 Welcome back in the last lecture we discussed some of the important decision procedure methods one such method which occupied the central position for this course that is the semantic tab-lux method. So using the semantic tab-lux method we discussed about validity of a given well-formed formula in the predicate logic and we also discussed something about when do we say that two statements are two sentences in the predicate logic are said to be consistent. So in this lecture what we will be doing is we will be taking up another important proof procedure method which also serves as a kind of decision procedure method so that is the natural deduction method. Natural deduction method simply involves some principles of logic this is what we have already discussed in the context of prepositional logic where we used some of the important valid principles of logic such as modus ponens, modus tollens constructive dilemma, destructive dilemma etc they are all valid principles which we have taken into consideration and then we proved some of the important theorems if something is a valid formula it has to find a proof. So in that context in order to prove that particular kind of valid formulas all the valid formulas in your formal system so we have used natural deduction as one of the important decision procedure method for proving a particular kind of well-formed formula. So natural deduction in the context of predicate logic is slightly different from that of natural deduction in case of prepositional logic although it is considered to be an extension of natural deduction in case of prepositional logic. So in the predicate logic we have quantifies and hence we have some new set of rules in the context of predicate logic so I will be discussing these rules first then we will be talking about some of the examples so that we will get ourselves familiarized with this particular kind of technique that is a natural deduction method. So another important thing which you need to note is that natural deduction is somewhat closer to the way humans reason because it involves simple principles of reasoning such as modus ponens etc and all which are which comes closer to the human common sensical reasoning and all. So that is why they find importance in importance especially in coming up with some important kind of decision procedure method. So all these methods has its own importance but there are some methods which are closer to human reasoning and there are some which close to the implementation of machine automated reasoning etc. So it is based on our convenience we will be using these particular kinds of methods at least four or five methods which we have discussed in this course we started with truth table method and then we moved to semantic tablox method and then we discussed something about resolution refutation method and then we also discussed something like reducing the given formula into conjunctive and disjunctive normal forms and then we can talk about whether that given formula is a tautology or not and then we also discussed about natural deduction method in the context of propositional logic. So now as natural deduction involves some kind to start with we have simple proofs and there are indirect proofs and then we have conditional proofs usually it is divided into two parts one is conditional proof and other one is based on another one is considered to be indirect proof which is based on redacture at absurd in the redacture at absurd of what you will do is you start with the given formula and what you do is you will negate the formula and then you will see whether it leads to contradiction or not if it leads to contradiction then the negation of the formula is unsatisfiable that means the original formula has to be true that means it has to be a valid formula. So we will be talking about these three important proofs in the context of natural deduction. So apart from all the rules of propositional natural deduction for the propositional logic where we have a list of rules sometimes it is difficult to remember these rules but in general they are simple valid principles of logic like modus ponens modus tolens etc apart from all the rules that are there for the propositional logic since we have quantifies for the predicate logic and we need to formulate some kind of rules for the rules with respect to the quantifies the first rule states like this which is called as universal instantiation so that means you are trying to find out one instance of a particular kind of sentence such as for all x there is a x in place px a x in place bx suppose if you have a formula like for all x px one instance of that one is just p where x is replaced by some kind of ground term C so that is PC is considered to be an instance of for all x px so universal instantiation allows us to replace universal quantifier that is for all x with any arbitrary constant so for any such kind of arbitrary constant that PC has to be true you can take PD PC P F anything for all kinds of that arbitrary variables that sentence for all x px is going to be true that is PC is going to be true that is p is true of everything then it is true of any individual thing we said for example if you say that all cross are black then if you find out a specific kind of crow and that crow also has to be black that is one instance of that particular kind of universal preposition and the second one is this thing which is a little bit tricky to use in particular suppose if you have a formula such as p of V then you can generalize it and say that for all x px so it allows us to assume an arbitrary individual V and you can establish some fact about that particular kind of thing so that means if something is true of that particular kind of variable V then you can say that it must be true of anything just like in our mortality is attributed to a single human being for example and then every human being has to die is some day or other so that is why mortality is attributed to all the human beings