 Welcome back in the last lecture we presented Bertrand Russell Whitehead's axiomatic system that is what we find it in the book famous book Principia Mathematica. So we started with the five axioms and then before that we have an important you can serve as an important kind of statement that is from any primitive kind of true preposition only true preposition implies so that means from tautologies you will generate only tautologies. So now in this lecture what I would be doing is that any formal axiomatic system that we come up with by using some simple kind of axioms etc and all so that axiomatic system I mean we should be in a position to derive at least some of the important laws of logic that are law of identity that is P implies P law of excluded middle P or not P or I mean there is another law which is called as law of non-contradiction it is not the case that both P and not P is the case. So how do we derive all this valid formulas that occurs in a given formal axiomatic system. So now for this one can start with by selecting some kind of axioms so now we will be talking about the Russell Wighted axiomatic system where it has only five axioms to start with we have these things suppose if anything which is a true preposition that implies only true preposition that means from true prepositions you will not get contradictions so that is the first thing so there is already there so this is called as law of the axiom related to tautology and second one is Q P R Q so this is called as addition and fourth one is of course you can say third one is P R Q implies Q R P is permutation either P R Q is the case or Q R P is the case which is called as permutation and the fifth one is law of association Q R P R R so this is what is called as association law and then sixth one is summation Q ? R is P R Q ? P R R so P R Q ? P R R so now to start with we have these axioms and then we have some kind of transformation rules if you apply this transformation rules on any one of these things it will retain its tautology good and the other important rule is that if you have P and if you have P ? Q where this implication is considered to be material implication then this P gets detached and then what you get is Q so that is all we have to begin with and from this particular kind of thing you can derive all the valid formulas so all the valid formulas means all the true prepositions in any given formal axiomatic system there are different ways to check whether a given formula is valid or given formula is a tautology so two important methods that we have already discussed that is we can check it with semantic tableaux method or one can use to table method and then we can find out whether a given formula is considered to be valid other means tautology or not so now what I will be doing is I will be deriving some theorems some important theorems in these are theorems in any given axiomatic system so the first theorem that I will be deriving is this thing so once you write to this thing in this way that means something is a theorem P ? Q ? P so this statement says that something is true then the truth is obtained from any kind of preposition and all this is also called as famously put it afterwards as paradox of material implication it is considered to be a theorem in Russell Wight-Ted system or any in any given axiomatic system it is considered to be a theorem so the only problem with this theorem is that if this preposition is true so that means using the semantics of implication P ? Q so this is the way we define the material implication T T F and F alternative T and alternative F so this is going to become false only in this case when P is T and Q is false in all other cases it becomes T so now in that sense if the consequent this is the antecedent and this is the consequent if the consequent is true irrespective of whether the antecedent is true or false that means the antecedent part is this one whether or not the antecedent we need to consider only those cases in which you have true consequent that means these two cases now irrespective of whether P is T whether P is false so this is going to be true only so in that sense a true preposition is implied by any kind of strange preposition that is even if this preposition is true preposition or it is a nonsense or it is a false any kind of a true preposition should not be implied by any kind of strange kind of preposition that is not what we are discussing at this moment when it when these theorems applied to day to day discourse here are some of the problems which we will talk about it as a limitation of this particular kinds of axiomatic systems especially when it applies to the day to day discourse but as far as analyzing the digital switching circuits are concerned as far as mathematical reasoning is concerned so these are the things which perfectly works all right. So now this is the one which we are trying to do P ? Q ? P by using only these five axioms so now one can use any one of these axioms to begin with and then ultimately our journey begins with these axioms all these axioms are absolutely true the truth of the proofs of these axioms cannot be questioned because already they are self-evident truths so now somehow you need to use one of these axioms so that you will generate this particular kind of thing. So in a way what we are trying what we are essentially trying to do is we take up one particular kind of axiom we trim it in such a way so that you will