 Welcome back in the last lecture we discussed one important decision procedure method that is the semantic tab-lux method, semantic tab-lux method phase better than the other semantic methods that we have that is the truth table method, truth table method will become quite difficult especially when the number of variables increases to 4 or 5 I may be more than that a computer can easily do it but as humans it is very difficult to process that much of information you know. So today what we will be doing is we will be talking about one syntactic method where we will be basically studying about some of the proofs of some important theorems in the propositional logic. So we started our journey with some kind of well-formed formulas and out of this well-formed formulas some are considered to be tautologies which are always true and some are considered to be always false they are considered to be contradictions and there are some other well-formed formulas which are sometimes true sometimes false these are called as contingent kind of statements. So what we will be doing today is we will be presenting a different kind of method which also serves as kind of decision procedure method with which you will come to know that I mean how to prove certain kinds of theorems. So anything which is there is one important theorem in propositional logic which tells us that whatever is provable is obviously true and whatever is true is also provable. So it is in that sense if we can prove that something is a theorem then you can already said to have shown that it is a tautology. So in this lecture we will be trying to study about the natural deduction and then we will talk about some of the rules of inference for proving certain theorems. This is considered with outline of this lecture first we will be talking about what we mean by natural deduction and then we will consider some examples of some of the important proofs which are proofs of some of the theorems. So then we will talk about conjunctive and disjunctive normal forms which we will talk about it little bit later but first we will focus our attention on natural deduction. So how did it come into existence? So it is introduced by two logicians in two different papers they were working independently to each other they are not aware of each of the works and all. So parallel is two papers were presented and each one is not knowing about others work and all. So the first is due to it is attributed to Gerhard Kensen and Jacos is a Polish logician they were working at the same time and then they came up with more or less similar kind of results and all. So it is in the year 1934. So what there of the view is this thing? This discovery is as important as other important discoveries such as discovery of resolution by Robinson which is there in 19 just come up little bit later in 1965 there are some of the important results that are very important in the prepositional logic. So these are like this are the discovery of logistical method by Gottlob Frege in the year 1879 and even I mean there are some discoveries like path breaking discoveries by the Greek philosopher such as Aristotle in the 4th century BC. So this natural deduction method is equally as important as one of these path breaking kind of decision procedure methods that are discovered in the history of logic. So why it is called as natural deduction? So logicians call this way of reasoning a natural deduction because it is meant to come as close as possible to the human reasoning which they use in the day to day discourse. So as far as possibility is closer to the intuitive reasoning or the reasoning that is employed in day to day discourse by us. So that is why it is considered to be a natural deduction method. See natural deductive approach is designed more for the ease of use and the closeness to intuitive methods of reasoning. So that is why it has got some kind of prominence in the literature of in the history of logic. So what is that we are going to do in this natural deduction method? So the founder himself says like this Gerhard Genshin remarks like this. My starting point was this the formulation of logical deduction especially as it has been developed by Fregge Russell and Hilbert is rather far removed from the forms of deduction used in practice of in practice in mathematical proofs. So we are going to see little bit later about some of the some of these proofs which are which we will be doing today in the context of axiomatic systems. And you yourself will note that how difficult it would be to prove simple theorems such as p ?p or law of excluded middle p or not p etc. So these things will become little bit simpler in natural deduction method. So there of the this view that mathematicians especially when they prove certain theorems in all they may not be following clear the method that is adopted by Russell and Hilbert. So they went on to say this thing that considerable formal advantages are achieved in written in contrast I intended first the Gerhard Genshin intended first to set up a formal system which comes as close as possible to the actual reasoning. So actual reasoning is the way we reason actually in day to day discourse it is a kind of some kind of linear process etc. So the result was this thing that a calculus of natural deduction. So it is called as Nj if you are an inclusionist you name it as Nj otherwise Nk especially the classical logicians belong to classical predicate logic we call it as Nk. It does not matter in what way you call it. So this natural deduction method is closer to the actual reasoning that we that a human being employs in day to day discourse. Then he went on to say that about this natural deduction that so there are two methods with the one method which we will be talking about little bit later that is the axiomatic method. So the difference between natural deduction method and the axiomatic method according to Genshin is as follows that the essential difference between natural deduction derivations and the derivations in the systems of Russell, Hilbert, Hitting etc. Hitting is an inclusionistic logician is the following. In the later systems that means in the Russell, Hilbert, Hitting axiomatic systems true formulas are true formulas in a sense they are theorems true formulas are