 Welcome back in the last lecture we presented Aristotelian theory of syllogism in that we have presented what we mean by a categorical syllogism. A categorical syllogism is a specific kind of argument which is considered to be a deductive argument in a sense that conclusion necessarily follows from the premises and these categorical syllogisms are formed in some certain way that all the prepositions that we commonly see in the categorical syllogisms are considered to be categorical prepositions. So what are categorical prepositions A, E, I and O are considered to be the categorical prepositions A stands for all men are mortal for example I preposition stands for some men are mortal and O preposition stands for some men are not mortal and E preposition stands for no men are mortal. So depending upon the quantity and quality Aristotle has classified these categorical prepositions into four different categories and then in a given syllogism at least two categorical prepositions will be there which will serve as premises and the other one will serve as a conclusion so which is also considered to be a categorical preposition. So what we are basically discussing is that how two categorical prepositions leads to another one which constitutes the problem of validity of syllogisms. In this class what we will be doing is we will be studying in detail the validity of syllogisms using Aristotle theory of logics and then we will talk about some of the rules of inferences some of the rules which validates these syllogisms and then we will move on to the reduction of syllogisms and then we will talk about some important operations which will help us in making some kind of immediate inferences. So then at the end we will talk about the limitations of Aristotle in logics. So to begin with in the last class we discussed in detail how Aristotle classified various syllogisms into different kinds of figures so this can be explained like this. So based on how the middle term is distributed Aristotle has classified various kinds of syllogisms into four different figures and all. So in the first figure the middle term occupies the position of a subject and in the second premise of figure number one the middle term occupies position of a predicate whereas in figure number two it occupies the position of predicates in both the things both the premises in the in figure number three the middle term occupies the position of a subject and in figure number four in the first premise the middle term occupies the position of a predicate of a sentence and it occupies the position of a subject in the second premise of figure number four. So according to Aristotle figure number three and figure number four are considered to be imperfect kind of figures or moods and all. So the moods that occur in figure number three figure number four are considered to be in perfect moods in perfect figures and these figures can be reduced to either figure number one or figure number two you will be seeing how the moods that occur in figure number three and figure number four can be reduced to figure number one which is considered to be the standard kind of figure perfect kind of figure according to Aristotle. So why he has why is of the view that figure number one the moods that fall under figure number one first of all what we mean by mood is like this that any triplet like a a a e a i etc all these things constitutes mood of an argument and then corresponding to the mood we have a figure then it is simply represented as for example if I say a a a one means is a mood which occurs in figure number one for example if I say a a a two then it occurs in figure number two in the mood first two letters stands for the premises and the third one stands for the conclusion. So this is what we have depending upon how the middle term is distributed we have four different kinds of figures and then there are four kinds of categorical statements which we have a e i and o and any any syllogism we have only three categorical statements. So therefore we have 64 moods possible in each and every figure. So four to the power of three that means 64 possible moods like a a a a e a e i i i all these things will constitutes different kinds of mood depending upon how the middle term is distributed. So here the middle term takes the position of a subject in the first premise and middle term takes the position of a predicate in the second premise in figure number one. So each and every figure has 64 different moods and that means we have four into four into 64 that means we have 256 syllogisms possible if you construct this thing in this way. So out of this 256 syllogisms according to Aristotle 15 are considered to be unconditionally 15 are considered to be unconditionally valid and 9 are considered to be conditionally valid. So in the first figure these are the syllogisms which are moods which are considered to be valid in all a a a and it has its own letters and all they named it with some kind of Latin names and all but the name Barbara suggests that we have to look at the overalls and all and even the consonants also convey some kind of information here we will talk about this thing in greater detail when I analyze this syllogistic poem which we will be talking about little bit later. So in Barbara the overalls that occur are a a a that means the first two proposed categorical prepositions are a prepositions and the conclusion is also an a preposition e a e in the same way it has its own name C lrent you have to observe the overalls that occur in this Latin word C lrent the first letter first oval which you come across after C is E and then after L you will find after the consonant L you will find a and after R you will find E and all. So that means e a e