 Welcome back in the in continuation to the last lecture where we discussed in some detail about one of the important decision procedure method which is called as natural deduction method. So natural deduction method the idea here is that we will be employing only some of the basic principles of logic such as modus ponens modus tollens etc and all and then we will be deriving some of the theorems. So the program is like this that so all the valid formulas and all we are trying to find a proof for that. So if all the provable things are true and all the true formulas are provable then your system is considered to be complete in this sense natural deduction as a formal axiomatic system is considered to be sound and is considered to be complete in a sense that all provable theorems are true all true things are provable and all and also it is the case that it is also said to be consistent. So in this class I will be considering some more examples so that you will understand this method in a better way. So natural deduction method has mainly two important methods and all so one is one in one of these things we will employ rule of conditional proofs in the second one we will be using reductio add absurdum method. So what is considered to be a rule of conditional proof in case of natural deduction it is like this supposing from an assumption a we obtain in a proof something called B. So now B is a tautological consequence from a that means the last step of your proof is considered to be a theorem so it is also considered to be a tautology. So then what you will do here is that you will discharge the assumption a and then you will start writing a implies B. So you have to note that B is not a tautological consequence from a then obviously a implies B is also not total over so what essentially we are trying to do here is that you start with a and then you got B and then you draw a line like this and you will say that a ? B is deduced from this thing. So this is one way of one method in which you can prove theorems in natural deduction what you will be doing here is you will be making use of some of the natural principles of logic and so together with that you have an assumption a and that led to B and since a led to B and all you draw a line from a to B and then you will say that a ? B is reduced so this is what is the first one and we will be trying to solve some problems by using this particular kind of method and the second one is what we call it as the lecture and absurdum method. So the basic idea of this one is like this suppose a given formula X and you derive contradiction and all X in place something falls or contradictory that means it should not be X and all but it should be not X. So the same thing can be represented in this way taking X assuming X into consideration you generated a contradiction. So when you generate a contradiction whenever you come across a literal and it is negation and all like P and not so that is called as consistent inconsistency or contradiction. So if that is a case then so this means X implies contradiction so that means you draw a line like this and then you will say that X implies contradiction that means it is not X is true so instead of X you have to assume not X and all. So this is what is called as a reduxio add absurdum kind of method these are the two things which you will employ in the natural deduction method apart from there are some natural principles of logic I mean simple rules which we employ in logic just like you know a person assuming that you are playing some game in all every player is supposed to learn some kind of rules and all if we if he knows the rules minimal rules then you will be able to play in a satisfactory or better way you will not make faults in the same way so if you know these rules and all you will employing these things these rules together with some assumptions leads to the conclusion that you are trying to derive. So basically what we are essentially doing is simply this that we are trying to prove some theorems so why we are doing it we want to have some kind of proof mechanism for all the valid formulas now so this is also considered as one of the important decision procedure method which is simple so the first rule is simple and straight forward whenever you have negation of negation of a you will replace it with a this is what is called as rule of double negation and rule of addition says that if a is true then since a is already true the semantics of for distinction allows us to add any other kind of proposition B without disturbing the value truth value of a so that means from a you can reduce a or b or from B you can reduce B or a so now the modus Tolens rule is this that in the conditional a implies B a is considered to be antecedent and B is considered to be consequent and if you deny the consequent you have to deny the antecedent as well so now the distinctive syllogism a or B and if you deny one of these possibilities then obviously the other one follows hypothetical syllogism like transitivity property a in plus B B in plus C then A has to go to see these are simple rules and all classical logic obeys these particular kinds of rules and all and you should note that this applies to classical logic and all but when you apply to day to day situations are some of the things which which you make use of it in day to day discourse then it might lead to some kind of counter intuitive you might generate a counter intuitive influences so that is not our interest at this moment but all these rules are truth preserving rules is obeyed by the classical logics that we are trying to talk about so there are some other rules and all these rules are called as assumption discharge rules so what are these assumption discharge rules they are like this let ? be a set of 0 or more assumptions if you start with one assumption it is something some assumption needs to be there or you can start with 0 assumptions also that means it is a tautology or something so that means already tautologies does not require any proof and all the self-evident truths you can take it for granted that they are all true absolutely true so now assumption rules are assumption discharging rules are like this given ? and for example B is deduced from a then we can discharge assumption a and then you can say that it is a ? B is deduced from ? so this is what is the conditional proof given ? and a suppose if you deduce B from it then you can discharge assumption a and then you can say that a ? B is deduced from that one ? the rule of disjunction tells us that given ? and a R B and you already deduce C from a if that is a case and you already deduce C from B given a ? so therefore you will also deduce C so these are some of the truth-preserving kind of rules and all then the fourth rule tells us that suppose if you deduce a contradiction from given assumption a and ? then obviously it should be not rather than a so these are some of the simple rules which we follow then there are some other logical equivalence relations which we employed in the natural deduction method their simple rules such as direct distributive law P and Q or R is P and Q or P and R then some demargin's laws especially especially when you are trying to translate conjunctions into disjunctions use the demargin's laws and then one of the surprising thing for us is this absorption law F P R P and Q will become P and the other law is other way around P and P R Q also becomes P so whenever you come across a formula P or P and Q we just replace it with its logical equivalent relation that is P so the contrapositive role is straight forward P ? Q ? ? Q ? ? so now we will make use of what is the what essentially we are trying to do is that we are trying to show whether or not the conclusion follows from the premises are not so let us assume that these are the three things premises are P and Q or not R and second one is P ? R and from that you are trying to reduce Q R R so now it is like this so what we will be doing is simply is that we will be using some principles to preserving principles plus the assumptions that are there in this particular kind of problem and then we will reduce P and Q R ? R is the first one and the second one is P and Q and from this you are reducing Q R ? R so now we are trying to prove this particular kind of thing using rule of conditional proof and we will also prove it with the help of RAA so now in this RCP you have to list out all the premises and then this is what you are trying to generate after using these assumptions taken together with the principles which are sitting at our background so if you add these things to it somehow you need to generate this particular kind of conclusion so now there may be n number of ways to come to this particular kind of conclusion sometimes you might imply four steps or maybe five steps etc and all so what constitutes an effective proof is this that whenever your proof ends in finite steps in finite intervals of time then it is considered to be an effective proof suppose if I showed that Q R R follows after 15 steps and all so that is served to be an effective proof and all but some others comes with a proof in which it includes only six or seven steps then that is definitely considered to be an effective kind of proof so now taking these assumptions into consideration the first thing that we will be doing is this thing so now P and Q are not R so this is the first assumption all either PR not Q is true or not R is true that is what it essentially says so now we are assuming that the first one is true because the moment you say that P and Q are not all then one of these things is true P and Q R P implies R so the second one is P implies R so now this is the assumption that we have began with so now P and Q the conjunction rule says that from P and Q we can deduce P and from P and Q we can even deduce Q also this simple law of conjunction so now fifth step this is law of conjunction the same thing so now six step now we need to observe these two things two and four more exponents will give us so now so now we need to observe one and six so now this is exactly opposite of this one now we have a rule distinctive syllogism suppose if X or Y is a case and not why is the case that means you are ruling out this possibility that means whatever is left is the one which is the one which you will be deducing X is the case so now here so we can take it as R so this is R and you are not R and all so this possibility goes out and then what we will have so P and Q again you will be deducing the same thing and all it is not making a big thing at all so now since Q is already true so we know that under the fifth step Q is already true so now you can safely add another kind of proposition any kind of strain kind of proposition without disturbing the truth value of it because Q is already true so why it is a case cause the semantics of this one PRQ is like this PQ so TTF and F TFTF so this PRQ will become false only in this case and both disdain are false in all other cases it becomes T so now it is in this sense Q is already true so now we need to observe this particular kind sorry so these these rows and all for example Q is already true that means these two things which you need to take into consideration so now irrespective of whether P is false or P is T and all so now PRQ is also going to be T only so it does not matter whether not R is true or false but it still holds you know so this rule is called as law of addition for example if we have P you can safely add another Q to this one without disturbing the truth value of this one so ultimately what we are trying to do is we are moving from truth to another truth and so we are not disturbing the truth values of this thing if P is true PRQ has to be true irrespective of whether Q is to true or Q is false it is in that sense we have written this particular kind of step Q or not R so now this is exactly the one which we are trying to prove now we need to write justification on the right hand side otherwise we will not be able to make out what exactly we have done so it is law of addition so that justifies why we are writing this particular kind of thing so now you draw a line from here to here and then so this is what you can say that rule of conditional proof now you can formulate the same problem and all you can say that P and Q are not R ? P ? R and then this leads to Q are not because we showed