 Welcome back in the last class we presented Aristotle theory of syllogism, where we discussed in extensively about the validity of syllogisms and we presented five rules for the validity of syllogism with which one can come to know what kind of syllogism is valid etcetera and all. So the rules are like this that you know how when middle term needs to be distributed at least once in the premises nothing no term is distributed in the conclusion which is distributed in the premises and if you have two negative premises nothing can be inferred in the same way if you have two particular propositions that means I propositions there is no way in which you can infer anything. And one of the final rules is that which is little bit controversial that is this that in Aristotleian logic if there are two universal propositions yet you can infer a particular kind of proposition and all. So this is not permitted in the modern logics because so we will be borrowing existential import into the conclusion which is actually not there in the premises and all. So this leads to existential fallacy. So we have been discussing Aristotleian logics which have dominated for more than two thousand years and then it served as a paradigm for this logics and all. So there are certain important features in Aristotleian logic they are this that they are closer to the natural language and then the rules are easy to apply but it has its own limitations and all. So in this class what we will be doing is we will be continuing our discussion with this famous syllogistic poem due to Aristotle and this poem conveys us lot of information and all. So what information it conveys is like this. So this is the syllogistic poem that we have. It is like this Barbaras, Celerand, Dary, Firiok, Q. So this stands for the four syllogisms that are valid, unconditionally valid in figure number one. Cesare, Chemistres, Festino, Baraco etc. they are all valid in figure number two. So these figures are these figures are formed just based on how the middle term is actually distributed and all. So based on how the middle term is distributed Aristotle classified into four figures and out of each figure there are 64 moods possible for each figure and out in total there are 256 such kind of moods are possible and out of that only 15 are unconditionally valid and 9 are said to be conditionally valid. So we will try to analyze this syllogistic poem with which the people in the ancient past in the Greek period they remembered everything based on this particular kind of poem. So each word let us say if I say Barbaras we need to look for the ovals and the consonants and all. For example in the case of Barbaras the ovals are A, A, A that means it is an A, A, A kind of syllogism. That means there are two universal prepositions A prepositions and we have another kind of preposition A which is considered as a conclusion. So for example all x are y, all y are z and all x are z that comes under A, A, A kind of preposition and then we a celerant means we have we have to look for the ovals here that is E, A, E and then not only that thing right from the second stanza and words that is cesare, chemistries etc. and all. So this consonants also conveys as some kind of information. According to Aristotle only the moods which fall under figure number 1 are considered to be perfect moods whereas the ones which fall under third and fourth figure are considered to be imperfect in all even the second figure as well. So that means so we will be talking in this class about the reduction of syllogism in all. Reduction of syllogism in the sense that whatever falls under figure number 3, 4, 2 etc. they all can be reduced to the moods in figure number 1. So there are some rules for reducing this syllogism into the syllogism of the first figure. So why Aristotle considers as figure number 1 as considered to be a perfect figure because it is in the sense that the middle term is nicely distributed in the first figure rather than the other figures in all. So this is the way in which the middle term is distributed and I will look into this syllogistic poem in greater detail. So the distribution of middle term is like this. In the first case in the first figure we have like this and these are some predicate and it is a subject here. This is figure number 1 and figure number 2 we have this thing we have mm here which occupies the position of predicate and then in figure number 3 we have middle term here which occupies the position of a subject and in figure number 4 we have mm. So these are the 4 figures which Aristotle could think of. So in each figure there are 50, 64 moods which are possible there are 256 such kind of syllogisms out of that only 15 are conditionally valid and 9 are unconditionally valid and 9 are conditionally valid. So now according to Aristotle so these are the things which are valid in figure number 1, Barbara, Celerand, Celerand, some D, A, R, I, I and then Ferry, E, U, Q. So that is E, I, O preposition. So now what we need to do here is to look for the ovals here, A here, A here, A here. So that means A, A, A and this is with respect to figure number 1 so that is why we have written 1 here. So this gives us complete information about what kind of mood it is I mean and then it falls under what kind of figure and all. So this is considered to be the perfect figures and all and there are some other things which fall under this one. All these things can be reduced to this particular kind of thing and all. So