 Welcome back in the last few lectures we talked about many important decision procedure methods starting with the truth table method and then semantic tableaux method and then we have come up with semantic resolution refutation method etc and all when we solve one particular problem with respect to all these methods and then we have seen which method to we need to employ and all it is all depends upon our convenience you know. So far we talked about the different decision procedure methods with which you will come to know whether a given well-formed formula is a valid formula or tautology or when two groups of statements are consistent to each other etc. So today I will be talking about another way of doing it which comes under the category of syntactic method so that is the axiomatic prepositional logic I will be talking about the axiomatic prepositional logic in that I will be talking about two different axiomatic systems one is due to the famous Russell and Whitehead there are two mathematicians and all this axiomatic system is due to Russell and Whitehead which is called as Russell and Whitehead axiomatic system which they proposed it in the historic book that is principia matematica. So I will be taking a selective portion of the principia matematica and I will be talking some of the proofs which are already there in that book so this comes under one particular chapter which is based on deduction in prepositional logic so I am restricting my attention to my focus on prepositional logic. So we will be mostly talking about Russell and Whitehead axiomatic system so now in this lecture what I will be doing is these are the things which I would be doing so first I will talk about what I mean by an axiomatic system how it has originated etc and then within the axiomatic system what occupies the central position is a proof of what a given theorem so we will be talking about these specific key terms such as axiom prove what you mean by deduction what you mean by saying that a particular preposition particular statement is considered to be a lemma or conjecture or corollary etc these are the things which you commonly come across within the proof theory. So usually a proof is considered to be a finite sequence of steps which ends in finite intervals of time so if your proof ends in finite steps in finite intervals of time then usually it is considered as an effective proof so how did we come to this particular kind of rigorous proofs and all so what is wrong with the proofs that are already existing in the Euclidean geometry Euclidean geometry is also considered to be one of the important axiomatic systems but still we do not treat it as a as rigorous as this that we find it in in the principia mathematical so what is the reason for that particular kind of thing I will talk about it in a in a nutshell and then I move on to a different kind of axiomatic system which derives its motivation from axiomatizing geometry principia mathematical is motivated by motivation has come from the arithmetic whereas Hilbert Ackerman axiomatic system the motivation comes from the geometry so then we will see with the help of deduction theorem will be simplifying some of the proofs that are there in either in Russell weighted axiomatic system or in the Hilbert Ackerman axiomatic system and then we will talk about some of the important meta theoretic theorems such as soundness so that is whether all the things that you are proving are going to be true or not usually in the proof the last step is considered to be a proof which is obvious obviously considered to be true if it is a true or tautology then it has to find a some kind of decision procedure we need to find a distant with the help of decision procedure method we should be in a position to check whether it is a valid formula or not the soundness ensures that all the things that you have derived are true and the completeness assures us that all the true propositions finds some kind of proof and consistency is a natural property which is in any given point of time you cannot derive both x and nautics so in this lecture I will be focusing my attention on the origin of axiomatic system and then partly I will be talking about Russell weighted axiomatic system and some of the important proofs within that axiomatic system due to Russell and Whitehead so the origin of axiomatic system is usually like this usually we find axiomatic system either in Aristotelian framework or even in the Euclid's in the Euclid in the works of Euclid that is elements so Euclid has come up with five postulates and then the few definitions and then there are some common notions and with the help of these common notions definitions and axioms he consider he use it as postulates with that all the other true propositions are true propositions in geometry are derived. So in the 90 in the mid 19th century the mathematicians started finding flaws in the Euclidian axioms so there is fifth postulates which is which is objectionable so the flaws in the Euclidian axioms in the geometry led to this particular kind of axiomatic systems so the work of cards in 1998 he found flaw in the Euclidian axioms is the postulates usually the fifth postulates is highly debated in on whether or not to replace that particular kind of postulate or change it a change it in a such a way that you can talk about a different kind of geometry. So after in the mid 19th century different geometries have come into existence Riemann geometry Cauchy's Gauss has come up with another kind of thing etc all this things comes under the category of non Euclidian geometry. So in addition to the independence of the parallel postulates established by another Russian mathematician Lopschewski mathematician discovered that certain theorems taken the certain theorems in Euclidian geometry are taken for granted by Euclid were not in fact provable by using his own axioms so there are two