that Q are not a follows from these two premises and so in this way one can prove this particular kind of whether or not the conclusion follows from the premises are not but there is another way of proving it that is what is called as reductio add absurdum method so this goes like this first you list out the premises and all are not R and then you list out the same thing now we are following the reductio add absurdum method so the idea here is that if X leads to its contradiction needs to some kind of contradiction that means it is not X and all but it should be not X so this is the idea which is commonly employed in proving many theorems in mathematics so instead of showing that something is true what you will you will start with the negation of that one and then you will show that assuming that the negation of particular thing leads to contradiction so that is why not X is false and all that means X has to be true so now PRR is the two premises now separated by that you have a conclusion so now in the third step what you will do is in the reductio add absurdum method what you will do is you will negate the conclusion and then you will construct it you will see whether or not it leads to contradiction or not so now if you simplify this one using De Morgan's laws it will become not Q and R so now this can be further simplified into not Q and R so this is four simplification you get this one and this is De Morgan's laws you have used here so that is why I wrote DEM so now observe this these so now PR and this one so what you will get is P and Q so how did we get this one 6 and 1 disenqueues you got this one so this is nothing but writing the same thing P and Q so now 7 simplification you get this one 7 simplification you got Q so now in this proof the problem is this that you have Q here and you have not Q here so that means in the 10th step what you need to write is Q and not Q so now what you do here is that starting from 1 to 9 1 to 9 reductio add absurdum so what has happened in the 11th step what you whenever you come across Q and not Q you mention it with this particular kind of symbol that is a contradiction so negation of the conclusion leads to contradiction now suppose if you take this as X so now X led to this one contradiction so we showed that negation of X leads to contradiction so that means it should be it should not be negation of X but it should be negation of negation of X so that means if negation of X leads to contradiction that means it is negation of negation of X that means X so what is our X here it is Q R ? so what essentially we did here is that we have taken into consideration that the conclusion does not follow from the premises and all then we used principles of natural principles of logic and then ultimately we came up with a contradiction and then since taking the negation of that one lead to the contradiction that is negation of X X is this one Q or not that leads to contradiction so that is why it is negation of negation of that particular kind of thing is true that means not not of X is using double negation rule of double negation you can say that X is true so what is X for us X is nothing but Q R ? the original conclusion remains in all so this is sometimes it is simpler than the first method that we have used using rule of conditional proofs sometimes it will be so difficult for example suppose if you are given if you are given something which is not a theorem and all so then you will keep on proving it proving it and all and then ultimately you will not be able to prove anything so instead of that maybe you can use redactio add absurdum method and things will become simpler so now any formal axiomatic system these three things should come as an outcome and all so they are law of identity law of excluded middle which says PR not be and law of non-contradiction that is it is not the case that simultaneously P and not not PR true so now this is considered to be some one of the trivial kind of proofs and all but still it holds it involves only two steps and all so is for proving P implies P you start with an assumption you need to note that all assumptions are obviously considered to be true if you take your assumption itself is to be false and all there is nothing we can prove it is already assumed that it is true it is true and all so now in this case what you will do is you write the state the assumption and then you will reiterate the same assumption P since you got P from P only so now you draw a line from P to P and you say that P ? P is proved by using rule of conditional proof nothing actually we did not do anything here but still this is considered to be effective proof in the natural deduction so what we have used is reiterated rule which we have used and then RCP you can also prove P ? P using reduction at absurdum method also you take into consideration negation of P ? P and then it is P and not P and that is a contradiction so that is why negation of P ? P leads to contradiction that means not not of P ? P is true that is P ? P so in the same way in your natural deduction system or any formal logical system and all these are the things we should come as an outcome and so later we are when we consider axiomatic prepositional logic there we try to prove these theorems using some of the important axiomatic systems such as veteran Russell whitehead axiomatic system or Hilbert occurments axiomatic system so now let us assume that let us consider a proof for law of excluded middle PR not P this we are trying to prove it with the help of reduction and absurdum method so as a first step what you will do is you negate the formula well form formula that is given to you that is not of PR not so now in that let us consider that P is your assumption that is there in the law of excluded middle and then so P is the assumption that we have and then you can add since P is already true you can add any preposition whether it is true or false that means you are adding not P here using law of addition so now 1 and 3 that is not of PR not P and PR not P these are contradict 1 and 3 in conjunction leads to contradiction in all so because it is not PR P PR not P and PR not P and all is XL X and not X so that for that what you