is considered to be the mood of this particular kind of thing and Aristotle could come up with these unconditionally you could come up with the view that only these kinds of syllogisms are unconditionally valid but how do we know that these are what do you mean by saying that they are unconditionally valid there are no specific external conditions which are imposed on this one so which also happens to be true in the case of modern logic as well. So later Boole has worked extensively on Aristotle theory of syllogisms which constitutes the modern logic so there now you will see that all these syllogisms that are listed under the category of unconditionally valid syllogism they are going to be valid even in modern logic as well but there are some conditionally valid syllogisms which depends upon whether or not the terms there Aristotle in the Aristotle in the logics are also called as term logics what are important in Aristotle logic the basic units are terms subject and middle and predicate terms middle term major term and the minor term. So these are considerably the three terms which are important in deciding whether a given syllogism is valid or not. So in Aristotle in theory by default it is taken for granted that all the terms are non-empty so that means there is no way in which you will consider an empty set in these sets corresponds to these terms which they refer to suppose if I say set of tigers and all is considered to be a non-empty kind of set. So what happens if you take into consideration there are many things which we talk about empty sets and all for example you can still one can still reason about world word 3 and then we can talk about how to prevent that world word 3 etc which does not exist at this moment which is considered to be an empty set I can still reason about those things you know. So in the same way what happens when you have unicorns or dinosaurs or something like that which are considered to be an empty set and all. So it sets some kind of limitation to Aristotle in logic but Aristotle according to Aristotle there are certain syllogisms which are considered to be valid based on whether or not the whatever term that occupies a subject position actually exists or in the other case whatever occupies the middle term whether that actually exist in the world etc and all based on that he made some other syllogisms valid. So in total there are 24 syllogisms out of 256 syllogism which are considered to be valid syllogisms according to Aristotle. So here is a poem with which they could remember what kind of syllogisms are valid with respect to what kind of figure and all it is this mnemonic poem which is also called as a syllogistic poem first appeared first appearance is like this and then it is let later been changed to the second part of this slide that is Barbara Celerent Dari-Firio-Q prior is means it is going to be the first figure and then the second one Cesare chemistry is a Festino-Borocco second means second figure and then there is something which this word refers to Tertia grandi sunas recita etc and all. So the idea here is that anyone who mucks up this particular kind of poem they can come to know what kind of syllogisms are considered to be valid and all. So this is a kind of coded kind of language and all in which for example if you take any particular Latin word and all let us say Barbara and all in that oval stands for the moods of a syllogism a, a, a for example and then the consonants also have some meaning in all. So in the first figure we do not find any such kind of thing because they are considered to be perfect moods and all. So whereas in the case of starting from the second figure onwards that means Cesare chemistry etc and all suppose if you observe Cesare C E S A R E E stands for a oval that means it is a E, A, E preposition and immediately after E we have a consonant S, S stands for some kind of unique kind of code and all which I will talk about a little bit later it stands for simple conversion etc and all. So consonants also expresses some kind of thing and all other letters the used in some kind of aesthetic sense and all. So when I analyze this poem and all when I talk about reduction of syllogisms I will go into the details of this syllogistic poem in greater detail. So now so far we have said that out of 256 syllogisms 15 are unconditionally valid and 9 are considered to be conditionally valid. So how do we know that these 15 syllogisms are unconditionally valid. So Aristotle has come up with some rules for the valid syllogisms after logic is all about the study of principles of valid reasoning. So there are some kind of rules for these valid syllogisms and the rules are like this. So the first rule is that the middle term of a valid syllogism if it is a valid syllogism then the middle term that occurs in a syllogism has to be distributed at least once in the premises. If it is not distributed at least once in the premises then it is considered to be an invalid kind of syllogism. So we need to know about something about what we mean by distribution and all. A term is distributed especially when it is referring to the whole class that it refers to and all. So we talked about this distribution in greater detail in the last few classes but if you want to remember using a mnemonic and all then there is a mnemonic which is widely used in most of the logic textbooks that is like this. Any student earning these is not on probation. So this is like this is the one which we need to remember if you can remember this one we will come to know what term is distributed etc. In what kind of categorical preposition any student earning these he got all these in all is not on academic probation that is simply write it as this thing. So what are