4.3 Venn Diagrams & The Modern Square of Opposition

modern score of opposition if you're following along with us we're using the Hurley textbook okay so let's start here Venn diagrams and the modern square of opposition let me start off here by doing a sort of quick review because you'll see that you probably noticed already that we're sort of slowly building up in terms of what categorical propositions are and how they relate to each other so what's the quick review now remember we looked at we ended we concluded last time by recognizing that there's four different propositions the eighth elion though and that they have different distribution patterns we're going to continue our discussion today by looking at how they can relate by looking at we're gonna sort of firm up what Venn diagrams are and how they relate in term by using a medieval device known as the square of opposition we're gonna be looking at the modern the more simplistic version of the square of opposition also refer to as the modern square of opposition so we have these four types of propositions so let's put that off to the side for a minute and let's quickly discuss existential import the well we're gonna be really be talking about here is the notion of the existential fallacy if we're gonna come to that the very end of our talk today but let's start timeing what is existential import well you can see the ROO word here of existential it sounds like it's a really sort of complicated term it's not really actually the root turn there is to exist so what is existential import that means an exercycle proposition has existential import if a claim of existence is made okay not all propositions categorical or otherwise necessarily make a claim of existence so we have existential support and whether or not a proposition has existential import the oft asked question doesn't make a claim about existence now of course think about it let me just flip here to the textbook and you can see here why this is sort of an important distinction to make right so let's take this example all Tom Cruise movies are hits print which is not true actually I don't think but you can see are all Tom Cruise movies or hits this does have existential for because we know that Tom Cruise exists and so we know that it's making a claim about something that exists whereas for instance if I say all unicorns are one-word animals this statement doesn't make a claim of non-existence does it and the reason it doesn't is because I know that unicorns are not something that are that's true right yeah so this doesn't make a claim of existence right so that's what existential import refers to or that's what the issue is about now there's two basic approaches to the question of looking at existential import the first is what we're gonna call or what your book refers to here as the Aristotelian perspective or the Aristotelian standpoint so let's put Aristotle up here another perspective is actually the perspective of evident contemporary well out of a contemporary put a modern location by the name of George Boole what's Aristotle's view something makes a claim of existence if it if the subject terms refer to real things right so think about like that if the terms refer to real things right then that means you have a claim of existence right if it refers to I'm sorry my handwriting it's hard to read here if it refers to real things okay what's bulls perspective that means the things that are imaginary things that are made up like unicorns or do not have existential import on Aristotle's view because it doesn't refer to things that we would consider to be real now what's Bulls view well bool we're gonna see as much is a little bit more nuanced and I think actually more sophisticated view where Poole says wait a second obviously things have does it have existential import well yes if it refers to real things to real particulars but no a statement doesn't have a claim of existence if it refers to universals never refers to universals now why is that even if we consider those universals real but that's the whole question because here's the thing we have to consider it do we have universal knowledge let's make a universal claim now obviously we would say that no it doesn't have existential import if it refers to imaginary objects all right so there's no existentially important they're like unicorns and stuff so air Istanbul both agree if you talked about unicorns you're not making an existential claim but bool and it did in addition and further than Aristotle he argues that if you make a universal claim let's say for instance all right let's see all worlds how are things with mass all right so let's make that our claim if we say all worlds are things with mass does this imply existence the answer bull says is no it doesn't why it goes to the question of can universals being known and this is actually a very deep and well-known problem is the problem of universals here can a universal be known that's sort of the question here Aristotle thought that you could in fact have universal knowledge meaning that he thought that because think about how do we gain knowledge we gain knowledge primarily through observation we observe things about the world so how do I know there's trees in the world because I see trees ok so if I think a claim like some trees are green and some trees are brown etc etc right those make claims about existence because I could know whether or not those things actually exist because if existence is what we're talking about how do you know something exists what if I make a claim like all trees or plants all trees are plants now that makes sense conceptually for us because we were sort of used to categorizing plants and types of plants and that sort of thing but wait a second that's a universal claim to say that all trees are plants now the question is do I know that that claim actually implies existence the answer bool says is no it doesn't because you don't have universal knowledge right you'd would have to literally go throughout the entire universe or rather the entire the universe of the entire class and double check each one of those particular elements to know if it exists so he argued that you can never have universal knowledge it's actually sort of almost by definition impossible right so if you can't know your Versalles can't universally know no which means that universals have no existential import that was bools inside there's no existential import all right there's no existential import okay and