now this is E, A, E. Now we need to look for the ovals here and then here in this case A, I, I, I and then in this case E, I, O and of course this is figure number 1. So now what we will be doing the next 10 minutes is this that we will be trying to reduce the syllogisms that fall under figure number 2, figure number 3, figure number 4 and we will try to convert it into the standard, the perfect moods which fall under figure number 1. So now this syllogistic poem conveys a lot of information and all. Starting from the second line let us say if we have something called Cisare for example we have this particular kind of thing Cisare. So this is E, A, E preposition, E, A, E preposition and it is in figure number 2. So now according to the syllogistic poem this over this consonants also conveys some kind of information. First you need to look for the ovals that is E, A, E preposition and it falls under figure number 2 that means middle term should be occupying the position of a predicate in all in this case. So now what needs to be a case is that if you find any word which starts with C in all this can only be reduced to a syllogism which starts with the letter C in figure number 1. That means Cisare can only be reduced to Cilerant. So now we are trying to see how Cisare can be reduced to Cilerant in all by using some kind of rules. So this is what happens in case of this thing. So I will work on this particular kind of thing here. So what is Cisare? So this is a Latin term in all we not have to worry much about it we need to worry about the ovals here so this is E, A, E and 2 and then these C, S etc conveys some kind of information in all. So the letters that are of importance to us are like this. So S stands for simple conversion in all for example if you have no X or Y you can convert it into no Ys or X and then suppose if you have a letter M that means you need to interchange the premises little bit and then the other letters that we have are S, M and then right now we do not have any such letters here. So then C is the letter which you will find it little bit later and then we will talk about that particular kind of thing. So this happens due to and there is one more letter which is called as P which talks about per accidents in all which we will talk about little bit later. So all the consonants also conveys some kind of information in this particular kind of Latin word that we are trying to use. So what is that we are saying we are saying simply this that Cisare is the term which occurs in the second stanza of your syllogistic poem that means it occurs in the second figure that means it can be reduced to the one which is having the same which is starts with the same kind of letter in all that means Cisare can only be reduced to Cilerin in all it cannot be reduced to any other Latin word that which occurs in figure number one that is Dari, Ferriocubar, Barbara etc. So that is the reason why they have chosen these letters carefully in all Cisare can only be reduced to this one. So now we will see how this can be reduced to Cilerin that is in figure number one. So now what is Cisare is like this no a's are these and all a's are these and then no Cisare no Cisare a so this is the way in which it is the middle term is occupying the position of some kind of predicate in all here. So the structure of this one is like this so here is the middle term and again so this is the one so that is the reason why this is in this falls under figure number two. So now we are trying to reduce this thing into this one and then how do we reduce it again we need to observe this Latin word carefully everything is hidden in this particular kind of information. So immediately following ith oval, ith oval means it here in this case the first oval immediately after this we have a letter called S, S stands for simple conversion. So simple conversion we will be talking about these three rules a little bit later conversion aversion and contraposition conversion applies to only e preposition and I preposition. So we will see a little bit later so what we will be doing is this thing so we have a letter S here and then we need to change this premises a little bit late I mean so now so observe the predicate of your conclusion this is the major term and C stands for minor term and all. So wherever you find major term in your premises and all that is considered to be a major premise and where ever this minor term occurs that is considered to be a minor premise. So here A occurs no one second no A's are B's no C's are B's sorry no C's are B's so this is the thing. So this occurs here this is a major premise minor premise and this is the conclusion now. So how this gets converted into sealer into now. So now the first thing which you need to note is this thing we are applying some simple conversion rule no A's are B's can be converted into no B's are A's because this C's are A in this C's are A it says clearly that you need to use S rule. So S rule is that you have to make some kind of simple conversion and then other things you keep it like this only now so this is same no B's are A's all C's are B's and this is no C's are A's. So now what we have done is we applied simple conversion rule here that kind of information is coded in this particular kind of word and all because immediately of starting after the ith oval ith oval means the second oval here the first oval here we have a letter called S stands for simple conversion we need to convert one of these premises into