hi this is an introduction to logic I'm Mark Thorsby and this course overviews the basics of categorical propositional and predicate logic today we're taking a look at some basic concepts in particular we're going to be taking a look at how to prove invalid arguments in fact you'll find out that a lot of what we do in logic is about proving invalid arguments and what we're going to be looking at today is really the barebones essentials for how to prove any argument involed though we though as the course progresses we're going to see there's many there's much more sophisticated precise and I think actually easier ways of proving arguments invalid but there's just the basic concept now let's start off reminding ourselves what's an invalid argument well an invalid argument is an argument that has true premises but yet can have it's possible for that argument to have a false conclusion okay true premises and a false conclusion so that basically means remember premises are the evidence and the conclusion is the takeaway so that means you have a true evidence but the takeaway the claim itself is false or can be false now how can you figure out whether or not an argument is invalid well you have to take a look at the argument's form right of course we're talking about formal logic here um that means that what we're going to do is we're going to look at arguments in their ordinary ordinary language State and then put them into their forms and then assess their form so let me give you some examples of that we've looked at this previously in other videos but as a reminder here because not everyone's seen all those previously previous videos let's take a look here here are two separate arguments let's analyze these real quickly all geese are migratory waterfall all migratory waterfall are birds that oops there's should be after are that fly south for the winter therefore all Gees are birds that fly south for the winter okay now this is an ordinary argument and if I want to put this into its form I just basically have to look for the key terms so you're going to look for key terms but notice that the key terms were here we mean The Logical terms not necessarily the uh the words because these are all each individual words so we're not looking at grammatical terms we're looking at logical terms right geese is one and migratory water file is another term you you can notice the the terms because they come up in the other premises so we have migratory waterfall again and then we have Birds fly south for the winner and then we have this this is an indicator conclusionary a conclusion indicator term and so with the conclusion is all geese so there's another one and then all birds that fly south for the winter so what we're going to do in order to put this into symbolic form is we're going to assign letters right uh we're going to assign letters to each of these so we basically we're going to give this a g usually it's typical to take the first letter of the word and we're going to give this an m and then we see M comes up here and then this since B we'll use this B and then we have G and B okay so then we're going to rewrite it into its formal structure right now this is a categorical um term here so we're going to keep that so it's going to look like this all G are M right all m Are B and then over here we have therefore so the therefore we usually symbolize as a line so it's going to be all grb okay so that's real quick how we symbolize an argument so maybe I can write that down here how to symbolize an argument maybe that'll help out for you how to symbolize an argument right first right determine the conclusion and most of these it's going to be easy the conclusion comes last but not always determine the conclusion and therefore of course you determine the premises so I'll put determine the conclusion in the premises right isolate The Logical terms isolate The Logical terms and then three rewrite rewrite um using um letter symbols letter let's say letter symbols okay it's pretty simple okay you determine the conclusion you isolate The Logical terms and then you basically rewrite it and then of course eventually number four you're going to analyze the form so we can throw that in there but um that's what you're that's the goal okay so let's do it again here okay maybe I'll change the color of my pen here just to help you see it differently so this one all daisies are flowers all flowers plants therefore all daisies are plants now here's a conclusionary indicator term so we know this is the conclusion right so this is premise one premise to conclusion okay we have all daisies or flowers so obviously D daisies and flowers are the terms so all D or F and then you have all flowers or plants so all f r p and then you have all D or P right it's pretty simple here so let's rewrite it right it's going to look like this all DF and of course you could you've already see how this going to go it's pretty simple all D ORF all f for p and then you have all D or P therefore all D or P okay and then we draw a line to symbolize the conclusion now what do you notice between these two arguments notice they have the exact same form now we use different letters here but the letters are irrelevant right because essentially the letters they're just symbols to symbolize anything that you could put there any possible term so think of it like this this is the actual argument because it has content right the content is things about geese and daisies and things like that and this is the form of the argument this is if you will the structure of the argument and remember at the very beginning of the course I said that John Lock argues that logic is something like the anatomy of human thought so you can see here that what the form of the argument is it's something like the skeletal structure to use that as an example it's like the skeleton upon which all of these ideas are organized okay and so we're trying to look for the skeleton to determine if the skeleton is a good a strong valid skeleton okay by looking at the form and if we notice all G or B right so this argument Works sort of like this you have these this middle term here it sort of links these because if all of the if all of the G's are M's and the M's are B's well obviously all the G's are B's you can make sort of a shortcut and in fact the conclusion is essentially the shortcut from G to B but notice it's the exact same form of the argument here so all D are F here we have the F and F and we can make a shortcut here from D to P okay so these is actually the same form this has the identical form and in fact this argument is valid which means that since this argument form is valid this argument is valid and this argument is valid they're both valid it doesn't matter if you're talking about daisies and flowers uh water foul Birds whatever if the form is valid the argument is valid too okay so in order to determine whether or not an argument is valid or invalid we always look to the form of the argument okay so what we're going to do here now is let me explain what's known as the counter example method okay and then we'll come to we'll talk about this oops let me scroll down here we'll give an example here in just a minute okay because we're today we're really talking about how to prove