problems with this Euclidian geometry the first thing is that the fifth postulates postulate has created problem and people have come up with various other kinds of geometry and all and the other thing is that there are so many common sensical notions direct observations etc which are not part of part and parcel of your proof they are already present in the proof and there are certain things which are outside the proof and unfortunately they are also part of the proof and so we need to separate all this common if you want to have a rigorous kind of proof everything needs to be stated explicitly and there should not be any notion which is not part of the proof that should play a role in deriving some kind of theorems. So among this is a theorem that this is the fifth postulates which tells us that a line contains at least two points are that circles of some same radius whose sentence are so whose centers are separated by that radius must intersect so this has created a problem in all and then it led to different kinds of theories whether the angles total sides of the angles of a triangle angle of a triangle is considered to be 180 or greater than 180 all these questions arose so this is one thing which led to think in a different way the non-euclidean geometry or making the proofs more rigorous and all that was Hilbert was taking up in his axiomatic system and the other thing is that there are concerns that mathematics had been built on not been built on proper foundations. So Fregge has built it built an axiomatic system so that led to some kind of paradoxes we usually mathematics was rest in on set the but set theory is plagued by important paradoxes which was discovered by Russell invited and this paradoxes plague this kind of axiomatic system that also led to the development of rigorous axiomatic system that is what you find it in principia mathematical there are better Russell and Whitehead's motivation has come from some of the paradoxes that arose in the work of Fregge a set theory is plagued with paradoxes right and there was issues in the geometry and then Hilbert Ackerman took geometry into consideration and axiomatize geometry and all these discovery of paradoxes in informal set theory cause someone to wonder whether mathematics itself is consistent or not and to look for proofs of consistency etc and all. So there are two things which we need to note the developments of Euclidean geometry led to some kind of rigorous kind of axiomatic system which is little bit different from what Euclidean has done that is in your proof if you proof has to be rigorous all the common sensical notions direct observations etc should not be it might be mistaken all we might it might be very well be the case that our intuitions might be misleading you know so in that sense everything has to be stated explicitly and then from that all the truth things can be derived in all so for that you need to come up with a few axioms as much as possible and then few transformation rules etc and then you derive all the other true things so that is what we have discussed so far we discussed syntactic and semantic decision procedures semantic decision presence of truth table semantic tableau etc and all syntactic procedures we have natural deduction etc and all and another method added to that we have this axiomatic propositional logic where we are not interested in talking about meaning of this formulas and all but we will be interested in some of the patterns etc and all so given the axioms we will be deriving some kind of theorems so now in this this is axiomatic propositional calculus comes under the category of syntactic kind of proofs and all which we will be talking about in greater detail so in this we intend that proofs in propositional logic should provide demonstrations of formal truths that means you are deriving all the true propositions from a given set of you can call it as self evident truths or things which cannot be proved etc and all they are all axioms and derivation should provide deductions of the formal sequences of assumptions if that is the case then that is what we are trying to do that means in a nutshell all the true propositions so that you come across all the valid formulas now we are trying to generate proofs for those things it is not that they came just like that but as an outcome of some of the axioms and the transformation rules you will generate these theorems in addition we would like our formal system to be rich enough for all formally true formulas of L to be provable that means all the two propositions that you come across in your formal axiomatic system have to be provable it has to find a proof you say that something is true and you do not have a proof that would not work in all so you need to have a kind of proof for all the two propositions that ensures that all the two propositions will find a proof and for a well-formed formula A to be derived from a set of formulas gamma whenever the argument from gamma to A is usually considered as formally valid all the two propositions also have to be valid so now these are some of the examples of axiomatic systems principia magnetic and Hilbert Ackermann axiomatic system there may be many other axiomatic systems but I will be talking about mainly these two axiomatic systems so to begin with what do you mean by an axiomatic system so an axiomatic system as you can take it as a Hilbert Ackermann axiomatic system or principia magnetic that is Russell whitehead axiomatic system it is defined as follows at least these are the four five things which essentially any axiomatic system should consist of the first thing is that a set of well-formed formulas are usually called as axioms these axioms need not have to be proved and all so there are considered to be self-evident truths which are obviously true always true and all and then these axioms will also serve as schemas schemas in a sense that