these sentence convey in all so now we need to look for the first letters of this particular kind of sentence. This is considered to be a preposition a preposition distributes subject the next one which is immediately there here and the first letter in this one this word is e preposition distributes both that means subject and predicate. So this is distribution this is the theory of Aristotle has come up with this view that in a preposition only subject term is distributed whereas in e preposition s and p are distributed and in the case of i preposition that is the case here in the third this is the word which we have n stands for neither of them that means this is the one neither subject nor predicated. So none of the terms are distributed in this preposition and the last one is o preposition and in o preposition this is the one which we have is not on probation this is on o preposition distributes only predicate. So this is the one which we need to remember and then there are other ways to know about this one if you can draw Venn diagrams or if you can draw Euler diagrams then also you will come to know which term is distributed etc. So the first thing which you need to know is which term is distributed etc. So now the second rule says that if any term in the conclusion of a valid syllogism is distributed that term has to be distributed in the premises that means nothing is distributed in the conclusion which is not distributed earlier in the premises in all if that is the case then if it is distributed in the conclusion but not distributed in the premises in all then the syllogism is considered to be valid invalid. So in the same way in any valid syllogism has at least one positive and one negative premise then its conclusion will always be negative and all you have one affirmative preposition let us say a preposition is there and then you have a negative preposition let us say E or O then the conclusion has to be with either E or O it cannot be an affirmative kind of preposition the affirmative prepositions are A and I prepositions are considered to be affirmative and E and O are considered to be negative prepositions. So this is one of the rules which makes some syllogisms valid and all and the fourth rule states that no syllogism is valid if it has two negative premises so that means if a categorical syllogism has two negative premises in all that means what are the negative premises E and O are considered to be negative premises suppose if you have E E and then you infer A and all first of all if you have two negative premises you cannot infer anything so this is a very important thing important observation is that if in order for a syllogism to valid at least one affirmative preposition should be there so in the premises so that is a fourth rule if two negative premises no inference can be possible and the final one is if any valid syllogism has only universal premises that means a kind of thing and its conclusion also should be universal but it is only in the case of Aristotle and logics if there are two universal prepositions as premises the conclusion can still be particular kind of preposition so that makes these nine prepositions categorical syllogisms conditionally valid so we will be seeing with some examples of course we will talk more something more about these rules of syllogism based on the observations that we have so when Aristotle has proposed this particular kind of theory then it has these particular kinds of things so there are only three terms in a syllogism so you might say if there are more than four terms and all what needs to be done so it has to be reduced to only three terms for example if you say all X are all A's are B's all B's are D all B's are C's all C's are D's etc that means A B C D there four terms in that one so in that case what we need to do is we need to reduce to categorical preposition to another one let us say all A's are B's all B's are C's that is reduced to all A's are C's and all so now the next proposition is all C's are D's that means all A's are D's but it is not as simple as the one which I am trying to express and all in actual practice it may not be the case so this again sets some kind of serious limitations to Aristotle and logics if there are more than three terms in all then things will become it becomes difficult to express in this simple theory of formal theory of syllogism which talks about the validity of syllogism so the second rule is that the middle term is not in the conclusion so middle term never occurs in the conclusion middle term occurs only in the premises by chance if the middle term occurs in the conclusion then there is something wrong with the arrangement of the propositions and all it is not considered to be it is not permitted actually first of all to be treated as valid or invalid. So the third rule is that the quantity of a term cannot become greater than greater in the conclusion and all so if there are two A propositions and all it cannot be E propositions and all the conclusion cannot be any preposition so for example if you have two particular kind of propositions and all I propositions it cannot be an universal preposition in the conclusion and all so the quantity of a term cannot become greater in the case of conclusion and all so in that sense you know Aristotle permits from two A propositions you can still infer an I proposition there the quantity of the terms that occurs in the conclusion is not greater in the conclusion and all when compared to the premises and all so this rule needs to be explained in greater detail but it is not that important at this moment so the fourth rule is that there are all some observations from the rules that we have already presented in the last slide so the