get this P ? Q ? P as an outcome so let us see let us take into consideration this particular kind of thing axiom 3 so that is P R Q now definitely this is not in this particular kind of format somehow you need to manipulate this one that means you need to transform this particular kind of things which we already know that it is true the axiom all axioms are obviously considered to be true so from the true proposition you need to get another true kind of proposition. So now if you can replace this thing let us say Q P wherever Q is there you replace it with P and wherever you have P you replace it with P these are the two substitutions which are uniformly making it in this particular kind of axiom 3 so you have to note that whatever substitutions you make in the given axiom if your substitution is uniform it is according to the transformation rule which we have discussed in the last class then its transformation is also corresponding to a tautology so that means what we are trying to do instead of Q we are putting P and wherever we have P we put it as ? Q and then Q stands for so we put ? Q so then this will become this ? Q wherever Q is there you are putting P so now what is that essentially what we have done here you replace Q with P and P with ? Q so that is what you have written this is the justification for this particular kind of thing since the substitution is uniform if this is a tautology this is true any substitution which it is done in a uniform manner should also be true the record needs to be closed here so this is the second step so now using the definition of material implication so that is a ? B you have P's Q's here that is ? A or B so that means this ? A or B is the same as A ? B so now this is as it is P and then ? Q or P is nothing but Q ? P so now this is what we are supposed to prove each step each stage we write this particular kind of thing and if you want to be more specific you can write PM PM stands for principium alamedica that particular kind of axiomatic system so now since we got this thing as a theorem now we can substitute uniformly anything into it that will also become a theorem for example if you substitute sorry not P for P wherever P is there you substitute with ? P and wherever Q is there you substituted with ? Q so now this kind of preposition will become ? P ? ? Q ? ? P if this is considered to be a theorem anything which if we can substitute anything into it uniformly for P is we substituted ? P for Q is we substituted ? Q then this is what is going to what you are going to get so this should also be a theorem if you are in doubt when one can check it with either semantic tableaux method or any particular kind of method so now let us try to check this particular kind of formula using simple tableaux method so this is not Q so now we have taken into consideration the negation of the formula and then we are constructing a tree so now this will become this thing ? Q ? P so now this will become ? Q ? P so now you will see here clearly ? P here P here there is a contradiction the branch clause that means what we showed or simply this that negation of X is unsatisfying so that means X has X is valid so in prepositional logic validity tautology they are one of the same so that is why this formula has to be true proposition and all the true propositions are also obviously considered to be theorems in the prepositional logic so in that sense this is also considered to be a theorem of prepositional logic if this is considered to be a paradox of material implication the one which I have written just now should also be considered as one of the instances of the paradox of material implication so when we discuss about paradox of material implication I will talk about these things in greater detail another instance of this one can be for example for P you just substitute ? P so then this will become Q ? ? P so there are thousands of instances like this so you keep P as it is instead of Q you substituted ? Q and then you keep it as it is this should also become a theorem one of the beautiful things that you will see here is that if something is a theorem something is a true proposition and it implies only true propositions so tautologies will lead to tautologies so there is no way in which you begin with tautologies and then you will end up with contradictions that should not be the case and all so that is the reason why logicians will be continuously insisting on the tautology rather than the contradictions if you begin with contradictions you can you will get any particular kind of proposition so that is the reason why we are not insisting on contradiction we are insist logicians will insist on only tautologies so now let us say you want to derive P ? P or P so this is another kind of our theorem P ? P or P this is different from what we have here so this is an axiom P or P ? P but we are we would like to derive P ? P or P so now again you can begin with the axiom 2 that is Q ? P or Q this is axiom 2 so now somehow you need to change this thing into this particular kind of format so now what substitutions you make into this particular kind of thing so that it will lead to this particular kind of thing which we require so what exactly we are trying to do in all these theorems is that we take these things axioms as ideal kind of situations and then these are all instances of this particular kind of axioms so now if you want to prove any one of these theorems in all so you