derived from a sequence of basic logical formulas usually they are considered to be axioms by means of few forms of inference rule and so what they achieved in the axiomatic system in case of Russell, Hilbert, Hitting is this. They started with the fundamental first evident self-evident truths and all they are called as axioms which cannot be questioned and all they are obviously true and they used very few rules of inference. The only rule of inference mostly they use is the modus ponens principle that is if A in place B is the case A and then B follows from that. Only that particular kind of inference rule very few inference rules they use and then rest of the things are only axioms and axioms are also very few in number. Sometimes in Fregge there are 5 axioms in Russell Wighthead there are 5 axioms whereas Hilbert Ackerman has 3 axioms to begin with and then they use modus ponens and then all the theorems I mean those statements which are obviously considered to be tautologies are all derived within that formal axiomatic system using some kind of transformation rules etc. On the other hand natural deduction however does not in general start from basic logical propositions they do not start with the fundamental principles or self-evident truths such as axioms but rather from the assumptions to which the logical directions are applied. So what they do is they start with there are some obvious rules of inference in used in logic like modus ponens, modus tolens, constructive dilemma and all these things law of conjunction, law of addition. So these are the principle they start with they begin with these assumptions truth preserving rules plus there are some assumptions which comes from the given formula and from that they will deduce some kind of theorems. So by means of a later inference the result is then again made independent of the assumptions there are no assumptions etc are used in the formal axiomatic system used by Russell Wighthead etc except that they used some kind of axioms which are considered to be self-evident truths which are obviously true etc. So now we will be considering some of the important proofs within the axiomatic system. So before that we need to the motivation for this thing for doing this natural deduction method is as follows. So far we have seen semantic tablox method truth table method etc in the truth table methods so they are very simple and they are mechanical and they are large proofs of validity for example if there are n propositions we have 2 to the power of n rows in the truth table. If n becomes large then it becomes difficult for us to handle because for checking the validity of a given formula for example if there are if n is 6 then that means 2 to the power of 6 entries will be there in the truth table that is around 64 entries you need to check all these 64 entries and then you need to check for a row in which your premises are true whether your premises are true and the conclusion is false if that is the case then you will say that the argument is invalid instead of checking all the 64 rows and then those rows in which you have two premises and false conclusion etc for proving the invalidity. So it becomes a little bit difficult for us to handle when n becomes large usually it is unsuited for human use because the once the number of rows increases then it thinks will complexity increases. So it is usually unsuited for the human kind of reasoning especially when n is large the entries in the truth table are large then computer can process it but it is very difficult for us to handle. So there are some other effective methods which we have introduced earlier that is the semantic tableaux method which is considered to be elegant method so far and then now we are going to see the natural deduction method it is suitable for human reasoning because it is in the beginning we said that it is closer to the actual reasoning the way we actually reason in day-to-day discourse it comes closer to that if not it is completely same as that one but it comes as far as possible it is closer to the human reasoning that we employ in day-to-day discourse. It often allows short proofs of validity for validity in particular we can prove it and that is considered to be a true statement and then all the true statements are obviously valid formulas. So it allows for shorter proofs of validity which are going to see how we are going to achieve this thing and it is also arguably too flexible for machines use it is not mechanical a large number of rules it involves large number of rules but it occupies it also occupies infinite such space is another limitation of natural deduction method. So what essentially we do in the natural deduction method is as follows we start with simple truth preserving rules like modus ponens modus tolens and the constructive dilemma they are all valid principles of reasoning in that we employ in logic. So we begin with these rules of inference and then we add these rules of inference to assumptions that we get from the formula which we are trying to prove and then from that we will get the desired kind of formula that we are going to derive. So natural deduction is used for constructing proofs of theorems theorems means there are obviously true formulas they are also considered to be tautologies etc. They are different from axioms axioms are considered to be self evident rules etc. So theorems will all be valid sentential formulas and all valid sentential formulas will obviously theorems of that particular kind of system any formal system you begin with all the valid kind of sentential formulas are obviously true tautologies etc. So all such valid formulas constitutes what we are calling it as sentential calculus. So what is that essentially we are saying is this thing you began with well formed formulas and then out of that there are some tautologies all these tautologies are obviously valid formulas and then this all these valid formulas constitutes what we mean by a sentential calculus. So it becomes a particular system the system sentential calculus is it can be viewed as a natural deduction system as it makes use of rules without any axioms. So we are going to see how we are going to achieve this particular kind of thing. So the derivations in natural deduction corresponds to