so it is under in that sense we are generalizing we are generalizing a particular instance to we are generalizing it and then forming universal generalization so universal instantiation can instantiate instantiate any constant including V whereas in this case so there is some kind of restriction which we need to impose on this particular kind of variable V that exists here so this is a second rule now the third rule is existential generalization suppose if you have a term PC all these things are considered to be terms and all we discussed what we mean by terms in the last few lectures so if you come across a term PC that means something is the case then you can generalize it and say that there exists something that px is the case suppose if you say that this dexter is yellow then you can say that there exists some x that is x is considered to be dexter and then that particular kind of dexter is yellow in color so that is what is existential generalization because there exists some x px is true especially when at least one object satisfies that particular kind of property and the fourth rule is existential instantiation which tells us that if you have a formula there exists some x px in the proof in particular where you come across this particular kind of formula in the proof you can always find one instance of that one that is you can replace it with pw in the sequence of your proofs but you need to be little bit cautious in using this rule whenever you replace the existential quantifier each time you play you replace the existential quantifier you need to use a different kind of parameter that means if you use w earlier in your proof you are not supposed to use the same w which comes next time when you replace this existential quantifier so each time use that w has to be a new one so this rule tells us that if you know that some property holds for at least one individual then we can we can we can say that we can name that particular kind of individual if you say that something is black in color and you can say that I can specify a particular kind of individual and say that this object is black in color. So these are the four rules that we have which are expressed in this sense universal instantiation for all xpx you substitute it as a substitute PC and if you have universal universe in the universal generation if you have pv then you generalize it and say that for all xpx with some restrictions and existential generation if you have a PC then p of C then you can replace it with there exists some xpx etc all these things which we have discussed just now so now how do we know that we have applied these rules correctly so let us consider some examples with which we can we will come to know whether we have applied these rules correctly or not so now the first thing is universal instantiation rule which tells us that for all xpx you can obtain PC now the correct application of that one is like this in the first case for all that fz so now in this one Z is replaced by a so then it becomes f a that it seems to be the correct kind of application in the same way for example if you have for all x dx and ex then one instance of that one is going to be db and eb here it should be read as for all x so what is considered to be the incorrect application of this particular kind of rule the rule tells us that for all xpx you replace it with PC now for example if you have a formula like this for all ypy you cannot simply say that it is not p y so why because you have to first transform this particular kind of formula into the corresponding formula then only you can substitute the variables with some kind of constants so this will change to there exists some y not p y then you can substitute an instance of this one and then you can say that PC or something like that so directly you are not supposed to substitute if you have if you come across not for all not ypy you are not supposed to substitute right away not p y you have to change it to the appropriate form then you make substitutions and that is going to be the correct one so that is one thing and then in the second case that you are seeing here for all xfx for all yzy in the only in the first instance you find some kind of universal instantiation and the next one you did not do we did not find any universal instance for that particular kind of thing and then that also it this way of applying the rule is also considered to be incorrect application of the rule that means you are not supposed to apply universal instantiation only to the part of it for example if you have some formula like a x implies B x and then you this is universal universal quantifier statement with the universal quantifier one instance of that one for example if you say that suppose if you have a formula like this for all x P x implies for all x Q x now suppose if you replace if you find only instance of this particular kind of thing and then you keep the other thing as it is then it is an incorrect application of that particular kind of rule that means universal instantiation does not apply to part of the sentences it applies to the whole sentence rather than parts now so in the second case it applied to only parts that is why it is considered to be incorrect application of universal instantiation. Now coming back to coming to the existential generalization if something is the case then you can generalize it and say that there exists some x P x the current applications are like this if you find that something is having some kind of property P let us say this duster is yellow in color then you can generalize it and say that there exists some x such that x is green in color so here capital letter capital F is F stands for predicate and X stands for individual objects within the domain the domain is considered to be inanimate objects such as chalk pieces pens etc. Pcs etc. So now if you have a term DB and EB then you generalize it and say that there exists some Y DY and EY so you are not supposed to say that there exists