 In continuation to the last lecture where we started with Aristotelian syllogistic logic there we discussed various kinds of categorical propositions this can be a proposition can be like this all men are mortal, I proposition is some men are mortal and O proposition is some men are not mortal and E proposition stands for in a no men are mortal. So depending upon what distributes what and all we have classified these categorical propositions in terms of quality and quantity. So these categorical propositions combine in certain way and they will form categorical syllogism and all categorical syllogism consist of at least two categorical propositions as premises and another categorical proposition as conclusion. So today we will be discussing about Aristotelian theory of categorical syllogisms and then we will also discuss about various valid rules of inference and then we also discuss about immediate inferences etc. So to start with what we mean by a categorical syllogism. So the definition of a categorical syllogism is like this it is obviously a deductive argument having a sequence of three and only three categorical propositions in this categorical syllogism there should be only three and only three propositions out of which two will serve as premises and the other one will serve as a conclusion. So it is a deductive argument in a sense that conclusion necessarily follows from the premises and all and there is no new information in the conclusion etc all the properties of deductive arguments works well here and not only that thing not only the thing that they have only three categorical propositions but they have three and only three terms appear in a sequence of statements for example any sentence any categorical proposition has two terms subject and predicate term but the three terms that we are referring to are like subject term and predicate term and the middle term. So each one of these terms occurs at least twice in this three propositions so each term appearing in exactly two propositions the way in which this term the terms of the syllogisms are arranged is what we call it as the figure of the syllogism. So the idea here is that middle term is the term which you will see only in the premises and whereas subject and predicate terms you will see in at least one of the premises and one in one of the conclusions and all. So these terms at least occurs twice in the whole argument and all the only thing which you need to notice is that middle term will not appear in the conclusion. So one example of a categorical syllogism is this thing suppose if you say all astronomers are scientists some astrologers are not scientists and from that the third proposition is a conclusion that is some astrologers are not astronomers. So this forms a kind of categorical syllogism now Aristotle has presented his theory of syllogism in which you know we will come to know how whether or not some astrologers are not astronomers follows from the two given categorical propositions that are all astronomers are scientists some astrologers are not scientists. So for that what we need to do is first we need to identify the terms that exist in this categorical syllogism so these are like this. So what I said in the beginning was middle term is the term which exist at least which exist only in the premises but you will not find it in the conclusion. So the middle term of a categorical syllogism is the term that occurs once in each premise it occurs only in the premises but it will not occur in the conclusion. So in this case the one which is in the red color is the one which you will see in the premises so all astronomers are scientists and some astrologers are not scientists they are considered to be premises and the third one here is a conclusion. So scientist is the term which occurs only in the premises that is why it is called as a middle term and first you need to observe the conclusion the conclusion here is some astrologers are not astronomers. So here the subject term is astrologer we are talking about the conclusion and the predicate term of the conclusion is astronomers. So whenever the predicate term occurs in the premises and that is considered to be the major premise whenever the subject term occurs in your premises then that is called as a minor premise the other way round you can say that astrologers is the subject term is called as a minor term and astronomers is considered as a major term whereas the middle term is scientist is not so how do why do we need to find out all these things we need to find out what is a major premise and what is a minor premise major premise is a premise in which you will find the predicate term the predicate term here is astronomers. So as you will see here clearly in this example the first premise astronomers figures out so that is why that preposition that categorical preposition all astronomers are scientists is considered to be a major premise and the subject of a conclusion that is the astrologers here is considered to be the minor term and whenever this minor term occurs that categorical preposition is considered to be the premise which is different to this categorical preposition is considered to be a minor premise. So one is a major premise and two is a minor premise and then of course third in third one is the conclusion it is a standard convention that you should ensure that always you state the major premise first followed by that minor premise and of course there is a conclusion so how do we know why are we studying all these things because we need to judge whether this particular kind of syllogism is a group of categorical propositions out of which two are serving as premises and the other one is serving as a conclusion so how do we know that the premises are leading to the conclusion that makes this syllogism valid that is what is our motivation and then Aristotle has come up with a wonderful theory which is very closer to the natural language and then Aristotle has come up with this theory of syllogism in this way the first and foremost thing