how we actually are naturally construct these theorems. So in the tree method especially we what we have done is simply for establishing the validity of a given formula well formed formula what we have done is simply this thing we listed out all the premises and we included the negation of the conclusion and then considering the negation of the conclusion leads to the branch closure that means it is unsatisfiable then we have said that negation of the conclusion is false that means the actual conclusion is true. So we list the premises but not the conclusion in the natural deduction method we do it in a different way we list out all the premises and then using all the truth preserving rules which we are going to talk about in a while from now we use those rules and we have premises which serves as assumptions and from that you will derive whatever you wanted to derive. So what we will do here is that we list out all the premises but not the conclusion is the one which you are trying to derive from these premises then we apply natural deduction rules which are true preserving and that will lead us to our destination that is the conclusion that is what we are trying to prove. So we are able to write the conclusion of the argument based on the assumptions plus natural deduction rules. The natural deduction rules are usually truth preserving thus we are able to construct the conclusion by applying them to premises and we know that the truth of the conclusion is entailed by the truth of the premises that is what we mean by a deductive argument the deductive argument is a one in which it is impossible for the premises to be true in the conclusion is false but here all the rules are all the steps of your proof are all truth preserving. So based on that there is some kind of truth preservation is achieved in this particular kind of proofs and obviously each step of your proof is true and ultimately the final step which is considered to be a theorem which is obviously true and all the true formulas are obviously they are considered to be tautologies and all tautologies are obviously valid formulas but the problem with this method is we talk at the end of this lecture is this that sometimes we may fail to derive the conclusion using these particular kinds of rules your derivation go on and on and on so then that sense it may not be a kind of effective method. So for that what we will do is we will follow another kind of method within the natural deduction that is called as reductio ad absurdum. So what we will do is we list out all the premises and we negate the conclusion and then we will derive a contradiction in sense that a and not a and that is the case then you cannot deny the conclusion that means the conclusion denial of the conclusion leads to the contradiction that means the actual conclusion is true. So now before doing natural deduction there are some rules which we will be following these rules are like this. So these are basically two different methods that we will be using first one is RCP conditional proof so what we will do here is this thing. So we have some kind of assumptions a which usually comes from the given formula and we obtained a line where you found B. So now B is a tautological consequence of A if that is the case then A ? B is considered to be a tautology. So what have what actually happens here is this thing so you have A here and then obviously after few steps after applying natural deduction rules you got B so that means A is derived from B is derived from A if that is the case then you can simply write like this A ? B you draw a line like this and then say that A ? B is derived from some kind of assumptions and natural deduction rules which you employed that helps us in moving from A to B that means A ? B is the one which you have derived so this is called as conditional proof. So that means in your derivation suppose if you have a step A and from that you got B now you draw a line like this then you say that you obtain A ? B because this says that B is derived from A so that means if this goes to the right hand side then this becomes A ? B suppose if I say this is a single turn style this means that it is a theorem this should be read as like this it is a theorem that A ? B A ? B is a theorem within that formal this is that system that is a natural deduction system so this is what is called as RCP that is conditional proof. So now the second thing which we will be using in the natural deduction method is this thing which is called as reductio and absurdum. So in this reductio and absurdum you started with some kind of assumptions it can be premises it can be hypothesis you can call it as a hypothesis and all the way down here in your proof you got some kind of contradiction so usually it is mentioned in this way so there are two symbols which we are trying to use this is the symbol which we use it for a formula which is always true and there is something which is called as a same formula which is always false this is top and this is what this is different from T and F for truth values. So now what happens here is that from a all the way down somewhere else here you got a contradiction so that means a implies this one so from a you got contradiction and all so that means if a implies contradiction then that means it is not it is not simply a and all it is it should be not so this particular kind of thing is called as reductio and absurdum and all so it essentially says that given some assumptions which leads to contradiction that means let us say if your assumption is a and that led to contradiction then the actual assumption is false that means it should be not a suppose if you began with not a and led to contradiction then that should be a so this can be also written in this way from not a it led to contradiction then it should be not not a that is so these are the two important methods that we will be using in the natural deduction so the idea of reductio and absurdum is this that if you derive a contradiction from a given assumption then that assumption is false that means it has to be not a so that is what we all said here. So these are the things which we follow but the actual founder Genshin he proves theorems in a totally different way and all so mostly only in few logic textbooks you will