some Y DY and just simply EB or something like that part of the sentence cannot be generalized in all if you generalize it has to be generalized test to be a test to apply on the whole sentence so then what are considered to be the incorrect application of this particular kind of rule so why we are discussing all these roles we will be making use of these rules in deriving some of the important valid formulas so the idea here is that all the valid formula should find some kind of proof one of the important and effective decision procedure method that we are discussing today is the natural deduction method and the rules that we employ in the natural deduction method are will come usually come closer to our human reason so here existential generalization the incorrect applications are like this suppose if you have not GC example then you cannot simply substitute it as not EZ and ZZ so here the rule incorrect application of this rule is like this you have a formula not GC that means something is not black for example then you cannot say that it does not exist some X GC so this needs to be ruled out in all so that is considered to be incorrect application of this rule in the same way part of the sentences you cannot apply universal existential generation G D implies HD only part of this thing you applied this particular kind of rule that is the first part of this sentence second part you did not do anything that is why it is considered to be incorrect application of this rule in the same way we can see what is considered to be correct application of this rule in case of existential instantiation and even existential generation etc. So this rule tells us that existential instantiation for all X sorry there exists some X B X from that you can you can specifically say something is having some kind of property so each time as if you replace this particular kind of thing then each time you replace this quantifier that term has to be it has to be new constant so correct application of this rule is there exists some Z FZ you simply replace it with FA and then there exists some X DX and EX it is simply you are replacing it with a letter B DX and EX will become DB and EB it is the universal instantiation now existential instantiation there exists some X B X PW and there are these are considered to be the incorrect applications of this rules there exists some X GB and H X and then you are replacing with X with B only second part is applied we applied this particular kind of rule so that will not work here so only to the parts you cannot apply this particular kind of rule if you apply existential instantiation you have to apply throughout this particular kind of formula in the same way there exists some X FX FB FB is the formula which is already found there now once you remove there exists some X FX in the sequence of this particular kind of thing one you already have FB here that means B is already the parameter B is already used in two so when you are removing there exists some X FX you have to choose another parameter that means it has to be FC rather than FB so for example if you have in the proof you have this particular kind of things there are two different quantifiers for example there exists some X P X and Q X there exists some X not Q X or something like that so now first time when you remove this thing you choose one particular kind of parameter these are ABC or the individual constants are also called as parameters first time when you remove this thing use PA and QA that is considered to be one instance of this particular kind of thing and the second time when you remove this particular kind of thing when you instantiate it you have to choose another parameter which is other than this a now this is going to become QB or any other you can use any other letter other than whatever letter that you have used in the earliest step of your proof suppose if you have used the same kind of parameter a then that is considered to be incorrect application now this existential instantiation rule so there is a there are three errors that are possible here that is existentially instantiation instantiating to a constant that occurs in the earlier line of the proof if you use the same kind of parameter then that leads to error and the second one second way this error results in is like this existentially instantiation to instantiating to a constant that occurs in the last line of the proof or third one applying existentially instantiation rule to only part of a line that also leads to error so now coming back to coming to the fourth rule universal generalization rule so we have P V and then you generalize it and say that for all V P V where PC is an instance of for all X P X so now correct applications of this rule are like this FA you generalize it and then you say that for all X FX some somebody is mortal means all human beings are mortal only with some kind of restriction we can use this particular kind of thing. So DB and EB and then correct application of this rule is for all Y DY and EY incorrect applications are like this so the above rule should be treated as a universal generation I have written existential generation that needs to be corrected here the rule needs to be read like this P V implies for all X P X so the rule should be read like this instance of this one is for all X P X so forget about the rule which is stated above so the incorrect applications are like this suppose if you have a formula AB that means something applies to only is some kind of specific situation then you cannot generalize it and say that for all X AX all animals for example if cats dogs have some four legs and all we cannot say that all animals have four legs and all they might be some animals which may be having two legs or maybe one leg except right now so you cannot generalize it and say for all X AX this is incorrect application of this rule in the same way there exists some Y Z Y GC and then you generalize it and say that for all Y G Y so that is also