 Welcome back, in the last few lectures, we discussed about how to recognize an argument, and then we also identified what kind of argument it is, based on how the conclusion follows from the premises. If it necessarily follows from the premises, then it is called as a deductive argument, the conclusion probably follows from the premises, then it is called as an inductive argument. So we also discussed about an important concept which is called as validity, and we said that a deductive argument is valid if and only if it is impossible for the conclusion to be false given the premises are true. You will not have any example where in which you have true premises and a false conclusion. If you can come across with such kind of instance, then it is the argument is automatically called as an invalid argument. All invalid arguments are unsound arguments. So in the last class, we discussed in greater detail about another important property which is called as soundness. Soundness is a kind of valid kind of argument. In addition to this that this argument is valid, it is also having true premises in it. A sound argument is a valid argument which has true premises. So then we discussed about some examples in which we showed that you can have true premises, you can have one of the premises false and the conclusion is still true or you can even have both the premises false and even the conclusion can also be false but it can be a valid argument or you can have conclusion false and one of both the premises are false then also it can be a valid argument except I know. One such argument we have seen is all squares are circles, all circles are parallelograms, all squares are parallelograms and all. So the premises are false but the conclusion is true but it is considered to be a valid argument but in day to day discourse we do not use this kind of argument because nobody will be in a position to believe that all squares are circles or something like that. It goes beyond our intuition and all. So that argument is valid but it is unsound argument. So today what we are going to do is this that we are going to talk about the strength of the inductive arguments, we can only talk about the strength of the inductive argument. So this often mistake will be committed by the logic students that we say that inductive argument is valid and inductive argument cannot be valid and all because the conclusion does not necessarily follow from the premises. In all inductive arguments however strong it is you can even come up with a single counter instance where you can show that the conclusion is false given the premises are true. So that is the reason why inductive arguments you cannot talk about talk of validity of inductive arguments you can only talk about the strength of the inductive argument. One example we have seen earlier 99% of the commercial flights that you took landed safely. So that means the next flight that you are going to take that is also going to land safely you know the next flight that you are going to take is what is going beyond what is stated in the premises. So the 99% of the flights that you have taken is the one side of the story and the other one is the next flight. So that information is not in the premises in all that means the information goes beyond what is stated in the premises in all that makes this inductive argument defeasible. So now today we will discuss about what do we mean by saying that a given inductive argument is strong given inductive argument is weak we can only talk about strength of the inductive arguments if it is a strong argument then we will talk about what kind of argument it is is it a cogent argument a cogent argument is a sound argument with true premises just like in the case of sound argument in the case of deductive arguments we said that a sound argument is a one in which it is a valid argument together with that we have true premises in all. So in the same way in the case of inductive argument a cogent argument is a strong argument with probably true premises in all if at least one of the premises is false then it is called as an uncogent argument the conclusion probably follows from the premises but at one of the premises may be probably be false in all that is called as an uncogent argument. So we will talk about these things with more examples then we will see a strong argument is a one in which it is probable that if the premises are true then the conclusion is also probably true. So a strong inductive argument is a one in which it is possible but definitely it is not improbable but it is improbable that the conclusion is false given the assumptions that the premises are probably true. So the idea here is that in a strong inductive argument it is very difficult to come across a situation in which the conclusion is probably false and the premises are probably true in all. So a weak inductive argument is a one in which it is not probable that if the premises are true the conclusion is also probably true in all. So I am just replacing the words necessity with probability in all so that is what you need to note here in the case of deductive arguments conclusion necessarily follows from the premises there is no single counter instance which shows that your premises are true and the conclusion is false but in the case of inductive arguments the necessity part is replaced with probability in all because inductive arguments there is no guarantee that if premises are true the conclusion is also necessarily true conclusion only probably follows from the premises so that is the characteristic of inductive argument. So another important thing which we need to note is that no valid arguments are strong and no strong arguments are valid in all even if it is 99.99 percent in all and it is treated as inductive argument for example 99 percent of the commercial fries that you took so far landed safely without any issue in all and from that you can infer that the next flight that you are going to take will also land safely in all. So even if it is 99.99 percent in all it might be very much possible that the next flight that you might take might end up with some kind of issue or problem in all it may not launch it may not land safely etc. So no valid arguments are strong you should not confuse our self with validity and