you substitute anything into it uniformly you will generate a theorem so that is the greatness of this axioms and these things does not require any proof that is why they are well self-evident kind of truths in addition to that there are some set of rules of inference which licenses some kind of operations on well-formed formulas of propositional calculus that means whatever you intend to have as an operation will not work in all but there are some set of rules like rules of inference such as modus ponens transformation rules such as if you uniformly substitute some formula into the formula then also you will generate theorems from the axioms so this is the second thing you need to have transformation rules such as modus ponens etc and all as minimal as possible in any given axiomatic system these rules would be very minimal and the third thing you need to have a set of well-formed formulas in the propositional logic propositional calculus which can be obtained from the axioms by means of rules of inference that is usually only one rule of inference is usually used that is the modus ponens rule so when you have a in plus b and a gets detached and whatever you whatever follows is b that is the main rule of inference which we commonly use in all the in throughout the axiomatic systems whatever axiomatic system you are trying to consider the fourth one these new well-formed formulas are usually called as theorems of the system that means you started with the axioms and then you transform those axioms are you trim those axioms in such a way that by applying transformation rules and truth-preserving rules such as modus ponens you generated another kind of inference so our path is like this that our path is like each and every step starts with the truth that is the axiom which is obviously true and then you if you apply transformation rules on these axioms and the resultant well-formed formula is also going to be true and then that gets transformed into another well-formed formula that is also going to be true each step is considered to be true so in that sense the last step of your proof is also going to be true if all the steps that are there in the deduction process is already true and the last step of your proof which is usually called as a theorem is obviously considered to be true so establishing the validity is another kind of issue but deriving kind of true formula from a given sequence of formulas is another issue so lastly that we use a particular kind of term that is what we call it as thesis thesis is like some kind of it can be taken as either axiom or it can be even taken as a theorem so in order to say that any well-formed formula is axiom or even a theorem sometimes we use this word thesis so every thesis is obviously considered to be a valid well-formed formula of a given field and every valid formula of that field is also considered to be a thesis of the system either it should be an axiom or it should be a theorem all the valid formulas it should be one of the one of the things should be true either it should be an axiom which doesn't require any proof or it can be proved by reducing the given axioms by using transformation rules and rules of inference to another kind of theorem and all. So this is what we mean by an axiomatic system so in a nutshell an axiomatic system should consist of at least set of axioms to start with either you can have three or you can have four or you can have five but as minimal as possible to start with and then you apply transformation rules substitution rules etc and all you transform these axioms into another thing which are also considered to be theorems and then you need to have some kind of rule of inference that is usually we use it modus ponens that's all you want to derive all the two propositions that occurs in your formal logical system. So now there are certain terms which needs to be defined clearly before moving further otherwise it will create some kind of confusion so the moment axioms the word axiom postulates etc they are one of the same Euclid has used this particular kind of phrase postulates which more or less means the same thing as axiom that is what we are trying to use in the modern mathematics in particular. So an axiom is considered to be a statement usually a statement is considered by the true or false that is assumed to be true without any proof and all it doesn't require any proof for example we will be presenting some this axiomatic system Russell and Whitehead axiomatic system suppose if you says that these are my five axioms that means any one of these axioms cannot be deduced from any other axiom that means one axioms should not reduce to another one so it so happened that Paul Bernes the logician which was come later contemporary to better Russell and Whitehead he came up with the proof of his fifth axiom of Russell Whitehead axiomatic system can be deduced from other axioms in that sense it loses its status of axiom it will become a theorem so when Russell and Whitehead formulated is axiomatic system it has five axioms. Suppose if it is proved any axiom is proved from any other thing then it is considered to be a theorem it will lose this axiomatic status so coming up with this axioms is the most difficult kind of thing so axiom is a sentence which doesn't require any proof you are like silk evident kind of truths in order to start any branch any field of enquiry as you are talking about physics or mathematics or anything one needs to start with some kind of metaphysical assumptions for example when you are studying physics in particular concept of matter and all these things you will be asking several questions in all how the matter has come into existence then you will date you will go back to some billions of years ago and then you start saying that there was a big bang and then from then due to the big bang then it has resulted in some kind of hot planets etcetera