 So far we have discussed the basic concepts and in that now we try to identify the arguments and we analyze the arguments in the sense that we evaluated the arguments and then we identified the various kinds of arguments such as deductive and inductive argument and then we evaluated those argument in the sense that we have seen when a deductive argument is valid when it is valid when it is sound and in the case of inductive arguments we came to know about the strength of the argument and then when they are cogent and when they are uncogent etc. So then we presented a model for an argumentation which is due to famous philosophers Stephen Toulmin and then we said that both inductive and deductive arguments can be fallacious when the deductive arguments are fallacious especially we will find it in the case of formal fallacies so inductive arguments can also be fallacious and then we discussed in greater detail with some examples about fallacies of weak induction except. So today we will be presenting Aristotle in Aristotle's logistic logic basically in this we will be presenting something about categorical propositions and then these categorical propositions combined together will form some specific pattern of reasoning which are called as syllogisms then we will present Aristotle theory of syllogisms which is presented way back in the year 3384 BC and 322 BC and then we will see with some kinds of rules there are some rules with which you will come to know the validity of syllogisms and then at the end we will see the merits and demerits of Aristotle logic. So as you see in this slide that Aristotle theory of syllogism was presented long back in 384 BC and the next question that comes to us is why are we studying at this moment and all. So one of the interesting thing is that Aristotleian logics have dominated for more than 1900 years in fact 2000 years so till the 19th century the beginning of the 19th or 20th century these logics are still used and all in various circumstances and all. So his theory of syllogism has had an unparalleled influence in the history of western thought. So he was the first to codify inferences into a system and to create rules for distinguishing correct and incorrect inferences for example if you say all men are mortal socrates is man socrates is mortal that seems to be a valid kind of inference and all and then not seems to be this is a valid inference and all. So on the other hand if you say all men are mortal socrates is man socrates is not mortal so that is a counter instance for this particular kind of thing that is an invalid kind of argument because the conclusion does not follow from the premises. Now how to distinguish the first argument with the second argument so then Aristotle has presented in his theory of syllogism he could codify inferences into a system and to create rules for distinguishing correct from incorrect inferences. So one of the interesting and important thing why we will be studying this Aristotleian logic is simply because of this reason that syllogistic logic is logic are called as syllogistic logic in a sense that he uses a specific pattern of reasoning which is called as syllogism I will come to what I mean by syllogism little bit later a syllogistic logics remained as a paradigm for logical reasoning for more than 2000 years and all. So right from 384 BC to till the advent of modern logics which are due to Frague et cetera Frague Russell Whitehead et cetera so till to that extent Aristotleian logics are still used and all even till to date there are some logicians who are interested in working in greater detail about Aristotleian logic and all. So why after all this is the case that you know we are still interested in Aristotleian syllogistic logics and all so it is considered to be an earliest formal study of logic and you can say that it is an origin of formal logics and all so there is a difference between when you analyze the form of the argumentation then we say that form is what is considered to be the most important thing and all men are mortal socrates is man socrates is mortal it exhibits some kind of valid form so that is why it is a valid argument in the same way all A's are B's all B's are C's so all A's are C's this is exhibit some kind of valid form so that is why it is a valid argument. So in that sense on the other hand we need to analyze the content of the argument to know whether there is any mistake in the argumentation and all for example this room is made up of atoms atoms are invisible so this room is invisible and all so unless until you analyze the content of the paragraph content of the argument there is no way in which you can find out what is wrong with that particular kind of argument. So it is considered to be the beginning of formal logic after all this course is about mostly about the formal logic so it is better to study at least in some detail about Aristotelian syllogistic logic because this is starting point for formal logics the same thing can be done later of course we are going to see in the case of in predicate logics also we can do the same thing so but it has its own modern interpretation etc and all. So one of the beautiful or fantastic thing about Aristotelian logic is that it is close to natural language not much jargons etc are used so it is not there are no high technical stuff involved in this particular kind of thing it is very closer to natural language and famous philosopher Immanuel Kant is the other of critic of pure reason which is considered an important book in the philosophy western philosophy 18th century philosopher is of the view that Aristotel had discovered everything there was to know about logic and everything that historians are pointing about logic you know he