which you need to know before presenting this Aristotle theory of syllogism is another concept which is called as mood, mood is not the one which we are talking about which mood you are in and all what kind of mood you are in it is a different kind of thing it is a form of argument in all so for example if you have no birds are mammals all bats are mammals so no bats are birds the first preposition is a preposition and the second one is a preposition that means all men are mortal like that and the third one is a preposition so the mood of this preposition is E A E so another kind of categorical syllogism which has which is like this no mammals are birds that is a preposition all mammals are bats that is a preposition and the other one no bats are birds that is usually served as a conclusion so that is a preposition so this is also called as E A E preposition but both of these syllogisms have the same mood but their logical form is different and all so although they have the same mood and all E A E but the logical form in a sense that how the middle term is distributed etc and all they are a little bit different from the second argument so in the first argument the middle term takes the position of predicate and in the case of the second categorical syllogism the middle term takes the position of the subject so these are the things you know although two categorical syllogisms have same kind of mood but the logical form might be different so this is the first thing which we need to know first thing we need to know is four kinds of categorical prepositions A E I and O and depending upon the quality and quantity we have classified into different kinds of categorical prepositions and then these categorical prepositions combine in certain way and they will form some kind of thing which is which we are calling it as a mood so same mode but you can have different logical forms so now based on how the middle term is distributed what is the middle term middle term occurs twice in the premises so that you will not find it in the conclusion suppose in the categorical proposition you find a middle term in the conclusion that is there is not a valid kind of argument first of all forget about the validity you should not use the middle term should not come in the figure out in the conclusion so now based on how the middle term is distributed according to Aristotle we have four possible figures so figures are in combination of different kinds of categorical prepositions and in these categorical prepositions what we need to look for is where the middle term is placed so there are four kinds of categorical statements are prepositions A, E, I and O and there are three categorical statements for categorical syllogism that means any categorical syllogism should have only three categorical prepositions you might ask what happens if I have more than three categorical prepositions in all so these will reduce to only two for example all men are motels all men all donkeys are cats all cats are dogs something like that three propositions are there then first two combine will form another categorical proposition and that categorical proposition combined with the fourth one and ultimately you will have only two categorical prepositions and then the fourth one is the conclusion if you have more than two categorical prepositions of course it presents problem to the Aristotle in model but it works nicely for categorical propositions three categorical prepositions out of which one is a conclusion so now we have four categorical propositions A, E, I and O and you have only three categorical propositions which will form a syllogism so that is why we have four cube that is four into four into four that is 64 possible moods which are possible for each and every figure so what is a figure he has nicely he has come up with a wonderful idea that depending upon how the middle term is distributed in these categorical prepositions he has classified into four different figures in all as you observed in these figures clearly that the conclusion is always in the format that subject and predicate but the only thing which is different here is the middle term for example in the first figure in the first premise middle term occupies the position of a subject and in the second premise of first figure number one middle term occupies the position of a predicate and in the figure two middle term occupies the position of predicate so each categorical preposition has subject and predicate in all in any sentence will have a subject and predicate for example if you say all men are mortal mortality is attributed to the middle to all men and all men is a considered to be subject all men are more mortality is nothing but a predicate some kind of property or something so as you will see clearly in these figure figures obviously the conclusion is always in the form in the form of subject and predicate but the only thing which is different here is the position of the middle term is different in all these figures so for example moods can be like this or any particular kind of combination will form a particular kind of mode it can be or all these things will form particular kind of mode so each figure will have 64 possible moods and then the first figure will have 64 mood second figure will have 64 and the figure 3 also 64 and 64 ultimately you will have 256 slogans possible is a wonderful construction of Aristotle there are few things which are exciting in all even the natural sciences also the periodic table which is developed by Mendeleev which is very aesthetic in nature and all it conveys lot of information so the one which you are going to see we are seeing right now that is based on how the middle term is distributed yes classified the syllogisms in nicely into these four groups some kind of analogy you might find it in Mendeleev's periodic table or maybe the origin of benzene structure etc these are wonderful innovations in all this there is something great about this kind of figures so only 256 syllogisms can be possible in all so