you can say that the argument is very very strong in all so instead of saying that you know argument is very very strong and you can say it is valid in all the conclusion necessarily follows from the premises use the language of deductive logic that is the validity you know. So if you want to incorporate necessity into consideration that is what we want to achieve in the case of mathematical reasoning. So in mathematics everything has to follow certainly from the premises in all the premises are also considered to be some kind of mathematical statements there is a kind of necessity which is required there. So in that case we call those arguments as valid in you cannot say they are strong in all. So strong and weak argument arises in day-to-day discourse you know in which you know nothing is 100 percent true in all. So in those cases we use arguments based on observations or beliefs etc. and all or will involve this inductive arguments where we invoke the strength of the argument but validity strength and weakness comes in degrees in all it can come up with for example in the conclusion you can say conclusion can be true by 70 percent or 90 percent or may be 85 percent etc. So you can calculate the probability values and then you can measure the strength of the argument. So strength and weakness comes in degrees that is anything between 0 and 1 the probability values but the validity and invalidity does not come with degrees in all. Suppose if you accept that all men are mortal socrates is man socrates is mortal. So the first statement all men are mortal is accepted with 100 percent certainty in all. So there is no way in which there are no exceptions to that particular kind of thing so it is 100 percent true. Socrates is man is also 100 percent true then from these certain truths you will automatically say that the conclusion also necessarily follows from the premises. So what are considered to be strong and inductive weak inductive arguments it is not easy to find what constitutes a strong inductive argument what constitutes a weak inductive argument it involves lot of additional factors in all. So it depends upon you can it is based on case to case basis you can say that a given argument is strong or given argument is weak. So when the evidence is hard to come by may be then in that case you can say that even 20 percent is also considered to be the strength of the argument in all. So let us consider some examples with which we can understand the strength of the argument. Suppose if you infer in this way many crores observed so far have been black you observed meticulously for some years and you are interested in non-ethology etc birds etc then you came to know that with your observations and repeated observations etc under various circumstances etc you observed in IITK observed in some other place etc you came to this many crores observed so far have been black therefore based on this observation you will say that probably the next crow to be observed will also be black and all because many crores are already in black and all. So this argument is a strong argument in a sense that it is talking about almost all the crores in all if not everything in all but it is talking about the entire class of crores in all. So it is considered to be a strong argument in all but for instance if you observed only 2, 3 crores in all and then you infer this particular kind of thing in all probably the next crow that you are going to observe is black and all of course we all know that in a crores will always be black and all but with few observations under some only few circumstances etc will not give us this will not lead us to the strength of the argument in all. You need to have repeated observations good enough in number and then it is tested in wide varieties of circumstances then only you will say that the given argument is strong in the same way we will come across arguments from analogy in day to day discourse. So this argument can also become a weak argument in this sense when a lighted match is slowly dunked into water the flame is snuffed out obviously put flame into the water and it will be snuffed out and all but gasoline is a liquid so imagining that gasoline is also liquid and then it is just like water in a sense that at least it appears to be like water and all kerosene or petrol or anything it has some special features of water just like water it is in appear to be the same as water. So therefore when the lighted match is slowly dunked into gasoline the flame will be snuffed out and all. So here you are trying to bring in similarity between 2 events so that is one one is this that putting matchbox into the water and then putting matchbox into the gasoline exposed to the gasoline. In the first case it will be snuffed out and in the second case it might flame may not be snuffed out. So based on one particular kind of instance if you say that using the similarity notion if you say that lighted match is slowly dunked into gasoline will also be snuffed out then this argument may not be a strong argument. So rather it will it will not be snuffed out so it is a weak analogy kind of argument. So analogies are very important in sciences in particular we want to reuse lot of metaphors to understand various phenomena and all. So I will talk in about these things in greater detail little bit later but at this moment this argument is considered to be a weak argument and if you go into the details of this argument then this argument is also considered under the category of weak analogy. Suppose if somebody convinces you with this particular kind of argument then that particular arguer is said to have committed a kind of mistake in the argumentation. So that mistake is called as the fallacy of weak induction which we are going to study little bit later and that fallacy of weak induction arises because of weak analogy. There is no appropriate analogy between matchbox into water and matchbox is in contact with gasoline and all behaving in two different ways. So in the same way in day to day discourse we usually we have lots of beliefs in all beliefs in God's beliefs in astrology or sometimes someone cat passes through as in all will infer that every time when cat pass through while you are going to your office something has happened in the office then