Indian Institute of Technology Kanpur NP-TEL National Programme On Technology Enhanced Learning Course Title Introduction to Logic Lecture-06 Strength of Inductive arguments, Counter example method by Prof. A.V. Ravishankar Sarma Dept. of Humanities and Social Sciences 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 in 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 it falls come true 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 someone else is a kind of a valid kind of argument in order in addition to in addition to this that this argument is valid it is also having true premises. A sound argument is a valid argument which has true premises so then we discussed about some examples. (Refer Slide Time: 01:41)
Which we showed that you have 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 circles are all squares or circles all circles are parallelograms all squares are parallelograms enough. So the premises false but the conclusion is true but at this it is considered to be a valid out but in dated a this course we do not use this kind of argument because nobody will be in a position to believe that all squares or 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 has will be committed by the logic students that we say that inductive argument is valid, in an inductive tea argument cannot be valid in 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 and you can only talk about the strength of the inductive argument.
So one example we have seen earlier 99% of the commercial flights that you took landed safely so that means the next way that you are going to take that is also going to land safely, the next way 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 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 defensible. So now today we will discuss about what we mean by saying that a given inductive argument is strong given intuitive 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 easy stay cogent argument, cogent argument is a sound argument with true premises in just like in the case of sound argument in the case of deductive arguments, we said that is a sound argument is 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 a nun cogent argument. The conclusion probably follows from the premises but it one of the premises maybe probably is false in that is called as a nun cogent 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 in, so a strong inductive argument is 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 enough. So a week inductive argument is one in which it is not probable that if the premises are true the conclusion also probably true in on, so I am just replacing the words necessity with probability and on 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 in the conclusion is false but in the case of inductive arguments the necessary part is replaced by probability and all.
Because inductive arguments there are no guarantee that if premises are true the conclusion is also necessarily true conclusion, only probably follows from the premises. So that is a characteristic of inductive argument. So another important thing which we need to notice is that no valid arguments are strong and no strong arguments are valid even if it is 99.99% and all and it is treated as inductive argument for example 99% of the commercial flights such to so far landed safely without any issue enough. And from that you can infer that the next part that you are going to take will also land safely safe you know. So even if it is 99.99% and all it is where it might be very much possible that the next file that you might take might end up with some kind of issue or problem at all, it might not lawns it may not land safely exit. So no valid arguments are strong it should not confuse us with the validity and you can say that the argument is very strong enough. So instead of saying that you know argument is very strong and I can say it is valued at the entire conclusion necessarily follows from the premises use the language of deductive logic that is the validity. So if you want to incorporate necessity into consideration that that is what we want to achieve in the case of mathematical reasoning, so in mathematics everything has to follow certainly from the premises and on are also considering some kind of mathematical statements. There is a kind of necessity which is required there, so in that case we call those arguments as valued in you cannot say they have strong. So strong and weak argument arises in dated a discourse you know in which you know nothing is hundred percent true in all, so in those cases we use arguments based on observations or believes exit I and all 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%, 90% maybe 85% you can calculate the probability values and then you can measure the strength of the argument. The strength and weakness comes in degrees that is anything between 0 and 1 the probability values but the validity 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 hundred percent certainty you know, so there is no way in which there are no exceptions to that particular kind of thing, so it is a hundred
percent true, so Socrates is man is all solid person true then from these certain truths you will automatically say that the conclusion also necessarily follows from the premises. (Refer Slide Time: 09:29) So what are considered to be strong and inductive week inductive arguments it is not easy to find what constitutes a strong in inductive argument, one what constitutes a week inductive argument it involves lot of four additional factors in all, so it depends upon you can it is based on case to case basis we can say that a given argument is stronger given argument is weak and so when the evidence is hard to come by maybe then in that case you can say that even twenty percent of the twenty percent is also considered to be the strength of the argument enough. So let us consider some examples with reach even we can understand the strength of the argument, suppose if you infer in this way many clothes observed, so far have been black observed meticulously for some years and you are interested in ornithology etcetera boards etc then you came to know that with your observations and repeated observations etcetera under various circumstances etc you observed in IDK observed in some other place etc you came to this thing. Many clothes 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 crows are already in black enough, so this argument is a strong argument in a sense that it is talking about almost all the crows if not everything in all but it is talking about the entire class of crows now.