is of the view that logic is complete in a in a sense that you know he has discussed most of the things in all because his contribution is enormous his contribution is not in not only in classical logics that we are going to talk about that is predicate and prepositional logic but his contribution is also there in the in the area such as modal logics etc and all. So it is a starting point for understanding these logics that we have that are available at this moment so syllogistic logics one of the important features of this one is that it is closer to natural language so as the name suggests a syllogistic logic what do we mean by a syllogism a syllogism is a logical argument where a quantified statement of a specific form usually it will be a conclusion which is informed from two other quantified statements and all so you should note that we are already using some modern concepts here quantified quantifiers are known only in the only in the 19th century or 20th century at the end of the 19th century or in the 20th century but for the sake of understanding we are using this word quantifiers quantifies are usually all it begins with the statements begins with the all some none etc so these are all statements which starts with the quantifiers for example if you say all Greeks are humans in that all is said to be a quantifier and all all humans are mortal therefore all Greeks are mortal so this exhibits specific form all yes or B's all B's are C's so that's why all A's are C's so a syllogism is a specific kind of logical argument in which it is combined by two logic two categorical propositions I will talk about what I mean by categorical propositions they are same as quantified statements and all so in the while studying the basic concepts we have seen that a proposition is a sentence which can be spoken as a true or false suppose if I say shut the door or dirty cockroach etc and all they are all are not statements in all suppose if you ask what is your name they are all not propositions a proposition can be clearly spoken as either true or false but a categorical propositions they are also propositions which can be stated as the true or false but they are slightly different from that in the sense that all these statements begin with statements propositions etc they all begin with all some no etc sometimes you know you may not come across all etc and all instead of all you might find every each any all these things all these phrases are same as that can be converted into appropriate categorical propositions for example in the for example if you say all Bollywood movie stars are rich no students are Bollywood movie stars no students are rich. So thing is that syllogism is a specific pattern of an argument in which you will find two categorical propositions are leading to another categorical proposition so now the how do we know that one categorical proposition which is which we are calling it as a conclusion of the syllogistic syllogism seems to follow is following from the other categorical propositions and all so in the case of all A's are B's all B's are C how do we know that all A's are C's follows that means how do we know that this argument is valid enough so Aristotle has come up with a theory of syllogism in which he has taken two categorical propositions into consideration and then from that he moved to another categorical proposition which we usually call it as the conclusion of the syllogism so before the little bit of background how did we how did Aristotle come up with this theory of syllogism and all actually is in Aristotle as divided natural sciences into three different categories based on what it is aiming at and all for example the three branches of science one is aiming at truth that is considered to be theoretical in nature mathematics natural science theology is to be dominant in those days so this is considered to be one category or one branch of science so another is based on the aim the purpose is this the action part that is practical in a sense that ethics politics etc commander this particular kind of category and the other one is production that is art rhetoric etc and all that basically productive in nature creativity is involved in this kind this particular kind of thing so now these are some of the branches of sciences and all as you see clearly here I mean you will not see logic anywhere in these things and all so where is logic in this particular kind of list and all Aristotle does not seem to have included anywhere because is of the view that it is there everywhere and all it might be there in the first one second one even the third one as well because it can be used as a tool in mathematics natural sciences ethics and even in art and rhetoric also we might use this particular kind of I mean logic can be used as some kind of tool for all these things so Aristotle has contributed contribution is enormous in varieties for branches etc and all has contributed in physics he contributed in metaphysics and all these things but as far as logic is concerned his works can be combined together can be called as different name which is called as Arganon Arganon is a can some kind of instrument and all so Aristotle use the word logic and his term is used in the sense of analytic sense equivalent to some kind of verbal reasoning and he calls it with the name Arganon Arganon means some kind of instrument which is considered to be prerequisite for all kinds of rational enquiry so that is a reason why in this case rational such kind of rational enquiry you will find it in all these disciplines and all I mean all these branches of science which Aristotle has classified the recent classification may be a little bit different natural sciences and mathematics and then within