considered to be an incorrect application of this particular kind of rule. So now errors with respect to universal generations are like this universally generalizing suppose if you universally generalizing from a constant that appears in the premise or it might result in because of this thing universally generalizing from a constant that occurs in a line derived earlier by an application of existential instantiation that is a rule number two incorrect use of rule number two which we have seen earlier and the third one universally generalizing from a constant that occurs in for all X P X and then fourth one applying universal generation to only part of the line rather than the full line full formula you need to apply universal generation if you do not apply it that is also considered to be incorrect application of this rule. So far we have discussed something about the rules now we apply these rules to some kind of examples to start with we use some simple kind of examples all the logic courses begin with this particular kind of example all humans are mortal socrates is human therefore you are trying to derive some human being is considered to be mortal. So here H X represents X is human M X represents X is mortal and then someone is considered to be socrates so how do we deduce this particular kind of thing using this rules. So now as a first step all humans are mortal for all X H X implies M X all humans are mortal and second one is HS there exists some person socrates is considered to be human being and now from this we need to deduce this thing there exists some X such that that particular kind of X is considered to be mortal. So now how do we use natural deduction method to solve this particular kind of problem so now we need to use only these two premises and then these two premises should lead to these two premises together with the rules that we have discussed earlier should lead to the conclusion that is one way of proving the this particular kind of thing that is called as direct proof there is another kind of method which we can use so we can negate the conclusion and see whether it results in a contradiction or not so that is considered to be reductio ad absurdum kind of proof which is considered to be indirect proof we will see both the proofs now. So now these are the things which are given to you one to your label it as one and two so now so this is H X implies M X happens for all X so one instance of that one is like this you take S you replace X with S that means one particular kind of individual S is considered to be socrates or anyone being so now this one instance of this one is like this HS MS so how did we get this one one you have used universal instantiation now fourth one we use all the basic principles that we have already used in the context of propositional logic like modus ponens modus tolens all these things actually will be using modus ponens is simply like this A implies B and then A then results in B and modus tolens A implies B not B and usually you have to deny the antecent as well and then there are some kind of other rules which are frequently used there list of rules which are we which I already mentioned it in the context of propositional logic when I discussed natural reduction in the context of propositional logic I discussed several rules just I am writing very few of these rules which frequently occur if you have A or B and you have not B then A etc there are some lots of other rules and all they are all considered to be valid principles in the sense that the conclusion necessarily follows from the premises so now coming back to this example so now HS and HS implies MS now two and three modus ponens will lead to M S so now this is not what exactly we are supposed to prove but we need to prove this thing there exist some X MX so S is having some kind of property M that means at least one one object in your domain is satisfying having this particular kind of property P so that means you can use for existential generalization and you can say that there exists some X MX so how did we get to this one suppose if you say that this talk piece is white in color and you can generalize it and say that there exists some X such that that X is considered to be white in color there is nothing wrong in saying that particular kind of it satisfies this particular kind of property there exists some X FX is going to be true if at least one object is having that particular kind of property F or G whatever so now this is what we got it in all for all X H X MX HS and these things we got what we wanted there exists some X MX so one thing which you need to know this is that you need to write justification here immediately following your sequence of your proof otherwise it doesn't make any sense to talk about it doesn't make any sense to talk about what we call it as a correct proof if it has to be correct and rigorous proof it has to find justification so usually we write it at the right hand side extreme right hand side of each term in your proof so now this is the direct method there is another way in which you can prove this particular kind of thing so that is indirect method indirect method that is reduction and absolute so what you will do here is that you take the premises into consideration you take the negation of this one and then see whether it leads to contradiction or not so for all X that means you are looking for a counter example where your premises are true and the conclusion is false so instead of looking for the validity what you are trying to do is you are ruling out instances which are considered to be invalid so when you say that an argument is invalid and you have two premises in a false conclusion when you can cook up some example where you have two premises in a false conclusion then obviously the argument is invalid if you rule out all the cases of invalid thing and all obviously you will end up with validity so now HX same thing you write it like this this is what is given to you premises