So it is considered to be a strong argument enough but for instance if you observe only two three crows and all and then you infer this particular kind of thing. You know probably the next crow that you are going to observe is black and all of course we all know that in our cross will always be black or not but with few observations under some only for few circumstances etc will not give us, it will not lead us to the strength of the argument in all you need how repeated observations a good enough in number and then it is tested in wide varieties of circumstances then only we will say that the given argument is strong. In the same way we will come across arguments from analogy in debt red is course, 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 the flame into the water and it will be snuff not at all but gasoline is a liquid, so imagining that gasoline is also liquid and then it is just like water in a sense that it at least it appears to be like water, Kerosene in our Patrol 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 in all. So here you are trying to bring in similarity between two events, so that is 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 the second gage might flame may not be snuffed out. So based on one particular kind of instance if you if you say that using the similarity in ocean 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 shall not be stuffed out so it is a weak analogy kind of argot. So analogies are very important in sciences in particular we want to use lot of metaphors to understand various phenomena and on, so we will talking I will talking 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 behave in a 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 believes in astrology are sometimes some cat passes through as in all will infer that you know 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 at all. Now there is somewhat a correlation in our correlation should not be confused with the causation, so here is an example which again tells us that this is a weak argument Austria logical calculations indicate that Indian stock market will crash in the years let us say 2014 is unpredictable behavior of stock market, somehow you have calculated the positions of planetary positions etc and all then you have also they also in a lot of calculations except I not meticulous calculations. So you could come up with some kind of thing that is applied planetary positions will lead to some kind of for business it 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 Mitch equity market is functioning are 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 for the market whatever you have invested it should get it out in, that makes some particular kind of sensor, it is very difficult for us to believe the conclusion to be true based on probably true based on the premises in all just based on has ta logical predictions you cannot we cannot come to this particular kind of conclusion.
(Refer Slide Time: 16:47) 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 for sound argument a valid argument is not just enough and on you need to invoke an additional feature, so the argument also has to be sounded. So just like that we have a strong argument can be cogent and a cogent argument. So what is a cogent argument a cogent argument is a strong argument in which all the premises it so happened 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. Let us consider a simple example to show that an argument is sound as a 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 in billions of years old universe there was some kind of a fireball or something like that it started cooling and then planet started forming you know that is what we know from this big bank, there was a huge big bang in on and how this big bang has come into existence that is not what we are interested in. You say that at least something used to be there based on scientific theories in all a scientific study suggested that there is a big bang and then after that some kind of fireball it started cooling and then all the planets formed it and if you argue that if the big bang theory is correct then the universes universe was definitely not created in six days. So that is what may be the Bible might be claiming that God has created entire universe in six days and the seventh day he took rest. So that is what we usually biblical belief or not of course you know we write to we have every right to believe something to be true in on 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 billion volt 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 scientific textbooks are some kind of theory of big bang theory. For example if you verified then all this statements that we have mentioned seems to be also true, so that makes this argument not only strong but also a cogent are you but if some of the argument some of the statements that you mentioned in this argument in the Big Bang Theory the one, which we are discussed is probably false in all our historical facts are maybe the scientific theories claims that that is not the case in all then obviously this argument may probably follow from the premises in all but still it may be one of the premises is false and on probably false. Then this is called as a nun cogent argument, so a cogent argument is a strong argument with true probably true premises, so we might confuse cogency with the soundness so a cogent argument can have false conclusion for its 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 is not a hundred percent falls in all but probably maybe you 99%true means one person falls 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 unity of arguments comes up with degrees of truth one 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 phase that you are going to take also land safely. And maybe the next flight may come under the category of 99.99%so that is why is probably to 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 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 help a false conclusion, false conclusion in the sense that 99% of the thing is true mean say at least one percent is false automatic. 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 you know if the premises are true the conclusion cannot be false enough you cannot say with the you cannot say that it is one percent Falls are ninety-nine percent true except I know if the conclusion is accepted it is accepted with hundred percent certainty there is no way in which you can have any such kind of four degrees of truth awarding a sound argument. A sound argument cannot have a false conclusion but a cogent argument can have a false conclusion the sense of the one which we are talking the case of deductive arguments, a sound argument cannot have a false conclusion because it is where it is valid argument and it is all premises are true in all but it is 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 you mean by un cogency so till now we talked about cogency is one in which you know it is a strong inductive argument with probability two premises a uncogent argument is the one in which it is it is either week we all weak arguments are automatically uncogent arguments means conclusion may not probably follow from the premises, automatically it is invalid sorry uncogent.