natural sciences there are a number of different kinds of disciplines and all but Aristotle has classified in terms of the purpose that it is trying to achieve if you are aiming at truth then it is called as mathematics natural sciences theologies etc theology action ethics politics etc there are different classification which we have the modern classification is slightly different from this particular kind of thing so the point here is that logic you will find it everywhere because this is a justificatory tool which can be used which is a prerequisite for all kinds of rational enquiry so Aristotle's works in logic can be classified into six different the six different works which are combined together will form what he calls it as an Arganon so this is a little bit important for us because we should know where what Aristotle has discussed and all so these are the works the first one is category where it is the analysis of terms were discussed the topic of that categories is the terms etc Aristotle in logics are also called as term logics etc in that in greater detail it was discussed and all second is the interpretation where the analysis of statements are made mainly categorical statements etc how to categorical statements combine together and form another leads to another kind of categorical statement that is that is what we find it in prior analytics where he has presented theory of inference and a post posterior analytics he has presented axiomatic structure of science and in the topics they are all so the works of Aristotle in topics he has presented in manual of argumentation analysis of argumentation what is a good argument what is a bad argument etc in these of Elange another work for Aristotle in which he has presented a manual on fallacies it discusses about fallacies you know it appears it is clear that we will be focusing our attention on prior analytics where he has presented theory of inference we are not going into the details of all the other things you mostly we will be studying about some of the things related to categories deinterpretation may be prior analytics in all posterior analytics topics de Sophie they are all interesting they are linked with what we study in this thing but our main purpose is to present Aristotle in the theory of syllogism and then there are rules to find out validity of syllogism and then we will discuss what are the limits of Aristotle in syllogistic logic so prior analytics is the one which we will be referring to to to continue further so what are the basic units of Aristotle in syllogistic logic so in the case of sentential logic the basic units are sentences etc. So in this case the basic units are terms so what is a term a term is a word or group of words which expresses verbally a concept or simple it can be also called as a simple apprehension you suppose if you say a term called heat in all it is corresponding to an object which is in some kind of fire or something like that so a term is considered to be a simple simplest unit into which a preposition and syllogism can be logically resolved and then you should note that not every word is can be called as a term in all. Suppose if I say aabra dabra timbuktu etc in all they are not called as terms in all for not every word by itself is an expression of some kind of concept in all so in that sense it is not called as a term but there are some co-significant words which can also be called as some kind of terms that co-significant words such as all but some because quickly all these things sometimes they can be adverbs sometimes it can be prepositions conjunctions articles articles such as the and etc. For example if you say the term women is an immediate expression of the concept of women so this these are some other things which can be which can come under the category of terms in all. So in the modern notation they represent some kind of sets or classes etc this will talk about little bit later. So if a term is employed in two widely different senses and then we call it as this problem of equivocation in all if there is a shift in the meaning of uses of the term then it leads to some kind of fallacy which is called as equivocation fallacy. So for Aristotelian syllogistic logics the basic units are terms and terms combined and form some kind of categorical prepositions in all so what is a categorical preposition or a statement or statement prepositions there are used in the same way a categorical statement is a statement that relates two classes are categories. So there are two categories in which you know for example if you say all men are mortal men and mortal there are two categories. So this categorical statement relates these two classes are classes of men and classes of mortal beings in all. So this is the modern notation which we are using it class or a set is a collection a class is a collection of collection or set of things in all for example some 50 or 60 students constitutes some in some PHI 142 class or something like that a class consists of some 50 students etc. So what are the categorical prepositions they are simply they are the prepositions which begin with all some none etc and all know some and all the statements begin with this specific kind of quantifier that is all know and some. So there are four kinds of categorical prepositions according to Aristotel they are like this each categorical preposition has specific kind of structure it has subject and it has predicate. So predicate is attributed to the subject for example if you say all men are mortal men men are called men falls under the category of subject and then being mortal is considered to be predicate of that particular kind of categorical statement. So they can be like this A E I O all I mean this