middle term must be distributed at least once in the premises so that is what we have stated already if it is not distributed at least once then the syllogism is considered to be invalid and all for example if you take two particular kind of propositions and all some dogs or animals or some animals are wise or intelligent something suppose if you say like that from the two particular premises you cannot infer anything you know so because of this that in the I proposition none of the terms are distributed in I proposition distributes neither of them so what we what is important here is that in a syllogism middle in especially when you take middle term into consideration it has to be distributed at least once in the premises and all so that means the middle term has to be either A or O kind of proposition and all where at least the terms are distributed and all at least once okay so the another rule is this that there are general observations and all the fifth rule is that at least one premise must be affirmative and all what are the affirmative propositions a categorical A and I are affirmative categorical propositions and all so now this observation we can see this one clearly now you see this unconditionally valid kind of syllogism and all in the first figure we have a in that at least one affirmative proposition is already there a proposition is an affirmative proposition so in the same way e a e again a is an affirmative kind of preposition and in the third one a i i there is one a is there even i is also an affirmative kind of preposition e i o again i is in affirmative preposition also in this way in all the unconditionally valid or even the conditionally valid syllogisms at least one affirmative preposition should be there in the premises and all in the same way e i o I proposition is considered to be an affirmative kind of preposition. So even the third figure for example if you see book book or that is o a i so a proposition is considered to be an affirmative preposition so in all the syllogisms if you do not have at least one affirmative preposition you cannot infer anything that is what we have been saying for example if you have two negative prepositions e e preposition you cannot infer anything no cats or dogs no dogs or donkeys so if you infer something else like no cats or donkeys etc and all even if you your conclusion is correct and all but you cannot infer anything in the sense that two negative premises you cannot infer anything so this is the fifth rule at least one affirmative preposition should be there in that syllogism if it has to be valid the sixth rule is that if one premises negative the conclusion will automatically be a negative preposition. So if the conclusion is negative the vice versa is also applies here if the conclusion is negative then at least one of the premises should be negative so then only so if the conclusion is negative the premise also should be negative and all if it is not the case then the syllogism is considered to be in value if both premises are affirmative the conclusion also should be affirmative and all suppose if you have a preposition I I preposition is ruled out because you cannot infer anything because the middle term is not distributed in that particular kind of prepositions and all so if you have a preposition you cannot infer e preposition so in the same way if you have I in the same way a preposition is there you cannot infer o preposition or e preposition and all. So eighth rule eighth rule states that at least one premise must be universal and all so this is another interesting observation which we can make out so at least one of the premises must be having some kind of it should be an universal preposition what are the universal prepositions a and e are considered to be universal preposition so look at the premises of all the valid syllogisms and all then you will find this that at least I mean all the valid syllogisms you will find either a or e in this valid syllogism so that is another interesting and important observation and the ninth rule states that if one premise is particular then the conclusion is also particular so for example if it becomes with all x or y and some y's are z and all then the conclusion also should be particular in extension logic that means the modern logics after bool if both premises are universal then the conclusion should also be universal it cannot be particular it cannot be particular in the sense that if it becomes particular then we are importing existence into the conclusion which is not there in the premises suppose if I say that all cats are dogs and all that does not mean that you know the cats and dogs have to be exist to say that it is true n can be assumed to be true also as it need not be the case that all cats are dogs should actually be true and all and the cats exist dogs exist etc but if you say some cats are dogs and all that means it talks about the existence of cats which are considered to be dogs so that means dogs actually exist so we are importing existence which is not there in the premises in the conclusion that leads to according to the modern logics or extension logics which is called as an existential fallacy which we will talk about it little bit later again it sets limitation to Aristotle in logics so this is considered to be kind of fallacy especially when you infer from two universal propositions you infer a particular proposition and that is considered to be an existential fallacy in modern logics. So suppose if you do not for rules does not satisfy and all then obviously there are mistakes in the argumentation that is one which was considered in the case of formal fallacies they are considered to be fallacies so these are some of the fallacies that we have