hot things and all it started cooling cooling and all and different planets have formed etcetera our universe has originated by means of big bang so now if you go back and ask how this big bang took place and all you are all silent and all these cases so we assume that there was a big bang and then from that all the universe has originated so in the same way when you are talking about theology or something like that you will have your metaphysical assumptions such as God has created this universe in six days and seventh day you took rest etcetera so you need to start with some kind of metaphysical assumptions to do anything to start any kind of enquiry and all so in the same way the axiomatic system you need to start with axioms which doesn't require any proof and all they are self evident kind of statements which are obviously true it's like 2 plus 2 is equal to 4 which cannot be questioned and all of course one can even question the foundations of any area and all you can start questioning the axioms but usually it is the case that they are assumed to be true without any proof so these are also considered to be basic building blocks from which all the other things can be constructed just like you are constructing a building you need to have bricks and bricks are arranged in certain way and you will build nice structures. Examples could be Euclid's five postulates general of frankle axioms or pianos axioms in arithmetic etcetera general of frankle axioms in the set theme so now what you mean by a theorem a theorem is a mathematical statement that is proved using some kind of rigorous mathematical reasoning that is what we spoke about it in extensive detail that is a deduction a deduction is considered to be a deductive argument is considered to be an argument in which if your premises are in certain they are absolutely true etcetera so that is what we mean by deduction in a mathematical any mathematical paper that you come across the term theorem is often reserved for the most important results if you consider any mathematical research paper and all is often considered as one of the important results that is considered to be theorem and that needs to have a proof in mathematics if you say that I have come up with a theorem and then if I do not show I mean that it is it can be proved in all that will never serve as a theorem you need to provide a rigorous proof for whatever result that you are trying to talk about in a mathematical paper so in a nutshell the last step of your proof is what is considered to be a true statement that is considered to be a theory a theorem so now most of the time we will be using these words lemma whenever you are reading some kind of mathematical mathematics books and all you will be seeing all these axioms theorems lemmas corollaries etcetera what do you mean by lemma it is a minor result whose sole purpose is to help in proving a particular kind of theorem they are supporting kind of theorems so it is a stepping stone on the path of proving certain kind of theorems is a minor kind of important steps that you will be taking up in proving the major theorems so that is what we call it as a lemma and there are some other terms which are also very important so they are like this so what these are the terms which you commonly come across when you are reading a mathematical paper or any other mathematical text book a corollary a corollary is an important result which the proof relies heavily on a given theorem so we often say that this is a corollary of a theorem a etcetera and all this is some kind of thing which comes as an outcome of your particular kind of theorem so this is what is called as a corollary and proposition is the one which we have been discussing right from the beginning of this course that is any sentence which is usually considered to be as a true or false that is called as a preposition so it can also in the context of mathematics it is a proved and often interesting result but generally less important than a theorem a theorem is considered to be the most rigorous kind of a statement and all in your enquiry and of course axioms are all which doesn't require any proof and all they are absolutely true and all which doesn't require any proofs so now the other term which you come across is the conjecture so conjecture is considered to be a statement that is said to be proved and all it is unproved as of now but it is believed to be true so there are many such conjectures which are our gut feeling says that they are obviously true and all they are all true statements but as of now they didn't find any proof in the same way when Fermat has proposed Fermat's last theorem it took almost 200 to 300 years to prove Fermat's last theorem and all till that time it remained as a conjecture till somebody some mathematician has come up with a rigorous proof for Fermat's last theorem conjectures are yet to be proved kind of theorems and all we are our gut feelings tells us that they are obviously true but still it didn't find any won't be in a position to find a rigorous proof for that so this is what we called as a conjecture so all these conjectures can be refuted or maybe it can be proved also so till we find a rigorous proof for this conjecture we will not accept it as a theorem but all conjectures may be potentially capable of becoming theorems once it finds a rigorous proof the last thing that we will be using in the in this context is the claim a claim is an assertion in any mathematical paper you will be coming across this particular kind of thing so in any research paper etc and all you will be claiming something so that is considered to be an assertion that is then obviously eventually it is going to be proved it is slightly different from the conjecture conjecture is the one which our gut feeling tells us that it is obviously true it is often used like some kind of informal lemma it is not lemma as