out of that Aristotle has come up with some kind of valid rules of syllogism with which he could come to the fact that the out of this 256 syllogisms 15 are unconditionally valid I will talk about what I mean by unconditional validity and 9 are considered to be conditionally valid in all so that means according to Aristotle's theory of syllogism 24 out of 256 syllogisms are considered to be valid syllogisms in all not all kinds of combinations will give us valid syllogism valid syllogism in a sense that for example if you have for example that may not be a valid kind of syllogism however well it falls into figure one figure two figure three or anything so now what we are going to do here is that we are just trying to see how Aristotle has classified these syllogisms into these four different figures so there is one more thing which we need to note the first figure is considered to be the most standard kind of thing so all the moods that are falling in figure two figure three figure four can be reduced to the figure one and all that means so this is considered to be the most standard kind of figure that you will commonly see Aristotle in theory of syllogism all the other things which fall in figure two figure three etc. For example if you have AAE two that means that stands for a categorical syllogism AAE and it falls in figure number two that means the middle term will be occupying the position of a predicate so these are some of the things which Aristotle has come up with now we need to find out how Aristotle has come up with the validity of only these 15 syllogisms which are considered to be unconditionally valid so I will go into the details of this little bit later but Aristotle has named these syllogisms with nice names and all Greek names so the first figure only these four are valid AAE there are some ways to remember this particular kind of syllogism these are called as mnemonics so they use lot of mnemonics to a kind of mugging up this whole kind of thing you know if you remember this mnemonics and all it is like a poem if you remember the poem you can understand everything about this validity of a syllogism so I will go into the details of this poem little bit later so which is considered to be a syllogistic poem which is quite popular. So the first one has a name Barbara and all so as you see here clearly the ovals that occur in this term Barbara that is A and A these are the ovals AE I OU or ovals and all whenever you find this ovals corresponding to this particular kind of thing so that is having AAA form another thing is Celerant that means in this Celerant the first oval is E second oval is A and the third oval that you that you come across in the particular order is E so that is why it is a EAE categorical preposition and Dari is this A the first oval is A and the second oval is I and I so not only the thing that these ovals corresponds to the mood of a syllogism which are considered the valid syllogism and all but the other letters consonants also are going to convey some kind of information here so which we will talk about it when we analyze the syllogistic poem into a greater detail but our concern now is to know when these syllogisms are going to be valid and when syllogism are going to be unvalid that means when the argument is valid when the argument is invalid in the same way second figure where the middle term occupies the predicate position EAE AE E I O A O O these are considered to be the moods which are is going to be unconditionally valid and all so the other thing is Bocardo O AI E I O I AI AI is considered to be valid in the third figure the fourth figure is E I AI and E I O so these are some of the things which we need to which Aristotle has come up with and these are unconditionally valid there are some other kind of conditionally valid syllogisms so they are like this the first figure in addition to AE AE AI etc and all these are also considered to be conditionally valid depending upon the subject term is empty or non-empty and all so Aristotle one important thing which you need to note is that is total takes it for granted that all the terms are considered to be non-empty they are not going to be empty sets example if you say unicorn and all that is not permitted in Aristotle in logics because it is an empty set set of unicorns which do not exist the set of goes set of vampires etc all these things are empty sets it would not exist so that is an empty set which we are not supposed to take into consideration but modern logic can take into consideration for example you can take into consideration all unicorns are intelligent that may be assumed to be true and all but according to Aristotle if you assume that thing to be true then it leads to a fact that there exist some unicorns which are considered to be intelligent that means unicorns actually exist which is not the case it is an empty set so Aristotle in theory of syllogism that is not permitted which sets limits to Aristotle in theory of syllogism which will talk about it when we discuss existential import in greater detail that will address this particular kind of problem. So what we are seeing at this moment is that he has classified these syllogisms into four different figures and then he says that 64 moods I mean that is corresponding to some kind of argument or syllogism categorical syllogism and then out of this 64 in figure one there are only four which are unconditionally valid and two are unconditionally valid so like this we have 15 plus 9 24 syllogisms that are going to be valid out of 256 kind of syllogism that are possible. So here is a poem with which in those old days they could remember the I mean the validity of the syllogisms with the help of the poem you know so the poem in the first appearance is like this and the second appearance is the one which we are going to take into consideration they are all latin or greek names so if you can buy hard this particular kind of poem then you can understand all the 24 kind of syllogism that you are seeing that you have seen earlier. So