(Refer Slide Time: 23:46) Are it can be this can still be a straw a strong argument but at least with the one probably false premise on, so an un cogent argument falls into at least one of these categories, so the category one is like this that it may be a strong argument but it is it as one false premise. In one of the premises is probably false and that leads to one cogent argument if both the premises are true in all then the conclusion also probably true then it is called as a cogent argument and the 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 an uncogent argument almost all weak arguments are automatically un cogent arguments. Another category in which the Sun cogent argument falls is this thing it is a week and it is 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 says that is a week but it has at least one false premise. (Refer Slide Time: 24:59)
So far we discussed about cogency, uncogency 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 need some is taken. Validity is something which is which invokes some kind of necessity in the connection between the premises and the conclusion the relationship between the premises and the conclusion 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 so 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 a 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 ultimate before that in a nutshell counterexample method is in which you will show that a given deductive argument is invalid. So when can you show that a 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 the 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 a false, for example if you say all cats are dogs then automatically the statement is false. Now it is not referring to the actual thing check this in the
world for example if it is a all dogs are animals of course dogs comes under the category of animals and that seems to be obviously it is a straightforward I suppose if you say that all cats are fish obviously that statement is wrong they all these things. So now it is person who is not having any knowledge of logic or anything can easily understand that the statement is false instrument is true with, suppose if you say all fish live in water and all so it makes some sense to believe that this is true enough. So all Cat fish if you say it is automatically falls, so we will be using a set of things which we are witch we are sure that they are true or sure that they are false except I not so you will take into consideration a set which consists of cats dogs mammals fish all these things you know. 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 you know you do not require any logical method to identify that this is an invalid argument, it only shows that the given in deductive argument is invalid first you will identify the form of the argument. Let us say for example if you say if the graph if we train the grass is wet and it rained and also that is the grass is wet. So the form here is a implies B and a then B follows in a me A stands for rain and then B stands for grass effect, so that seems to be perfectly valid arguments it has valid form in 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 falsehood enough, first what you will do is you will extract the form of the argument. And then 16 for example if you have a form like a implies B & B then a 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. (Refer Slide Time: 29:11)
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 A+ B and not A and then if you input not B. So given an English language argument for example if you said that it rained then the grass is wet grass is wet, so that is much rained at all. So obviously you can say that this argument is invalid or not in a sense that you can have a counter example in which if you rain the grass is wet the class is indeed wet and all. But this may be falls in that means it rain may be false alarm instead of it rained and all it might be might be the case that the sprinkler might be on or maybe some other means in which the grass was wait in. So instead of talking about this thing what you will do is here is a sec you know you always thinks cats, dogs, animals the different classifications in all fish or mammals all these things which we are obviously. We know that they are obviously you can say tables are anything which you can look up and on, so if you say all cats are dogs and the statement is false in if you if you want to make this particular kind of thing true in 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 the next is an animal at all suppose if you say x is a cat then x is an animal obviously, all cats are animals on it cannot be fish 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 animal, so we have substitute one of these terms it depends upon our creativity. So what we what we have done here
is that given an argument we extracted the form in all, so once we extracted the form we forget forgot about what is mentioned in the content of the argument. So now instead of for these thing obvious things we take into consideration cats dogs mammals except on our retreat or any other thing which comes to your mind and all. So x is an animal so now what is he here in X is it cap you can easily see that this argument is invalid in a 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 at all. So what we have done here is that you have true premises but it you can have a false conclusion, 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 in all that means is necessarily follows from these two things in all then there is some kind of problem. Here X need not have to be cat in or it can be donkey or it can be some other thing in are dog or any other thing you