that's sort of a really critical discovery we're gonna see why here in a little bit here so so that's the first thing to recognize here between there's the Aristotelian perspective and there's the boolean perspective both agree again that if you talk about things that aren't real there's no existential claim the difference is that Aristotle thinks that yes Universal statements can have existential import they do make claims about existence whereas George Boole says no they don't right and we're gonna see that it's ultimately I think we have to go I think bool is the person to trust in this regard but that's sort of a really critical juncture okay and we're gonna talk more about that a little bit and a little later from now okay so they do make claims by existence so now we don't want to talk to you about here's Venn diagrams and we've already been using Venn diagrams so and most of you probably even encounter Venn diagrams when you took that your mathematics courses Venn diagrams is essentially their ways of diagramming pictorially or graphically a categorical proposition okay what you do is you draw two overlapping circles which represents different possible States now we're gonna have to write a key up here if something here's the rules you're gonna see it'll it's gonna be a little bit confusing but if you fill something in that means there's empty space all right but if you put an X that means at least one exists okay so this is the key and these are the tools what we're allowed to do in the Venn diagrams is either fill things in or put X's in okay now let's take our four propositions we have an a proposition and the a proposition would look like y'all quick video again s and the P all the a proposition says that all the s is our P's so how do we fill represent that well the answer is what we would do is we fill it in we fill in this space here okay because what this means is that that all of the esses are here and out here is emptiness there's nothing out there right or at least there's no s is out there so all the esses are contained in here and you can see they're nicely sort of drawn in within the peas so that means all of the esses are Peas you can see it has to deal with the distribution the distribution of an a statement is the S to the P and you can see here we sort of literally pushed over the asses into the P circle now what about the e proposition the proposition to remind you is I should write them out for you is no SRP so first thing we would diagram that by drawing two circles put in an S and a P down I'm gonna change my color here and in this case we would draw actually we'd fill in this little football spot right this means the empty space that's what our rule is when we filled things in so what this says is that none of the esses are peas and by the way look at it all the esses are separate from the peas so this represents no SRP that's how you tiger on the east a bit you'll notice that they that you only use the fill you fill when you have Universal stainless you're going to see you use an X when you have particular statements okay so let me scroll down here and let's show you the I though okay so the I seen it says that some SRP and the O statement says that some s are not P so again we're just going to draw the circle here for some S or P the P you should always label and again it's standard for to always put the s first and the P second I mean that's the general rule we want to apply in the midst of this course because that way we all have a universal way of sort of assessing each other's work that makes them so since this is a particular claim I have to use write up an ax right so that means that where should I put thanks there's three plate there's only two places I can either put tags like with ax here I put the X in the circle in the in the overlap area well this is some of the s is our P's now remember the distribution of what gets distributed none neither gets distributed right so that means that I actually put the S right here the X right there rather because it says that there is at least one s and that ass is also at P it's within both circles but you can see it's not distributed because there's no way of knowing anything about any particular circle by itself okay now what about over here world would I put thanks here for sub s or not being sorry my handwriting is horrible on this the answers I put it right here right because this is that there are some asses in the universe are there some asses but those asses are not P so I can't put it in the football those sort of middles section that overlaps I have to put it over here with the essence okay so this is the this is essentially the four ways you can diagram you can diagram what these propositions say and mean now you'll see later that it's actually a really sort of critical feature because the it will actually allow us and sorry about that the Venn diagrams will see will actually allow to test whether or not arguments are valid eventually so the Venn diagrams actually been very helpful and there's sort of two types of people in the world I find there's the there's the people who are good at geometry and those people sort of need graphic illustrations they sort of understand things spatially and I think those people do really well with that diagrams but we'll see there's there's other other tools to help assess arguments and one of them is known as the square of opposition and I think the square of opposition helps a lot of people who are maybe less pictorial a declares if you're not a geometric person maybe you're in algebra person so this is the square of opposition and we'll see we're gonna relate them all together okay now the square of opposition today we're only looking at what we were referred to as the modern square of opposition which means that this is the more rudimentary the simpler one or modern in the sense that it's the boolean square of opposition rather than the aristotelian going back again to the X essential important thing so Venn diagrams here I can show you if you have the book on page two was at 209 right okay so you can see here they're just talking about Venn diagrams okay so here are the four possible ways of doing the Venn diagrams so the modern square of opposition okay here's what the modern square of opposition looks like we're gonna take each of our four propositions the a the e the I and the oh we're gonna arrange them into a square then we're gonna draw lines in between each of those those statements which represents any of