its converted kind of thing and all that is E kind of rule and all simple conversion rule we need to use. So now this becomes like this now you will observe the middle term here so middle term is here and you have a subject something here and now middle term occupies the position of subject here and it occupies the position of a predicate here. So now we have converted this C's are A 2 to Cilerin 1 by using the simple conversion rule. So this is how this figure number 1 the moods which fall under figure number 2 which are considered to be in perfect moods can be reduced to the perfect moods suppose if it so happen that you will come across another kind of thing let us say chemistry or something like that. So this is the one which you have then this again can be reduced to only Cilerin and all because the corresponding letter that you find it in figure number 1 is Cilerin so chemistry can also be converted into the Cilerin kind of thing and all. So what do we get out of these things so it tells us how something which is considered to be an imperfect mood can be converted into a perfect mood by using some kind of rules which are simple conversion per accidents and some other kind of rules which is called as may M rule which is talking about some kind of simple conversions and all. So here is how we make these conversions and we will go into the examples little bit later. So the first letter of the Latin word that you have seen earlier corresponds to one of the perfect moods that is needs to be reduced and all for example if we have letter B then it will be reduced to Barbara if you find letter C in the second and third kind of stanza that you have seen there in the poem then it can be reduced to Cilerin. Suppose if you find any Latin word which starts with D that can be reduced to Dari and if you have any letter F it can be reduced to this one for example in this Cileristic poem let us say you consider that is Dati si so that can be reduced to only Dari i that means the one which is in the fourth figure can be reduced to the first figure that is a ai proposition with respect to the first figure in the same way if you find Cami any yes and all so that can be reduced to only Cilerin and all the first word tells us to which it can be reduced and all suppose if you find Daimaris for example it can only be reduced to Dari i so that is the first impression that we get from this Cileristic poem this is a very interesting poem and all it conveys as lot of information and all see tells us not only what kind of mood that Cilerism has and it also tells us I mean how this can be converted into the perfect moods and all which occur in figure number one so now the letter s after ith oval it can be first oval it can be second oval that occurs in that particular kind of Latin word indicates that corresponding preposition needs to be simply converted and all so that means no x or y can be converted into no vice or x some x or y are converted into some vice or x and all however it does it won't apply for all x or y all x or y is different from all vice or x in the same way some x or not y is different from some not x or y it will not apply to o preposition and e o preposition a preposition it will not apply there so now if you find a letter p after ith oval and the corresponding proportion has to be accidentally converted so this rule will talk about it a little bit later so that p rule is this that for example if you have all x or y you can change it to some vice or x and all cats or dogs that means some dogs or cats and all so this is a little bit objectionable to us but still Aristotle follows these things from all x or y you can say that some vice or x and all that is the case it is called as per accidents kind of rule suppose if you come across after ith oval maybe second or third kind of thing if you find a letter C not in the beginning and all but after some ith oval once you come across an oval and after that you find a letter C and the second oval indicates that the mood has to be proved indirectly by using contradictory of the corresponding premise so what you will do is you take the conclusion you will take the negation of the conclusion and you will add it to the major premise and then you will come across a contradiction so if you come across a contradiction then whatever you assumed is wrong and all so in that case conclusion has to follow from the premises it is like some kind of reduction add absurdum method so what you will do if you are asked to prove something first you will take the negation of the conclusion and then you will show that some contradiction arises out of it if the contradiction arises then you will say that negation of the conclusion is false that means the conclusion has to be correct so this is the one which we use in mathematics that is direct show add absurdum method so so this is what we do when you come across letter C after ith oval not in the beginning so now what will happen if you come across M in the case of chemistries C A M E N E S where you will come across M after A so then what you need to do so this realistic poem again tells us the coded language and all it tells us the letter M indicates that the premise have to be interchanged so that means you will see where the major premise occurs in all major premise always it should be stated first and followed by that you have a minor premise and then you will have a conclusion in all so usually it is an interchange of premises in all and nothing much is