the bottom left. And so in this case that's cats. The predicate of the conclusion or the major term goes on the bottom right. And in this case that's dogs. And then finally the middle term, in this case felines goes up top. Okay, so we'll put all that together now. So you have felines, cats and dogs are the three terms we're dealing with. Okay, now we'll move onto plotting our premises. So we have spaces one, two, three, four, five, six, and seven. The first move you're going to make is to fill in your universal premises. In this case both premises are universal. So it doesn't matter which one you do first. Really if you're ambidextrous you can do them both at the same time. I'm not, so I always just pick the, if they're both universal I just do the top premise first. So we're going to do no felines are dogs. In this case we're going to shade out the part where felines and dogs overlap because we want to say that no felines are dogs. The old school Venn diagram you'd have felines on the left side, dogs on the right side and you'd shade out the middle. But to match this to the three part Venn diagram we're going to have to rotate this up so that felines is up top and dogs is going to be bottom right. So you have felines up top, dogs bottom right, keep your shading in the middle. Now if I had transparency paper I'd take the transparency paper and make overlapping circles. So I'd do the three part Venn diagram. And then do two circles on the transparency paper. And then you can slide the two circles over the top of the three part Venn diagram and see exactly where things fit. In this case just pretend that we're sliding this rotated old school Venn diagram over the three part Venn diagram. And as you slide it over you see that it's going to overlap the green circle and the red circle. And specifically the shaded part is going to be right where spaces three and four are found. And that says no felines are dogs. Now we're going to move onto the next premise. So remember the universals are always shading in two spaces. We're either looking for footballs or crescent moons. For the no felines are dogs we did a football shading. For the A form we're going to do a crescent moon shading. So in this case we're doing all cats are felines. So the old school Venn diagram you have cats on the left and felines on the right. You're going to shade in the left side. But now felines is up top, cats on the bottom. For the three part Venn diagram, so we need to rotate the circle so that felines is up top, cats is on the bottom. And then we're going to go ahead and shade in that cat section again. Now as we slide the old school Venn diagram over the three part Venn diagram we would see that space, spaces five and six are going to get shaded here. Spaces five and six get shaded. And there you have, remember we're looking for either the crescent moon or the Pacman, spaces five and six make that crescent moon. So now the only space left for cats to be in is also in the feline circle. And so that does say that all cats are felines. And finally we move into the conclusion phase. At this point we put down our pen. We're done plotting out anything on the Venn diagram. So we don't tough the Venn diagrams again. And the questions that's asked is can you read your conclusion in what you've already plotted? More information is okay, less is not. The space where dogs and cats overlap is shaded out. So there's no way to be both a cat and a dog. In other words, it does say no cats are dogs. Specifically two, three, four, five, six, seven, spaces three and six are the spaces where cats and dogs overlap. And that would be where you'd have a cat/dog hybrid. And spaces three and six are overlap or shaded out which means nothing can exist there. And so that does say that no cats are dogs. So this argument is valid. Now when we go back to that more information is okay, less is not. Space five and space four is more information. And that's okay. Because space, really what's going on is between spaces two and seven, that's, space two is where all your cats occur. And space seven is where all of your dogs occur. And so it still does say no cats are dogs even though there's more information like space five discussion and space four discussion. What wouldn't be acceptable is if say space six wasn't shaded. Because space six if it wasn't shaded would be a space where you are allowing a cat to be a dog which violates that statement, no cats are dogs. So that's what we mean by more information is okay, less is not. But once again because in this argument, spaces three and six are shaded. The argument no cats are dogs is valid. Really quick, to look at the old school Venn diagram, if you had cats on the left, dogs on the right, the middle space would be shaded. And if you carry that over and look, yep, the middle space is shaded. But once again we didn't write anything down for the conclusion. We only wrote stuff down for the premises. And then the question is is can we read the conclusion in the premises. So what happened is if these premises are true, we do know this conclusion with absolute certainty. And that tells us then that the argument is valid. Let's try another one. So this argument says no cats are nice animals. Some pets are cats. Therefore some pets are not nice animals. Step one, we're going to plot out our universal premises first. In this case the universal is the top one, no cats are nice animals. So we'll go the old school Venn diagram. We have cats on the left, nice animals on the right, and we want to say no cats are nice animals. But then we have to rotate the Venn diagram. So we're going to have cats up top, nice animals bottom right and we're going to shade in the