if the rules are not the rules are violated and all suppose if you have it is the first rule is pretty straight forward and all and it is also taken for granted that there is no equivocation in the present in the argument and all that means they should be exactly three terms and these terms should be used in the same sense and all they should not be used in two different senses in all for example if you say all this room is made up of atoms, atoms are invisible and this room is invisible and atoms are used in the premises in two different ways in all the first premise it is used in some sense in the second premise atoms are invisible in that it is atoms are used in a different sense. So this is what is equivocation fallacy it has all the terms should be used in the same sense and all and there is no shift in the meaning of the words that you have used in the premises and all if there is shift in the meaning of the usage of these words in the premises then there is something wrong with this argument that is equivocation fallacy. So middle term must be distributed at least once in the premises that is what we have said in the rules and all if it is not distributed then it is called as fallacy of undistributed middle and in any syllogism we have major term, minor term and middle term and all suppose if it is a case that no term can be distributed in the conclusion which is not distributed in the premises that is what we have said in the rules suppose it is distributed in the conclusion but it is not distributed in the premises and all. So depending upon what is a major premise or minor premise etc and all so this rule is violated with respect to a major premise it is called as illicit major its rule is violated with respect to the minor premise is the one in which where you will find the minor term major premise is a premise in which you will find the major term. So major term is considered to be the predicate of the conclusion and minor term is considered to be the subject of a conclusion in a syllogism. So wherever the subject term of a conclusion occurs in the premises that is considered to be minor premise and wherever the major term occurs that is the predicate of the conclusion that is considered to be the major term and all. So when the problem lies with the major term or minor term is the one which we we need to look for where this undistribution is taking place that is the not this that is the that is where this kind of fallacy arise. So the fourth fallacy is fallacy of exclusive premises that means if you have two negative premises nothing can be inferred in the same way if you have two particular premises nothing can be inferred and the fifth one affirmative conclusion from negative premises is not allowed and either premises negative then the conclusion also has to be negative if the conclusion is negative and the premises one of the premises also have to be negative. So this is the fallacy which I talked about in the last slide is no particular conclusion from the universal premises if you have two universal premises or categorical propositions you cannot infer a particular categorical proposition but for Aristotle depending upon whether or not the term which occupies position of a subject term occupies the position of a predicate or the middle term they actually exist in the world and all if it is non-empty and all then there is no problem for the validity of a solution it makes it conditionally valid and all for example unicons dinosaurs etc they would not exist and all they do not actually exist and all so in that case I mean it is difficult to say whether Aristotle in theory applies are not. So Aristotle in theory in general it takes into consideration that all terms are non-empty and all so whatever term you take into the syllogism that is already taken for granted or by default it is it is referring to it is not referring to any empty class and all empty sets are those sets in which you do not have any elements and all for example if you say set of unicons etc and all then there is no unicons exist in the world or set of ghosts etc and all then empty set. So Aristotle in theory is a little bit silent about this particular kind of thing but Aristotle compromise with this particular kind of thing and he says that he is of the view that depending upon whether or not SPM etc actually exist and all it makes this syllogism conditionally valid. So now so far we have seen different rules and all so now we will like to see whether these following syllogisms are valid or invalid. So now observe the first syllogism and all and then we will work on one or two examples and then we will move on to some other kind of things that is a reduction of syllogism and all which we can talk about the immediate inference and all immediate inference are those inference in which from one particular kind of categorical preposition another categorical preposition follows. So now let us consider some examples which are there in all and then we will analyze this example in greater some fish are tasty and then of course all fish can swim then what is the conclusion here some tasty things can swim, some tasty things whatever things that are tasty and all can similarly forget about what it means and etc and all. So as long as form is there and all we do not have to worry much about it. So this can be translated into it does not matter whether fish are tasty things are swimming etc all these things whether they exist or not. Now we can translate it into some x or y where x stands for this thing and y stands for this and then this we fixed x for fish and all it is already there and then swimming is considered to be z and all x or z and this is a form of this one and then some tasty things some y is r