begin with one of these axioms and apply transformation rules and you trim these axioms in such a way till such a way that you will get whatever you decide so now this is the axiom which we began with this is axiom number 2 so now this is not in this particular kind of format somehow you need to translate it change it into this is a corresponding form that is for example if you substitute P for Q then this will become Q will become P and then P is as it is and Q is this one so now it is converted into P ? P R P with one substitution we got this particular kind of formula is also considered to be a theorem so now in this way one can prove all the valid formulas suppose whatever is considered to be a kind of true proposition or a tautology it should find a proof in a given axiomatic system sometimes the proofs might be very lengthy sometimes say you might get a proof in simple four or five steps so for instance let us try to prove the particular thing which we commonly know in logic that is law of identity so using one of these axioms particular so we will be proving this particular kind of thing how do we prove law of identity using a given kind of axioms so now just now we proved this particular kind of thing so Q ? P R Q is already a theorem substitute Q for P then you will get P ? P R P how did you get this one you substituted P for Q you get this particular kind of thing which is we already showed it in the last slide so now this is the first one to begin with now the second thing is this so we know that one of the axioms states that P R P ? P we know that this is obviously considered to be theorem and all is axiom so that is axiom number to it is tautology axiom of course you know if you are bored with this piece Q's etc. Now you can view this piece Q's R's etc as some kind of propositions may be in the branch of in the field of arithmetic or any other kind of field which you can think of or you can simply treat this piece Q's etc as some kind of switching switches in particular simple switching in the simple switching circuits this piece Q's are represents some kind of switches that means if P is if I write just simply P you can interpret it as switches on or if I write not be usually it is written in this way P bar that means which is off in the switch when the switches are on for example when both switches are on and all the current passes through this particular thing so that is end gate if it is in the if they are arranged in a parallel kind of connection it can be an R kind of gate so all these things which we we can visualize the same thing in the context of simple analysis of simple digital switching circuits so now coming back to our particular kind of thing so we are trying to show that law of identity will come as an outcome of these five axioms so now the first one which we have shown is this thing we substituted P for Q then you got P ? PRP so now we have this is the first step and the second step is this and the third step so we have a rule which says that the rule of syllogism which is obviously is a case and all so if suppose if X ? Y and Y ? Z then X ? Z in that sense so here it is P ? PRP and PRP ? P so that means simply P ? PRP ? P then the justification for this one is one two syllogism you will get this P ? P so it involves at least two or three steps to prove this particular kind of thing something is equivalent to itself want to show it then this is what is the case this is law of identity so now for example in the case of the same thing in the context of natural deduction if you remember it when we discussed about this particular kind of method natural deduction we showed the same thing how to prove this thing using natural deduction so this involves maybe one maybe less than two steps and so in this you take the first one antecedent of this one as assumption or hypothesis so now what you will do here is this is since this is already true you reiterate this one you reiterate the same proposition again and then that means you got P so that means you draw a line like this then you say that P ? P is deduced so in this natural deduction method this will come come in just two steps but the same kind of thing for in vessel righted axiomatic system maybe it involves three or four steps are maybe in the other axiomatic system which we will be thinking of which will be studying that is a Hilbert Ackermann axiomatic system it might involve more than five steps to prove simply speed implies speed but here what we have used is just a principles natural principles of logic that is you used principle of reiteration so we just reiterated the same thing and then you draw a line like this and then one and two three one and two conditional proof you will get this P ? P so what you will do here is that you will discharge your assumptions P and then you will talk about P ? P here using the conditional proof is also called as a rule of conditional proof so for we deduced law of identity and now let us consider the law of excluded middle so how to deduce the law of excluded middle so law of excluded middle states that either the sentence either a proposition is P or it is not P but not the other way around not P or P is not considered to be law of excluded middle but P or not P is considered to be the law of excluded middle so this holds in classical logics but this may not hold in many other situations when it comes to day to day discourse examples related to day to day discourse this may not