find the proofs which are given by Genshin but usually we follow just go see ski's proofs in particular which are which seems to be a little bit simpler and simpler to understand but both the proofs can be used so now what are the rules that we will be using in this natural deduction method so this these this can be compared with playing a game and on example let us say you are playing cricket you do not have to know how how and when it is going to existence etc you do not have to know the foundational principles etc what you exactly need to know is the rules of the game you have to know about white ball you have to know about no ball etc now so once you know these rules then things would be relatively easier so then you there will not be any defects in your in our game so are mistakes in the mistakes which can happen in the same way natural deduction you start with the assumptions and then you have some kind of rules and these rules are like this particular kind of thing so this some rules are considered to be discharging some kind of assumptions and then in some of the rules you will introduce some kind of implication etc and so these are some of the rules that we will be using in the natural deduction method you might ask do I need to remember all these rules etc and all it comes you know through practice will not be much to remember and all these things obviously follows so these are some of the rules that we will be using in the natural deduction method so first let me list out these rules and then we will talk about something about these rules and then we will move on to some other I mean improving certain kinds of theorems so these are some let us say ? is some set of assumptions that we have and from this let us say union there is another formula B and from this you go so this is a and then from this you got B this says that ? is something like set of formulas which are sitting at the background so it can be some kind of natural deduction rule a principle or whatever it is and these things which you are adding it to the assumption that is there which occurs in a given formula and from this you got B so now what you will do is your assumptions are a and B and all here now you discharge these assumptions and then you will write simply this thing from ? you obtain a ? B so now earlier it was only a proposition a and proposition B and all now you discharge those assumptions and all instead of saying that it is a B and all you will say now a ? B is the case so now you discharge these assumptions and all so this is one of the rules of which come under the category of discharging the assumptions and all in the same way if you have a formula like this thing not a so you might ask what is this ? and all ? involves all this truth preserving principles and all it can be this thing a ? B which I will talk about it a little bit later so they are all truth preserving principles and all you can be considered as a kind of tautology you know but it is not axioms in this case or in the case of natural deduction method something which is true now from this let us say you got something called a contradiction so that means what you got is this thing now not a place contradiction so this is what you got so that means not a implies this is the case then it should be a so now you simply write a so this is one thing which we will be using there is something called reiteration rule which is some kind of redundant kind of rule and all suppose if you have a formula a and then you can simply say it is a nothing great about this particular kind of formula if you have a you can derive obviously a so this is a reiteration kind of rule so now there are some other kinds of rules which are there for conjunction disjunction etc so then I will talk about some kind of rational for these particular kinds of rules so in all these things gamma is already there means something which is true and all so now so this stands for conjunction introduction I stands for introduction so when you will do that particular kind of thing you have obtained a and you have obtained b in your proof let us say you have gamma and then all the way down in your proof you got a and you got b then you will simply write you will introduce the conjunction and then you will write a in plus b so now this rule is called as conjunction introduction so now conjunction elimination so each natural reduction rule comes up in some kind of pairs and all one is introduction another one is the elimination so each connect you will have this particular kind of rule for the conjunction if you have a b you have proved a and you have also proved b then a and b can also be proved from this particular kind of using particular kind of truth preserving rules conjunction elimination is like this if you have a formula a and b you can simply write a it is like it is raining and the grass is wet from that you can prove grass is wet or you can even prove it is raining so what we have done here is that we have eliminated this conjunction and then in your proof you are substituting it with simply a and in this way in this sense a and b b also can be derived so this is called as conjunction elimination rule so now this stands for disjunction disjunction introduction suppose if you have a formula a that means is true and you can safely add any other things without disturbing the truth value of this one so in your proof each and every step has to be true so without disturbing the truth value of your propositions you can safely add another b because if the b value even if it is true or false it is not going to affect the truth value of the compound statement even if b is t and this all become true even if b is false then a or b is again true so you have to ensure that in your proof all the steps are obviously true then only the final step is going to be true that is a theorem so in that sense that is going to be a tautology and obviously that formula is going to be valid so our effort is this that ultimately we have generated some kind of valid formulas and then we are trying to prove those valid formulas and for proving there is some kind of decision procedure method that is the natural deduction method so this is what is the case and disjunction elimination is like this so now you have shown that a or b is true