need to do is you whatever you assume is wrong and all that means not P is the case so now again you add for 5 you add P to it because not P is already true you add P to it it will become PR not P and again 1 and 6 1 is not of PR not P that is assumption that we have and then we got PR not P these two are in contradiction to each other and that means it should be not of PR not P is the case so that means by double negation you can prove that PR not P is the case so these are some of the ways to prove these theorems in all so this is considered to be one of the important instances of paradox of material implication so it says that a true preposition is implied by any kind of strange kind of preposition in P ? Q ? P when P is true that means the consequent is true our semantics allows us that obviously Q ? P is all going to be true irrespective of whether Q is true or false and that makes since in P ? Q ? P Q ? P is already true then P ? Q ? P will also become true so that makes the whole conditional true so that means any true preposition is implied by any strange kind of preposition like Q here in this case so how do you prove this particular kind of thing this is considered to be a valid formula in classical logic so all the valid formulas needs to find a proof you have to find a proof for those valid formulas so again you start with so whenever you have a formula like this you assume the antecedent part of your conditional that is P and then you will also consider the inner conditional and all that is Q ? P in that Q is considered to be the antecedent so you assume these two things so now we already proved that P ? P is the case earlier so now one and three modus ponens you will get P and then since P is proved from Q that means from two and four if you observe it then from Q you got P in your proof after traveling certain distance you got P since you got P from Q you write it as Q ? P and then you will state from where you got this particular kind of thing from two to four using rule of conditional proof you get Q ? P since Q ? P you got it from one that is P so that means P ? Q ? P is the case so this is the way to prove some theorems in this way so now what we will do here is that so we can extend it extend the natural reductions is principles this method to solving some kind of problems which we commonly come across in our day-to-day discourse so here is the English language sentence first what you will do is you will translate it into the language of propositional logic and then you will talk about whether or not the argument is valid or not so this argument goes like this God is omnibenevolent provided that he is perfect suppose if you represent God is omnibenevolent is as P and he is perfect is represented as Q for example if you say that he is not perfect it becomes not Q now the second premise if God is both perfect and creator of the world that is a conjunction and followed by that is a conditional then there is no evil in the world there is a evil in the world is yes no evil in the world is represented here as notice now the third proposition is it is supported by some other things premises but it is an incontestable fact that there is evil in the world that means S is the case so now furthermore it is supported by some other statement that is it is usually claimed that God created the world that is represented as R so from that you go the conclusion is that if God is imperfect that is not P or he is not been omnibenevolent so now how to show that this argument is valid using natural deduction method so there are two ways to show that whether or not the conclusion follows from the premises are not so either we can use the rule of conditional proof or you can use reduxio add absurdum so this is the first premise and second one is Q and R Q and R implies not S and then SNR and followed by that there is a conclusion not P R not so now we translated the English language sentence into simply the symbolic form and all now we will forget about God and all these things and all so now we will be manipulating these symbols and all so in the reduxio add absurdum method I will be using the second method and all which is relatively simpler so what you will do is you start with the denial of the conclusion so the what is the conclusion here so this is the conclusion not PR not so now you deny the negation of the conclusion so there are two ways of showing that whether or not whether this argument is valid or not so we are not sure whether this argument is valid or not that's why we are taking into consideration the reduxio add absurdum method so taking the negation of the conclusion whether or not this leads to contradiction or not is the one which we are trying to see so now this if you can simplify it this will become P and Q using demarcans loss negation of negation of PSP and negation of dissension will become conjunction negation of negation of Q is Q so now this can be written as P Q this is six simplification you will get this so now 1 and 7 modus ponens you will get Q 1 and 7 you will get Q here so now observe this particular kind of thing 2 and 3 so it is like x implies y and then not wise so this is a rule called as modus tolens so if not why is the case you need to deny x also so is the modus tolens rule so that means 10th one you have not s here and s here that means you have to deny the if you deny the antecedent consequent you have to deny the antecedent also that means it is Q and R so how did we get this one from 3 2 and 3 modus tolens you got this one so now this can be written as not Q or not R so now so you have Q here and you have not Q here so this 9 and 11 dissident use Syllowsum leads to 11 is this one it leads to not all so now under in the fourth step and all we have R here I mean above so now we are going like this so now observe in the fourth step we have R here and in the 12th step you came across not so that means it is R and not in the 14th step it is a contradiction X and not X is a contradiction so how did we generate this contradiction in the 15th step what you will do here is this that negation of not P or not Q led to contradiction so since it leads to contradiction that is X implies this contradiction then it is