 Welcome back in the last lecture we just presented one axiomatic system we just introduced what we mean by an axiomatic system and what should an axiomatic system should consist of. So these are the things which we have discussed in the last class today I will be presenting an axiomatic system in the prepositional logic that is due to Bertrand Russell and Whitehead. So we will be taking into consideration a portion of the famous book philosophy of Mathematica where Bertrand Russell and Whitehead talked about deduction. So we will be focusing our attention on that particular portion of that book and then we will be trying to present this axiomatic system in the best possible manner. So any axiomatic system should have these three things at least. So it should have to begin with it should have some axioms which does not require any proofs so they are like self-evident truths etc. And then these axioms I mean if you use some kind of substitution rules and transformation rules these axioms transforms into another kind of statement which you call it as theorems. So what we have are first to start with axioms and then these axioms are changing into theorems with the help of transformation rules etc and then only one rule that we will be employing here that is the modus ponens rule which is also called as rule of detachment from alpha and alpha and plus beta you can obtain beta. So now you will clearly see here what we are essentially using is as many few rules as possible and you should note that these rules are truth preserving kind of rules and axioms are obviously two statements which are obviously absolutely true and whatever you substitute in that one uniformly and that axiom will generate a particular kind of theorem and then we will be applying the modus ponens rule which is also considered to be a truth preserving kind of rule. So it will generate you will generate only theorems so that means what essentially we are trying to do is that in a given formal language let us say L we are trying to find out proofs for all the valid formulas so now you know that some formulas are valid maybe just by means of some truth table method etc and all. So now we are trying to generate proofs of this formulas by means of a particular kind of syntactic method which is called as axiomatic prepositional logic. So now Russell and Whitehead in their in his famous book Principia Mathematica he has presented this particular kind of axiomatic system so this grand project is called as constructivist project his focus is on arithmetic was an arithmetic so he suit to show Principia Mathematica suit to show that all of arithmetic can be reduced to logic. This is famously popularly known as logicism so what is logicism it is a thesis that mathematical concepts we take any concepts in arithmetic so we will be focusing our attention on arithmetic the same things can be extended to even geometry also. So it is a thesis that mathematical concepts are definable in terms of logical concepts all the mathematical concepts will find some kind of notation in the logic and also that mathematical truths are reducible to logical truths and mathematical modes of inference are also reduced to logical modes of inference and mathematical knowledge is in that sense essentially a logical knowledge. So if you can reduce mathematics to logic there is a main thesis statement of logicism this can be reduced to logic then in that sense mathematics will serve as a branch of logic so there are other kinds of constructivist constructivist axiomatic systems which you will find in the literature of history of logic so they are due to initially to start with we have got to the axiomatic system and David Hilbert and Paul Bernays and PN etc all these are examples of constructivist camp and they belong to a particular kind of program called as either logicism or formalism. So now all of mathematics if we can be developed through appropriate definitions in the system of logic as defined in the principia principia the main thing which which you will find it uses that essentially the project is all about reducing arithmetic to logic so that means all the statements of arithmetic etc will find some kind of corresponding translation in the language of logic. So arithmetic analysis set theory all the branches of mathematics will now become part of pure logic so now we will be folk we will not be focusing on entire book of principia Mathematica but we will be focusing our attention on part one of the book where he mentions about Russell and Wighted mentions about theory of deduction. So Russell and Wighted axiomatic system is like this he presented an axiomatic system of preposition logic with only two variables so that means two logical constants so different logical systems have different kind of they use different kind of symbols for example in the case of Russell and Wighted the only primitive symbols that you will find the logical symbols that you will find are disjunction and negation but in the axioms that I am going to mention in a while from now you will find mostly this particular kind of symbol so this stands for according to Russell it stands for material implication since it is easy to write in terms of implication so it is better to it is easy to use material implication so now that particular kind of material implication Russell and Wighted uses this particular kind of horseshoe I am using this particular kind of symbol so this by definition is same as not a or B so not a or B means a impress B only so how did he come to this particular