can be like this all SRP that means all dogs are animals that is a preposition a categorical preposition e preposition is no humans are donkeys suppose if you say some SRP that is some soldiers are cowards some cells are brave etc suppose O preposition is some SR not P some supporting subatomic particles are not electrons so lots of mnemonics are used in understanding this particular kind of thing based on what the categorical purpose is trying to achieve the quality of the categorical preposition a and I prepositions are affirmative and E and O prepositions are negative. So as this mnemonic says that affirmation ego where you need to see the ovals that are there in the in these two words the first oval is a that means a preposition and the second one second oval that you find is I a and I prepositions are affirmative that is why affirmation and E and O prepositions are negative so with this you can say that you know suppose if you forgotten some somewhere other which one is affirmative and which one is negative then you can use this mnemonic to find out a and I prepositions are affirmative and E and O prepositions are negative. So what do we do with this categorical prepositions in all categorical prepositions is that you know it is a preposition which it links to categories you know so not all the time you will find categorical prepositions in the standard format sometimes it is used in a non-standard format then what you need to do is you have to take some pain to convert these things into the standard format for example if you say saints are prayerful persons always they pray a lot saints always pray for some or for themselves or something so this is not in the actual standard format where your categorical prepositions begin with the all some no etc. So you have to convert it into the appropriate standard format that is converted into the standard format it will become all saints are prayerful persons now this seems to be the standard format of a categorical preposition sometimes we need to take a lot of pain to convert these particular kind of things into categorical that might set some kind of limitation to this Aristotelian logics but in most of the cases you can easily converted convert from non-standard format to the standard format for example if I say a standard chemical substance never is a prologist on so it is talking about only one particular kind of chemical substance a substance a standard chemical substance never is a prologist so this can be translated as no standard chemical substance are prologist and simple things like a thief is caught is that is also not in the standard format in the same way now all men are mortal socrates is man socrates is mortal socrates man is not in the standard format but you need to convert it into appropriate form and all there is someone X that X is a mortal and all suppose if you say so a thief is caught means corresponding to only one thief and all at least one person is caught and all here so that is why we use some thief is caught and all you should not say that all thieves are caught and all from this particular kind of thing or you cannot say that no thief is caught because it is saying a thief is caught and all so this translation will sometimes be simple sometimes with complex sometimes will be painful to translate into standard format so once you convert it into the standard format things will be easier and all one of the important constituent of syllogism is the categorical preposition so what are the parts of different parts of a categorical preposition we are trying to analyze what we mean by categorical preposition so first it begins with a quantifier it all the categorical preposition should begin with all no some etc so sometimes you may not find these things sometimes it may be instead of all you might find every each etc no sometimes can be used in never or something like that some can be used in at least some of the things etc so every categorical preposition has a subject term all men are mortal that means men is considered to be a subject term and it is also predicate term that is mortality mortality is attributed to the subject that is a predicate term and in addition to that we have something called a copula copula is a some kind of Latin word which means binding something here what it seems to be binding is two categories you know so all men are mortal men is one category another mortal beings is another category and what is binding them is what is called as a copula there are Latin words tying up fastening etc these are the meanings of copula so these are the words which you commonly use it as a copula to be was were will be etc all men are mortal are is considered to be a copula or in the same way all men are not mortal are not is considered to be a copula suppose if you say water boils at 144 degree centigrade and on 104 degree centigrade water yes that is a subject is water and then water is such that it boils at 104 degrees centigrade and it boils at that that it boils at 104 degrees is a predicate and water yes is what is called as a subject so every categorical preposition has a subject and predicate and it has a copula as well for example the categorical prepositions whales breathe I mean there is a not in a standard format and all suppose if you say whales breathe it can be written as it is talking about the whole class of whales and all so that is why we can write it as all whales are breathing things so then you know your seems to be converting this nonstandard format whales breathe into the standard format so these are you can analyze categorical preposition in this particular way so a categorical preposition is defined as a statement which unites two terms by verb which is called as a copula and