and then from a independently you proved this thing see that is what you write it here and then from b also you got C I mean C is deduced from a and C is deduced from b and a or b is already there if that is the case then you remove this disjunction and you will simply state here so you simply say it is C so this is what is called as disjunction elimination so you have a or b which is already true and then from a independently you proved C and from be you proved C and that means from all these things you can prove C you can eliminate this disjunction and then show that this is the case suppose if these are not there then you cannot say that a or b C can be deduced so this is another rule the rational of this rule is that one which we have discussed so now as far as this implication is concerned again we have introduction rule so especially from a let us say and you prove b then what you will do is you will introduce this implication and you will say that a implies b is reduced from this thing so this is what we mean by introduction so there only assumptions a and assumption b is there and then now we are introducing this thing because b is reduced from a so that is why a ? b will come as a theorem of this one so now this is what is the introduction of implication and in the same way on the other side introduction elimination when I write e is stands for elimination I means introduction this will be like this for example if you have a formula like this thing this is usually modus ponens rule and all so now you will eliminate this implication and you will say that it is simply b so what is that we are trying to do is simply making use of some kind of rules so what are these two there are all truth preserving enough you see any one of these rules obviously for example if you take this one if a is true any ? b is also absolutely true 100% true then obviously b has to follow from this this is the principle of valid reasoning so instead of starting with axioms etc and all we start with only these rules but the problem here is that one has to remember these some of these rules of course with some kind of practice with some kind of strategies which we can use with that you can say you can usually remember these simple rules or if you do not know some of these rules are you can derive these rules from some other kinds of things which you have already know so this is one thing and then we are talking about conjunction disjunction and implication and the negation of this one is like this for example if you have a formula a then you will derive only this one and then negation elimination is like this if you have not not a then you will negate this elimination and you will simply write a in your proof so that is what is with respect to negation so now there are some other rules for example if you have not a and then from this you got a contradiction then this means it is not a it is not not a but it is a in the same way if you have a formula a and then you got contradiction contradiction in the sense that you got x and not x that is the case it is called as a contradiction then since a led to contradiction it should be not so that is a rational of this particular kind of rules it is just like you know in playing while playing cricket you will be discussing about what you mean by no ball what you mean by wide ball or when you will get five runs etc and all these things you will be discussing what you mean by a boundary or six or something like that all these things are some kind of rules which game in the game subsequently we evolved with this particular kind of rules just like that traditions while practicing with the number of theorems and all proving number of them they came up with this particular kind of rules they are all truth preserving rules there is no way in which you can say that if this is true this is true then it cannot be the case that a and b can be false enough if that is the case it is not a directive argument at all so these are some of the rules which we have and then there is one more thing which is important so whenever you have a and then you came up with not a then so this led to this contradiction you derived a and you derive not here also then obviously that is a contradiction you cannot simultaneously say that it is raining and it is not rain so then this leads to this one what is happening here is that you again you removed the conjunction and then it is actually a and not now we can simply remove this conjunction then you say that we simply the contradiction and in the same way if I this is what we have already discussed so these are some of the rules that we need to remember before solving some kind of problems and there are some other obvious rules such as modal stolons etc and all they are already there sitting at the background so so there are some other rules such as replacement the rules so what are these replacement rules so example if you have formula a or b by commutative property it will become b or and this happens for conjunction also so this double colon stands for equivalence and all this means these two are logically equivalent to each other it can be b and a suppose if you find a and b in your proof you can easily substituted with b and a so these are some kind of replacement rules they come in pairs and all conjunction and disjunction this is a commutative kind of rule so now we have another kind of thing a and b RC suppose if you find something like a and b RC then you replace it with this particular kind of thing this is a and b RC you can write it in this particular kind of way so in the same way if you have a R B and C and this is same as this particular kind of thing a or b associative law and then see that means you know whenever you have this particular kind of formula you can simply replace it with this thing that is what it says and then third set of rule is this thing suppose if you have a formula aim plus v and you can this is by definition is not a or b you can simply substitute for aim place b not a or b and then there are some kind of D Morgan rules which are obviously we know for example if we have a or b negation of a or b and you can simply substitute it as negation of a and negation of b because negation of disjunction is conjunction in the same way negation of a and b is