kind of definition he was looking for a solution for this particular kind of thing when when can we say that a materially implies B so what kind of substitution one needs to make here so that we can move from a and then some more statement to B in that sense is of the view that B can be reduced from a so now he was looking for it was experimenting on various kinds of things here substituting it in place of the missing thing here missing blank here so when you substitute this particular kind of thing there is a possibility of moving from a to B it is in this sense a materially implies B so that is the reason why this particular thing not a or B has served as definition so this can be written using De Morgan's laws as this thing it is not the case that a and not so this is same as this so now in the Russell Wighted axiomatic system you will find only disjunction and the negation so there the minimal kind of logical concepts that you will find it in the Russell Wighted axiomatic system another choice could be simply implication and negation this is what where you will find it in the next axiomatic system that you will be talking about we will be talking about which is due to another great mathematician Hilbert and a current he makes use of these two logical symbols in his axiomatic system whereas Russell's choice was this disjunction and negation but mostly you will find this axioms in this form implication form because it is easy for us to write it and all so but actual translation should be in the form of disjunction you will find out you should find only disjunction and negation and that is it in all the axioms. So he presented an axiomatication of propositional logic with only disjunction and negation as primitive logical operators a symbolic logic according to him consists of all these three things but we will not be focusing our attention all these things first is the calculus of proposition calculus of propositional sense that when a one proposition changes to another one let us say P ? Q ? P if you substitute something into it it will change to another statement so it is in that sense change of propositions is nothing but a calculus of propositions and the other one is about calculus of glasses something like set theory and the other one is the calculus of relations but we will be focusing our attention on calculus of relation calculus of propositions basically we will be talking about a particular kind of method which is called as deduction what we will be reducing we will be reducing some theorems based on the axioms that were presented by Russell and Whitehead. So what essentially we are trying to talk about is simply like this to start with you have four or five axioms and then you have some kind of transformation rules and you have modus ponens and now you can deduce whatever you think is truth in the arithmetic can be deduced by using the logical notations you can deduce the truths of all these things that means you know deducing the truth means you are proving that particular kind of obviously valid statements valid truth that exists in your formal logical system. So now according to Russell the prepositional calculus is characterized by the fact that all his propositions have as hypothesis and as consequent the assertion of material implication. So what is central to Russell and Whitehead axiomatic system is this particular kind of material implication so a materially implies B for Russell and Whitehead is like this in all the case at A or B is the case so this is the way he came up with this particular kind of thing a materially implies B only when you can make this particular kind of substitution of course this substitution is same as this one is not the case at A and not B. So what is central to Russell Whitehead axiomatic system is the material implication if this is missing then there is no way in which you can move from one proposition to another one so one the one in your proof each step is considered to be a part of the proof and all you cannot move from one step to another step without invoking this particular kind of concept that is the material implication. So now it is in that sense all of its propositions have as hypothesis and is consequent means the next step that follows from that particular kind of proposition is considered to be an assertion of some kind of applying material implication so the definition of that one is that what you will see on the blackboard. So when you see the original work of Wett and Russell and Whitehead book Principia Mathematica which has three volumes the notation would be very difficult to follow but we are using a different kind of notation but it is more or less we are conveying the same kind of information but anyone who is interested in the actual notation and all they should look into Principia Mathematica but just for the sake of our understanding I am mentioning the notation that is used by but Russell and Whitehead in his part breaking book the Principia Mathematica usually you will find some of these symbols and all but you might find some more symbols but at this moment we will be restricting our attention on propositional logic axiomatic propositional logic so that is why you do not see any quantifiers except right now so now the first one is that when he mentions star it indicates some kind of number or sometimes it is also used as some kind of chapter example if you say star 1 it is some theorem in chapter 1 if it says 20 then it is in chapter 20 and then some theorem followed by that we are using this particular