Valid and Invalid Arguments in Logic using Euler diagram

okay this is our last video for the topic valid and invalid arguments and for this last video we are going to consider arguments which are for which the premises are not in the form that uses connectives like the implies Q P n Q or P or Q instead we have here arguments for which the premises and the conclusion are expressed using quantifiers 20 fires are those phrases [Music] normally appears now in the statement like all every non no those are vertical Universal quantifiers then there's a third type of quantifier called as existential those are quantifies that uses some there exist okay and so on and so forth so you'll notice here that the argument we have in this example no the arguments or the premises are in the form that uses quantifiers pay for the statement now before we determine whether this argument is valid or invalid we will use the Euler diagram as our guide for for that matter okay this is the Euler diagram this is actually similar to the Venn diagram okay where you have the square or rectangle and sets are placed not inside that the rectangle okay and this are statements that are expressed using quantifiers so what do we mean by the statement all xry so when you when you have all xry according to the diagram it means that X is a subset of Y okay where X is inside Y so all X belongs to Y right okay that's what we mean by all X are Y or you can also express that that as every X is y okay so same idea another statement here is the statement that says no X are Y it means that you have two sets which are disjoint there's no intersection so no X are Y in other words a particular element of X it's not found in Y the second the third case is the statement some X are Y this time it uses existential quantifier by the way that all the every the know the Nano are called us universal quantifiers okay for some X are Y it means that the set X and the set Y have a common read John okay so this what we mean by some X are Y if that dot here belongs to X and belongs also to Y that means that they are not disjoint so there's a common point or common element within depth to and when you say some X are not Y it means that x and y have a common factor but there is one particular element that belongs to X but does not belong to why in particular this dot here which is although it belongs to X but it does not belong to Y so that's what I mean by the statement some X are not Y okay and using the our guide let's consider our first example okay we will refer to the diagram from time to time okay okay so the idea here is you have to construct the diagram Euler diagram for this argument and based on the premises and then from the diagram that you have made you have to check whether your conclusion is consistent with that diagram okay so let's start with the first premise so draw rectangle here and then you say all mathematicians are logician so when you say all mathematicians are logician it means that you have a set of mathematicians okay and then you have a set of logician where the set of mathematician is inside the set of logicians right okay from our diagram here all xry so all mathematicians so this is the set mathematicians well this one the bigger set is the set of logicians okay that's for your first preemies now your second premise is it an is a mathematician okay that means that that's a single point actually that's a single person so we represent that by at that okay that's you that's it at okay it an is a mathematician that must be inside the set of mathematician oh okay okay so this is the diagram representing the two the two premises now look at now your conclusion is your conclusion consistent with this diagram is it true that Etan is a logician meaning is it true that Etan which is represented by that that there is inside the set of logician yes that particular point that belongs to the set of mathematicians also belongs to the set of logician so since your conclusion is consistent with the diagram then you say that the argument is valid okay so that's how to determine whether the argument is valid or not valid whenever your arguments are expressed using quantifiers let's move to the second example okay sample number 8 determine whether the argument is valid or invalid using the Euler diagram okay let's start with the first one so first three Me's let's make a diagram and you have no prime numbers are negative but then by this no prime numbers are negative going back to our diagram when you say no X are Y it means that there's our two sets with no common point okay so that means that we draw here circle and this represents the set of prime numbers and we draw another set here which represents the set of negative numbers so no prime numbers are negative so that's your first premise what about your second premise your second premise is the number n is not negative it means that your n is not inside the set of negative numbers but that statement actually implies that your n is either here right okay that's one possibility or your n is okay we have a case here where and can be found not only inside the prime number but also outside the prime number so if this is your prime set of prime number and this is your set of negative numbers okay aside from n inside the set of prime n can also be found outside prime and negative right because your second premise is the number n is not negative so when you say it is not negative it means that it's not found inside the negative set of negative numbers which brings you to two possibilities that is either your n is inside the set of prime number or outside the set of prime numbers okay so let's now look at our conclusion is the conclusion consistent with the two diagrams okay it says the number N is a prime number oh that's your conclusion is it consistent with the first diagram the number n is a prime number yes it's consistent but the question is is it consistent to the second diagram no it's not consistent anymore so therefore because it's not consistent to the second diagram although it is consistent to the first diagram but we say that the argument is invalid because your conclusion is not consistent with the two diagrams take note you don't always get two diagrams now it depends on the implication of your premises actually using the first premise you have this diagram but the second premise will lead you to creating another diagram okay and therefore you have to check whether your conclusion is consistent without two diagrams since it is not consistent with one of the diagrams then we say that the argument is invalid last some a student so again we draw diagram here some a student sit in the front row okay when you say some this is represented by a diagram where the two sets intersect right okay so when you say some students sit in the front row we have two sets here the first set is the set of h2 dates and the second set is the set of students sitting at the front row so this is a students well this one is students sitting in the front okay front row and okay that's for your first preemies what about the second premise the second premises all those who sit in the front row okay referring to this set add to this particular circle all those who sit in the front row are attractive so you have another set here another circle that is actually bigger than the set of students sitting in the front row right because all those who sit in the front row are attractive so what doing by that it means that we will draw another circle where the circle four students sitting in the front row is inside okay so the bigger circle here is for attracting students okay or you can also make your circle Biggers to contain the a students that this is still consistent with the the second previous but let's look at now the conclusion is the conclusion consistent with our diagram the the conclusion says some a students are attractive okay these are the a students and the bigger set the bigger circle here is attractive students so is it true that some a students are attracted obviously yes because the set for a students intersect with attractive students so since our conclusion is consistent with the diagram that we say that this argument is valid okay so thank you and God bless

Channel: levskt

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Length: 14min 25sec (865 seconds)

Published: Tue Apr 14 2020