know, so you have true premises but you could construct a false conclusion. So that makes this particular argument in valued at all because you could come up with a counter instance in which both are true and this is false in all today circumstances you come up with this particular kind of example. So like this I 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 ABC etc on all and once you substitute it that means that becomes a form of the argument. You forget about what is ABC 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 if B is the one which is the case then is we have representing it as animal. So animal is the thing which has to be uniformly substituted enough suppose if you substitute B for A and all that then something wrong here it 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 in 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 the assumption which has him to be true then based on that if you infer that X has we cater X is X is a cat necessarily follows from these things. Then nobody will be in a position to believe that the conclusion to be true in X can be a donkey or is can be a big but it can be an animal at all, so we already constructed a counterexample it which you could show that the given argument is invalid. (Refer Slide Time: 35:09) So counterexample to an argument form is a substitution instance in which the premises are true and the conclusion is false in. So what you are trying to do here is you are trying to come up with some kind of counter in instance in all, so things which we obviously known to be true obviously known to be faults 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 maybe in plus B not a and not B all these things are invalid forms. So once see how this form you forget about what is B 6 a tie on, so now for A B etc uniformly substitute with the things which we obviously know I not it can be cats donkeys, snakes, rats us or anything. So that is why the big the sect is not completed on you can involve the snakes or some other important features that you can add in all here. So a good outer a counterexample to an argument form is the substitution instance in which the premises are well-known truths like all cats are animals are well known faults like statements like all cats are donkeys all these things obvious things are. Well known truths and the conclusion obviously a well-known fall really not like a cat is a snake or something it does not like to be big sense if you can con e 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 do not have to have any knowledge of logic to know that this argument is invalid once you extract into the form you substitute the instances that you obviously know to be true obviously no you know them to be false. One form is like this no as a B's some C's are not be so some C's are a one counterexample is this thing that, now we are taking fish cat s paw malls etc in that particular set we have mentioned there. So No is our B for instead of we substituted fish and then B you substituted with cats and we are not disturbing the truth value of this particular kind of thing now is I mean obviously it is the case then no Fisher Cats and all no donkeys are cats are no cats are monkey sex a trial, so obviously the case which we know that this the case. So the first one is satisfied with this particular kind of thing and some C's are not beasts instead of C we substituted mammals some mammals are not cats I mean album walls need not have to be cat on it might be a tiger it might be something else a pig or something else some other kind of form, so it 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 fishing on obviously this summer moles cannot be cannot be fish in a 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 or B's some C' or B and no C sorry this is also an invalid argument you can think of one example which is already there here in the same way you can take a fish, cats mammals walls etc and then you can cook up a counter example and then say that this argument is invalid. (Refer Slide Time: 38:56) 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 mobiles will not rise. So anyone know 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 have 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 or not 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 form of the argument and 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 it 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 GPX a time. So this is safely can be safely represented as if G then p.m. 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 piano this is more or less coming under the category of this one a implies B and not a and then you got not be you know. 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 a more or less same as this one implies the instead of a we have G and then instead of be we have, so this is not let us consider a in plus V not a and not B so how to show that this argument is invalid, so there are obvious things which we know indebted a discourse in our we can verify with historical facts you may not use this particular kind of example you can use cats donkeys snakes etcetera also they can come up with the same kind of counter example. So here is a counterexample it depends upon our creativity to construct a counter example, so we all of us know that for example G stands for a tall Hitler committed suicide and then p stands for in this case it is be a doll Hitler is dead now so we all know that historical fighter is dead they died actually already. So now we substitute G&P into this particular kind of argument if g then p exit i know, so suppose if you is in this case stands for if at all filter committed suicide.