such but it is considered to be a kind of informal kind of lemma so that is what is called as a claim so now given this set of keywords etc and all what is that we are essentially trying to do is that we are presenting one particular kind of axiomatic system which is in which the proofs are considered to be rigorous as well as there are no other kind of hidden kind of assumptions which will become part and parcel of your proof I mean just as in the case of Euclidean axioms there are certain extra assumptions etc which has come which which also played a crucial role in formulating the proofs within the geometry we want to be we want to avoid such kind of thing so our proof have proofs have to be rigorous so what do you mean by a proof so now we have axioms theorems lemmas corollaries conjectures etc and all claims prepositions now with using these things what do you mean by a proof so usually a proof is considered to be a sequence of formulas and all these formulas will find some kind of justification so each step needs to be justified by some kind of a statement that is either whether you used modus ponens rule or use transformation rule or it has come from axiom or it has it is just another kind of theorem all these things needs to be addressed so each line of the proof in axiomatic system L it should have this particular kind of things so in a proof either the particular kind of thing has to be an axiom that means it does not need any proof and all you just write it and end it like that you say that that is an axiom that itself is a since there is not require any proof and all so that itself will be some kind of thesis and all thesis is the one which we have introduced earlier in that context we say that it is a theorem it is as well as an axiom also because the last step of your proof is considered to be a theorem so if you are given axiom and all you don't it won't require any proof and all then obviously it is considered to be a thesis so that is an exceptional kind of thing and all it's obviously a tautology etc and all so an axiom doesn't require any proof suppose if we come up with the proof of that one it will lose its axiomatic status so that is the first thing that you need to have and the second thing is that the next step whatever step that you are trying to consider whether or not it is a result of applying some kind of transformation applying some kind of rules of influence one rule of influence that we will be commonly using is the modus ponens rule that is a a and b follows from that particular kind of or you can begin your proof by assuming some kind of thing like it can be considered as an hypothesis which is already a part of you the given formula that needs to be proved so we discussed in the context of natural reduction method suppose if you are trying to prove a in plus b b in plus c and a in plus c so now what you will do is you list out all the hypothesis that is a in plus b b in plus c and in addition to that you will assume the antecedent of the conclusion that is a and from these three whether or not c can be deduced is the one which we are trying to see either it should be a hypothesis which comes from the given formula or it must be some kind of a lemma which is supporting kind of theorem it's also considered to be true so our journey starts from truth that is a axiom and transfer apply transformation rules to it is also true and then we are using truth preserving rules such as modus ponens etc and all it's a deductive kind of reasoning so that is also obviously absolutely true etc every step is considered to be truth so that's why our journey ends with the final step of your proof that is considered to be a theorem so the last step of your proof is usually considered to be a theorem so you need to note that one can come to the destination in finite steps and all some sometimes you can come up with the 15 steps to find the final kind of reach the destination that is the theorem or sometimes you might prove the same thing within some 10 steps and all so the proof which consists of cancer is obviously considered to be a rigorous kind of effective kind of proof when compared to a proof which consists of 15 steps are maybe more than that so as far as possible or in excessive information should not be there to maintain informational economy and in order to make this proofs effective your proof has to be the steps of your proof has to be as minimal as possible so when we write this thing l single term style a that means a is considered to be deduced from l since a is considered to be the last step of your proof so a is called as a theorem so that means we write z1 z2 zn etc the conjunction of all these formulas usually it can be it can be called as premises etc and a is considered to be the conclusion and that is deduced within the formal system l if and only if if a finds a proof in a given formal system from a given set of formulas z1 z2 zn etc it says that if we have a proof of for this thing q from p then we are guaranteed that we have a proof of p plus q as well so this is a sort of deduction theorem which was later introduced by her brand etc and all so we will talk about this particular kind of deduction theorem a little bit later so what essentially we are trying to do is from q if q is obtained from p then you discharge your assumptions like pq etc and all and you will start talking about p implies q rather than just p and just q so now there are in the elementary I mean suppose if you go back to our elementary schooling etc and all in the in our childhood we might have derived certain proofs which are considered to be bogus proofs but it might have convinced us at that stage so let us assuming let us assume that whether or not this constitute to be a genuine proof or it is a bogus proof or it requires some kind of fixing etc and all fixing the problem that arises in one of this steps in all so what is considered