you will infer that cat is responsible for all my problems and all. There is somewhat a correlation and all correlation should not be confused with causation. So here is an example which again tells us that this is a weak argument. Astrological calculations indicate that Indian stock market will crash in the years let us say 2014 is unpredictable behavior of a stock market somehow you have calculated the positions of planetary positions etc and all when you are astrology also you know lot of calculations etc and all meticulous calculations. So you could come up with some kind of thing that is a bad planetary positions will lead to some kind of business impacts business and all in that particular kind of country. So then suppose if you infer that you should get your money out of the market before that year because stock market is going to crash there is no proper evidence for this particular kind of thing it is very difficult for us to believe just based on astrological predictions and you will be taking out your money. Suppose if it is based on some other kind of behavior in all suppose if you have seen how the equity market is functioning or some other means GDP growth all some of the important factors which are responsible for the stock market growth if you observe it then it makes sense for us to believe that you should get all the money out of the market whatever you have invested you should get it out that makes some particular kind of sense it is very difficult for us to believe the conclusion to be true based on probably true based on the premises you know just be based on astrological predictions you cannot we cannot come to this particular kind of conclusion so then the next question that comes to us is once the inductive argument is strong enough then there is one extra feature which will be adding to it that is what is called as cogency just in the case of sound argument a valid argument is not just enough and on we need to invoke an additional feature so the argument also has to be sound so just like that we have a strong argument can be cogent and cogent argument so what is a cogent argument a cogent argument is a strong argument in which all the premises it so happen that all the premises are probably true so probably true means it is based on some facts of experience your experience suggests that it is true or maybe it is a scientific fact or maybe it may be historical fact etc that makes this statement probably true let us consider a simple example to show that an argument is sound as in not strong argument as well as cogent argument so all of us know about big bang theory universe has come into existence somewhere sometime in all suppose if you argue like this if the big bang theory is correct then the universe is billions of years old billions of years old universe there was some kind of fireball or something like that it started cooling and then planet started forming you know that is what we know from this big bang there was a huge big bang you know and how this big bang has come into existence that is not what we are interested in we say that at least something used to be there based on scientific theories you know scientific studies suggest us that there is a big bang and then after that some kind of fireball it started cooling and then all the planets formed and if you argue that if the big bang theory is correct then the universe as universe was definitely not created in six days you know so that is what maybe the Bible might be claiming that God has created entire universe in six days and the seventh day it took rest that is what usually biblical belief in all of course you know we have every right to believe something to be true in all but we might believe so many other things in all that may be false also so it is our subjective kind of opinion or some kind of so does it based on that if the universe if you argue that the universe is billions of years of old then it was not created in six days probably not created in six days it makes sense for us to say that this argument is strong of course you can verify with some good evidential source in science scientific textbooks are some kind of theory theory of big bang big bang theory for example if you verify it then all these statements that we have mentioned seems to be also true so that makes this argument not only strong but also a cogent argument but if some of the argument some of the statements that you have mentioned in this argument in the big bang theory the one which we have discussed is probably false in all our historical facts are maybe the scientific theories claims that that is not the case then obviously this argument may probably follow from the premises in all but still it may be one of the premises is false in all probably false then this is called as an uncogent argument so a cogent argument is a strong argument with true probably true premises. So we might confuse cogency with soundness so a cogent argument can have false conclusion first premises do not absolutely guarantee the truth of the conclusion so it can still be called as a cogent argument even if it has a false conclusion false conclusion in the sense that it is not 100% false in all but probably maybe 99% true means 1% false only that means at least one counter instance is already there so in inductive arguments you cannot talk about this is necessity in all in that all inductive arguments comes up with degrees of truth of the conclusion will be accepting the conclusion with some degree of truth 99% of the commercial flights landed safely implies that probably the next flight that you are going to take also land safely maybe the next flight may come under the category of 99% so that is why it is probably true that does not mean that it is a strong argument and probably it is it is also a cogent argument because you verified with lots of facts and all your repeated observations tells you that that is the case are you gut feeling tells us that it cannot be false enough but even in that case so it might very well happen that a cogent argument can have a false conclusion false conclusion in the sense that 99% of the thing is true means at least 1% is false automatically but in case of a sound argument it cannot have a false conclusion because what is the sound argument it is a valid