can swim is referred as z. So now we transform this thing into this one it does not matter what we mean by fish donkey is cats it does not matter so once we transform it into form and all now we can see whether this argument is valid or not. So now the first thing which need to know about this one is this particular kind of thing so what is the this is a sentence in which y is considered to be subject and z is considered to be predicated the predicate this is the conclusion and these two are premises. So now the predicate of the conclusion is called as a major term major term here is z so now the subject of a conclusion is called as minor term so this is the minor term minor term is this is the minor y so now whatever term occurs twice in the premises so that is considered to be the middle terminal so these constitutes x is a middle term. So this is the first thing which we need to find out before knowing whether this particular kind of syllogism is valid or invalid so now there are some rules which we need to apply and then we need to see whether this particular kind of thing is valid or invalid so these are some of the rules that we will be applying and then we will see whether it does not mean that if one of the rules satisfies then the syllogism is valid and all but it has to satisfy all the rules and all it is a conjunction of all these rules and all. So now what is the first thing which need to see is the distribution of a middle term so what is the middle term here x so now wherever you have x in all the middle term should be distributed at least once in the premises in all so here it is not distributed in all because it is a high preposition high preposition distributes neither of them neither subject nor predicate it won't distribute in all so you not have to worry much about it but observe this is the second statement in all all x are z in all where x is considered to be the middle term so if it is in a preposition it distributes subject in all so what occupies the subject position here is x so middle term is distributed distribution of middle term in premises so this is rule number one and all so this satisfies now we need to look for other rules and all so the other rule is this that if any term in the conclusion of a valid syllogism is distributed that term has to be distributed in the premises and all. So now look at the conclusion in all conclusion is an high preposition high preposition it distributes neither of them in all so nothing is distributed in the conclusion in all so either y is not distributed even z is also not distributed because it is high preposition high preposition distributes neither subject term or not even the predicate so now we need to see whether this y and z are distributed in the premises and all so since it is in high preposition the first one there is no question of the distribution of the term y and since it is in a preposition a preposition distributes only s and all but not the predicate and all so it is also not distributed so the idea here is that nothing is distributed in the conclusion which is not distributed in the premises and all here it is not distributed even in the premises also it is not distributed. So rule number 2 also applies here and then rule number 3 is this that if it has one positive and one negative premise the conclusion has to be negative in all. So here you would not find any positive and negative kind of conclusions in all so that rule would not apply in all so that is also follows in all automatically we do not this rule will not apply on this particular kind of thing and there is a fourth rule also is this that no syllogism is valid if it has two negative premises we do not have any negative premises here all are affirmative propositions it is an I proposition it is an A proposition and this is an I proposition. So even fourth rule also satisfies and the fifth one if any variationalism as two universal premises it does not have any two universal premises in all so it is in this sense more or less all the rules applies in all in this particular kind of thing. So some of the rules may not even directly apply to this one so we do not have to bother much about it if it applies then we need to see whether that rule is followed or not. So it is in this sense this particular kind of argument is considered to be valid this argument is considered to be a valid kind of argument. So now let us consider some more examples one or two examples with which you will come to know whether a particular kind of syllogism is valid or invalid just for the sake of this thing we write something in all some X or Y some Y's are Z's some they are pretty simple things in all so now the problem with this particular kind of thing you might say that is a straightforward thing in all you can say some cats are animals some animals are animals bark and all so that means some dogs barks and it might seem to be a sensible for you in all but the problem here is that according to the theory of syllogisms all are I propositions in all first of all when you have two particular kind of propositions I propositions you cannot infer anything why because again the middle term here is why so middle term has to be distributed at least once in the premises so that means it has to be the middle term has to the proposition which consists of middle term should be at least either a proposition or at least e proposition or at least even o proposition but it should definitely should not be an I proposition I proposition middle term is not distributed this is a middle term because it occurs twice in this premises in all so this conclusion this categorical syllogism is clearly invalid because of this particular kind of thing middle term is not distributed at least