involved in that particular kind of thing so all other letters such as T other letters P etc all these things which you have seen in the syllogistic poems they are only used for some kind of acidic for position all is only for remembering that particular kind of word we will be using this particular kind of thing and so although Aristotle has no formal axiomatic system and all but still you know it is a beginning starting point of formal logics and all Aristotle system still has some kind of axiomatic it can be called as a axiomatic system in a in a weaker sense so it has these four axioms four axioms in a sense that you know whatever falls under perfect mood that means figure number one Barbara Cillarendry and Ferriot in corresponding to that so these four are considered to be axioms of Aristotelian syllogistic logic so what are considered to be axioms axioms are considered to be self-evident truths which do not have to be proved in but you have seen till now that all the syllogisms that fall under figure number 2 figure number 3 4 and all there all can be reduced to figure number 1 but whatever occurs in figure number 1 whatever the syllogisms that you observed in figure number 1 they cannot be further reduced and all so it is in that sense it retains this axiom status in all axioms cannot be reduced further it cannot be needed how to be proved they are all self-evident kind of truths so these are like this Barbara means a preposition all A's are B's all B's are C's then all A's are C's and the other kind of axiom is this thing Cillarendry that means no A's are B's and all B's are C's means no A's are C's Dury that rule says that all A's are B's some B's are C's and then some A's are C's in the sense Ferriot can also be read like this no A's are B's some B's are C's and then some A's are not C's so this is what considered to be some kind of axiomatic system of Aristotelian logic but it is not so rigorous like the one which you will see later in the case of Russell Whitehead axiomatic system or Hilbert Ackerman axiomatic system which we are going to see while dealing with the meta logic a little bit late so these are considered to be some of the axioms of Aristotelian logic because it cannot be reduced further into any other kind of axioms in all so these are the conversion rules which we were talking about so this is a simple conversion rule X I Y that means some X are Y which is similar to some Y's are X some cats are animals that means some animals are cats is one of the same so in the same way no cats are dogs that is X E Y is similar to Y E X that means no dogs are cats so this is a rule which we use that is P rule per accidents kind of rule that is for all X are Y you can convert it into some Y's are X and this is little bit difficult to follow but this is a rule which Aristotle allows and the fourth rule is simple conversion that is no X or Y can be converted into some X are not Y so these are the conversion rules so which we use and these rules which we will be using for converting this thing into this particular kind of thing so let us consider one simple example how this particular kind of thing can be reduced to another kind of thing so now we will see how chemistries for example which occur chemistries can be reduced to which occurs in figure number 2 can be reduced to celerant of figure number 1 so this is what is chemistries so now we need to observe the ovals here AE E this is AE E preposition and then based on how the middle term is distributed we need to say what kind of figure belongs to so it appears that this falls under this figure number 2 where the middle term occupies the position of a predicated so this is like this all is our bees no seas are these now you will see middle term here in the occupying the position of a predicated so now this conclusion is no seas are AE is AE so this is what we have so now you will see clearly here that this is the middle term and whatever occupies the whatever occupies predicate of the conclusion should be the major term this is a minor term and wherever you will find C that is considered to be a minor premise that means no CRB is a minor premise and then wherever this term A occurs here so that is why it is a major permission so it is arranged in this particular kind of form and so what is that we are trying to do with this particular kind of thing why we have taken this thing this occurs in figure number 2 so now we should be in a position to reduce this thing into the corresponding kind of word which occurs in figure number 1 so the first letter is C that means this can be reduced to only Celerant what is Celerant here this is EAE with respect to figure number 1 so now this AEE should reduce to EAE 1 but EAE 1 cannot be reduced to any other thing in all it is in that sense they are considered to be axioms in all where these first two are premises and the other one is called as a conclusion it is always valid kind of thing so so now what are the what are the consonants that occurs after the letter A this is the first oval that means ith oval after that we have a letter M so now M rule says that we need to interchange the premises in all so what we need to do here is like this so this cannot be changed in all so for that what we need to do first is you need to look for a proposition where you can apply some kind of simple conversion and based on that you can change the premises in all so now first what you will do is you will apply some kind of simple conversion rule because it