not X so that means if that is the case if this whole thing is considered as X X implies contradiction then it should be not of not of X so that is the 17th step we can show that not of not Q is the case that means this is the conclusion that we are trying to achieve so this is the actual thing which follows and all that is the conclusion actual and original conclusion so what is that we have achieved in this one simply this that we denied the conclusion that is we denied that it is not the case that God is in perfect or he is not omnibonivalent now you have to translate it into the original argument and all then we showed that it led to contradiction I mean negation of X leads to contradiction that means X is the case that means the conclusion remains the same the conclusion holds provided you follow this conclusion follows from the premises that are stated here that is God is omnivalent provided that is for fate etc so in this way we can translate the English language sentences in appropriately into the language of propositional logic and then you can apply this natural deduction method but the problem here is that for example that the conclusion does not follow from the premises and all then suppose if you are using RCP that rule then you will be working rigorously and ultimately you may not generate the conclusion that you are supposed to get so if in an invalid arguments this may not work in there you have to use semantic tableaux method or maybe Redux you add absurdum method is the one which you need to employ sometimes proving a contradiction itself might find you might find it very very difficult in all so in that case complexity increases so in the one hand you are verifying it on the other hand you are showing that something is not false enough so that is what you are trying to do in the redux show at absurdum method so like this we can talk about several examples where we can employ either rule of conditional proof or redux show add absurdum method suppose if you are trying to prove this particular kind of thing whether P ? q follows from ? of P and ? q and here is the array a proof redux you add absurdum kind of proof you start with an assumption so that is ? of P and ? q that is assumption and then and you take the antecedent part of the consequent that a conditional that is P in the right hand side you will find that one so now our assumption is that ? q is our assumption so what essentially we are trying to do is we are considering a case in which you have true premise in a false conclusion and all if you take the first one not of P and ? q as true and then P ? q as false and then that means you are taken into consideration a counter example in all so that that creates us a kind of counter example so now from this P and ? q is our assumption already so 2 and 3 if you use law of conjunction you will get P and ? q so now we have from 1 and 4 on the one hand we have ? of P and ? q and we have P and ? q these two leads to contradiction so that means we started with ? q as our assumption that has to be false that means q has to be true that is what we got it in the seventh step or from the sixth step by using double negation rule you will get Q so now since you got from 2 you got 7 so that is Q Q is obtained from P that means you can draw a line from P to Q all the way from 2 to 7 and you say that P ? q can deduce by using rule of conditional proof in that what you will do is you will discharge your assumptions P Q etc and all you will start talking about P ? q there are some other notions which are important that is syntactic validity any argument a 1 a 2 to a n and from that B follows is valid in a syntactic sense especially when a conjunction of all the formulas a 1 and a 2 a 2 a n is valid in a particular kind of language here so what essentially we are trying to show is is that when you show that all whatever you proved is also valid that means true tautology valid are all one of the same and on all the tautologies are valid formulas and all valid formulas are obviously tautologies so if we show that all your proofs the theorems that is the last step of your proof so that is a valid formula is a tautology then then also you are said to have talked about your user to talk about syntactic validity so so these are some of the important things your direction natural direction system is also considered to be consistent that means you will not be able to derive both X and not X and all so if you derive that particular kind of thing X and not X then your natural deduction system is inconsistent so with this we will end this particular kind of natural deduction method so what essentially we did here is like just like some players are playing game in all suppose if you are playing a cricket etc you one must be in a position to know all the rules in all as far as possible many rules then you will be able to play without any faults suppose if you are playing a cricket and if you don't know what is a no ball and what is a wide ball etc and all and they will not be in a position to play without any faults and all so in the same way natural deduction has as far as possible very limited set of axioms and all that means there are no self-evident truths etc to begin with but you start with some assumptions which are also always considered to be true and then you added to that you have some basic principles of logic such as modus ponens modus torans etc and all all these rules are truth preserving rules truth preserving in a sense that conclusion necessarily follows from the premises and all and you will take those particular kind of truth preserving rules and then you will deduce some other theorems so what essentially we are we are trying to do in the natural deduction is that all the valid formulas should find a proof so here is a formal logical system which makes use of natural deduction which is considered to be consistent complete and even sound so that takes care of our inquiry that all the valid formula should find a proof so here is a proof which we can produce it with the help of natural deduction system so in the next class we will be talking about another decision procedure method which is called as conjunctive and disjunctive normal forms.