the possible in fact all the possible relationships between those two statements given that only two of them can interact at the same time so we have the a the e the I and the oh okay now what's there's four it's possible for these two to relate to these two these two to register these the please okay oh and finally the bottom line so these are the four possible relationships now what is it that can be known now what how do these two relate to tire we're gonna mainly focus today on the diagonals in fact that's all we're gonna focus on right what are the diagonals what's the relationship between the diagonals we're gonna see that the answer is that the diagonals are always contradictory they're always contradictory now what does contradictory mean what is a contradiction I mean that's an important philosophical question here especially in logic what is a contradiction a contradiction is when you say something both is and is not at the same time in the same way and in the same respect right so say something both isn't is not it's to say something that's it's for instance to say that I'm in the room and I'm not in the room that's a contradiction right only one of those can be true that they can't both be true right a contradiction is any error in reasoning can always be traced back to a contradiction sort of a very very critical concept going all the way back to Aristotle and his discussion of the law of non-contradiction or law of contradiction okay so these are contradicting now what about these relationships here these horizontal and vertical relationships well since we're talking at the modern square of opposition the answer is we don't know okay that means that these are logically undetermined they're unknown we don't know what the relationship is and part of it has to do with the existential fallacy but it has to do with fools interpretation but we're not gonna I don't want to go too deep because I want you to understand it we're gonna ignore these relationships right now the only relations we care about in 4.3 here are these diagonal relation okay so the modern square of opposition looks like this again we go back over here okay so that means that these are contradictory and this should make sense actually when you think about it because think about what is in a statement right and a statement says that all SRP let's say all SRP now an a statement says that on zippy or let's give an example all dogs are animals we can say that Vaz's our dogs and peas or animals and let's compare them right because up here we said that right that the a is contradictory to the Oh so let's what's an O statement look like a no statement says that some s are not P right which says that some dogs are not animals now just think about it for a moment could it be the case that all dogs are animals therefore some dogs are not animals no that doesn't make any sense actually that's a contradiction in fact look here at the what the Venn diagrams look like the a Venn diagram looks like this where we've got this circle filled in and what is the O diagram look like it looks like this you can see here these things can't be true at the same time because this says there's empty space and this says something exists there how can both of those be the simultaneous the answer is they can't right it's a contradiction right so that's what how the a is a contradiction to the you know simultaneously think about the e in the I how did the e in the I statements look Annie statement says what no SRP what does the diagram look like it looks like this right and what does the I state that look like the I seen it says that some SRP if we did the diagram for it that's ope SNP we have an X in the middle you can see here these can't be true simultaneously right because the this is supposed to be empty space and this percent says something is not yes they can't be true simultaneously the E and the AI are contradictory terms okay so now what we can do is let's combine the Venn diagrams we've discussed with the square of opposition and what we're gonna get it's actually a very powerful tool that's gonna really help us make sense of a lot of the stuff so let's do the a the e the eye and the o right now what do we say here let me let me just draw what they look like here circles and when you do your homework I would suggest you draw something out that looks just like this so you have the a and then the e state it looks like this and then you have the I statement which means which if we draw it looks like this sorry and then if we drew the O's tame it we have this okay now let's do this how are the I&R related these are contradictory and how are these two related these are also contradictory okay so now but what about these this relationship unknown unknown unknown okay so we can use this to now take a look at what are known as immediate inferences and then we can test whether or not those immunity inferences actually are logically valid now let's quickly sort of maybe do a quick primer we'll come back to this thing I just draw here there remember here that what is validity validity is when if the premise is true the conclusion must necessarily follow or in other words that there's truth functional proposition where the conclusion will always be true if the premise is true right now there's different types of validity there's what we're going to call conditional validity and there's going to be what we call unconditional validity now unconditional validity we're gonna see is are those propositions we're both Poole and Aristotle agree and conditional validity or propositions where Aristotle would say they're valid and then Providence's are invalid they propositions can be invalid but then they can also be interestingly enough unknown law or what we would your book refers to here is logically undetermined so those are the possibilities when we test an immediate inference yeah so what is in a mediate inference how do we test immediate difference now immediate inference is when you take when you make one when you take one categorical proposition and then conclude immediately something from that categorical proposition it's not a full-fledged Arda because there's only one premise and one conclusion so they're just sort of immediate inferences but you'll see me give you let me let's go here to take a look at the book okay scroll down here testing immediate inferences let's go back here so here's an example of our media inference some T are not M therefore it's false that all t RM right so you can