is same as this one no is sees no sees are is the same as no is are sees so now what happened here is this thing that this is a major term now this is a minor term so now M rule is the one which we need to apply of course immediately followed by a e preposition we need to apply this S rule so immediately followed by this e preposition we need to apply S rule here that is what we have done here that means we have converted no sees are is to this one so then we look back and then we will apply M rule M rule says that now we need to interchange this premises in all interchanging the premises in essence that major premise should always come first followed by that you have a minor premise so this one wherever C occurs it is a major premise now right now that means this should go first and this should come late so now no sees are these now this will become all is all is are these so this is step number two step number two so now what we have done we have applied M rule here and S rule here and again you have to apply there is one more letter here yes that means you need to apply S rule again so now we need to apply S rule for this particular kind of thing so now this changes to now not this one so it goes like this so now this changes to no bees are sees and then the rest is same and all all is are be and then you keep it like this only no is are sees so now you will see here clearly this is an e preposition and e preposition now we need to check whether this falls under figure number one or not how do we know that it falls under figure number one it is based on how the middle term is distributed and all so now you will see clearly here middle term is like this of course there is a term here the term is C and there is one more term here that is a and all which occupies the subject position so now what is that we have done based on the information that is coded in this one is simply this thing first in the first step what we have done is we change we applied some kind of simple conversion rule to whatever follows after the e preposition and all so e preposition occurs here and then after that you need to apply S rule and all here so with that no sees are is are converted into no is are sees so now once you converted into this thing then you need to reshuffle this premises and all why we need to reshuffle the premises because it is a convention that major premise always should come first so now based on this information that means C is the major term right now wherever see occur that should come first but in this case it came second and all but interchanging the premises that is what we mean by M rule so we reshuffle the premises and all without violating the truth of this categorical preposition and now this becomes like this no sees are bees all is are bees so now again there is one more operation here one more consonant S here we need to look for only these four letters and all S M P and C so these are the letters that we need to look for especially the consonants that we need to look for and the other things which you need to look for our ovals and all which tells us what kind of mode the preposition is in so then what we did was you can still apply simple conversion rule to this one so now no sees are bees are converted into no bees are sees it is like no cats or dogs that means no dogs or cats let us say one of the same and all so now we kept this thing as it is now we reduce this thing into this format and all so now what we have achieved is simply this that chemistry is considered to be the one which occurs in figure number 2 that means EAE2 can be reduced to EAE1 so like this many things can be converted into this particular kind of thing and all there are some other examples which you which we can take into consideration so these things can be reduced to this particular kind of thing and all so this is the M rule so M rule cells tells us that shifting of major premise in the place of minor premise and all so then we apply S rule to e preposition which occurs in the major premise that is what we have done in the case of chemistry is in all for example in this case how do we reduce from Festino to Ferio, Ferio occurs in figure number 1 Festino occurs in figure number 2 again so because middle term is occupying the position of a predicate in both the things and all no is is be some sees be et cetera some sees is not a then what you need to do here is that immediately following e we have a letter called S stands for simple conversion rule and then after that there are no other consonants that we can will be interested in that means T and N does not convey any information now we need to look for only S M P and C and all so now no is is be in the first premise is by simple conversion is converted into no B's are a's and all and you keep the same thing some sees are be and some sees are not a and then it changes into the Ferio part which is which falls under figure number 1 so like this we can convert things into corresponding Latin word in which occurs in figure number 1 so in some cases things would be little bit difficult and all like suppose what happens when you come across a word let us say C rather than this one so now let us consider one more example in which instead of you know you come across instead of S and M you come across a word a letter C and all so that means the move the syllogism needs to be proved by using contradiction rule indirect method we can use in particular so now we are trying to convert to something like this