those things which are there are some kind of hypothetical prepositions and all if p then q kind of things it has non-verb copula and all we will go into the details of this thing when we while we are talking about limitations of Aristotle and logics why it fails for some hypothetical prepositions why it is not easy to infer aim place B and A and from that B follows in all using Aristotle and logics because it is difficult to convert sometimes in these prepositions into the actual categorical prepositions so categorical prepositions can further be divided into different different categories can be further divided into singular in a sense that suppose if you are referring to a singular class of objects then it is called as singular suppose if you say this man is a liar socrates mortal or something like that you know this doctor does not give good medicine all these things comes under singular term singular categorical prepositions referring to only one particular object particular things are like this some men are selfish some includes may be at least one and all how it may be more also 10 people 15 20 maybe 30 also but at least one person is selfish and all we can say that some manner selfish or you can say not all men are cowards etc. And universal universal sense every man is fallible I mean everyone makes mistakes etc then it is referring to the entire class of human beings and also it is an universal categorical preposition so it is all these things are based on the extension of subject terminal subject term in every man is fallible is man so this man is referring to all the class of man is in all the class of every man is fallible means fallibility is attributed to whole class of men and all so that is why it is called as universal preposition and there are some other kinds of preposition which poses problem for us that is in translating it into the standard format they are indefinite categorical prepositions suppose if you say woman is fickle men are shellfish beauty is truth all these things comes under indefinite kind of categorical purpose mostly we will use singular particular universal categorical prepositions. So this kind of distinction is based on the extension of subject term it is very difficult to say in the case of indefinite categorical prepositions the extension of the subject terminal so based on the quality categorical preposition will also have subject and predicate and copula these are the things which we have in addition to that every categorical preposition is having some kind of quality as well every categorical statement is a quality and it can be that quality can be affirmative or negative in the case of the last few slides we have said that affirmation ego a preposition and I prepositions are affirmative and O preposition and E prepositions are negative if a statement affirms that one class is wholly or partly partially included in another class then the statement's quality is what is called as affirmative if a statement denies that one class is wholly or part partially included in another its quality is called as negative so it depends upon whether or not one particular kind of class is included in other class partially or fully enough based on that we have different kind of things affirmative and negative categorical prepositions. So these are the things which we have been talking about a preposition simply all SRP and the quantity is universal because all SRP that means it is referring to the whole class so that is why universal preposition A and E are universal categorical prepositions I and O are particular prepositions so and the quality of A and E A and O A and I are affirmative whereas E and O are negative and all yes somewhat it is not written properly but so usually a preposition and I preposition are considered to be affirmative all men are mortal some men are mortal so you are affirming something so if you suppose if you say no men are mortal you are negating this particular kind of thing that means there is no one who is considered to be mortal and all in that particular kind of case some men are not mortal means at least there is one who is not considered to be mortal. So these first we are analyzing the categorical prepositions and then we will make use of it in forming the rules in we will try to understand the rules of syllogism etc a little bit later so a preposition can be represented as all SSP or no SRP etc so there are some variants of A which are not in the standard format whenever you come across these kinds of categorical prepositions you need to translate it into all SRP for example if you have every SSP every IITK student is an intelligent person so suppose if you say that thing we will say all IITK students are intelligent each SSP or if you come across any SSP then also you can translate it into all SSP if anything is an S and that is also P example if I say all cats are animals if anything is in a cat then it has to be an animal also. So things are S only if there are P or only PRS all these things can be translated into the standard form as all SRP in the same way in the case of E you do not find in all the time you do not come across no SRP etc and all but sometimes it will be used in a different sense such as nothing that is an S is a P is the same as no SSP I think is S only if it is not P or if anything is an S then it is not P all cats are dogs for example no cat is a dog if anything is a cat it cannot be dog in the same way nothing is an S unless it is not P all these things comes under the same thing the translation of that one is nothing but no SSP and the same way I and O propositions which you come across sometimes they may not be in the standard format I proposition can simply be represented as some SRP and it can