snip this so we can use this and sort of diagram this right now all right so this says that some T or not M right so the first thing to do what are the steps the first thing to do is number one assume the premise is true because we're looking to see whether or not they're logically valid not sound all right so we're gonna assume the first premise is true and then we're going to I number two then you need to identify each proposition right so some T or not what kind of proposition is that that is in a proposition and then here therefore it's false that all T are M this is an a proposition so what this is saying is that I'm gonna assume this is true and this conclusion is saying that a is false so the question is does that inference work is it true now go back to our square of opposition here scroll up here right we had an oath statement we're assuming the Oh statements true which means that if it's cut with the contradiction of the Oh would be an a statement the eighteen would have to be false right it would have to be false which means that this is a valid inference this is a valid inference now is it conditionally valid or as a conditionally valid or is it universally valid the answer is it is going to be conditionally God it's conditionally value because we're concluding with the universal right we've got four here we've gone from a particular to a universal remember what did fool say he said that Universal claims don't to know it existence right which means that we can't know what else is true this true would be true for Aristotle but not for bullets therefore we have what we would call conditional validity okay so let's go back here are things so how do we test a medium for instance again one we assume that the product that first purpose is true there were two we identify what the statements are and then we use the square of opposition or we use a Venn diagram to ascertain whether or not it's true or not okay so let me go back here to your book because you're gonna see that your what about if you want to just use Venn diagrams I think it's helpful to use either one of them the first thing to diagram is you have to always diagram what is the case what happens if you get a statement like this it is false that all AR be how the heck am I gonna do it because if it's false then what should it look like well here's where we can go back to the square of opposition right let's take our example again it is false they're all a or B right so that means that all a or B this is we're supposed to assume this is I'm sorry we're supposed to assume that this is false so I don't we diagram how do I diagram something it's false here's the thing you should never diagram something that's false so what let's take let's to use the square of opposition right I know that if let's go back up to here I have an a proposition if it's false then the auto proposition has to be true right so that means I should diagram this in order to diagram a false proposition then I'm gonna diagram this right which basically says sub s or not p but using because it's contradictory if this is false and this is true right so then let's go back here so you go to a book because what was the example a game it is false that all are B so that's how we diagram what's false so that means if you have a proposition that starts with something like it is false at that you should immediately think oh I have to diagram the opposite the contradictory version so you can use the square of opposition to deter than that now you can sort of go through the problems right so let's take this example here it is false that all a that looks like this actually okay so here we go it is false that all AR see how should we do this first this is obviously an a proposition but since that's an a proposition that's false right then that means I use the square of opposition right AEI oh all right and I say oh if this is false then this must be if the a is false then the oh must be true so I'm diagram the a proposition which lo and behold looks like this right it looks just like this one now what's that therefore no MRC so then the next thing to do to test any inferences diagram what the conclusion looks like right what is the conclusion look like well no ever see looks like this so here's the question can you read this can you gather this from this and the answer is no you can't why it's let's go right so you can see art says this inference is invalid you're the conclusion the conclusion diagram is sirs that the overlapping area is empty since infant this information is not contained in the premise diagram the inference is andhad the question is can you read the conclusion from the premise I can't in this case so it's invalid right which means you can't go why is it if not ultimately you can't go from a particular to a universal okay take a look at this one let's try this one okay all cellphones are wireless devices I'm gonna assume it's true and that's an a statement so here it's just draw it out for me notice you're supposed to put the C in the W first C for cellphones and W for wireless devices and so the question is can I read this from this all CH over there for some c RW well you can see here that the C's have to be in here that's the only place they ever could be right and look we have an X here which means I can read this diagram from this one right so this is a valid form right let's see what is the book say one of the things I find that's helpful is that when you do this is to is to diagram yourself and then see if it makes sense so you can see here this would these as the information of the conclusion diagram is not contained in the premise diagram so the inferences in daaad however if the premise were interpreted as having existential import then the C circled in the prose diagram would not be in an empty specifically there will be members in the overlap area this would make the inference invalid okay so maybe I should sit here and explain this one of the things we need to do here is Aristotle assumes that universals have truth how actually refer to things so for Aristotle there's actually X's up here right and the way we represent actually is by putting the X with a circle around it which means that we're taking Aristotle's assumption here the universals do make truth claims existential claims whereas down here the X so the X always means that something exists you can see here if we put these here then we can read the diagrams accordingly let's do another example here just so you can see what I have