and the second one you will show the same thing third one this is the conclusion separated by an oblique there exist some X MX so now what you will do is you look for the counter example you assume that these two premises are true and then the conclusion is false MX then this leads to you cannot directly apply existential instruction rule and then say that it is MB and all this is wrong so first you need to transform it into the appropriate form that is not of there exists some X MX leads to this thing for all X this negation goes inside and then this will be like this so this is 3 by definition so this leads to this one so now one strategy of again in the natural deduction is this that first you deal with the existential quantifiers and then you move to the universal quantifiers so there are no existential quantifiers here so now we need to find instances of this one and then you can find out contradiction in this one so now one instance of this one is like this H A so you can take S also but I am take I have taken into consideration it does not make any big difference in so it makes a difference because we have used S so we use capital letter stands for the predicate and S for the individual object here S refers to Socrates so one instance of this one is this another instance of this one this not MX holds for all X that means it might be true even for Socrates also now how did we get this one five one universal instantiation and then four universal instantiation we got this particular kind of thing now seven we have a rule which we have discussed just now X implies Y and not Y so this is X implies Y and not of this thing denial of the consequent leads to denial of the antecedent so now this needs to H S how did we get this one five and six modus tools so this is the rule that we have used here so now observe this thing you draw a line like this and then in the eighth step what we got is we have HS here and you are not HS here so that means socrates is human and then socrates is not human that is what we got if you deny this particular kind of conclusion so now from this you draw a line like this you say that using reductio add absurdum method what you got is the contradiction because HS and not HS is contradictory to each other so now the denial of the denial of the conclusion leads to contradiction that means this is what is unsatisfiable there exists some X M X leads to this one that means the actual thing that has to be true is not of not of there exists some X M X that means there exists some X M X has to be true this is this is considered be the original conclusion so what essentially we have done here is this that we are taking a simple example then we applied natural the principles of natural deduction and then we showed that the conclusion follows from the premises that means the argument is considered to be a valid argument by using both direct method and the indirect method so indirect method is considered to be sometimes it will be more effective in a sense that suppose if the argument is invalid then you keep on applying the rules and all you may not end up with because it is an invalid formula you will never be able to derive that particular kind of formula and so indirect method will come to our rescue so in many occasions indirect method is the one which is often widely used but more or less the both the proofs are having make or making use of these rules universal instantiation existential instantiation etc. So let us consider some more proofs so that you know we will understand this thing is formula in a better way just I will go through this proof these are the things which we have already used in the context of theory of syllogisms Aristotle has come up with this theory of syllogisms where all the sentences begin with some kind of special kind of propositions which are considered to be categorical propositions so they all begin with all some none etc they are also like this only so but not all the sentences can be in this particular kind of form that is that sets limit to Aristotle in theory of syllogism but in the predicate logic one can express relations all these things in a better way so we can overcome some problems which we have faced in the context of Aristotle in theory of syllogism so now let us consider this particular kind of example all trees are plants all plants are living things so all trees are living things it is simple since the moment you see this particular kind of formula it is clear that it is some kind of transitivity property is Tx implies Px and Px implies Lx and then obviously Tx has to be Lx so now what we have done is we listed out the two premises for all x Tx implies Px for all x Px implies Lx and then the conclusion is for all x Tx implies Lx now first you applied this universal instantiation rule for number one then it has become PA implies PA now next time again you applied the universal instantiation rule on to that is for all x Px implies Lx since Px implies Lx is true for all x so you can use the same parameter but if you have a different quantifier here then you have to use a different kind of thing first you need to handle that thing and then you move on to universal kind of quantifier so now if you apply universal instantiation rule to to it has become PA implies LA and now used again the valid principles of logic in the context of natural reduction then there is a rule called as x implies y y implies z and x implies z and all this is this rule is called as hypothetical silo so that is what we have used in the fifth step so now once we have this particular kind of thing PA implies LA which did which you did not come across with an application of existential with the elimination of existential quantifier but if we got it as an instance of universal statement with the universal generation so in that context PA implies LA can be generalized and you can say that for all x Tx implies Lx that is precisely what we wanted to prove same thing can be proved by using indirect