also be at least one SSP there exists an S that is P if you say some something is both S and P and there is an S that is not P etc all these things comes under the I proposition that is some SRP some SR not P is like this at least one S which is not P and all not all SRP that means some which are not P or not every SSP and something is an S but it is not P and there is an S that is not P all these things can be safe translated into an O proposition which are which does not seem to be in the standard format but non-standard forms can be translated into the standard format how to interpret this categorical propositions we so far we have seen A E I and O we classified it according to the quality and we said A and I propositions are affirmative and E and O propositions are negative etc. So how to represent it in terms of this is the modern notation sets and all sets are collection of well defined things in all which can be distinguished in all. So A E I and O may be interpreted as assertions about sets sets are what classes or collection group or universe etc all these things are important for defining the set and a relation between sets and all sets are well defined collection of distinguishable and individual things and all set of cad set of animals etc set of dogs etc where it consists of only dogs and all. So this is the modern notation that we can use we can represent this categorical propositions in this particular way and then I will try to draw a diagram to show that in the modern notation we can represent this A E I and O in a certain way. So when we say that one set is included in the other one one set is included in the other set if the members of the first set are also the members of the second set and all that is the case and you can say that A is included in B usually represented as A is a subset of B one set for example that is all men are mortal mortality is there included in some kind of all men and all. So one set is excluded from another set if two sets have no common members and all no cats are dogs so cats and dogs are different entities so this is quite simple to understand this particular kind of thing they are disjoint kind of sets and all there is no connection between these things and instead of being fully included or fully excluded from each another one can also have partial inclusion and all one set is partially included in another one if some members of the first set are also members of the second set some means at least one is there it is good enough for us to say that it is partially included in the other set in the same way one set is partially excluded from the other set if some members of the first set are also not the members of the second set. So based on this thing the recent notation John when has come up with when diagrams usually we say pictures says thousand words and all so let us try to represent these categorical propositions with when diagrams and then we try to see in what sense they are different from each other so when has used this particular kind of diagrams which are very intuitive which you are using these diagrams one can show whether a syllogism is valid or invalid so let us talk a little bit about when diagram our interest is not in analyzing this when diagrams but you know we are trying to represent this particular kind of categorical propositions in terms of when diagrams so it is like this so when has used this particular kind of classes any two classes when they intersect with each other then it will be like this a and b so now we are trying to talk about a I and O using this when diagram usually a picture says thousand words and if you know about these things with the help of diagrams then things will make give us make our life simpler and so it has four portions first one is this second third and there is someone outside this one usually we draw like this so this is an universal set usually you know it is excluded and all so we do not take into consideration this box in general we do not write this particular kind of thing so that means you know it is bounded by some kind of domain and all so which we do not state it explicitly here so in terms of set theoretic notation suppose if you have this particular kind of thing then what you say is this first portion this first we will talk about the second portion second one is nothing but a intersection b and then it is the first portion is a intersection b complement and the third one is a and then the fourth one is a bar and b bar so these are some of the things which are there in this particular kind of thing so first one is a intersection b because for example if you say all men are mortal then the first one a is referring to men and b is referring to motority these are the two categories which we are trying to relate with the help of some kind of when diagrams you know so this portion referring to a intersection this portion is referring to a intersection b and this is this particular kind of portion till here one so till here that is referring to a intersection b bar that means what it says is a intersection b bar is what is called as emptiness and all emptiness is the one which we are shading it in this particular kind of way so in the same way if you take a bar and intersection b bar and all if that is an empty set then you have to shade this particular kind of portion three needs to be shaded and all so in the same way a complement and b complement if that is an empty set then that is referring to the fourth one then nothing which is this is designed from all these things so based on this particular kind of idea we can draw a E I and O in this particular way so so the first one a preposition can be this is a preposition which is shown as this particular kind of thing so since