hold simply this is going to be applicable in only those cases in which you can draw a clear boundary between X and 0x. For example you are talking about motel and non motel it is easy to draw a line between whosoever is considered to be motel the living beings and the dead beings in easily draw a line the same way you can easily draw a line between black and white so it everything is not in this particular kind of situation when it comes to day to day discourse you can very well have both P true and not P is also true this is also contested in the context of intuitionistic intuitionistic mathematicians like brover he argues against this law of excluded middle he is of the view that law of excluded middle need not be not come as a theorem in a given logical system so how it is the case is the case because suppose if you are so brover accepts P or not P only when you are able to deduce you are able to prove either P or you are able to prove not P and suppose if you are not able to prove both of them P and even not P and all then that can be questioned so improbability leads to the fact that you need not have to have this law of excluded middle as one of the theorems in your axiomatic system so that is intuitionistic logic and then in many deviant logics like fuzzy logics etc. and all we do not have this law of excluded middle as an outcome of it will not come as a byproduct in particular from using your axioms that means you know you are talking about some kind of deviant logics in the beginning of this classical propositional logic we already we clearly stated that in all the classical logics at least this law of identity law of excluded middle and law of non-contradiction are necessarily one of the rules of one of the important theorems in any given axiomatic system which follows classical logic so forget about this thing let us try to show this law of excluded middle using the Russell Wighted axiomatic system so now in the same way proving this thing one can begin with any one of these axioms so it is not hard and fast that you know begin with any one of these things but you have to use little bit of creativity while proving theorems also so a proof is considered to be an effective proof only when it ends in finite steps in finite intervals of time suppose suppose if you prove involves some 150 steps in all and some other proof somebody else came up with a simple proof where it involves only five or six steps then the second thing is considered to be a better proof than first so what constitutes an effective proof is the one in which which has which has less length that is number of steps are less and it and it ends in finite intervals of time your proof should not go forever and ever so now what is that we are trying to prove we are trying to prove this thing so now take into consideration the fourth axiom P R Q implies Q R P so now you somehow you need to transform this thing in such a way that you will have only the letters P because in our final result that is P R not P only two only one literal up yes that is P that means you need to get away this Q's and all so now what you have done here is somehow this Q needs to be eliminated and all so it has to be substituted with P so now what you will do here is this is to begin with axiom for so now you substitute not P for P and then not Q for Q so then this will this will become is the first step and second step is this thing this is justification which we are trying to provide for this one this will become not P R Q so now this you will change it as so what exactly you have done here is this not P and for Q wherever Q is there you substitute with P so now this will become P R P for P what you have substituted not P so that is what you have so now this transforms to by definition this goes to not P R P implies P R not now somehow you have to use something so that this gets detached and you will get this thing as your outcome so now for that you can take into consideration this one P R P not once again so somehow you need to get away from this particular kind of thing for example if you substitute in this one let us see how we do it in all in this one so earlier we proved not P R so earlier we have proved this particular kind of thing so just now we have proved this particular kind of statement so this is identity so that is written as ID so this is the third step so now by definition P ? P can be written as not P R P by definition so what is the definition definition of material implication so now P ? P R P is the one which has come as an outcome so now here you should note that so till here it is fine and all but in order to prove this thing P ? P all the steps which are there in P ? P has to be there before that since we already proved it so we are just straight away inserting that particular kind of thing in this proof there are many proofs which you already proved so that can serve as the starting point for proving other kinds of theorems in that sense this is we already proved that this is to be true as a theorem so from this by definition it you will get this one so now somehow you need to detach this particular kind of antecedent so now observe these two things not P R P here not here so these two that means two and four modus phonens are rule of detachment you can use so this is what you P R not P so how did we get this P R ? P we started with axiom number four one can start with any one of these axioms but ultimately it will be very