argument with true premises in all if the premises are true the conclusion cannot be false enough you cannot say with we cannot say that it is 1% false or 99% true etc. If the conclusion is accepted it is accepted with 100% certainty there is no way in which you can have any such kind of degrees of truth allowed in a sound argument a sound argument cannot have a false conclusion but a cogent argument can have a false conclusion in the sense of the one which we are talking in the case of deductive arguments a sound argument cannot have a false conclusion because it is a valid argument and it is all premises are true and all but if it is a valid argument it is automatically the case that it is impossible for the premises to be true and the conclusion is false so it is in this sense a sound argument cannot have a false conclusion but a cogent argument can have a kind of false conclusion. So what do we mean by uncoagency you know so till now we talked about coagency coagency is one in which you know it is a strong deductive argument with probably true premises an uncoagent argument is a one in which it is either weak we all weak arguments are automatically uncoagent arguments means conclusion may not probably follow from the premises automatically it is invalid sorry uncoagent or it can be it can still be a strong argument but at least with one probably false premise. So an uncoagent argument falls into at least one of these categories so the category one is like this that it may be a strong argument but at least it has one false premise in one of the premises is probably false and that leads to uncoagent argument if both the premises are true and all then the conclusion is also probably true then it is called as a cogent argument and a strong argument. Category two is that it may be a weak argument but all these premises are true probably true in that case it may be a weak or an uncoagent argument almost all weak arguments are automatically uncoagent arguments another category in which this uncoagent argument falls is this it is a weak and it has at least one false premise. So it is slightly different from the category one category one says that strong but one of the premises falls category three suggests that it is a weak but it has at least one false premise. So far we discussed about cogency and uncoagency of an inductive argument we said that we cannot talk about validity of an inductive argument if we talk about validity of an inductive argument it is a mistake. Validity is something which is which invoke some kind of necessity it is a connection between the premises and the conclusion the relationship between premises and the conclusion in the case of validity is a kind of necessity relation and all but in the case of inductive argument conclusion probably follows from the premises. So let us consider one simple method it is basically a common sensical method and all this is not a formal kind of method we will be entering into different kinds of formal methods little bit later but let us talk about one simple method with which you can show that a given deductive argument is invalid. So here is a method which is called as counter intuitive method so what is a counter intuitive method and what you are going to do is simply like this. So what you will be doing is first you will be identifying the form of an argument before that in a nutshell counter example method is the one in which you will show that a given deductive argument is invalid. So when can you show that deductive argument is invalid if you can cook up a single counter example that means cooking up with the counter example means you are coming up with true premises and a false conclusion. So there are certain things obvious things which we know them to be true there are obvious things which we know them to be false and all for example if you say all cats are dogs then automatically the statement is false it is not referring to the actual thing which exist in the world. For example if you say all dogs are animals of course dogs comes under the category of animals then that seems to be obviously it is a straight forward and suppose if you say that all cats are fish obviously that statement is wrong all these things you know at least person who is not having any knowledge of logic or anything can easily understand that the statement is false this statement is true. Suppose if you say all fish live in water and all that is okay it makes some sense to believe that this is true and all so all cat are fish if you say that is automatically false. So we will be using a set of things which we are which we are sure that they are true or sure that they are false etc and all. So you will take into consideration a set which consist of cats dogs mammals fish all these things and all then you will cook up a counter example once you extract the form of a given deductive argument. So here is a method which is very interesting in all you do not require any logical method to identify that this is an invalid argument it only shows that the given deductive argument is invalid. First you will identify the form of the argument let us say for example if you say if we train the grass is wet and it rained and all so that is why grass is wet. So the form here is a implies b and a then b follows a me a stands for it rain and then b stands for grass is wet. So that seems to be perfectly valid arguments it has valid form and all. So now what you will do in this method is you will find some English statements are terms that substituted for the capital letters in the conclusion of the argument form produce a well known false pseudonym first what you will do is you will extract the form of the argument and then once it for example if you have a form like a implies b and b then a for b follows a follows then what you will do is you will substitute some of the English terms into this thing. So for example here are the two things which you will commonly come across for example is you extracted the form and then you have written in this particular kind of sense. So this is one particular kind of thing you will be talking about and the second one is a implies b and not a and then if you infer not b and so given an English language argument for example if you said that it rained then the grass is wet so that is