middle. Let me write the numbers on this, three, four, five, six, seven. So as we slide the old school Venn diagram over the three part Venn diagram we'd see that it goes right over the top of spaces three and four, cats being the green circle, nice animals being the red circle. So spaces three and four are what we overlap. And that's how you plot no cats are nice animals. Okay, now for the second premise. We're going to do some pets are cats. The old school Venn diagram, you have pets and you have cats, some pets are cats. You're going to put an x in the middle there. We're going to rotate the C up to the top and keep the pets down on the bottom. So we'll have cats up top, pets down on the bottom, we put an x in the middle there. Now as slid this Venn diagram, old school one, over the top of the three part Venn diagram we see spaces two and three are the two spots where that x would need to go. But because we already shaded out space three and shading means that that space doesn't exist, the only space left for that x to go is in space two. And so that is the second premise for the argument. The x could have gone in two spaces but one has been shaded out so now it can only go in one space. So now we move into the evaluation phase. You put down your pen. And the question is do you see the conclusion in what you've already charted? In this case, yes. There's an x in the pet circle but not in the nice animal circle. And so that argument does say some pets are not nice animals. So it's valid. Let me show you with the old school Venn diagram what that looks like. You have pets, nice animals, if you were saying some pets are not nice animals, you'd put an x in the far left. In this case there is an x on the far left. It's in the top of the far left but it is on the far left. That would be the equivalent of just putting an x on the top part of your old school Venn diagram where just the P part of the circle, so the far left space. It does say that some pets are not nice animals. And so as you write out your premises the conclusion is already there. You didn't have to write anything new for that conclusion. The two spaces that would have worked is space two where you did see the x. The other space that you could have, if you saw an x it would have been valid is in space five. Now at this point the question might come up, what if you had an x on the line between space two and five? This argument, that's not the case but let's say we had an argument where you had an x between two and five like that on the line. And the conclusion was some pets are not nice animals. Would that be a valid argument? For whatever reason, so go back to that hint, an x on the line means the argument is always invalid. For whatever reason this situation will never arise. Likewise if you have a double shaded space and the question comes up, what if it's a double shaded space but it gives us the conclusion? For whatever reason double shaded space will never give you the conclusion. If you see a double shaded space the argument is always going to be invalid. So it's kind of a nice way of double checking your work. So that's it for these practice arguments. Let's go ahead and practice getting the component parts figured out so as we go through arguments we're going to be a little more fluid with them. We're going to use this Venn diagram where the spaces are one, two, three, four, five, six, seven. You have S on the bottom left, P on the bottom right and M up top. And we're going to answer some questions. So question one, how would you show N M's are P's? As I go through these I'm going to go at a natural pace. But I encourage you to hit pause and try it on your own before you see me solve this through. So in this case no M's are P's. We're going to shade in the space three and four between the M and the P circles. And so there's the football. Next question, how would you show all S's are P's? In this case we're going to shade out spaces two and five and there's the Pacman or the crescent moon. As I'm going through this if you happen to get one or two of them wrong, just go ahead and rewind it, to finish it all out rewind it and try it again in about ten minutes . You'll probably forget all of this and then you're just going to be re-practicing from scratch. You could probably get through about four runs of, you know, waiting ten minutes, rewind, start again before you have this stuff memorized. So you'll be able to practice it again and again and again as though it's brand new to you. Okay, so that one we shaded in two and five. The next one, how would you show no S are P? In this case we're going to shade in the space between the S and the P, that's three and six. That says no S are P. How about some M's are S's. Here an M that's an S is space two, an M that's an S is also space three. And so because we're not sure if it goes in space two or three we're going to put it on the line between the two. So there are x on the line of two and three. Some P are not M. So seven is a P that's not an M, six is a P that's not an M. It's not clear which of these it is, so we're going to put an x on the line between six and seven that says some P are not M. This time we're going to have spaces three and four already shaded in and we go through another series of questions. So how would you show some P are not S? So spaces four and seven are P's that are not S's. But space four is already shaded in. So you can't go there. The only space left to go is in space seven. How about some S's are not M, some S's are not M. Let's see, so space five is an S that's not an M, space six is an S that's not an M. It's not clear which space it goes it so we're going to put that x on the line, so x goes