kind of thing V dash but Russell and Whitehead uses a particular kind of symbol which is called as assertion sign so in the modern notation we will be using this particular kind of symbol in the modern notation is most convenient so we use use this particular kind of symbol this means suppose if two variables are there on both sides this means Y is deduced from X so Russell and Whitehead in this book they use this particular kind of symbol colon so this stands for asserting some kind of a preposition this is also called as assertion sign obtained by employing the uses of material implication so that particular thing anything which follows after this assertion sign it needs it has to be either simply an axiom or it can be a primitive preposition with whose to truth cannot be questioned and on so there obviously are absolutely true which is denoted as PP or it should be a theorem so that means an axiom if it is an axiom it do not have to have any proof if it is a primitive kind of preposition already true preposition like 2 plus 2 is equal to 4 also does not require any proof or it can be some kind of theorem and all is all always true so in principia Mathematica uses DF stands for definition and then these are the symbols that you will you will see in that book full stop colon semi colon and there are two colons followed each follows each other each and a follows so that is used for some kind of punctuation so in the contemporary modern logic textbooks this stands for single colon stands for brackets or sometimes some other symbol two colons follows each other may stand for square brackets etc so usually they convey some kind of punctuation marks for example punctuation marks are very important in the sense in the last few classes we have seen that for example if you have P R Q and R so what do you mean by saying that it is it P R Q R R there may be some confusion which arises in our mind whether it should be read as P R Q and R or whether it should be read as Q and R so now in that sense we come out of this particular kind of ambiguity in this way that we give some kind of reference to this particular kind of logical constants so first you will give reference to and and then or this is negation and then implies and then if and only if so in this sense suppose if there is no bracket which is given and all and this means you need to use some kind of convention other from this first you need to bracket this thing because conjunction needs to be given first reference and then followed by that the whole thing so now this is what we mean by this P R Q and R so now you can you can eliminate this bracket you can still say this without loss of generality we can even remove the outer bracket also so this is what we mean by this one so it is in that sense Russell Whitehead uses this particular kind of symbols colon sometimes he uses two colons or these four dots and all followed by this thing it stands for left bracket or if we if you find both the things may be it is like closing by a bracket and all so it is it helps us in reading the formulas and all since we are not doing the way principium the way you find it in principium alphabetical but we have slightly changed our proofs and all which fits in our convenience so which with our convenience so now as usual P Q's are etc are propositional variables in any formal language is one of the same there are infinitely many number of such kind of variables if P Q's are all for bits are exhaust you can use P 1 P 2 P 3 etc. So now there are some other individual variables such as X Y Z etc so they all represent propositional variables represent some kind of prepositions individual variables may represent some kind of individual names etc. So this is the axiomatic system due to Russell and Whitehead in their this is also called as PM this is principium alphabetical so you have to call with some name and all so we are calling it as PM. So is principium alphabetica consist of first of all propositional variables like you know it is raining this is a duster this is a chalk piece etc all these things are represented by some kind of propositional variables and he mixes of only two logical constants that is negation which is a monodic operator it operates on only one particular kind of preposition and the next one is a dyadic operator he chooses one monodic operator and one dyadic operator it operates on two propositions it requires at least two propositions. So disjunction is the one which he has chosen and the other brackets are for him like colon and semicolon etc and all are two colons following each other so this stands for some kind of brackets which is very important for punctuation and these are the formation rules not any kind of well-formed formula cannot be any kind of formula which you generate is not a well-formed formula. So the formation rules are like this a prepositional variable standing alone if you write just PQR etc and all simple prepositional variable that itself is a formula or suppose if x is a well-formed formula not x is obviously is going to be well-formed formula but this is the formation rules and all suppose if x is a well-formed formula and then obviously not x is also well-formed formula suppose you are not supposed to write like this x following negation and all this is not a well-formed formula so this is the first thing very simple kind of rules which we have discussed when we have introduced the language of prepositional logic in the same way first we need to define our language that is a syntax so now suppose if x