why it rained and all. So obviously you can say that this argument is invalid and all in the sense that you can have a counter example in which if it rained the grass is wet the grass is indeed wet and all but this may be false and all that means it rained may be false and all instead of it rained and all it might be the case that the sprinkler might be on or maybe some other means in which the grass was wet. So instead of talking about this thing what you will do is here is a set you have all these things cats dogs animals the different classification fish or mommas all these things which we are obvious we know that they are obviously you can say tables or anything which you can cook up and all so if you say all cats are dogs in the set statement is false enough if you think if you want to make this particular kind of thing true and all so then you have to cook up an example in which you have to say that all cats are animals if x is a cat then x is a animal suppose if you say x is a cat then x is an animal obviously all cats are animals it cannot be fish or it cannot be any other kind of species like any other thing. So this is what we have transformed into this thing so now B stands for x is an animal so we have substituted one of these terms it depends upon our creativity so what we have done here is that given an argument we extracted the form and all so once we extracted the form we forget forgot about what is mentioned in the content of the argument and all. So now instead of this thing obvious things we take into consideration cats dogs mommas etc and all or trees or any other thing which comes to your mind and all so x is an animal so now what is a here and x is a cat you can easily see that this argument is invalid in the sense that there are so many animals which are not cats and all maybe leopards are some other things which come under the category of cats and all but it might be a pig or it might be dog it may be any other thing and all. So what we have done here is that you have true premises but yet you can have a false conclusion all so x is a cat then x is an animal assume that this is true and x is also an animal is also true then if you infer that x is x has to be cat and all that means necessarily follows from these two things and all then there is some kind of problem here x need not have to be cat and all it can be donkey or it can be some other thing or dog or any other thing and all. So you have true premises but you could construct a false conclusion so that makes this particular argument invalid and all because you could come up with a count counter instance in which both are true and this is false and all in day to day circumstances you come up with this particular kind of example. So like this counter example method only establishes that a given deductive argument is invalid but when you have valid arguments then this kind of technique will not help us. So now what we have done here is simply this that we substituted English statements are terms with the relevant capital letters that is a b c is etc and all and once you substitute it that means that becomes the form of the argument you will forget about what is a b c etc and all then there are certain things which you obviously know to be true obviously known to be false. So now you have to find English language statements are terms that if substituted uniformly that means if a is there you need to substitute it for x is a cat only if b is the one which is the case then it is we are representing it as animal. So animal is the thing which has to be uniformly substituted and all suppose if you substitute b for a and all that then something wrong here is not in uniform substitution. So now once you substituted that one you have to check your work and then see whether if you have succeeded that means you have come up with a counter example you have shown that argument is invalid and all here we clearly we showed that the argument is invalid if x is a cat then x is an animal it seems to be okay for us all cats are obviously animals that is true x is an animal this is assumption which assume to be true then based on that if you infer that x has to be cat or x is a cat necessarily follows from these things then nobody will be in a position to believe that the conclusion to be true and all x can be donkey or it is can be a pig but it can be an animal and all. So we already constructed a counter example with which you could show that the given argument is invalid. So counter example to an argument form is a substitution instance in which the premises are true and the conclusion is false. So what you are trying to do here is you are trying to come up with some kind of counter instance in all. So things which we obviously known to be true which is known to be false are the ones which you are trying to substitute here the first thing which we have done is we extracted the form of the argument it can be a implies b or b or a or may be a implies b not a and not b all these things are invalid forms. So once you have this form you forget about what a is b is etc and all. So now for a for b etc uniformly substitute with the things which we obviously know and all it can be cats donkeys snakes rats or anything. So that is why the big the set is not complete and all you can involve snakes or some other important features that you can add in all here. So a good counter example to an argument form is a substitution instance in which the premises are well known truths like all cats are animals or well known false like statements like all cats are donkeys all these things obvious things in all well known truths and the conclusion obviously a well known false would be known like a cat is a snake or something it will make big sense if you can continue with this example and also it will be boring for us to go into the greater depth of this thing. So it only establishes that a given argument is invalid you need not have to have any knowledge of logic to know that this argument is invalid and once you extracted the form you substitute the instances that you obviously know to be true or obviously know you know them to be false. One form is like this no a's or b's some c's are not b so some c's are a one counter example is this thing that now we are taking fish cats mammals etc in that