first one is Barbara Ciller and Dary Ferriyoku priorities that means the first kind of figure, Cesare, chemistry, Festino, Baroko, second kind of second figure and Terchia Grandi Sonas etc. there is some kind of things actual translation we do not know what it is. So in that third figure the Ropthi, Philopton, Desames, that EC and all these latin names which are going to be varied syllogisms and then followed by that we have Kaminis, Deymaris, Fisapo, Fresi son etc. So this is not what is of importance to us what is important for us is you can come up with your own mnemonic and all but the idea here is that we need to focus on the ovals that exist in these latin words and all. So they tell us what kind of syllogism is valid in what kind of figure and all. This poem we will analyze it little bit later but so before that we are going to consider just for the sake of this thing we will consider some examples and then we will see whether this particular kind of syllogism is valid or invalid with the help of this particular kind of thing. For example if you have an argument like this thing no B is A some examples we are trying to consider no B is A the first one and the second one is let us say all C is B and then this is the third categorical proposition so that is no C is. So now suppose if you are given this particular kind of syllogism how do we know that this is valid or invalid. So now the first thing which you need to note is to identify the terms in this particular kind of syllogism so now first come to the conclusion so that is the subject term of a conclusion this is the conclusion the subject term is called as a minor term and the predicate term of our conclusion is a major term and then whatever occurs twice in the premises that is B so this is called as middle term this is the first one which we need to identify. So whenever you find a minor term in this premises that is called as a minor premise wherever you will see this major term the major term here is A so wherever you will see this major term that is called as a major premise. So now wherever you find the term A this is called as major premise because you will find major term in A and now this is called as minor premise so now there is a convention that in the standard format you always state the major premise first for example in a categorical preposition this comes first so you need to change it to change to in this particular kind of order based on the middle the minor and major terms of your conclusion so this is in a standard format only and this so now the second one which you need to note is this particular kind of thing so based on how the middle term is distributed so we have this particular kind of thing so what we have said was in the first figure figure one so the middle term is like this middle term occupies the subject position here and of course it does not matter whether it is predicate or subject in all MP S and this is P M S M figure number 2 just to state it like this so now middle term occupies the position of a predicate in all so now the second one is third one is MP M S it occupies the position of a subject and this is figure 3 and figure 4 figure 4 middle term is like this P M and M S so now there are some ways to remember it in all so this diagram goes like this so this is where you have your middle terms are there in the first one and then middle term is here and then followed by that again there is something these two and then of course the middle term occupies this particular kind of position so this is what is considered to be the thing which you take into consideration so this is a diagonal and this is go this goes like this and then you have middle term here and then this and then followed by that you have another diagonal here so with the help of diagrams you can also understand where the middle term is distributed in all so this goes like this diagram goes M M and then this is M that is what we have done two lines in all first of all and then after that this goes like this okay forget about this particular kind of diagram now we need to find out what kind of mode it this particular thing has so now we have to identify the middle term first middle term occupies this position and this one so now this is what is called as middle term is like this M and then this is also here M so that means it falls under the figure one because the middle term occupies this position here and this one and of course in all these cases the conclusion is always S and P subject and predicated because each sentence has obviously the subject and predicate so this one sees we know sees a so this is what is the preposition here this first of all this is an e preposition because no a no B's are a's are no cats are dogs like that and then this is a preposition and again this is a e preposition so now the mood mood of this thing is e a e and then we need to state whether it falls under figure 1 or figure 2 or figure 3 or figure 4 so e a e since the middle term middle term occupies the subject position here and here in the case here it is a predicate position so it looks like that this is the thing and all e a e 1 so now you will see here e a e 1 for example in the first figure a a a e a e 1 that is obviously a valid kind of kind of argument and all this is what is called a Celerant and all so as you clearly see that e a e 1 is a valid kind of argument for example if you change this thing into so that is why this is called as valid according to Aristotle but we did not come to know how it is valid and all for that we need to state rules a little bit later we will state these rules and all in a minute from now so this e a e 1 will tell us the entire thing about this particular kind of Syllows and all so e a e is a mode and then it falls under figure number 1 as you have seen in the thing Aristotle makes this thing as a valid kind of Syllows and all for example for the sake of argument you try to change this particular kinds of things the words here this you keep it like this only instead of this say C and b a