invalidity okay so how do you prove invalidity we're going to be looking at the counter example method the counter example method the counter example method here is really quite simple okay and I'm just going to write down the steps of it for you here are the steps of the method and so follow these steps and you'll be able to do the counter example method to determine to prove invalidity okay but I should mention this the counter examle method can only prove invalidity it cannot prove validity okay so this argument is not going to work for pro inv validity only for invalidity okay the first step is to determine the conclusion all put to determine the conclusion and the premises C and P there for conclusion premises that's the first thing you do it's always the first thing you do the second thing you're do is you're going to isolate the form or you're going to symbolize the form using letters just like we did earlier uh oops isolate the form of the argument number three after you've isolated the form you're going to create a substitution instance and I'm going to explain what a substitution is here in just a moment it's basically well maybe I can sort of explain it real quick okay notice that when we looked at these two arguments right we had the form of the argument but we had different contents right a substitution instance is when you try to create different content in which the premises are false I'm sorry the conclusion is false and the premises are true if you can create an instance of that using the same form of the argument you'll prove an argument in B okay and you'll get a sense of that here when I do it in just a moment um some helpful hints when you create a substitution inance start with the conclusion start with the conclusion and then work to the premises start with the conclusion work to the premises um and then you're going to look you remember the goal of substitution is to find a false conclusion and then you're going to um oops I made a little error let me just erase that then you're going to look for True premises because remember remember an invalid argument is and I know I sound sort of like a broken record invalid argument is an argument that has where it's possible for the form of the argument to contain true premises and a false conclusion look for True premises all right once you've done that number four then you will if you can create a substit instance in which you have true premises and a false conclusion then you've proven the form invalid and if the form's invalid number five then the original argument is invalid the original argument is proven invalid okay so that's the real quick steps here and I'm going to demonstrate this here in just a moment okay so the first thing you do is determine the conclusion the premises isolate the form of the argument create a substitution instance and if you can do the create a substition substitution instance in which you have false conclusion and true premises then you'll have proven the form in valid and by default you'll prove the original argument inv Val okay so let's take a look at this argument here that we have okay maybe I can bring it up here okay all romantic novels are literary pieces okay that's true all works of fiction are literary pieces that's true therefore all romantic novels are works of fiction that's true too this sounds like a good argument actually doesn't it right um because look look the premises are true and the conclusions true well let's analyze it is it really a good argument okay let's use the sub the counter example method and look for a substitution instance what do we do first step again look for the conclusion I have this conclusion indic indicator term right here therefore so I know that this is the conclusion okay so let's symbol let's start off by symbolizing that first well actually hold on um well you're going to notice this is a term romantic novel so we're going to make that R works of fiction we're going to make that W so that's going to be all r r w okay that's the conclusion now let's do the premises okay now remember since we already used R here for romantic novels then that means we need to stick with R in literary pieces we'll use an L for that works of fiction we already did that so we have a w there and this should be l so it looks like this all r r w I'm sorry r l all w r l Okay so we've now moved here we did the first thing is we found conclusion we've now isolated this is the form of the argument okay this is the form of the argument now let's see can we create a substitution instance remember substitution instance is can I fig can I use different terms and replace the r and the L here with other things right these are just sort of variables that stand in for things think like an algebra it's like an X it can be anything in the universe so think about it can I come up with a possibility in which this form will have true premises and a false conclusion this is my goal okay I want true premises and a false conclusion can I do it well let's think about it here I said you now we have to create a substitution instance where I'm substituting these variables with new ideas okay so let's write it to the side let's change our color here let's say we rewrote it where R I always say well here's what I say was helpful is always start with the conclusion since I want an obviously false conclusion let's start with something that's crazy what if I said that R equals fish right and I said that um W is people okay so this would say something like this all fish are people all fish are people that's obviously false right because not all Fisher people in fact no Fisher people right so that's not true that's false all Fisher people but what happens if I rewrite up here what about L what if I say that so here I have an obviously false conclusion let's right here all Fisher people and I'm following the form that's a false conclusion can I come up with true premises well R what if I say the L here are things that swim things that swim and I or things that can swim let's say okay so I said that all fish right are things that can swim I'll just write swim here but I really mean things that can swim just I don't want to spend the time writing it all fish are things that can swim and then I say all people are things that can swim as well now that's true all fish are things that can swim and all people are things that can swim well in general human beings can swim you might quibble here and say that well some people can't swim because they maybe they don't have arms and legs U and maybe but actually I think people well I don't know if you don't have arms you probably can't swim um probably need but either arms or legs but in general we'd say that these premises are true and this this conclusion is false right and this would be an example of true premises false conclusion thus the form is invelop if you don't like that argument I can make another one up for you right you can make a lot of them right what if we said here that r instead of is dogs and we say l are animals and we say that W are cats so it would read something like this right all dogs are animals all cats are animals therefore all dogs are cats that's obviously a false conclusion with true premises so this may actually be more simpler um substitution