particular set we have mentioned there. So no a's or b's for instead of a's we substituted fish and then b for b is substituted with cats and we are not disturbing the truth value of this particular kind of thing no a's or b's I mean obviously it is a case that no fish or cats and no donkeys or cats or no cats or monkeys etc obviously the case which we know that is the case. So the first one is satisfied with this particular kind of thing and some c's are not b's instead of c we substituted mammals some mammals are not cats I mean all mammals need not have to be cat and all it might be a tiger or it might be something else a pig or something else some other kind of mom that also satisfied by this example and then the other example the conclusion which we are represented is in the uniform substitution we came up with some mammals are fish in all obviously this some mammals cannot be fish in all fish does not come under the category of mammals. So obviously we have a counter example the premises are true the conclusion is false that means you could cook up a single instance where your premises are true and the conclusion is false that makes this argument automatically invalid if you take like this you can consider lots of examples in which you know you can say there are no a's or b's some c's are b and no c's are a this is also an invalid argument you can think of one example which is already there here in the same way you can take fish cats mammals etc and then you can cook up a counter example and then say that this argument is invalid. So there are some other instances in which you can show that a given argument is purely invalid enough so that is like suppose if you take into consideration this argument if the government imposes import restrictions the price of automobiles will rise therefore since the government will not impose import restrictions it follows that the price of automobiles will not rise. So anyone who is not having any knowledge of imposing restrictions why how this automobiles price will increase I am not interested in all these things in all our government will not impose restrictions all these things I am nothing to do with anything I am a layman for example if you say that then being a logician you can still talk about validity of this argument in all. So the first thing which you need to do is the English language sentences which are mentioned in this argument you need to extract the form of the argument it is not easy to come up with the form of the argument in many arguments that you come across in day to day life but in most of the cases suppose if you could come up with the form of the argument in all then you can test the validity automatically. So that is the reason why we said in the beginning of beginning that this method only works for deductive arguments and then you can only establish that the given argument is invalid. So now if the government imposes import restrictions stands for G letter G and then price of automobiles will rise is represented as P. So when we talk about propositional logic in greater detail then we will enter into the details of this thing how to represent a given sentence in terms of sentential letters like G P etc. So this is safely can be safely represented as if G then P and all and then the next statement the government will not impose restrictions is what is represented as not G. So therefore it has to be not P and all this is more or less coming under the category of this one A implies B and not A and then you got not B and all. So for this you can think of some kind of counter example to show that this is invalid argument. So in this case if G then P and not G and not P it is more or less same as this one A implies B instead of A we have G and then instead of B we have P and all. So this is not P. Let us consider A in plus B not A and not P. So how to show that this argument is invalid. So there are obvious things which we know in day to day discourse and all we can verify with historical facts you may not use this particular kind of example you can use cats, donkeys, snakes etc also they can come up with the same kind of counter example. So here is a counter example it depends upon our creativity to construct a counter example. So we all of us know that for example G stands for Adolf Hitler committed suicide and then P stands for in this case it is B and Adolf Hitler is dead and all. So we all know that a historical failure is dead it died actually already. So now we substitute G and P into this particular kind of argument if G then P etc. So suppose if A in this case stands for if Adolf Hitler committed suicide so this is the one if Adolf Hitler committed suicide then Adolf Hitler is dead, B stands for dead and then A stands for suicide. If he had committed suicide obviously he has to be dead if it is successful enough. So Adolf Hitler did not commit suicide that means the second one that means A does not commit suicide enough. So that follows from this it follows that Adolf Hitler is not dead enough. It is because he has not committed suicide that does not mean that you know everyone has to die in some day or other enough. So even if you know he had not committed suicide but he might have died in some other ways he might have died in natural death or he might have died in some other ways now some plane crash or something like that. There are many ways one can die and all but if you are not committed suicide that does not mean that he had not died does not seem to be acceptable to us it is counter intuitive to us. In the same way you can say that if X is a cat then X is an animal so X is not a cat that means X is not an animal. So then there clearly shows that the argument is invalid if X is a cat then X is an animal obviously that is true and all then something like X is not a cat and all you came across one kind of let us say pig and all obviously that is not a cat and all. So from that you cannot infer that X is not an animal and all. So if you infer that X is not an animal then your it is clearly an instance where your true premises and a false conclusion that means you can always come up with an instance where your true premises and a false conclusion. You must note that even if you can come up with a single counter example then that makes this argument automatically