simpler if you can start with this particular kind of thing so what exactly we have done here the strategy is that the last step of your this thing so this will come as this particular kind of thing somehow you need to transform this thing into this particular kind of format that is what we have here and rest of the things what we are trying to do here is that we are trying to detach whatever is there in the antecedent part then you will get this particular kind of thing so how did we get this one using rule of transformation or modus ponens etc modus ponens plays an important role here so there are two things which are important in the quotation that we have seen the last class I think which is central to this derivation is the material implication and not only the material implication but also this particular kind of rule of detachment that is if you have P and if you have P ? Q and you will get Q so this is the way of showing PRP as a theorem in principia mathematical that means you used the Russell white it axioms and then you have deduced this law of excluded mid so in the same way one can show whether or not law of non-contradiction is holds are not so now there is an important thing that is law of double negation let us try to show whether I mean how we can prove this double negation kind of this P ? ? P and ? P ? P is considered to be the rule of double negation so now since we got this PR not P that is law of excluded middle so now from that PR P ? ? not not you will get it as an outcome so this is what you have already shown to be true that is PR not just now we proved this particular kind of thing PR not so what is that we are trying to prove P ? not so now this is already a theorem which we have shown just now so now what you do here is to begin with this is a theorem just now we have proved so now substitute not P for P that is if you substitute wherever P is there you substituted with not P so now this will become not P ? P so now this by definition suppose if you view it as P as X and this whole thing as Y so then this is in the form not X or Y so this is same as by definition X ? Y so now what is X here for us this is not P and Y is not not so that means what you got here is the third step this by definition leads to P ? this is by definition is same as P ? not not P but that is not what we want to get in all but we want to prove the double negation double negation rule that is not P ? P so one will be tempted to say that if X ? Y and Y ? X and all that is not the case here X ? Y and Y ? X are totally different things so now this is what we have already proved shown to be the case so now we are trying to show this particular kind of thing not P ? P we will make use of this particular kind of thing little bit later so now let us try to prove this particular kind of thing what we are trying to prove we are trying to prove not P ? P so now one needs to start with any one of these axioms and all so now somehow if we can use this particular kind of axiom and then somehow you replace this in such a way that so this will be like somehow if you substitute ? ? P and then if you substitute R for P and all ? for ? P or for P itself then this is what you get ? ? P ? P so with this somehow we will get some kind of clue of what to substitute for P and what to substitute for R in this whole formula so that the last this consequent of this conditional will have this particular kind of format and this other things what you will be doing is you will be trying to detach whatever is there before that by using rule of syllogism or modus ponens or maybe two other transformations etc. So that is what we will be doing here so now you take into consideration the first step Q ? R so that is what we have taken into consideration P ? Q ? P R so this is what axiom number 6 or summation axiom but you need to note that this summation axiom later it was shown to be the case that it is no longer an axiom in Russell Whitehead axiomatic system because this will come as an outcome of one of these axioms and so it loses is axiomatics it will last is axiomatic stated but at this moment Russell invited in their book Principia Mathematica proves by considering this also is also an ax and axiom but later Paul Bernes student of Russell Whitehead I think he showed that this is no longer an axiom an axiom is a one which is considered to be a self-evident truth which does not require any proof. So now you start with this particular kind of thing so now you substitute not P for Q and not not P for R so why you have substituted this thing because somehow you need to generate this particular kind of thing so now this will become so wherever Q is there your not P here so now this will become not P R is thing not of not of not three knots are there and then P and now P R Q P means as it is Q for Q we have substituted not P implies P R R so we have substituted not P for R so this is same as P and R for R you have substituted not of not so this is what you will get from this one so now we have already proved this particular kind of thing this is what you have already we have shown that P in place not not P that is what we are using here theorem proved earlier just now we have shown that this is the case so how did you get this one your P R not P and you substitute not P into this one and this will become not not P then this is same as P in place not not P so in that way you proved this particular kind of thing so you make use of this