can only be reduced to because the first letter is B and then it should be reduced to this thing only just let me go into the details of this one Bacardo occurs in the fourth figure so this is the fourth one and then it should be reduced to it can only be reduced to the first letter which occurs in figure number 1 that means the letter that starts with B is only Barbara kind of syllogism so now this can be written like this aob that means some is are not bees and then this is Cab and a just second so aob and Cab this is not in this particular kind of format this will look into some kind of example with which we can come to so let us try to convert actually this should be in this particular kind of format if it is in the fourth figure the middle term should be here but here it occurs in this one it occurs in this one we will change it a little bit and see what is the case this is okay aob the fine and then this should be Bac so Bac and then this converts into this one so now what we need to do here is this thing so now what you came across after this oval is the C whenever you come across a letter C that means this syllogism can only be proved by means of indirect methods that means you can only prove with the help of a contradiction so now so what you will do here is this particular kind of thing so just one second now this is a major term and this is a minor term and whenever wherever major term occurs that is a major premise and minor premise minor term occurs that is considered to be minor premise so now what you will do is there is a some kind of thing which we follow A E I and O in this one so A and O are opposite to contradictory to each other and E and I are contradictory to each other means diagonals are contradictory to each other so now what is contradiction in contradiction to AOC that is a preposition so a preposition is the one which we take into consideration a stands for all is a season so now what we have taken is this that you have denied the contradiction and all and that is what we have taken into consideration and then added to that then you have to add it to the major premise here the major premise is the one in which the major term occurs so now this is what you need to see all A's are C's all B's are C's so now so what is the conclusion that we get let us say all A's are C's for example all A's are because C is a middle term it should not occur so now this should be the case all A's are B's so now this one all A's are B's is wherever the minor term occurs minor term occurs in this one this particular kind of premise AOB so now this and these are incompatible to each other so denial of the contradiction leads to some kind of conclusion which is contradictory to the minor premise so what is that we have done we are just trying to prove by contradiction that so this is the conclusion that follows from this particular kind of thing AOB and BAC only this follows from this one so since we have come across rule C we are stating that this can only be proved by some kind of indirect method now what is the indirect method first what we have done is we take we took the negation of the conclusion as your first premise and added it to the major premise and then we let to it leads to some kind of conclusion which is incompatible with this particular kind of thing AOB that means your premise is wrong and all that should be this one AOB and BAC and AOC so now how it gets reduced to this particular kind of thing is what we need to find out so so in one particular kind of thing we will try to prove this particular kind of thing you know so in the case of this particular kind of thing we can prove this is what is a bocado which can be proved in this particular kind of way so this already can we have converted into some kind of a proposition and all so now the only thing is that we need to see so now let us just let me finish this particular kind of thing this is not still in a preposition one and all so now we need to apply some kind of rules which we need to use so that is this thing instead of bees you take into consideration letter C and instead of C you take into consideration what happens here this should be like this middle term should be like this you take C as a and then B wherever B is there you replace it with C and all so now this becomes this a now so once again now this will not apply here so this is an a preposition but somehow this has to be converted into a a a 1 we need to use some kind of rule so that we can convert this thing into a preposition now so but in this case what we have done simply is this that first we have taken the negation of the conclusion and this is what it is the case and then we added it to the major premise and we showed that we got this particular kind of thing which seems to be contradictory to your minor premise that means AOC should be wrong and all it should be that means a negation of the conclusion leads to contradiction that means you cannot negate the conclusion and all AOC follows and all from this particular kind of thing that means we showed that this conclusion follows from the premises by using some kind of indirect method and now so now we will move on further little bit and then we will see what are the other things which we can do based on this particular kind of thing so this is what we have done already so now all B's are C's all A's are C's now all A's are B's so now ultimately this reduces to this particular kind of thing all A's are B's okay so now so far we discussed