to be a proof so far we have discussed that each step has to be true and it should find a justification if a justification is wrong then you cannot move to the further step so there means some this proof is considered to be defective or there seems to be some problem with the proof so now observe this particular kind of proof to is equivalent to one so for that you start with some assumptions let us assume that a is equivalent to b then you multiply a both sides and the second step you will get a square is equal to a b now you add a square to both the both sides and all LHS and RHS then it will become the third step that is a square plus a square is equal to a square plus a b because you have added a square to both sides LHS and RHS so now the in the step 3 LHS will become 2 a square equivalent to a square plus a b so now you subtract minus 2 a b both sides this is what you get so that is 2 a square minus the six step 2 a square minus 2 a b and a square minus a b so now you take two common of these things 2 into a square minus a b and 1 into a square minus a b so it is like 2 into a square minus a b is equal to 1 into a square minus a b so this is what you got it from a is equal to b all the way down you got this particular kind of step till here it does not seem to be a little bit problematic in all because you started with all these assumptions etc and all they are not just done anything etc except that we are added subtracting from both sides etc so now the problem real real problem comes from is that the moment you cancel these two things then suppose if you say that so this is the n minus one step and this is the nth step you cancel a square minus a b both sides and then you will infer that 2 is equal to 1 and you can say that a square minus a b and a square minus a b it will become 2 is equal to 1 but one of the important principles in the logic is that you cannot cancel in such a way that you know a square minus a b from both sides because when especially when a is equal to b then a square minus a b usually it will become so when a is equivalent to b it is substitute for a b and all becomes b square because it is b into b is b so now it will become 0 so 0 cannot be cancelled with another 0 and all so it is in that sense our principles of mathematics will not permit us to move from this it will not allow us to cancel this particular kind of step and all so now this will not lead us to the next step because there was some problem with the justification our justification is defective and all because 0 cannot be cancelled by 0 so that is not permitted in at least in the principles of arithmetic so that will not let us allow that 2 is equal to 1 of course if you do not notice it properly and all sideline this particular kind of thing it appears that you know it appears to be a wonderful or nice proof and all but you can you can stop it step number 7 and you can question how did you cancel a square minus a b both sides and all 0 cannot be cancelled by 0 0 cannot be cancelled by 0 so this step is not allowed so that is the reason why this itself the proof itself starts stops here and then you need not have to talk about this particular kind of so now each step of your proof needs to be justified so here there was a problem with the justification it goes against the principles of arithmetic so you will not generate a proof for 2 is equal to 1 if you generate a proof for 2 is equal to 1 and then say that this is your effective proof and all and this is not considered to be an effective proof it is a bogus kind of proof so then the next question that comes to us is that what constitutes an effective proof so as it appears that from this particular kind of thing what we can learn from this particular kind of proof is that if your proof has to be effective etc and all each step should find a justification that should be according to the principles of logic etc in the same way when you are talking about axiomatic system which is based on set of axioms and transformation rules etc that thing has to when each step needs to be justified either by using the axiom or transformation rules applied on axiom or it should be it should result in from by up in from applying some kind of modus ponens principle or I mean any one of these things should be there then only the next step is going to be justified but here the seventh step is a problematic step your proof ends there itself and you can you can clearly say that 2 cannot be equivalent to 1 so there are certain things which are very important mathematicians would always be interested in starting with the tautologies that means the true statement a mathematician will never begin with contradictions the contradiction is a one sentence which can be spoken as both true and both false and x and not x is kind of inconsistent statement is unsatisfiable also considered to be contradictory to each other p and not p is contradictory to each other so why mathematicians think that this the contradictions are going to be considered to be as a hell so here is one of the interesting and funny kind of example which is given by a famous mathematician again they can't since we are talking about Russell white had axiomatic system so this example is also due to Russell invited Russell Betten Russell in one of the parties which he attended he funnily proved that a contradiction implies anything so that is the reason why you know one mathematicians will always be loving this tautology is etc and all from truths you generate truths etc and all but when you start with the contradiction you can derive anything so this is what is the thing he just gives a funny kind of proof so that is like this is what Betten Russell says if 2 plus 2 is equal to 5 that is a false statement he false statement implies anything then he is trying to prove any strange kind of proposition that you know that he is going to be a pope he wants to