is a well-formed formula and y is also a well-formed formula then he makes use of only one particular kind of logical concept that is a disjunction so x are not way is also a well-formed formula so since he has used used only negation and disjunction there are no other formula such as all the other things can be defined with the help of negation and disjunction by using material implication. So these are the only three formation rules none of the formula is a well-formed formula if it does it does not follow one of these three rules you can add fourth rule this states that you need to follow these three rules meticulously so now any axiomatic system should have to begin with axioms followed by that you need to have some kind of transformation substitution rules and third one is as minimal rules of inference will are going to be present in an axiomatic system in this case we use only material implication all the other rules will come as an outcome of these axioms except I know it is like a capsule and then we are trying to derive everything from particular kind of so everything is hidden in this particular kind of capsule which consists of five axioms. So now the first one is pretty obvious in all so that is we start with tautologies and you will end with tautologies there is no way in which you will get if you start from tautology that means always a tautology in the prepositional logic is considered to be a statement which is always under all possible interpretations so you start with the tautology and you will you will transform it into some other thing it is going to be a tautology only if you use uniform substitution or truth preserving rules etc. On that you will generate only tautologies from tautologies you will generate tautology so there is no way in which you will get contradiction so that is the first statement anything for implied by a true elementary proposition obviously it has to be true so that is a primitive kind of post postality calls so now these are the five axioms with which you can talk about entire arithmetic in all the statements of arithmetic true statements of arithmetic are valid formulas of arithmetic we will find proofs by using only one of this any one of these theorems are maybe more also so the first principle says that every axiom has his name and all it says law of axiom related to tautology PRP it is raining or it is not it is raining the street is raining you are not saying anything great about this so now the addition if you have Q you can say if that is already true you can add another P to it without disturbing the truth value of that so now you have to note that here implication is somehow serving as kind of deduction here but later it was questioned by CI Lewis his work survey of symbolic logic he questions this particular kind of whether or not material implication would serve as what Russell invited thought the thought of as deduction deduction according to CI Lewis the later works you will find it that so this is somewhat different from the material implication and all so in that context CI Lewis has come up with another kind of implication which he calls it as strict implication and that strict implication has led to modal logics etc and all so early that led to non classical logic that is not what we are trying to talk about so this principia Mathematica has served as a starting point for many other kinds of non classical logics etc so but how did this prince principia Mathematica has come into existence there were some problems it is related to freguese axiomatic system because freguese axiomatic system is based on set theory and set theory is plagued by paradoxes such as we need to talk about that particular kind of paradox that is Russell's paradox so in order to avoid or avoid this particular kind of paradoxes Russell invited has come up with a grand axiomatic system so which is which you find it in the principia Mathematica it is it is considered to be a grand kind of program which tries to reduce mathematics that means arithmetic to logic so now all the arithmetic statements can be translated into one of these axioms etc and all and from that you will generate lots of theorems that reflects one of the statements two statements of arithmetic this p skews etc stands for truths of arithmetic or if you are if you are not happy with this particular kind of thing if you are interested in analyzing the simple switch switching digital circuits this p skews etc you mean some kind of switching circuits and so when I represent suppose if I can represent p means p is closed the switch is closed or not p means p is switch is open it is in that sense one can can can you this particular kinds of theorems particular kind of formulas so permutation is like this it is like some kind of commutative property PRQ implies QRP association PRQR implies QRPRR and the summation rule which was later questioned by a famous logician Paul Bernes and he showed that this axiom can come as an outcome of one of these four axioms which you have stated that is from two to five using one of these things one can deduce the six one that is the summation axiom so it is in that context later in the later works of Russell pointed axiomatic system you will not find this particular kind of axiom because it will anything which is reduced from some other kind of axiom which will lose this axiom status and all so it will no longer service an axiom so these are some of the simple axioms but you must note that coming up with this axioms is the most difficult part so one of the