same as negation of a and then negation of conjunction is disjunction now it is not here so like this we can list out all these rules and all another rule is not not a is there in your proof you substitute it as simply a and there are some other kinds of rules like a implies b implies c if you have this particular kind of thing you can simply substitute it as a and b implies that means whenever you come across a implies b implies c that can be replaced by this particular kind of thing that is why these are all called as replacement rules and finally without this thing alpha and not alpha a and b are c and this is same as this is a distribution law and all distribution law it says that a and b first one or it is a and c this is distribution law used on conjunction and distribution law in the same way it can be used for disjunction or disjunction and suppose if you have this thing and this is same as this particular kind of thing a or b and a and c a or c so now you might be asking written so many rules etc and all so now what is that we are going to do with this particular kind of thing so now we try to prove some simple kind of formulas and then we will try to see how we can prove these particular kinds of theorems using the rules of inference that we have seen so far so now this is the one which we are trying to we will try to begin with obvious things a implies b and b implies c and now this is what we know that this is the one which comes as an outcome a in place c so now what we will do in the natural deduction method is simply we write these things as assumptions and all this is assumption one and this is another kind of assumption one two so now this is what we are going to get a in place see so now what we will do here is this thing that you add principles of natural deduction the one which we have seen rules of natural deduction then ultimately you will generate a in place see so now how do we get this a in place see is the one which we are trying to see so now we start with these assumptions and you will work out till you generate this particular kind of thing using the principles of natural deduction so now what you will do here is you will assume the antecedent part of the conclusion so this is antecedent and this is the conclusion so now what you will do here you will assume the antecedent of a conclusion again this is an assumption in some test books it is assumption is written as premise in some other test books it is written as hypothesis does not make a big difference so now this is a thing so now we assume these things now we need to use rules of inference truth preserving rules so now observe one and three one and three modus ponens what is modus ponens if a in place b is the case and a is the case and then you can get b so that is the truth preserving rule which we will be using a in place v and a you you get b so this is one and three modus ponens so now observe two and four two and four modus ponens will give us C b in place C and b C is the case so now we have used some truth preserving rules now from a you got C so now what is that there is something called in the beginning we discussed about RCP rule of conditional proof rule of conditional proof this is what is RCP so from a suppose you got C that is what is the case here from me you got C then you can discharge these assumptions can give up this a and you can give up C and all and then ultimately you say that it is a in place so for that one what you will do here in the natural reduction method is that you draw a line like this from a to C and then the six step what you will say is from three to five RCP according to the rule of conditional proof which is there here then you will simply write a in place C so now a in place C is the one which we wanted to do that so now you draw another line here from the whole thing a in place b and b in place C you got a in place C so now what you will do here in the step to a in place B and B in place C and from this you got a in place C so this is what we have deduced so like this one can derive many formulas for example let us try to talk about some simple things in all like which will be quite difficult in the axiomatic system which we are going to see little bit later so now we are trying to prove P implies P so which this involves in any axiomatic system you take into consideration it involves at least four five steps but in natural reduction method it is quite simple so this looks very strange for us but actually this is considered to be the proof so now you started with this assumption P what you need to prove is P only so now you write again P only so why you have written this thing if P is there so you can deduce P automatically that is reiteration you can n number of times you can write the same P again and again does not make any sense and all but so this is reiteration suppose P is deduced to be true when P is derived and again P can also be derived now you draw a line like this and you will say that P implies P is derived and all this is the most simplistic kind of proof that one can use it seems does not make any sense and all but it involves but according to the rules of inference and natural reduction it involves only two methods two steps and the same P implies P in the given axiomatic system sometimes it might take eight steps or maybe nine steps and all if you start you can start with axioms and then you can use modus ponens and rules of transformation and then maybe after six seven steps you will get the desired thing P implies P in the same way if you want to prove this thing P implies not not P so if you want to derive P implies not not P again in the natural deduction method you will start with an assumption P now so to whenever you have P you can substitute this particular kind of thing in the same way not not P is there you can substitute P so now this is not not P 1 and double negation so DN stands for double negation so P is there you can substitute not not P I can even substitute four not also it is going to be P so now since you have derived P from not not P you draw a line like this and in the third step you will say that P implies not not P that is one two one to two conditional proof because not not P is reduced from P so that is why P implies not not P and then you will write it