establish himself that if 2 plus is going to be 5 then he is going to establish that I am going to be a pope I mean in Betten Russell here so now so these are this is the proof that he tries to give is a funny kind of proof and all this not be taken very seriously and all but usually mathematicians would love to start with truth tautologies are rather than the contradictions because contradiction needs to help so now the proof goes like this if 2 plus 2 is equal to 5 then if you subtract 3 from both sides so that is like this first you take the antecedent of this particular kind of thing so now our assumption is that 2 plus 2 is equal to 5 so now 2 plus 2 minus 3 you substituted minus 3 from both sides subtract sorry not substitutes subtract 3 from both sides so this is what you get so now 4 minus 3 is 1 and this is equivalent to 2 so this is the proof and all so this is your hypothesis 2 plus 2 is equal to 5 you are assuming that that is true if that is true then the next step that is 2 plus 2 minus 3 5 minus 3 also has to be true that means 1 is 2 is also true now the third step is that is funnily he has used it is not a rigorous kind of proof and all it is only is this proof is only used for the sake of fun so now better Russell says that better Russell and Pope are two different people better Russell suppose one is taken as a better Russell and two means that better Russell and Pope are two different things so now this equation tells us that better Russell and better Russell and Pope viewed it in different way they are one of the same since one is already equivalent to better Russell better Russell and Pope also has to be same even if there is another person exist in all Pope the two has to be equivalent to one so it is in that sense better Russell and Pope are considered to be one person so that means he has to be none other than Pope himself that means he proved that he is the Pope so this is a funny kind of example in all so which this is come from this thing that falsity implies anything so this the same thing which can be proved in classical logic in a different way by using the principles of logic so that is the reason why we do not begin with contradictions in a given proof for example you start with P and not so this is we know that it is an apparent contradiction a statement cannot be both true and both false so now this is what is given if you assume that this is true and even these are also true because of this particular kind of principles so this eliminate this conjunction then P will become true and in P and not P not P is also going to be true so what we have done is we have eliminated this conjunction so this is what is called as impossibility or something which is a contradiction so now fourth one since not P is already true is also always considered to be true then irrespective of whether or not Q is true or Q is false the whole statement is going to be true because of this particular kind of formula this we know that the semantics of disjunction is like this that if it is F and F T F T F and all then it is going to be false only in this case in all of the cases is going to be true so now we know that this is already true so that means these are the two things which you need to take into consideration instead of P we have not P here so now irrespective of whether Q is P whether Q is false the P or Q is going to be true only so in that sense this statement is also true so now in the fifth step you have P here not P or Q now so these two so how did we come to this one this is law of addition so now this and this 3 and 4 sorry 2 and 4 disjunctive syllogism leads to this one Q so that means any particular kind of strange preposition one can prove so starting from the contradiction suppose if you say that it is raining and it is not raining then the statement is going to be pigs flies and all Q is considered to be pigs flies so this is one way of proving this thing so now the problem here is that you can prove either Q and even you can even prove not Q also so how can you get this not P again the same proof first step you take the same assumptions and 4 so instead of this one what we try to do is since P is already true you can add any strange kind of preposition Q and this is also going to be true so now fifth one so these this is law of addition this is the case of this particular kind of thing so now observe these two 3 and 4 disjunctive syllogism will lead to not Q so now with the contradiction you proved not Q and you with the contradiction even you proved Q and all so that makes your system inconsistent since you are derived Q and not Q is part of your formal system whatever system you are trying to talk about L so it is in that sense your formal logical system is going to be inconsistent that is what we are trying to avoid so this eventually led to different kinds of theorems which are little bit counterintuitive and all like paradox of material implication which we talk about it after we introduce Russell whitehead axiomatic system so now with this particular kind of note I will try to end it and the all other proofs etc we will try to do it in the next lecture so now what is that Russell and whitehead as achieved in the book principia mathematical lots of things which he did it is a voluminous kind of work and all which consists of three books ranging from 400 to 500 pages each book has 400 pages so now better Russell and white in the book principia mathematical which is considered to be a path breaking kind of book one of the greatest books of this 19th 20th century in that it is a kind of some kind of constructivist project constructed proofs etc it suit to show that all arithmetic can be reduced to logic that means there are two things which are important here when you whenever you talk about some statement in the arithmetic that can find an appropriate translation in the logic that means if you if you utter any statement