important characteristics of this axioms is that whatever you substitute for PQ etc and all uniformly you will generate only tautologies because that is obviously tautology and if you all tautologies with uniform substitution or transformation will lead to tautologies it is a machine that generates tautologies so that is the reason why logicians are always interested in tautologies in the sense that all tautologies are considered to be valid formulas and all valid formulas obviously have to have a proof so these are some of the five axioms and all this is some of the important transformation rules so now in what way this axiomatic system is different from the ones which we have presented earlier that is a natural deduction etc and all in the natural deduction system just like when you are playing some kind of game you need to know you need to familiarize yourself with all the rules of the game and all just like that in the natural deduction we are familiarized ourselves with all the rules etc like there are truth preserving rules etc and these truth preserving rules are added to some of the hypothesis and assumptions which are also assumed to be true and then we have generated various kinds of truths so that is what we have done in the case of natural deduction. So here we use as many minimal number of rules as possible obviously one or two at most and then mostly you will use as many less number of axioms and then rest of the things are all derived from of course one important rule that we use it is a rule of detection so now what we mean by rule of substitution suppose if x I will be using the word thesis that it can be a theorem or it can be an axiom and axiom is a self evident kind of truth a theorem is one which is generated out of transforming this axiom into some other kind of state the substitution you might one axiom might lead to another kind of proposition so that might lead to theorem if x is a thesis thesis can be considered as a theorem or even axiom so in that sense we have some kind of flexibility in using this phrase that is thesis means it can be either axiom or it can be even theorem if there is only one line for example in this case let us say consider to PRP implies P so that does not need any proof in all because it is already an axiom and all so the proof of that one is simply the same statement you reiterate the same thing PRP implies P it is already an axiom axiom does not require any proofs that is in that sense PRP implies P can be called as a thesis in that particular kind of sense it can be called as an axiom it can be called as a theorem so if x is a thesis in that sense containing propositional variables P1 to Pn and y1 to yn is considered to be a well-formed formula well-formed formulas then how did we get this x is obtained from substituting y1 with P1 and y2 with P2 and yn with Pn then you will generate some kind of statement that statement is also considered to be a thesis this means is already a theorem or it should be an axiom so it is like this particular kind of thing so this essentially says that for example if you take into consideration one particular kind of axiom here PRQ or QRP this is what is in Russell Wighted axiomatic system as permutation so now this is the formula that we have so now you can substitute P implies P for P wherever P is there let us say you can substitute P implies P uniformly you need to substitute and then you substitute not Q for Q so this can be this should be read as not Q is substituted for Q that means uniformly you are substituting into this particular kind of axiom then the resulting statement that means this is what is uniform substitution control so now this will become P implies P or Q means not Q is the first statement implies now Q means you have to substitute RQ or P means this P implies P now so this is what we got by substituting this thing uniformly so now if you substitute in this way uniformly then the resultant statement will obviously become true so one can check it with the help of any methods that we have learned so far so let us consider this as X so so now we use semantic tableaux method for this not X is this one so you put negation behind this and then see whether negation of this formula leads to branch closer or not so what essentially we are trying to show is is that in a given axiom whatever you substitute uniformly and the resultant statement is also considered to be theorem that means it has to be a valid formula so how can we show that this whole formula is a valid formula you deny this for problem deny this formula and then you try to construct a tree and if all the branches closes that means not of X is unsatisfiable that means X has to be valid so now this will become P implies P or not Q and then not of not Q or P implies P so now this you will get Q not not Q is Q negation of disjunction is conjunction and then negation of P implies P so now negation of P implies P is nothing but P and not P so this can be written as Q followed by that not of P P since you have P and not P this branch closes and you might have to worry much about the above state so what essentially we showed is simply this that when you substitute anything uniformly for Q anything uniformly for P it will retain this tautology hood so that is what we mean by uniform substitution so if you can substitute some complex kind of thing into this one but still it will turn out to be a theorem for example let us say P the substituted P or P P or Q or R for P and then let us say not P implies Q not P implies R for Q so if you