instance but regardless either one of these proves that this argument form can allow for True premises and a false conclusion thus it is invalid it's an invalid argumentative form if it's an invalid argumentative form then that means this original argument is also invalid okay so that's the counter example method in works so let's do another example here okay and you're going to see here that's basically all we're going to do in this video is basically finding isolate the form of an argument right find a substitution instance prove the inval that's really it's very quite simple okay and in many ways it's just sort of building on things so here's another argument let's take this since some employees are not social climbers and all vice president are employees we may conclude that some vice presidents are not social climbers now this is a good argument in the sense that a good example because this sounds something like someone may actually argue this right people don't usually talk like this right all fish or swimmers and stuff like that people don't talk like that but people do talk like this so let's start off with our counter example first thing we got to do remind ourselves find the conclusion okay in this case we have an indicator term we may conclude that right so this is the conclusion so what follows from this term is this is the conclusion so let's rewrite it right U so sum V for vice president this is our first term are not social climers our second one here is going to be S so let's go look here these are obviously the premises you can see I have a conjunctive term and which means this is my second premise and this is my first premise so this is going to be uh since some e employees are not um social climbers you can see social climers there one here right social climers and we've got employees and all vice presidents and we already had vice presidents at term so that's going to be v r e employees therefore some Vice pres are not social clim so let's rewrite it into its formal structure it's skeletal structure if you will it's going to look like this some e are not s right Su I'm sorry that's not some it should be all let's erase that right all v r e therefore some V are not s now we're going to find out because this is actually uh let me clean that up a little bit this is actually a categorical argument and so we're going to see that there's actually some rules that govern how you rewrite categorical arguments and one of the things you typically do is you always put um a universal statement first um but don't worry about that right now it's not a big deal right at this point for the substitution instance it doesn't really matter okay so but here's the argument form some e or not s all V or E some V or not s therefore some V or not s is this argument form valid okay well let's see can we come up with substitution instance again substitution is false conclusion true premises it's usually easier to start with the conclusion and work your way upward but if that's not helpful start with the premises it doesn't really matter right you just want to stick to to to this sort of rule here true premises and a false conclusion so let's sort of rewrite it like this we have these are our variables e s and V so let's think a false conclusion some V or not s so what if it was something like this uh what if it was some cats are not animals okay some cats are Nots what if V was cats and S was animals that's obviously false there's no cats that are not animals that doesn't make any sense cats by definition are animals right so this will be a false conclusion so would it work up here with the premises so let's start with this this is an obviously false conclusion can we find out true premises though so this would be some blank are not animals okay it could be maybe um and this would be all cats are e so I need something I need e here all cats are mammals some mammals are not animals no that doesn't work what if I said that all let's see all cats are felines some felines are not animals that doesn't work let's see here you can see here the with the subs part of the difficulty is sort of figuring out true premises and it may take you some while to wrap your head around it right so what if you said something like this um let's see animal what if we changed it around here what if we said let's try this what if I said that some dogs are not mammals okay and I'm actually using Hurley how Hurley does it here so what if we turn this into dogs and we turn this to mammals and what if we may eat animals so it looks like this some animals are not mammals that's true right because lizards are not mammals okay well then we said that all V all dogs are mammals that's true so therefore some dogs are not mammals that's false so this would be an example of a substitution instance you can see in by the way when you're doing these in your homework there's nothing wrong with starting and starting over again and that's why I sort of did that there to show you right um I could have sat here but I didn't want to sit here in the video trying to think through it um rather I just wanted to sort of give you an example quick so don't waste your time okay so you can see here that in this case this would be again some animals are not mammals all dogs are animals therefore some dogs are not mammals that's a false conclusion with true premises which means that this argument is invalid right and that's the counter example method okay so what I want you to do now is assuming if you're just watching the videos that's great um but if you're taking the course and you have the textbook you're going to see here that the next thing you're going to do here and please read through these and give an example he gives lots of examples here I'm scrolling down here is go through these exercises right use the counter example method to prove the following categorical syllogisms invalid okay so in in doing so follow the suggestions given in the text okay so you can see here's another example we'll just start with this first one all GX are structures that contain black holes in the center so all galaxies are quazar since all quazar are structures that contain black holes in the center first thing you got to do there is figure out the conclusion and I'm not going to tell you what the conclusion is though I know where it is but I'll tell you this it's not a normal in its normal order so use those indicator terms put it into its form and then figure out if you can come up with an Institute in a substitution instance and thereby prove it in dog okay um and so that's exact ly what you have to do in this um in these exercises for 1.5 okay so that is the counter example method and that's where we're going to conclude our session today on how you can prove invalid arguments so okay in our next you're going to see that from here we're going to be done with chapter one and we're moving right into chapter four next where we're looking at categorical syllogisms and categorical propositions um for arguments and so we're looking at we're going to get it's going to get a little bit more uh specific detailed and I think a little funner too okay so thanks a lot for watching um and good luck finding those counter example methods okay talk to you later see you online bye