in the arithmetic it will have its corresponding language you can discuss everything with the help of only this axioms transformation rules and other things so this grand program is what is considered to be a program which is called as the logistic so logisticism is like this so the thesis of logisticism is that all the mathematical concepts are definable in terms of logical concepts so that means you express any kind of plus operation or anything in arithmetic etc that has its corresponding logical operation that means it is in that sense you can talk all the mathematical truths in terms of logical truths logical truths are axioms etc and all so any kind of mathematical statement can be converted into an appropriate appropriately into a corresponding logical statement and all the mathematical modes of inference are reduced to logical mode of modes of inference and it is in that sense all mathematical knowledge is nothing but a real logical knowledge so so the basic idea here is that mathematics is a branch of logic or all the mathematical concepts can be reduced into the concepts of logic for example if you if you come up with an axiomatic system which consists of only a set of axioms and transformation rules and the rules of inference as minimal as possible rules are as minimal as possible and then you start talking everything in the language of this axiomatic system so what is there in our language of axiomatic system with respect to principia mathematical we have some Russell invited has five axioms and then there is a transformation rules and substitution rules and then the modus ponens principle that's it that's all we have and all the truths of mathematical arithmetic etc they can be appropriately translated into this one of this axioms all the p in plus q in plus etc will become statements of arithmetic and if we can do that particular kind of thing then it attains is rigor and all because whenever you are proving certain kind of theorem such as simple theorems like p in plus p or law of excluded middle etc and all that can be this p skews ours can be some of the statements in mathematics so now we are generating all the mathematical theorems we are trying to show that they will come as logical theorems now so mathematical knowledge will now turns out to be a logical knowledge because we are using only axioms and modus ponens etc and all so this is what we do in the principia mathematical so mathematics can be appropriately reduced to logic so now there are other things which Russell invited is trying to achieve the other major historical figures in the constructivist camp so how did they construct they constructed various kinds of theorems based on few set of rules which are few set of axioms rules and the transformation substitution rules there are lots of other axiomatic systems which are already there before Russell invited you have freguese axiomatic system in canter and of course after that David Hilbert were trying to axiometric geometry and he has come up with axiomatic system David Hilbert axiomatic system and Paul Bernays especially showed that one of the axioms of Russell invited can no longer have that particular kind of status we have different axiomatic systems and Pano Sardimetic also comes under this particular kind of category all the mathematics should be developed through appropriate definitions in the systems of logic defined in the principia if you can achieve that particular kind of task and all then you are said to have reduced mathematics to logic in the sense mathematics is considered to be a branch of logic so all the arithmetic analysis the other things which you commonly come across in the mathematics set 3D etc if this can be reduced to a set of concepts of logic then it can be called as developments of pure logic rather than the development of mathematics so in the next class what we will be doing is we will be focusing our attention on the main book the principia of Mathematica due to better Russell invited where there is a chapter specific chapter on deduction where using his set of axioms set of axioms he showed that for example law of contradiction law of excluded middle law of identity etc can be derived from this set of axioms so you should note that we should note that any formal axiomatic system that you are going to come up with at least we have to ensure that one should be in a position to derive the minimal things such as law of excluded middle that is a sentence is either true or false or law of identity such as P implies P etc all these things should come as theorems in your axiomatic system in the next class using the principles using the axioms of Russell whitehead will be proving certain important theorems such as law of identity law of excluded middle law of contra position all these things are valid theorems so now what is that we are trying to achieve in a nutshell is that we know that certain kinds of valid formulas exist in our formal logical system so now we are trying to find a rigorous proof for this valid formulas valid well-formed formulas so that is what we are trying to achieve and this will constitute a part of the proof theory so once we introduce this axiomatic systems then we will discuss about whether or not your when a given axiomatic system is going to be consistent that means when you will be in a position whether or not you are in a position to derive both x and not x if you are if you are able to derive x and not x as your theorem your axiomatic system that axiomatic system is going to be inconsistent or we will be talking about all the valid formulas whether it finds proof etc and all so that is all the two statements are provable and all the provable statements are true that is a soundness property or whether you are a formal axiomatic system is going to be complete etc and what are the limitations etc all these things we will discuss in the forthcoming classes.