uniform uniformly substituted with any kind of preposition then the resultant formula is also going to be a theorem but you need to ensure that the substitution should be uniform for example if you substitute P or Q here and then you substituted not Q or P and all this is not an uniform substitution because for Q you have substituted Q here only but here you change it and you have used not Q then you yourself will see that this is not going to be a tautology and that means it is not going to be a theorem so a substitution has to be uniform so then only your axiom will turn to another thing which is considered to be true so that is what we mean by rule of substitution and the second rule is simple rule which is called as a modus ponens rule or it can also be called as rule of detachment etc if X and X ? Y are thesis they are already assumed to be true or at least this is this then then why the resultant one is also called as a thesis that is also considered to be true so these are the minimal transformation rule one requires so now in any axiomatic system we need to have some kind of definitions so since Russell and Whitehead has used only disjunction and negation so now we need to talk about other important connectors that is implication by implication and conjunction and since he has used only negation and disjunction all the other things should come as an outcome of that how do you come up with this particular kind of definition so he made use of the concept of material implication so implication is defined in this sense X ? Y means by definition it is not X or Y or you can even write it in the form of a conjunction that is it is not the case that X and not Y so now the conjunction is written in this sense so using some kind of de Morgan's laws which are already there available to us X and Y is equivalent to it is not the case that not X or not Y if you transform the same thing it will become X and Y so now conjunction is defined in the sense of disjunction by using only negation and disjunction the one which is there on the right hand side you will find only negation and disjunction so now X if and only if Y is same as X ? Y and Y ? X where X ? Y is defined as the first one that is not X or Y so these are the definitions that are already there and according to Russell and Whitehead this is one of the important quotations which he has used it in the book principles of mathematical philosophy page number 14 so the axioms that we have presented here the five axioms are considered to be the formal principles of deduction employed in the principle of Mathematica a formal principle of deduction has kind of double rule what is the double rule it has the use of premises of an inference and use as establishing the fact that this premises leads to some kind of conclusion in the schema of inference that means one preposition is transformed to another preposition we have a preposition P and a preposition P ? Q and from these two you will generate Q that means you are already using the material implication so now when we are concerned with the principle of deduction our apparatus of primitive prepositions has to yield both P and P ? Q as our inferences so now what you are going to see in the proofs that follows from now is that when you are moving from one preposition to another preposition somehow in some stage you need to have a preposition is in the form of P and another preposition in the is in the form of P ? Q that allows us to infer Q by using the definition of material implication or the rule of detachment so that is to say that our rules of deduction are to be used not only as rules which is there use in establishing P ? Q but also as substantive premises that is as P of our particular kind of scheme so this is what he tells us in his book principia mathematical so what essentially he says is that so a proof is considered to be finite sequence of four steps etc and all three etc. So now you have reduced Y from this one let us say so now that means X1 implies Y and now you have reduced this thing X1 implies Y and all so now from one and four there is a way in which you can move to the X1 detaches and then you will generate Y so these are all considered to be premises etc and all in addition to that we can generate some kind of statements like this by using the material implication I mean by using the definition of material implication as well as rule of detachment so this is what is the rule of detached so now what is that better Russell invited us trying to show so now you have formulated an axiomatic system which consists of disjunction and negation and we have some set of axioms and then we also know that a two statement will lead to two statement only that is the first kind of propulsion is talking about and then we have transformation rules which preserves the tautology in is of your axioms if your axioms are trimmed in such a way that the fine trimming will lead to only tautologies only so now you are in we ensured ourselves that what way we can generate tautologies so now all the varied formulas which you can think of should come as a theorem of theorem by using only these rules and only by using these rules and the axioms that I have given so now here is one of the examples which is I produced it as it is which is used in the context of principia Mathematica but we use a different kind of notation so star means in chapter one is a second kind of proposition or something like that so now that single turnstile followed by colon I mean that should be written as PRP the whole