invalid. So in the same way some other set of examples in which you can show that these arguments are obviously invalid. So they are like this all A's are B's all C's are B's so all A's are C's. So the actual valid form for this one is like this all A's are B's all B's are C's then you will say that all A's are C's. So this is considered to be a valid kind of form and all obviously whatever you substitute for A, B and C and all it can be donkey or it can be cat or anything and all you will not be able to come up with any counter example to this one. For example if it is not used in the correct form if you say all B's are A's or all C's are A's something like that change it a little bit then it becomes an invalid form obviously it will become an invalid argument. So the one which we have is not in that particular kind of form it is slightly different all A's are B's all C's are B's instead of all B's are C's we have all C's are B's. So if you infer that from that all A's are C's then you can obviously construct a counter example like this all dogs are animals it satisfies the first premise and then all cats are animals where C stands for cats and B stands for animals and A stands for dogs here. So from that uniform substitution you came up with a conclusion that all dogs are cats so nobody will be in a position to accept this conclusion to be true. So what did we do we constructed premises which are obviously true but the conclusion is false like all dogs are cats. So from a given form invalid form is substituted some instances which you obviously know them to be true then it will lead to obviously false conclusion. So some other examples can be all A's are B's no C's is A no C's is B and here is a counter example every cat is an animal here is represented by cat and B is represented by animal then the second premise no C's is A that means no dog is a cat that is also obviously the case in all no donkey can be a horse or something like that. So obviously the conclusion is no dog is an animal so if you say that no dog is an animal then obviously dogs are animals only so it is not any other kind of reptiles or something like this thing. So now we have discussed an important method in which we showed that a given argument is can only be invalid it shows that it works only for deductive arguments which are obviously invalid. So all invalid forms suppose if we can come across invalid forms automatically the arguments will be invalid. So here are the limitations of this counter example method which seems to be working nice for establishing the invalidity of deductive arguments but it has its own limitations the counter example method cannot prove validity. So that is why you know it cannot be used for establishing the validity of a given argument but it only shows that a given argument is invalid. So it is a decision kind of procedure only for establishing the invalidity of a given deductive argument of course it will not work for the inductive arguments it is a completely another story. So sometimes constructing a counter example in some situations will be extremely difficult in all. So we have to use lot of creativity is involved in constructing the counter examples if the argument is having so many variables etc. and all we just handled with simple examples and all like modus tonens modus tolens are the one which we are use is a transitivity property etc. and all are not all arguments can be as simple as the one which we have talked about sometimes constructing counter example may be kind of difficult task and all. So sometimes it will be too difficult. So in this lecture what we discussed was simply this that we first we spoke about the strength of the inductive arguments. So we said that inductive argument can only be strong or weak and all because the conclusion only probably follows from the premises there is no guarantee that the conclusion necessarily follows from the premises in the case of inductive arguments. So once we identified that it is a strong argument then we questioned ourselves is this argument is having some kind of false probably false premise and all that is the case then we said that is a weak that is a uncogent argument and all and uncogent argument is a one inductive argument strong inductive argument in which it has one of the premises probably false and all. So in a sense all weak inductive arguments can automatically be uncogent arguments and all. So then once we identified that given inductive argument is strong or weak we talked about cogency and uncogency and then we introduced an important method which only establishes the invalidity of a deductive argument it will not work for the validity of a deductive argument it only shows that a given argument is invalid. So that method is called as counter example method in the counter example method what we have seen is simply this that given an argument we transformed it into appropriate form and then since it is automatically invalid form you substituted with some instances which we obviously know them to be true or obviously know them to be false like in all donkey or cats is obviously false state all cats are reptiles to say that is also false state you construct some examples like this and then we showed that in all the invalid forms we could come up with some kind of counter examples we could come up with counter example means we could come up with an instance or example where you are true you have two premises but at you have a false conclusion. So this is the one which we have discussed and then in the next lecture what we are going to talk about is a different kind of model for an argumentation which is due to a famous British logician philosopher and of course he is also considered to be an historian of science is Stephen Toulmin. So we will discuss Stephen Toulmin's model so why in the sense that he is totally dissatisfied with the formal kind of logic in which is dissatisfied with the models of formal logic it is failing to capture the day-to-day argumentation that we use in obviously in the day-to-day discourse. So in the next lecture we will be talking about model of argumentation due to Stephen Toulmin it is widely used as one of the important models for the argumentation which is called as Toulmin's model of argumentation. So we will continue with this