about how the syllogistic poem behaves and all there are certain things which we will still need to be discussed in greater detail especially whenever the letter C occurs and all how do we prove how Bocchardo kind of thing can be reduced to Barbara etcetera all it needs to be dealt in greater detail etcetera so now there are some kind of immediate inference rules that means if you have A, E, I and O how this A is converted into some kind of I proposition or E is converted into I proposition etcetera and all so that is what we come to know in these three rules and all so they are these two three rules are like this conversion rule the converse of a standard form of a categorical proposition is formed simply by interchanging the subject to predicate thing and all wherever you have a subject you replace it with the predicate then it will become a conversion kind of rule only E and I propositions can be converted that means suppose if you have a proposition called no cats or dogs it can be converted into no dogs or cats and all in the same way some cats or dogs means some dogs or cats and all but it cannot be applied to O proposition and S proposition because the meaning changes in all so that is what happens in all this is what we have in the standard form all SRP so this no SRP and some SRP and all these things are converted into this particular kind of thing all SRP is different from all P's are S that is why it cannot be reduced to all P's are S some P's are some SRP is converted into some P's are S that means same as that particular kind of thing and in the same way some SR not P is converted into some P's are not SNR which is which is totally different from some SR not so now conversion applies to only E and I proposition now there is an important operation which is very important especially when whenever your syllogism C is not in standard format we need to apply these rules in all for example if you say all cats are non fish and all for example if you say that particular kind of thing non fish is referring to some kind particular kind of class which are completely excluded from what we call it as fish and so 1 minus whatever you consider as fish and all that it constitutes non fish and all so the aversion consist of two steps and all so what you will do in the aversion is first you will change the quality and all for example if you have all X or Y you change it to no X or Y and all in the same way if you have some X or Y you change the quality to some X or not Y and all in the second step what you will do is you replace the predicate term with this corresponding complement and all for example if you have a letter called fish you replace the letter fish that is in the predicate with non fish and all are non cats non dogs etc the complement of X is this thing complement of X is a class containing all things that are not members of X and all 1 minus X is considered to be the complement of this one. So the term complement is a word which our phrase which denotes the class called complement and all for instance donkeys if you say and its complement is non donkeys and all in the same way cats means non cats and all the ones which are not cats is considered to be non cats and all 1 minus that particular kind of class and all so aversion consist of two steps first you change the quality all X or Y is to no X or Y and then it changes to no X non Y and all so this is what happens all SRP changes to first to no SRP and in the second step it changes to no SR non P in the same way E preposition no SRP in the first step it changes to all SRP and all in the second step it changes to all SR non P because predicate is replaced by its complement and all so non P replaces P in the same way in the case of some SRP in the first step of of the aversion it changes to some SR not P now in the second step we need to replace P the letter P with non P and all so that is why it becomes some SR not non piece and all so this is the one which happens and aversion applies to all the categorical prepositions and all that means you can have immediate inferences based on aversion in this format you can be changed to aversion that is no SR non P it is one of the same in all aversion applies to all kinds of categorical prepositions there is a third rule which is called as contra position rule contra position rule is formed just by replacing the subject term with the term complement of its predicate term and we replace the predicate term with this complement of its subject term so it includes two steps and all for example it is simply like in all of contra position and all P implies Q means implies not Q implies not P and all so for example if you have something called all SRP so there are four steps for a to its corresponding contra position and all so this changes to first thing is used aversion rule so then it becomes you have to change the quality of this one then you have to put complementary of this particular kind of thing no SR non piece in all so now we use some kind of conversion on this particular kind of thing because no SR non P same as no non piece are SNL so that is why the step number three no non piece are SNL now we have converted piece into SNL I mean we replace subject term with the predicate predicate term with the subject in all so now in the fourth step here all non piece are non SNL now again we used aversion rule and then we converted into this particular kind of thing and all non piece are non yes so what is that we have done here there are