all SRP means S intersection all SRP means S intersection P should be empty it should be an empty set so what is S intersection P bar this is yes and this will be the subject term and predicate term and this is the thing which is considered to be this S intersection P prime so that has to be empty then we should be shaded this particular kind of portion and all so that is what is referring to all SRP all cats are animals means this particular kind of thing so there cannot be a thing which is considered to be cat and complement of this one is one minus P so that intersection should be empty and all that is referring to all SRP so in the same way this is a preposition and I preposition is like this is I a what is called as O preposition and then you have I preposition in this sense so this is called as I preposition so you put some kind of dot here in between this particular kind of thing this shows that there is at least one X which is S and which is P and all some cats are some dogs bark and all that means if there is at least one dog which is barking and all which is a cat and which also box and all and that will serve your purpose you might ask 100 dogs bark etc and all but what is satisfied by this particular kind of thing that some SRP is this particular kind of thing so this whole class is yes and this is P some X or Y you need to put one particular kind of cross in the in this area so this is what is called as I preposition and so this is called as so now this is called as E preposition I am sorry for this so this is a E preposition so in which is again same thing or no SSP and all so if you if you shade this particular kind of portion then this will be like this intersection P is empty so if that is the case then it is called as no SRP for example if you say no cats are dogs so that means cats if you take the intersection of cats and dogs that should be an empty set that means there should not be any particular kind of element which should be there in that particular kind of set these sets are considered to be distant collection of objects which are arranged in a certain way so AE I and what this is called as O some SR not P so this is represented in this particular kind of set so there should be at least one X which is not in the P and that means you need to put here you should not put this star here that makes it all P are not S some piece are not SNL so that is not we are trying to talk about some SR not P means this particular kind of thing so you should put some star in this particular kind of thing so what is that we have done we are trying to represent categorical propositions with the help of some diagrams and all so now what we are going to see is some kind of square of opposition which is considered to be most important in this particular kind of thing so you have a proposition and e preposition and I preposition and O preposition why I have mentioned is mentioned this in this particular way we are going to see in detail the diagonals in particular the direct arrows are also important so these diagonals are contradictory to each other contradictory to each other and then these are contrary to each other so I will talk about what I mean what we mean by contrary contradictory and this is what is called as subcontrary subcontrary this is what is called as implies that means a preposition implies high preposition in some sense so and thus this is also what is called as implication from no SR no SRP we can derive some SR not P and no cats are dogs I mean some cats are not dogs so why I am mentioning this particular kind of thing because I want to show that a preposition is contradictory to O preposition and e preposition is contradictory to I preposition for example if you say no cats are dogs then suppose if you come up with a preposition in which some cats are dogs that means at least one cat is a dog and all then it is contradicting this no cats are dogs in the same way in this particular kind of case all men are mortal if we can come up with the one particular case in which some men are not mortal at least one person is there is good enough to show that this is false and all so that is why in this particular kind of sense these two are diagonals are contradictory to each other and then we will talk about what we mean by contrary and subcontrary you know so so these are some of the relations which we will commonly come across so why I am drawing this diagram this diagram tells us lots of things and all so we will come to know how these categorical prepositions are related to each other so if you allow for this particular kind of thing from a you infer I and all then you call it this particular kind of thing as immediate inference so immediate inferences are those kinds of inferences in which the conclusion follows from only one particular kind of categorical proposition. So suppose if you this if you say that a implies I and all for example if you say all men are mortal from that you infer some men are mortal to traditional logic that particular kind of inference is allowed so in the same way suppose if you say no cats are dogs from that if you say some cats are not dogs and that is also called as another kind of inference in all and then these two are contradictory to each other so we will get into the details of this one little bit later when I discuss about the relation between a e I and O in greater detail so what we have done is we have represented the categorical prepositions in terms of some kind of diagrams in all so now how do we know that a particular thing is distributed and particular term is not distributed and all so this is the book which we are following Patrick Hurley concepts introduction to logic in 194 page so it is like this that if a certain term