thing in brackets PRP ? P so now this you will obtain it with the help of three steps so now how do you generate this particular kind of thing so we have we need to generate this PRP ? P so now these are some of the axioms that we have so now let us say that you are generating P ? PRP so now this is what is the axiom one so that is PRP ? P I mean we are what we what essentially I am trying to say is is that this is a notation which is used by Betten Russell and Whitehead is not a theorem of anything it is considered to be an axiom axiom one is this one so this can be obtained in our modern notation as follows so first we will eliminate this colon and we will put some kind of square bracket and then this will become PRP and you will use the dot symbol dot there and then we do not disturb this dot and all so now in the second step we are assuming we are assuming the dot means this the brackets so now PRP is in brackets and then P is also in brackets and then we are removing the excessive kind of brackets and all unnecessary kind of things and all outer brackets we removed but still we can retain the same thing even in the second statement P in brackets does not make any big difference in all same as P so now the formula becomes PRP ? P so like this when you are trying to translate the ones which you are which you see in the principle of mathematical into our modern notation some steps are involved in it so first we need to you can eliminate this colons etc and all and then put it into the brackets etc and all this will become our modern notation so but Russell has used as in the heading that is a singleton style followed by that colon PRP and dot I mean see you have to stop there you have to give a pause there and this symbol horseshoe is the one which has used but we are using the straight arrow in particular so written Russell has used this one and then this means some kind of square brackets and then dot means a kind of a bracket example a dot is before this one means the left bracket a dot before after this thing the right hand side is considered to be a right bracket so like this we have used and then we have translated into some kind of convenient kind of notation so now what we will be doing now is simply this that any axiomatic system that you are trying to talk about we have three laws of logic that is law of identity P ? P and law of excluded middle that is PR not P and of non-contradiction it is not the case that P and not P at the same time so now at least these three pin loss of logic should come as an outcome of course there are many it is also expected that all the valid formulas should find a proof and all but the bare minimum is that at least these three things should come as an outcome you formulated a grand formal axiomatic system and you should now we need to ensure that at least these three laws of logic will come as an outcome because all the other things are constructions of these three fundamental laws of logic. So now let us try to prove some kind of theorems so that is like P ? P etc so now let us try to prove this particular kind of thing which is considered to be kind of transitive property that is Q ? R ? P ? Q and P ? R so now how do you prove this particular kind of thing so now you need to use only the axioms that that are listed there that is the five axioms that we have and you have to use transformation rules and you need to use only modus ponens rule and ultimately that proposition which is appearing in the blue color first one Q ? R and P ? Q and P ? R so you are trimming this axioms in such a way that it will lead us to this particular kind of theorem that is Q ? R ? P ? Q and P ? R so now in these cases you must note that what kind of axiom that you will be choosing I mean that depends upon our kind of some kind of creativity in choosing these axioms you might choose any one of these axioms but if you take some selective kind of axioms and all then your proof might be consisting of less number of steps so that depends upon only our creativity etc. So in a way it is proving some kind of the proving theorems is also kind of an art just like programming is a kind of art so this proving these theorems is also considered to be kind of art because in somebody's proof it will have only four steps are in someone might struggle and then you will come up with a 15 page proof and so you might ask what is that we will be getting from these things because so in the reason why we are meticulously working on these proofs is because of this particular kind of thing is in the Euclidean geometry this is also considered to be an axiomatic system so there are many things which are not part of the proof are also outside the proof are also taking part in the proof so that we should avoid it on few if a proof has to be rigorous then everything needs to be stated explicitly that is what we mean by axioms and then from the axioms we transform it in by using transformation rules and the modus ponens etc. And we will transform it into some other thing so now how do we prove q ? r ? t ? q and p ? r so now to start with we used axiom number five so that is this one q ? r ? p r q ? p r r so this is what we have began with so let us consider this particular kind of thing by using this which is what we are trying to prove q ? r ? p ? q ? p ? this is one second r is that we are trying to prove q ? r means p ? q ? p ? r so this is what we are trying to prove so now you have began with this axiom number five so that is p r q ? that axiom p r so now you begin with this particular kind