some four steps for a to be converted into its corresponding contra position but in simple terms we have used all kinds of operations here ultimately we converted all SRP into all non piece are non is which is considered to be the contra position of that particular kind of thing so contra position is valid for only a and O preposition that means suppose if you infer from a you infer all non piece are non is that is considered to be valid and from the O preposition if you infer some something called some non piece are non SNL for example from a preposition some SRPs you infer some non piece are non SNL so then also it applies in some non piece are not non SNL in that case it is considered to be valid in all so there is a way to memorize this particular kind of thing the dots over that particular kind of thing is the one which you need to take into consideration which is the first one is considered in our rule which works for all kinds of four standard forms in all so any proportion can be reduced to its corresponding abortion that is considered to be a kind of valid inference and conversion we need to see the ones the letters that are with double dots and all it applies to only E and I prepositions there is a way to remember it so we need to observe those ovals which are with stars and which dots are there in the contraposition it applies to a preposition and O preposition so in the letter contraposition you need to observe the ovals which occurs there of course the O is the O which occurs here but you need to ignore that one it just for the sake of remembrance only there is no criteria which is used here for the sake of memory we are using this particular kind of thing in all so this is what is considered to be a square of opposition where the diagonals are considered to be contrary to each other whereas the ones which are at the same level are considered to be the first level it is considered to be contrary to each other and then the square of opposition will be like this so it can be explained in this particular kind of thing in all if A is true then obviously E has to be false and all that means A and E are contradictory to each other if I is true that is some X or Y is true then obviously it is negation some X or not Y is obviously it has to be false and all so if A is false then obviously O has to be true because A and O are contradictory to each other so in the same way this tells us how A E I and O are related to each other in the same way if E is true then obviously its contradiction A has to be false if I is false then O has to be true and if E is false then its negation its contradiction that is I which occupies the position of a diagonal which has to be true and if A is true I if A and O A is true and O in the case of O it is unknown so this is what happens in the case of square of opposition in simple nutshell I will end this lecture by stating that these are the some of the important relations between A E I and O so this is the famous square of opposition so which is like this first we need to write all the universal prepositions like this A E and we have I and O and so A and O are contradictory to each other I and E are again contradictory to each other and then there are some other kinds of relations between these two things so these are counter is contrary to A and E are contrary to each other and I and O are called as subcontrary and then this is called as implication in all is subaltern superaltern etc depending upon the arrow which is there here so that means all these things are related in this particular kind of way so now we quickly need to know what we mean by contrary contradictory etc this tells us how A and E are related to each other so now in a quick nutshell so two statements are considered to be contradictory if both cannot be both can contradictory prepositions cannot be both true they cannot be both false as well that means one preposition is true another proportion has to be false contrary prepositions cannot be both true but they can both be false and so these things can both be false and if one is false another one can also be false but both cannot be true and but in the case of subcontrary prepositions I and O so the in this case it can both be false but it cannot be both be false but can be both be true and that is some cats are dogs and some cats are not dogs can be both true and all but both cannot be false and subaltern must be true if it superaltern is true and the superaltern is false if the subaltern is false and all so these are the relationship between A E I and O this tells us how these categorical prepositions are related to each other so in this lecture what we have seen is simply this that we have seen we have analyzed the syllogistic poem in greater detail and then we have seen how one in perfect mood can be reduced to another one and then we also discussed about three important operations aversion conversion and contraposition and then we have seen what we mean by contradictory contrary and how when we say that it is subcontrary and when it says subaltern etc and all so Aristotle in theory of logic gives us some kind of greater analysis of this categorical propositions but it has its own limitations when it when it comes to hypothetical syllogisms are some kind of complex kind of syllogisms which involves more than three terms Aristotle in logics may not work modern logics there are certain things which are easy easy to do in modern logics in all so with this we will end this lecture.