what we have in mind and I know this is a lot to take in this chapter is actually fairly lengthy right so let's do a couple of exercises here where it says use the moderate square of opposition yeah use Venn diagrams but yeah we'll just use a Venn diagram anyway it doesn't matter let's take three problems here let's take a look at these are these problems okay the person here is no sculptures by row Tanner boring creech creations so this is an e statement and this is therefore all scholars by rote Anna or pouring creatures that's an a statement moving from the e to an a is logically undetermined right because that's it remember we have the a the e the I and though we can only make claims here using the moderate or opposition by using the diagonals these horizontal that ye the a to the e is a horizontal relation which means that it's unknown so from our perspective it's invalid it's logically undetermined all right so that would be answer number one right now second thing let's take this segment it is false that some lunar craters are volcanic formations therefore no lunar craters are volcanic formations here it's this isn't it some s I'm sorry some LRV that's what that says so we can maybe helpful just to diagram it and this is supposed to be false so some LRV and then the conclusion therefore is that it is - as soon as true is that noel RV ok so the question is so if this is false that this is a pie statement and this is a key statement alright so if this is false the e statement has to be true using the square of opposition this is actually a valid inference okay never wait a second you're thinking does that work because what if we do the Venn diagrams let's do it down here how would I diagram the conclusion some LRV do it's false well it's false that's a diagram with actually it's true right and what is true is that if some LRV is false then the East damn it has to be true which is this right and we'll put an X in the middle here which represents the Aristotelian assumption of validity but not the boolean right and what's the conclusion the conclusion is that Noel RV you can see I can read the conclusion from the premise so this means that this is valid and you can see this is actually unconditionally valid because it doesn't I don't I can read the conclusion from the premises without this thing this thing doesn't matter this existential marker right so let's look at problem number three all trial lawyers let me write it over here and you'll see that the best way to do these problems in your home area is to write them out like this all TRP all trial lawyers are people with stressful jobs therefore the three dots means therefore therefore some T R P so let's take a look at this go back over here to our diagram all at all TR p right that's an a statement right here let me check well this is an a statement and some T or P is an i statement so we've gone from an a to an i right we haven't covered that yet so this is an invalid argument why it's invalid because the relationship is logically undetermined it's an unknown value right so you can't go right you can't make that downward relationship we can only use the cross take a look at it like this too if we diagram the first premise it would say that all the teaser pees and then we also diagram the conclusion some T or P is an egg's okay now wait a second let's put our existential marker in here where this represents Aristotle's assumption so no take a look here can I read the conclusion from the premise you can see I can read it actually I can read it if I use Aristotle's assumption because look the X is here which means I can read this from here so that means that actually this is conditionally valid if we by using the Venn diagrams we realized that it has conditionally involved now why is that that's because we're going to see in our next section that all of these lines Aristotle worked out the truth conditions for these lines but in all assumes on Aristotle's view that we assume that universals make existential claims so that's why the existential fallacy here that's why I think your author earlier is combining Venn diagrams existential fallacies and all this stuff all in one okay we're gonna see that what is the last sort of element in this chapter that I'll close off here and what you're probably have to do is read maybe watch this video again or read the book again because the last thing we'll see is what is the existential fallacy well the existential fallacy occurs when you make a claim when you conclude with the claim of existence where you didn't have one to start right so that means the Aristotle has a different view on what counts as an existential fallacy and bool has a different view Aristotle's view is that if you start with the universal and move to a particular that you'd haven't broken in you haven't broken the rule because universals include existence so all rb2 therefore some ERP this is conditionally valid for Aristotle it's valid for him or it's conditionally Valley because it's conditioned upon whether or not we assume with Erised all the universals make claims about existence whereas by contrast bull would say that this is invalid so there's sort of a debate here about whether or not this is a valid claim or an invalid claim typically I think we should side with pool but you're gonna see here that the book is gonna tell you in the exercises whether or not to assume Aristotle or Bulls perspective and you can use Venn diagrams to evaluate the median for instance of exercise and in identify any that commits the existential fallacy okay we're gonna sort of ended there there's a lot of stuff to cover and I honestly am not sure if I did a very good job with this video but let me know I may post a new one you'll see that go ahead and take a look and try to use the combine these different the square of opposition the existential fallacy in this and Venn diagrams do you put it all together you're gonna build up a pretty powerful tool set to evaluate these arguments next time in our next chapter here we're going to take a look at four point four which is conversion of version and contra position these are operations that can be formed to change the to change a proposition but keep its truth functionality so we'll take a look at this in our next section four point four okay thank you very much for watching and we'll see you guys there bye

Channel: Mark Thorsby

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Length: 40min 19sec (2419 seconds)

Published: Tue Jul 31 2012