randomly you take into consideration this one and of course so this C should not come here because middle term should not come in the conclusion now this is the one which you have so now we need to find out what is the major premise and what is the minor premise etc of course in this case a is a major term and then b is a minor term so whenever you have this minor term that is considered to be a minor premise and whenever you have this major term in the premises that proposition is called as a major premise is called as a major premise so a occurs here so this is a major premise and then minor premise wherever it occurs is usually called as a minor so usually our convention is that you state the major premise first and the minor premise late so now this is all C is a that is the first one we need to re-change it a little bit and the second one is no C is b and then of course the conclusion is same that is no b is a what is b c is a can be anything donkey's cats are any other thing anything you substitute it you will come to know whether it is a valid or invalid kind of argument when you have true premises in a false conclusion obviously it is an invalid kind of argument we are trying to see whether which figure it falls in so now we have this particular kind of standard kind of format so now we need to observe the middle term middle term is occupying the subject position here and then of course whether it is predicate or not it does not matter and subject predicate is always going to be the conclusion and all this can be a subject term or it can be a predicate term and all so it does not matter much and all but we are interested in how the middle term is distributed and now middle term is occupying the subject position in both the premises and all so now this is not the one which you are looking for this is also not the one because it occupied the predicate position and the one which you are looking for is because this seems to be closer to this particular kind of thing so now of course this is a minor term this is a major term and of course now we have written it in order major term major premise and minor premise major premise and minor premise and then of course so now we need to find out the mood of this particular kind of thing so now the first one is an A preposition all C's are A's etc and the second one is no C C is B that is a E preposition these two are serving as premises to us and the third one is obviously the conclusion that is also E and all so now the middle term is distributed like this and something else here some other term so this is closer to this particular kind of thing MM and PS so now this is figure number three so now Aristotle says that now we need to look for whether AEE is going to be valid or not in the third figure so now you will see here in the third figure only OAAI EIO IAAI these are going to be valid kind of forms you do not find AEE corresponding to figure three which is going to be valid and of course AEE is valid in the fourth figure but we are not getting that particular kind of thing the one which we are having is AEE corresponding to the third figure based on how the middle term is distributed so you will not find AEE here so that is why it is an invalid kind of argument maybe you can look for the conditional validity you need to see whether the third figure third figure also you will not find this particular kind of thing AEE kind of thing you find only AIAO etc they are considered to be conditionally valid it is not even conditionally also valid I did not talk anything about conditional validity I will talk about it little bit later when I talk about existential import in a simple nutshell at this moment conditional validity means it presupposes that you are subject for example in the first case first figure AAI and EAO they are considered to be conditionally valid in the sense that the subject term is non-empty so there means there are some kind of subject terms which are actually existing in the world and all so in the same way in the second figure AEO, EAO are considered to be again conditionally valid again based on whether or not the subject term actually exist that means whether or not it is empty or non-empty you will judge you can judge whether they are going to be conditionally valid or not. So now how do we know that these particular kind of categorical syllogisms are valid and some particular kind of categorical syllogisms are invalid so Aristotle has come up with the five interesting rules with which you can judge whether or not the given categorical syllogism is valid or invalid for all these things what is important here is to identify these terms in the syllogism first is the middle term of course the major term and the minor term so these are the things which you need to identify and then you should ensure that your middle term does not occur in the conclusion it occurs only in the premises it occurs twice in the premises. So the first rule is that the middle term of a valid syllogism is distributed at least once in the premises so for distribution we have come up with one particular kind of a mnemonic so that is like any student earning I am writing it in capital letters for the sake of identifying this particular kind of thing. Ending B B grade for example is not on probation so before that we have another mnemonic that is a firmony all these things are very important and then we will use these things a little bit later that means universal propositions like A and I are affirmative and negative propositions are E and O they are considered to be negative kind of propositions so this is the one which is going to be very important in all with this you can come to know what proposition is distributing what so a proposition now you will write it here a proposition distributes subject and whereas that means a distributes yes and now e proposition distributes both that is subject and predicate this is the one way of remembering it in all there is no standard rational kind of judgment in all this is for the sake of remembering