particular kind of thing so now you substitute not P into this one not P for P in three so now this you will get not of not of P so now observe this two and four not P in place not not not P and this is same as this one so now these two modus ponens you will get whatever is remained here this gets detached and then you will get P R not P implies P R not so this is considered with a fifth step till now we did not get to this particular kind of thing so now how did we get this one two and four modus ponens two is what this one this portion and this portion is same so that's why this gets detached it's like X and X implies why so you will get why so this is the fifth step till now we could not trim this one in such a way that it is translated into this particular kind of thing so now we showed that P R not P is the case this is a theorem which we have proved law of excluded middle so there are so many things which we are embedding into this particular kind of theorem and all improve by proving this particular kind of thing we made use of all the things which are already shown to be true earlier that means shown to be theorems earlier so now this is what is the case now so this is theorem number 8 now so why we are using these things because somehow we need to detach this thing we have to get to the last one which is there in the consequent is occupies the consequent of your condition so now these two five and six modus ponens you will get this P R not not so that is what you get so now till now this did not get over so now in the eighth step we use axiom number four that is P R Q implies Q R P so now somehow this should be in this particular kind of format so that it gets detached so that means for Q you need to substitute this not not not P so now this with the transformation you will get P R now Q is so now you need to write like Q is nothing but not not so now this you will get not not three knots P implies Q is this thing not not not P so now observe this 8 and 9 sorry 7 and 9 is what I am writing it here so now the 10th step is here from 7 and 9 modus ponens because this P are not not not P is same as this one P are not not not so these two gets detached and then this is what you get not not not P implies P so now till now we did not get what we wanted we what we wanted is this thing not not P implies so this is what we are trying to get so now this by definition suppose if you view it as this thing the whole thing as X and this thing as Y so this is in this form not X so just so not not not P or sorry R P so this is not implication so this is R P so this is like not X or Y so that means it is same as by definition X implies Y what is X here not not P and what is Y here speed so that is what you get by your definition so now in the 11th step so this by definition 10 by definition you will get not not P so now what is that we have achieved in this particular kind of thing we showed that for example you want to show that it is not the case that it is not the case that this is duster means that it is a dust so that is what you two times you negated that one negation leads to the same thing so now why this proof is considered to be sometimes it makes very silly to talk about proofs like P ? P or P or not P etc and all but one thing one should note and all here in all these proofs our proofs are considered to be very rigorous in a sense that we started with the axioms which are considered to be obviously true and then transformation rules which preserves the truth and then the rule of detachment that is also considered to be truth preserving kind of rules and everything is stated explicitly on the board and all from that you got this not not P implies P there is no there is no step in this proof which can be that can be questioned and all just like in the case of proof of 2 is equal to 1 in the one of the funniest proofs that we have seen in the last class we clearly we have seen that after following some 6 or 7 steps in the 8th step there was some problem and all that is cancellation of a2-ab both sides and I mean 0 by 0 is not permitted in that particular kind of proof that means the proof is considered to be was considered to be defective in that case so you cannot move further because that step is wrong so but here in this case each step is we are making our journey in such a way that we started with the truths and the next step is also going to be true and then we are moving to we are closing moving closer to whatever we wanted to direct in this case we showed that not not be impressed P by following some kind of rigorous kind of method that is so will be rigorous method is used employed here why because everything is stated explicitly and from that you have derived is not not be impressed P. So finally we one can also derive some of the important other theorems and all like one of the important theorems that you employ in classical logic is law of contra position that is if P ? Q is the case then ? Q ? ? P is the case so how do we get this law of non contra position law of contra position within this Russell pointed axiomatic system so for that also you begin with some axioms that means they are obvious truths and all then you trim those axioms in such a way and then by using transformation rules and rule of modus ponens rule and then you will generate so you will generate law of contra position so again the axioms remains the same so what essentially we are trying to prove is this so from P ? Q ? Q ? P since it is a theorem you write it in this way so since it is a theorem in principia Mathematica you write it as P M or even Russell pointed R w also one right so now again we start with the summation axiom so that is the one which you stated here clearly implies P ? R now in this particular kind of thing somehow you need to generate this thing suppose if you can substitute P for not being you will get P ? Q and if you substitute for P for R you substitute something else maybe you might get something closer to this one if not the same formula so now what substitutions one can make is like this so you substitute ? Q for R so now this you will get if you substitute ? ? Q for R you will get ? ? Q so P R Q is and what else one can substitute here is this ? P for these are the three two things which you are substituting here so wherever P is there you are substituting with ? P and wherever R is there you are substituting with where R is there ? Q so this will become ? P R Q implies P means ? P or R means ? Q so why we have done this thing because somehow we transform this thing as much as possible closer to our destination our destination is this one so now three we already so now we write this one first of all Q ? ? Q so this by definition is same as P ? Q so there is a reason why we have transformed this thing into this particular kind of format and now this is nothing but P ? ? ? so this is what is the case so now we already showed that P ? ? ? P but the same as Q ? ? Q so this is what is called as ID means law of identity so now these three and four modus ponens you will get this particular kind of thing so these two gets detached and you will get P ? Q ? P ? ? Q so now till now it is not transformed completely into this one we have to trim it a little bit more so that you know this will become ? Q ? P so for that what you will do is we will be making use of another axiom so each step is considered to be a kind of truth in all either you have to use axiom or you have to use a theorem which you showed earlier or it should be it should come as an outcome of some kind of transformation rule that means with a uniform substitution we should get this particular kind of thing so now we have P ? Q ? Q or P so this is what we already have here axiom number three some of this you need to transform this axiom in this way so now in this axiom suppose if you substitute ? P for P and ? Q for Q this is what you have done here so now this gets transformed to P will become ? P or Q means ? ? Q for Q your substituted not Q and for P your substituted not P so now this by definition you will get P ? ? Q and this is same as ? Q ? ? P so now so this is what we have so now P ? ? Q ? Q or not so six and eight P ? Q means once again so now observe this particular kind of thing five and eight P P ? Q ? P ? ? ? Q the same P ? ? Q ? Q ? P ? P this is like X ? Y which is in the step number five and Y ? Z in step number eight if you find a similar kind of thing like this then you can infer X ? Z by using property called as syllogism now we can again once again see here P ? ? ? Q that is this five P ? ? Q is this one sorry P ? Q is this one and P ? ? Q is this one that means P ? Q should go to this one P ? Q ? ? Q ? ? P ? P ? Q ? so this is what we are trying to get so that means in some kind of nine steps we got whatever we wanted to prove so in this way one can derive many theorems whatever you consider to be a kind of valid formula or a tautology I mean that should find a proof in this principia aromatica system so in the next class what we will be doing is whether all the valid formulas finds a proof or not so there are three important properties that we should talk about now the question the immediate question that comes to our mind is that is this system consistent consistent in the sense that suppose if you derive both X and ? X from within the given axiomatic system that means the system is considered to be inconsistent in that sense principia aromatica is considered to be consistent and another important property is that all the provable things that you are just now proved have to be true that means it has to support the property of soundness all the provable theorems are true and in the same way the other way round if all the valid formulas are also also find a proof and all any your system is going to be complete so we are going to say in the next class that the principia aromatica the system which was introduced by West Betten Russell and Whitehead in that many valid formulas can be derived in all all these things corresponds to some kind of statements in arithmetic so it is in that sense all the statements of arithmetic are translated into one of these axioms and all and then these axioms can further be transformed into a corresponding theorems and all that also corresponds to some of the statements in arithmetic so in this class what essentially we have seen is a we have started with the Betten Russell Whitehead axiomatic systems axiomatic system where it has only disjunction and negation although in the axioms you have implication sign here but actually it should be read as actual things should have only disjunction and negation and from that using transformation rules and the rule of detachment we derived many theorems. So in the next class we are going to see whether principia aromatica is considered to be consistent is complete sound all these important properties which we will discuss it in the next class.