So this is the one if at all Hitler committed suicide then Adolph us Larry is dead B stands for dead and then E stands for suicide if they had committed Susan obviously has to be greater if it is successful in, so adult Hitler did not commit suicide that means the second one let me say a does not commit suicide enough. So that follows from this it follows that at all fitted is not dated is because he has not committed suicide that does not mean that everyone has to die in some day or other. So even if you know you had not committed suicide but he might have died in some other ways he might have died a natural death are you my to hide in some other ways now some plane crash or something like that there are many ways with one king died and all but if you had not committed said, that does not mean that he is it not died does not seem to be acceptable to us it is counterintuitive to us. Now 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 at all. So then that they are clearly shows that the argument is invalid if x is a cat then x is an animal obviously that is true in all then something like x is not a cat and all you came across one kind of let us say pig you know obviously that is not a cat in all, so from that you cannot infer that x is not an animal in all so if you infer that X is not an animal then you are it is clearly an instance very of true premises and a false conclusion that means you can always come up with an instance where your true premises in 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. (Refer Slide Time: 44:48)
So they are like this all A are B all these are B so all B is C so the actual valid form for this one is like this all A are B all B are C then you will say that all is all A are C. So this is considered to be a valid kind of forming on obviously whatever you substitute for a B and C and all it can be donkey or it can be carrot at anything in all you will not be able to come up with any counterexample to this one. For example if it is not used in the current form if you say all be sorry is are all see sorry a something I change it a little bit then it becomes in invalid form of the street 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 surprise all C service instead of all resources we are all see surprise, so if you infer that from that A are C then you can obviously construct a counter example like all dogs are animals it satisfies the first premise. And then all cats are animals we have C stands for cats & B stands for animals and a stands for dogs here, so from that uniform substitution you came up with a conclusion that all dogs or cats so nobody will be in a position to accept its conclusion to be true. So what did we do we constructed a premises which are obviously true but the conclusion is false like all dogs or cats so from a given form invalid form is substituted some instances which we obviously know them to be true then it will lead to obviously false conclusion at all. So some other examples can be all A are B and here is a counter exam every cat is an animal ear is represented by cat and B is represented by animal then the second premise no C is A that
means no dog is a cat that is also obviously the case in all know donkey can be horse or something like them. So obviously you know the conclusion is no dog is an animal, so if you say that no dog is an animal then obviously docks are animals only, so it is not any other kind of reptiles or something like. So now we have discussed an important method in which we showed that a given argument is can only be invalid in, it shows that it works only for deductive arguments which are obviously invalid in all. So all invalid forms suppose if you can come across invalid forms automatically the arguments will be invalided out. 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 his own limitations. The counterexample 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 B that you 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 enough. So we have to use lot of creativity is involved in constructing the counter examples if the argument is having, so many variables except I not we just handled with the simple examples in all like modus ponens modus tollens, are the one which we are used as a transitivity property exit on all or not all arguments can be as simple as the one which talked about sometimes constructing counter example may be kind of difficult task and all 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 a weak and all because the conclusion only probably follows from the premises in 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.
You know if that is the case then we said that that is a week that is a uncogent argument and a nun cogent argument is a one inductive argument strong inductive argument in which it has one of the premises probably falls in on. So in a sense all week inductive arguments can automatically be unkind arguments enough, so then once we identified that given inductive argument is strong or weak we talked about cochin see around currency 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 an invalid form you substituted with some instances which we obviously know them to be true or obviously know them to be false like you know all donkeys are cats so obviously false state. All cats are reptiles to say that is also fall straight you constructed 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 an example where your true you have true premises but it 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 we also constitute any storing of science stiffen to Lynn. So we will discuss different tool means model, so why in the sense that 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 leg stretch next lecture we will be talking about a model of argumentation do too steep and tool it is widely used as one of the important models for the argumentation which is called as tool means model of argumentation, so we will cone with this lecture next thank you. Acknowledgement Ministry of Human Resource & Development Prof. Phalguni Gupta Co-ordinator, NPTEL IIT Kanpur Prof. Satyaki Roy
Co Co-ordinator, NPTEL IIT Kanpur Camera Ram Chandra Dilip Tripathi Padam Shukla Manoj Shrivastava Sanjay Mishra Editing Ashish Singh Badal Pradhan Tapobrata Das Shubham Rawat Shikha Gupta Pradeep Kumar K. K Mishra Jai Singh Sweety Kanaujia Aradhana Singh Sweta Preeti Sachan Ashutosh Gairola Dilip Katiyar Ashutosh Kumar Light & Sound Sharwan Hari Ram Production Crew Bhadra Rao Puneet Kumar Bajpai Priyanka Singh Office Lalty Dutta Ajay Kanaujia Shivendra Kumar Tiwari Saurabh Shukla Direction Sanjay Pal Production Manager Bharat Lal
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