this thing we are using this particular kind of mnemonics so e proposition distributes both that is both subject and predicate and then I proposition distributes neither I proposition in a sense some men are mortal some X or Y some cats are animals I proposition distributes neither subject nor predicate and then O proposition in this case distributes only predicate p stands for predicate and O stands for O proposition distributes predicate so now you need to note that this whether or not this particular kind of distribution is satisfactory or not is a very difficult question to answer there are some propositions categorical propositions in which for example it is a universal affirmative proposition a proposition all bachelors are unmarried people and for example if you say that thing it appears to be the case that of course it is an a proposition it will distribute subject only because unmarried people are referring to the whole of it is talking something about the whole class of bachelors and all so it means a s is distributed no doubt about it but unmarried people is nothing but bachelors only so bachelors also seems to be distributed to the whole of unmarried kind of people and all so it appears to us that here both subject and predicate are distributed and all so this is what is going to set some kind of limits to Aristotle in theory of distribution distribution of terms there are some cases in which although it is an a proposition but it distributes both predicate and subject so that is not what we are going to look for but in the in most of the cases what happens is a proposition distributes s and e proposition distributes both subject and predicate I proposition distributes neither of them whereas o proposition distributes predicate that is what is the actual standard theory of Aristotle theory of syllogism so now we apply these five different kind of rules to these two syllogisms and we will see what syllogism is valid and what is invalid so now the first rule is this particular kind of distribution of middle term so this is the first rule so now we look for this particular kind of rules there are five rules which Aristotle could come up with and based on that you can judge whether a given syllogism is valid or not so now we are applying on these two this is one and this is two which we discussed it already but you are applying these rules here so now coming back to this particular kind of thing so now here the middle term first you need to identify these things middle term middle term is what see and then so the other terms are like this so we have other terms a and b which you forget about it you know whether it occupies a subject position or predicate position based on that we have other terms you know first you identified the middle term you know so this rule says that the middle term should be distributed at least once in the premises you know because middle term does not occur in the conclusion so no question of distribution in the conclusion so the middle term should be distributed at least once in the premises so now so what is this proposition this is an e-proposition so now e-proposition distributes both subject and predicated that means the middle term is distributed here so now that means we have satisfied this particular kind of criteria that a middle term should be distributed at least once in the premises and all so it is already satisfied and of course you can look for other thing and all all Cs are a that means here C is the middle term and all now if it is a subject a proposition a proposition distributes only yes that means what is the subject term here C is the subject term so that so C is said to be distributed here since it is a a proposition a term is said to be distributed if it is talking about something which is referring to the whole of that particular kind of class which the term is referring to so C is referring to in a is distributed to the whole of C and all so that is why C is said to be distributed suppose if it is partially distributed etc. and all then that is said to be non-distribution this is the one which we have explained it in the last lecture so now the first rule seems to be satisfied because middle term is distributed at least once in the premises and all so it is distributed here and here also so now the second rule is that no term is distributed in the conclusion which is not distributed in the premises enough so now here B is the term and then a is the term which you are seeing it in the conclusion so no term is distributed in the conclusion which is not distributed in the premises enough. If that means if it is distributed in the conclusion it has to be distributed in the premises as well so now this is a e proposition that means both terms both subject and predicate are distributed and all that means B is distributed and EA is also distributed so now that means in the premises at least these two terms I mean at least one of these things should be distributed at least once and all so now here in this case once so in this particular kind of thing so no C is B again it is an e proposition it distributes both of them so that means B is also distributed so distribution conclusion and premises so if your term is distributed in the conclusion it has to be distributed in the premises also at least once and all so that means B is here and you are distributed and all so this rule is also satisfied and all and there are some other kinds of rules which we need to look for so that is if any valid syllogism has one positive and one negative premises premises then its conclusion is negative and all so here at least one negative proposition is there here so this is because A and I propositions are affirmative and E and E and O are negative propositions so you find one negative proposition here that means your conclusion also should be negative and all in the same way vice versa also the same thing suppose if you find a negative kind of conclusion here then you should have at least one negative kind of propositions in the categorical rule.