method as well so what you do is for all x Tx implies Px for all x Px implies Lx you distort the same things and you negate the conclusion and then you can see whether it leads to contradiction or not so let us consider this particular kind of thing and we apply indirect method on this particular kind of thing and then see whether it follows are now so what we have is like this for all x Tx all trees are plants and then for all x Px what is there here for all x Px Lx or living beings then for all x Tx Tx implies Lx so what we are trying to do is we are applying edex show add absurdum method which is considered with the indirect method so now what you do here is so now you deny the conclusion Tx implies Lx so this is denial of conclusion denial of conclusion so now as a fourth step what you do is we have some rules for all x Px is same as there exists some x not Px in the same way there exists some x Px is same as not for all x not Px so now this will become there exists some x not of Tx implies Lx so now this is 3 by definition so now we need to look for the quantifies this statement starts with the existential quantifier first you handle that particular kind of thing you eliminate this one and then you move to this particular kind of thing ta where x is replaced by a so this is what we have and then you further simplify it then it will become ta and not so now now we need to talk about universe this is for existential instantiation so we need to write justification here otherwise it does not make any sense nobody will understand what you have done here if you do not write the justification for this one here on the right hand side so now 8 you find one instance of this one any one of these things you can handle now so one instance of this one is since it happens for all x it happens for even a else ta implies PA and then in the second case for all x Px implies Lx that means one instance of this one can be this one PA implies La so now what we have is like this so it goes this ta not La now sorry you are not supposed to use this particular thing so now you have ta and ta implies PA that means 6 and 6 and 7 modus ponens you will get you have ta here and ta implies PA so you will get PA so now and we have ta implies La in the 8th step so now these two modus ponens you will get La so 7 9 10 for example 9 and 10 modus ponens you will get this one so now you observe here in this proof what is the proof first of all a proof is a sequence of steps and all it ends in finite steps in finite intervals of time so the each step is considered to be true and all so the final step is also considered to be true so now you have La here and not La here so you draw a line like this and then from whatever it is 6 to 9 it led to contradiction so how did we end up with a contradiction since we denied this conclusion if you deny the conclusion you end up with a contradiction if you are not denied it did not have led to contradiction that means negation of the conclusion leads to unsatisfactory unsatisfactory in the sense that it leads to contradiction so in that sense the original conclusion is the one which holds so like this so these are the simple examples with which one can solve this particular kind of thing but one can use for the complex case also one can use this natural deduction method and then one can solve the problems so there is one particular kind of prescription one uses one make use of it as a strategy that is like this one should only universally generalize from a constant that is introduced by universal instantiation for example if you have this is one of the important strategies that we need to use so let us say for example you have a formula like this for all x P x implies Q x for all y sorry there exists some y P y and Q y for example these are the two formulas that are there in your proof just we are trying to talk about some kind of strategy so that you can make use of these rules correctly so now one instance of this one could be like this PA implies QA etc and another instance of this one for example if you say it is PB and firstly usually we handle this existential quantifier so let us say this is PA and Q y so this is an instance of PA and QA existential instantiation so now this can be written as PA and QA so now the strategy tells us that if you got this formula out of existential instantiation you cannot generalize it and say that it is for all x P x so this is wrong the same way you cannot generalize QA and then say that P x by using this particular kind of rule this is incorrect application of rule because it you got this PA and QA out of the existential instantiation rather than the universal instantiation you need to apply this universal generation only when you come across you came across that particular kind of instance through the application of universal instantiation rule otherwise you are not supposed to do it so how does an individual constant get into the proof in the first place if we limit our attention to direct proofs there are only three possibilities that is an individual constant can be introduced in the premise of the argument that means if universal instantiation you will you might do it or you might apply existential instantiation and you come across that particular kind of thing the third one is universal instantiation that is what we have done in the first step so several there are some other examples complex examples one can take into consideration and then you apply this particular kind of rules and you can deduce this particular kind of theorems so one can use both direct and indirect proofs to handle this particular kind of situation we will end up with one simple example then we will see will end this lecture so we talked about distribution of quantifiers with respect to conjunction so we know that this particular thing holds there exist some x fx or there exist some x gx and from this to get fx or gx so this is what we are trying to prove again one can use any one of these methods you can deny