once middle term not distributed at least once in the premises but in both the premises it is not distributed so in this sense we can talk about several other kinds of argument this X and Y's refer to cats dogs donkeys anything you replace it with something then you will see whatever you replace it with then that argument is obviously going to be an invalid kind of another example for you this thing you take into consideration no X or Y all Y's are Z that is for the sake of consideration and then three is some X are not so now this is an e proposition negative proposition and then this is an a proposition according to our rules of syllogism it is clear that even when you have a negative proposition your conclusion also has to be negative enough so it seems to be the case that you know conclusion is also a negative preposition because so positive prepositions are a and I because these are all affirmative it affirms something and all and negative prepositions are E and O so there is a there is a mnemonic that we have used earlier that is affirmo affirmo and Nego so you observe this ovals that occurs in this word a and I that means a and I are affirmative and E and O are considered to be negative so this is called as a mnemonic so mnemonic we use and all and the other mnemonic which is quite useful for us is this particular kind of thing okay so now it appears that your premises are at least one of the premises is negative the conclusion is also negative and all the third or fourth rule applies and all now we need to see the first thing we need to find out is what is the predicate and what is the subject of your conclusion so this is considered to be minor term and this is considered to be major term and wherever this Z occurs the term Z occurs that is a major term that is considered to be a major premise and all this is a major premise and this is called as minor premise because X occurs here whereas Z is the predicate of the conclusion because it occurs here it is a major premise and also the convention is that we always state this major premise first followed by that you have a minor premise and then you have a conclusion that is the style which is followed in most of the syllogisms so okay it will be a little bit boring if you entering into greater analysis of this one which some of the things which I already covered it so now we need to talk about the distribution of middle term and all whether this rule applies to this one or not so what is the middle term here why is considered to be the middle term here so the rule says that the middle term should be distributed at least once so in both the cases it is distributed because it is an e preposition e preposition distributes both of them both of them in subject and predicate whatever occupies the subject and predicate positions and all so that the terms and all that is said to be distributed both X is distributed and Y is also said to be distributed so I mean this middle term is distributed even at least once is satisfied and all not only that thing even in the second premise also categorical preposition the term Y is said to be distributed and all so middle term distribution of middle term is having no problem and now the second rule is that nothing is distributed in the conclusion which is not distributed in the premises and all suppose if it is distributed the term is distributed in the conclusion it has to be distributed at least once in the premises and all so now in this case it is an O preposition O preposition distributes only the term which occurs in the predicate position so that means it distributes only Z here so Z is distributed here so now we need to see whether Z is distributed in this one so now in this preposition only Y is considered to be distributed and all and in this preposition X and Y are said to be distributed but not Z and all so now this is leading to a problem that something is distributed in the conclusion here that is the term Z but it is not distributed anywhere in the premises and all although it occurs here but in it is an A preposition A preposition only S is considered to be distributed so the problem here is that in conclusion the word Z is the term Z is distributed but it is not distributed in the premises and all so that means it violates this particular kind of rule that is this one second rule it violates so that means this particular kind of argument is in value so like this we can find out whether or not a given syllogism is considered to be valid or not and out of this 256 syllogism Aristotle could come up with 15 syllogisms which are considered to be conditionally valid and 9 are considered to be conditionally valid and all so we will talk about one instance of conditional syllogism and all so that is a I so this is like this suppose if you have any argument in which you have this particular kind of thing all X or Y all all Y's are Z's and then from this you infer some XR Z's this is with respect to figure number one in figure number one the middle term should be like this M and M in the first frame is the middle term should occupy the subject position and in the second preposition categorical preposition middle term should occupy the position of a predicate so now you have to change it a little bit and all this one it is not in the standard format so now first you identify the major term and the minor term so this is the major term Z is considered to be the major term and this is considered to be the minor term so wherever this Z occurs so that needs to be stated first and followed by that we have this particular kind of thing minor premise and all so this becomes this thing all all Y's are Z's becomes all X or Y so why we have done like this because the predicate of the conclusion is a