on the line of five and six. All right, what about some S's are P's? So S on the bottom left, P on the bottom right. Space three is an S that's a P, space six is an S that's a P. But space six or space 3 is an S that's a P, space six is an S that's a P. But space three doesn't exist so the x has to go in space six, that's the only place it can go. Okay and finally all M's are P's. Let's see, so space one is an M that's not a P and we want to say all M's are P's. Space two is an M that's not a P and we want to say all M's are P's. So space one and two have to go. If this was a real argument it'd be very odd because it's basically saying that M doesn't exist because you already shaded out three and four. So nothing exists anywhere for space M which is kind of weird. Anyways, so all M's are P's, space one and two get shaded in. And there's your Pacman shape or your crescent moon shape. All right, next set of questions. The, what we're trying to figure out now is what you would need to see to get the following conclusions. So if we want, if we had a conclusion in an argument that said some S's are P's. The question is, what would you need to see happen where for that conclusion to occur? In this case some S's are P's. You would need to see an x in either space three or six. If you saw an x in either three or six the argument would be valid. What about for some S are not P? In that case an x in either space is two or five would be what you'd need to see to make the argument valid. But again because these are your conclusions you're not going to write those x's in. It's only if you can read them. If the x is actually there, the argument is valid. If the x is not there, the argument is invalid. So in this case because there's no x anywhere the argument would be invalid if that was your conclusion. What if you had no S are P? You'd have to have spaces three and six shaded in. And what about all S are P? You'd have to have space two and five shaded in. Both of those spaces would need to be shaded in. Okay, bonus round. I'm going to draw this so you can see exactly what's being talked about. If two, three and five were shaded, so there's two, three and five. If that was shaded when an argument with the conclusion of all S are P be valid? The answer to that question is yes, it would be valid. It has more than the conclusion says. But remember that hint in the very beginning, more information is okay, less is not. And if you look visually space six is the only space where your S's can exist. And space six are part of the P circle. And so every single S really is a P which is what that conclusion says, all S's are P's. So this argument would be valid. So let me back up a minute and walk you through one more time the reading these conclusions. I'm going to set up the old school Venn diagrams right beside it so you can be, we can be sure that you're really clear on what you're looking for. So the old school Venn diagram, if you have your S on your left side, your P on the right, and what's really going on here is space two is up there, it's cut off, space three, space four, space five, space six, space seven and then this is up where your M circle. We're going to get rid of that M circle. We don't need it. And that's what we're looking at. So space two and five, three and six, four and seven, that's where the conclusion exists. So if you saw some S are P, an x in space three or six are both spaces that do say some S are P for the old school Venn diagram. If you said some S are not P, an x in space, either space two or five is what it would take to make the old school Venn diagram work. And so space, x in space two or five is what it would take to make the three part Venn diagram work. If you were saying no S are P then you'd need to shade in both three and six. And if you said all S are P then you'd need to shade in both space two and five. Okay, so that's it for the three part Venn diagrams for analyzing arguments. It's a lot to take in. I really recommend going through the practice section a few times before you really start working through exercises. But if you've got a grip of it go ahead and get going on it now. I think the tricky thing is is that because there's that third circle in there a lot of times it confuses students. And the trick is is if you go back to those color slides that I had in the beginning, let me back up to that part. Here as you're plotting out your premises it helps to think in terms of these three colors. And so when you're going no felines are dogs, that's the top premise. You're pretending the blue cat circle doesn't exist. And so when you say no felines are dogs. If the blue cat circle doesn't exist and you're saying no F are D, well that space is the space that needs to be shaded. And then when you say all cats are felines. That's between the blue and the green circle. You need to pretend the red circle doesn't exist. And so all C are F. If the red circle didn't exist you'd shade in this space here. Remember the red circle doesn't exist right now. And so the trick is is you're always focusing only on the two circles that matter. Then when you're reading your conclusion, no cats are dogs, pretend the green circle doesn't exist, the felines. And is the space between cats and dogs shaded? In this case it is. So I think if you think in terms of like those are three separate circles and you are only focusing on two circles at any given time, the thing that helps to practice then is to figure out what circle do I need to pretend isn't there. And that's kind of a tricky thing to get the hang of. But as you get there it comes with practice, just keep working on it. Okay, that's it for today. Take care and see you next time.