like this to say that this particular thing is considered to be a theorem so you must take care of this particular kind of justification these are the steps and all on the right hand side whatever you find it here is a justification for writing this particular step so how we are justifying this thing based on the two preserving rules that we already have so this says that one we have applied double negation to it and then this is what we got and then since not not P is reduced from P using the conditional proof you can say that it is P implies not not P so this is the derivation of a theorem that is P implies not not the same thing which you can do it in semantic tab looks method in semantic tab looks method what you will do is semantic tab looks method so in this what you will do is you will take this as X and then you negate the conclusion so that is P implies not not P and then you will see whether the branch closes are not so now this is P the first one this is not of X implies Y is simply X and not Y so now you write the same thing and then you substitute another not P so not of not not P this is not not P is P and negation of P so this is not P so now you are derived P and not P and all so hence this branch closes because of this particular kind of conflicting information is there your P and your not P so the branch closes so that means what we have said not of X is false are usually you write it in this way so that means X has to be T the original formula should be T that means it is a valid formula this is another way of showing it using semantic tab looks method but in the natural deduction method of course things are little bit simpler you just use one devil negation rule and then you got the answer but here little bit of some more rules which we are using of course this is also very efficient method which also involves 1 2 3 3 steps so these are considered to be effective kind of methods for proving that this particular kind of formula is a valid formula so how are we showing that this particular kind of thing is a valid formula because we started with the truth and then we used the principle of natural deduction which is obviously truth preserving kind of rule truth preserving truth preserving and ultimately conditional proof also preserves the truth and all so obviously the final step of your theorem is obviously true so that means so all the tautologies are obviously valid formulas so there is one important theorem which tells us that if something is probable like in this way in the natural deduction method and that has to be true that is a tautology and all tautologies are obviously valid formulas so completeness soundness etc takes care of this particular kind of thing propositional logics are all complete sound and even consistent as well these are some of the things which we will be doing and then some more proofs which will try to see then things will be little bit clear so the more and more you solve these problems the more and more today we will get expertise with this particular kind of method so we all know that this is a law of contra position from p ? q you will get not q ? not p so this is the one which we need to prove first you will write the assumption p ? q this is the assumption and then this can be written as in this one p ? q and then not q ? not p now you take the antecedent part of this conditional not q as your assumption this is also an assumption so now three there is a rule again in the natural deduction so that is if a ? b is the case and not b is the case then it should be not that means you deny the consequent here in the second step you have to deny the antecedent also this is called as modus Tolens root modus Tolens so now with these two things you can derive not p ? q and you are denied the consequent you have to deny this antecedent also so now this is what is the case so now you draw a line like this what did you get from q you got not p so not q and not p so now you need to write justification for this one one and two modus Tolens you will get this you have to write justification otherwise nobody will be able to understand this particular kind of step and usually that is not considered to be an effective proof and justification used to be given after all logic is used as a justificator a tool so now not q ? not p is the one which we got so now you draw another line like this and say that from p ? q so how did you get this one ? q ? not p 2 3 from two steps 2 to 3 conditional proof rule of conditional proof you got wrote this one and now you say that in the fifth one you write like this because it is a theorem so from p ? q you got this one from p ? q you got not q ? not p so this is what we wanted to prove so how did we get this one so 1 4 conditional proof or you can write rule of condition proof you got this one so this is the way in which you can show that law of contra position can be derived in your natural reduction method so now let us talk about law of excluded middles so in your formal axiomatic system you should ensure that at least you know all these laws fundamental laws such as p ? p that is a law of identity and p are not p is the law of excluded middle and law of contra position law of non-contradiction all these things should come as you should be in a position to derive these things first and then rest of the things obviously follow rest of the complex theorems etc will automatically follow now so in the next class we will be talking about some more complex proofs based on natural reduction and then we will see how this method is can be considered as an effective kind of method for proving certain theorems and all one of the advantages of this method is this that your proofs can be simpler and it might involve very few steps and all so in this lecture we have presented natural deduction method natural deduction method is based on some kind of truth preserving rules and we stated those two preserving rules and we discussed the rational for using this to preserving kind of rules and then we proved some simple theorems such as p ? p and the law of contra position etc in the next class we will begin with we will we will consider some more complex proofs and all based on natural deduction so that we will become well acquainted with this particular kind of method.