of axiom so now it appears to be more or less somehow we need to change these things in such a way this axiom needs to be trimmed in such a way so that you will generate this particular kind of thing so now there are many ways which you can think of so that this will transform to a particular kind of thing for example what kind of substitution one need to make so that you transform these things into this particular kind of thing q ? r is same as this one and then some kind of substitutions you need to make so that it will be this thing so now in axiom number five suppose if you substitute not p for p that means wherever p occurs you substituted with not p then what will happen axiom five is transformed into this thing q ? r implies so now p will become not p r q and this will become not p r so now this one let us say this is one and two and three two you have to use definition what is the definition that you will be using so p implies q is nothing but by definition not p r q so now wherever not p r q this means q ? r this means this so now the justification of this one is that this by definition is nothing but this p ? q so you need to put a bracket here so now not p r r implies p ? r so this is what we are trying to generate so now in this proof you have only two steps with one substitution you can transform this axiom you are trimming this axiom into another kind of thing statement which is usually considered as theorem so why it is called as a theorem each step of your proof is considered to be true then obviously the final step of your proof is what we mean by a theorem that is what we have defined earlier in the beginning of this axiomatic systems when we are talking about a system the last step of your proof is considered to a theorem so this is the way to prove this particular kind of proposition so there are other theorems which we can take up and we can prove these things in this way so let us say you are trying to prove this particular kind of thing p ? q ? r ? q ? p ? r so now for this again you need to think in a certain way what kind of axiom which you can use so that you can come closer to this particular kind of theorem if not in one step at least by transforming into some other kind of steps using transformation rules and modus ponens etc so now you started with axiom number four that is law of association in this case you started with p r q r ? q r p r r so now in this axiom for your substituted q for not q wherever q is there not q and so then this will become this is what we have done in the second step that is what you find it here not p for p and wherever you find q is substituted with not q so then this association kind of principle you will get this thing so now let us consider this thing p ? q r r ? q r p r r so this is what is called as axiom for and this is also called as association so now what you have done here is this that wherever he is there you are substituted with not p p and wherever q is there you are substituted with not q in a4 so now this will become not p r not q r r implies this is q means not p you put it in the square bracket so that you will avoid the confusion so now so this by definition is this thing in the second step we will write it q ? r so now again this by definition you will get p ? r so now again you invoke definition on this one so this is not x or y that means it is x ? where x is here p not p x is p and then y is q ? r so that means it is p ? q ? r ? q ? p ? r so this is the way to prove this particular kind of theorem so now one might ask many questions here so how do we generate an effective kind of proof so now it depends upon what kind of axiom that you are going to take into consideration so in principle you can take any axiom into consideration and then you can generate proof for this one but if I have chosen the a4 axiom then my proof would be simpler and I can generate a proof in two steps in the same way you can generate proof in even six or seven steps also by using maybe you can start with a1 and ultimately if you did not it will not work out and then you move to some other axiom then work in such a way that somehow the or trimming this axioms in such a way that you will generate whatever is considered to be a theorem so in this lecture what we have done is that we have presented but in Russell invited axiomatic system and then we have seen that any axiomatic system should consist of a set of axioms transformation rules and the rule of inference in the case of principia mathematical you will find only disjunction and negation as primitive symbols and then transformation rules and the rule of detachment and making use of these things definition of material implication that is nothing but not here I mean you could you could talk about all the other connectives based on these two primitive connectives by using the definitions so now we have seen some simple kind of proofs in which we have transformed the given axiom we started with an axiom and then we applied some kind of transformation rules that means in a way we have used uniform substitution etc and then we trim these axioms in such a way that we generated whatever we decide to prove so in the next class we will be talking about some more proofs in principia mathematical like at least this law of identity law of excluded material etc and then we are going to see whether or not this principia mathematical is going to be consistent or whether this system formal axiomatic system is going to be complete etc the things which we will be talking about in the next class.