major term major term occurs wherever the major term occurs that is considered to be major premise and you should have major premise and followed by that your minor premise and you should have a conclusion so now it is in this particular kind of format so the middle term occupies the position of a subject here because this is considered to be the middle term because it occurs twice in the premises so this is exactly the same as this instead of barbara it is barbari so this is a and I proposition so now for Aristotle this is considered to be kind of valid kind of argument depending upon whether or not the term for example this is going to be valid according to him especially when the subject term that you are referring to that is this one some X so whatever you are referring to that actually exist in the world now suppose if it is referring to some unicorns some dinosaurs some kind of other things which are non-existing kind of thing goes etc. and all then Aristotle is silent about those things you know so it Aristotle in the theory of syllogism directly applies to those things in which these things are X Y Z etc. and all are considered to be non-empty and all but in modern logic in particular if this kind of problem is there and all for example you can take into consideration some example all forget about seeing all all unicorns or dogs forget about whether it is true or false it does not matter you can assume this thing to be true and all dogs bark let us say is also again wrong and all but most of the dogs may not bark but they when we bite and all so from this if you infer that some X are what some dogs some dogs are Z Z means one second this is not the one which so this is why he and this is considered to be Z and now some X one second what is this some unicorns or dogs let us say Y is this thing and Z is this one and then some X are Y X means it should be some other thing okay forget about this particular kind of example we will talk about little bit later but the idea here is that in modern logic in particular whenever you have all Y's are Z all X are Y etc. and all from that you infer this particular kind of thing some X are Z this presupposes that this some X are Z means so let us say so you say that some cats are animals in all there are some cats which are considered to be animals in all that means it leads to some kind of thing which is called as the existence of this particular kind of thing the object that is referring to the cats or animals or whatever it is dogs donkeys etc which presupposes the existence of these things in the conclusion and all suppose if you say all Y's are Z does not mean that you know that it has to be exist it has to be existent in the actual world and all but if you say that some dogs are some some dogs bark and all then it has it is it is it is it means that there are some dogs which actually bark and all so that leads to the existence of dogs dogs and all so this leads to what we call it as some kind of conditional kind of validity and all so depending upon whether or not SPM terms are empty or non-empty is non-empty then only we can talk about validity of the syllogism so suppose if they are empty and all like unicorns etc goes goes etc and all dinosaurs etc then Aristotle theory fails and all in this particular kind of case because it presupposes that all the terms that you are referring to in a syllogism are considered to be empty. So now we look into the other aspect that is this that in the ancient past during the Greek period so they came up with this particular kind of syllogistic poem and with this poem they could identify what kind of syllogism is considered to be valid with respect to what kind of figure and all so quickly we can analyze this syllogistic point in this way all the vowels in a syllogistic poem corresponds to the moods and then all the consonants etc are corresponding to some kind of operations that we can use etc and all so that all the moods that occur in the second third fourth figures can be reduced to the first figure and all. So in this class what we discussed is we presented some kind of rules of syllogism which makes a particular kind of syllogism valid or invalid so the rules are like this that there are five rules which are followed out of that four are four rules are followed even in modern logics as well but only in the with respect to the fifth rule so that is if you have two universal premises according to modern logic you need to have any universal preposition on if you say that if you infer a particular kind of preposition from two universal prepositions then it leads to some kind of fallacy is a mistake in the argumentation because we are importing existence in the conclusion which is not there in the premises that according to modern logic leads to fallacy which is called as existential fallacy. So Aristotle in theory of syllogism more or less works for this 15 both Aristotle in logic as well as the modern logic which followed after that one works for this first 15 syllogisms which are considered to be unconditionally valid so there are few problems with respect to Aristotle in theory of syllogism for example if you have more than three terms there is a problem then if it is referring to some other kinds of propositions which are not in the standard format then also it presents some kind of problem in the next class we will be analyzing the syllogistic poem and then we will talk about some of the important rules of immediate inference such as conversion of version and contra position rules which helps us in transforming from one categorical proposition into another one or it tells us how this A, E, I and O propositions are related to each other so we will continue the same discussion in the next class.