is distributed in a proposition this simply means that the proposition says something about every member of the class that the term is designating in all if it is not talking about the every member of the class then the term is said to be undistributed in all if a term is undistributed the propose and does not say something about every member of the class in all, so the term refers to every member of the set it designates in all. All soldiers are brave, that means braveness is referring to all the soldiers in all, distributed amongst all the soldiers in all. So here braveness refers to the whole class of soldiers, but not in the vice versa in all, for example if you say all brave people are soldiers in all, that is not all soldiers are brave is totally different from all brave people are soldiers in all. This here is a term refers to the whole class of soldiers, for example if you say no reptiles are warm-blooded things in all. So any reptile at all is not a warm-blooded thing and there are no warm-blooded things that are reptiles in all. So in both cases subject and predicate are distributed here, so what is the subject here? Reptiles and warm-bloodedness is a predicate, reptiles are referring to whole class of warm-blooded beings and the same way warm-blooded beings are referring to the whole of reptiles in all. So this is what is called as distributed, distributedness in all if it is referring to the whole class, preposition is say something about every member of the class that the term denotes then it is called as distributed and if it is referring to only something about every member of the class it is called as undistributed in all. So why are we talking about this distribution and undistribution, how do we know that what preposition in a preposition whether subject is distributed or predicate is distributed etcetera and all. So there are few mnemonics that we have used earlier Afirmo and Ego that tells us that A and I prepositions are affirmative and E and O prepositions are negative. And then there are other prepositions in which you can talk about this concept of distribution and this is like this unprepared students never pass, it is only for our understanding and then we need to find out the ovals here. So in the first thing U stands for universal prepositions and S stands for subject, N stands for neither of them, P means N stands for negative prepositions, categorical prepositions, P stands for predicate. So universal prepositions distributes only subject and negative prepositions distribute predicate that is not good enough and is not giving us full details of what in a categorical preposition what term is distributed and all. So the best thing would be this particular kind of thing, any student earning these is not on probation. Now we need to look into the ovals here, the first letter is A and immediately followed by that S is there that means A preposition distributes subject. Now that is the first two words and all with which you can find out that A preposition distributes subject and E preposition distributes B means both and I preposition distributes neither some men are mortal means there is no term is distributed in that particular kind of thing. And O preposition distributes predicate that means E preposition distributes both of them both subject and predicate and O preposition distributes only P and all this is the way in which you know when ancient past they have used lots of mnemonics to identify all these shortcuts and all are hidden in this particular kind of thing we are going to see some more mnemonics little bit later. So far we have defined what is categorical, now we have seen what is a categorical preposition what it consists of etc and all how to represent it in terms of diagrams etc and all so that we can easily understand these things etc and all. So now coming back to a specific pattern of argument in which you will find only categorical prepositions, if it is not in standard format you need to convert it into a standard format. So a categorical cellulism is a deductive argument having a sequence of three and only three categorical prepositions that means two will serve as premises and other one will serve as a conclusion such that three and only three terms appear in a sequence of statements each term appearing in exactly two prepositions in all. So these are some of the things which you will find it in categorical prepositions you will find subject term predicate term and the middle term and all. So the way in which these terms of cellulism are arranged is what is called as figure of that particular kind of cellulism. So in this lecture what we have done is simply is that we are trying to talk about Aristotelian cellulistic logics what is considered to be important is the cellulism a cellulism consists of categorical prepositions now. So in this lecture we analyzed what we mean by categorical prepositions and we represented this categorical prepositions in terms of we have given some kind of diagrammatic representation which is due to John Wen is a mathematician on with which you know how this A, E, I and O prepositions are related is the one which we have discussed and we have said that A and E prepositions A and O prepositions are contradicted to each other I and O prepositions are contradicted to each other and then we have also said that sometimes you know from A preposition you can infer I preposition and the same way E preposition you can infer I preposition and all in the some limited kind of sense. In the next lecture we will be talking about Aristotelian actual theory of cellulism and then we will try to find out some kind of rules for the validity of cellulism. And we will continue with the validity of cellulisms in the next lecture.