this particular kind of conclusion and then start constructing using the universal instantiation etc all these rules then you will come up with a contradiction and you say that negation of this one leads to contradiction hence negation of negation of this formula is going to be the case that means the actual consequent remains so now let us consider this proof of this one quickly and then we will end this lecture so we had to solve many problems to get ourselves familiarize with this particular kind of technique so in that context so I could only discuss how to judiciously use this particular kind of rules and when it comes to the applications in particular and while solving the problems that is going to help us so now you list out this thing they are all x for all sorry gx is what is given to us this is given or you can write it as assumption etc. So now from this to take into consideration there exist some x fx just one part of it you take into consideration that is also considered to be an assumption now fourth one just as quickly write it and all so now one instance of this one is this three existential instantiation is this one now fifth step fa ga so how did we do this thing there is something called law of addition since fa is already true you can add anything to this particular kind of thing without disturbing the truth value of that so that means you can write fa R g so now you can apply existential generalization rule and then this can be this holds for at least one particular kind of situation so you can generalize it and say that some x for some x this is the case that means fx R gx so now this is what we got it by taking this particular kind of thing there exist some x fx but even if you take this particular kind of thing you will get the same kind of result so this is what we have we are supposed to prove and then we proved it but it might this proof you might get coming across this particular kind of thing even by using by taking this particular kind of assumption also we have either P or Q kind of situation first you have taken this into consideration and you prove this one and in the same way you take if you take this also you should be in a position to derive this particular kind of thing so that is you take this particular kind of there exist some x gx this is again assumption and all now this you take into consideration again you will prove the same thing in that ends that gives us the complete description of your proof so now one instance of this one is GA 7 existential instantiation now 9 again you can add the same thing FA R GA this is addition law of addition so like you know suppose if you have a formula a which is already true and all then you can add a or B of course you can use the commutative property and you can say that the same as B or a then make any big difference actually it should be GA or FA and that is same as FA R GA so now since you got this one one instance of if you apply existential generalization rule then this will become FX or GX so now even if you take this into consideration as your assumption you could prove this particular kind of thing hence there exists some X FX there exists some X DX and you got this particular kind of thing one final thing remark is that you can use indirect proof also to solve this particular kind of problem so that is like this so this is considered to be anti-strain this is considered to be the consequent now what you will do here is that you list out the premises like this there exists some X GX and now you deny the consequent and then you will end up with a contradiction FX or GX this is denial of consequence something one this is given so now simplify this particular kind of thing this will become for all X not of FX or GX now first you need to handle the existential quantifies then you move to the universal quantifies now once you replace this thing to take this as your assumption there exists some X FX then this will become F of a you replace X with a the same way you can take there exists some X GX and then you can start with GA and then you can find the proof of this now fifth one not of FX GX one instance of that one is this thing not of FA negation of the extension is conjunction so it will be not GA since you have FA not FA it closes here itself so now this is when you take this into consideration now you can take this also to consideration the proof will be a little bit different so now it will be G entire thing will be same it will be GA not GA some not a fair something like that so now GA not G it closes and all any one of these things you take into consideration it leads to branch closure so this is another way of proving the same kind of formula by using natural deduction method so we will stop here and then we will what essentially we did in this lecture is simply this that we discussed some of the important principles of natural deduction method with respect to quantifiers in the context of predicate logic we introduced for the universal quantifier we introduce universal generation and universal instantiation and with respect to existential quantifier we introduced existential instantiation and existential generalization and then we discuss two different kinds of proofs one is considered to be a conditional proof another one is based on indirect proof which is called as reduction add absurd of kind of method so this method is closer to human reasoning in a sense that we are well familiar well familiar with modus ponens modus tolens etc and all rather than proving a formula nobody usually we do not prove particular kind of thing by denying the formula and then see the contradiction etc and all so in a sense natural deduction method comes closer to our humans and human reasoning but sometimes some other methods might be may fair may fair better than this natural deduction method so one important method which unfortunately we are not able to discuss that is in the context of predicate logic which is very essential in the context of computer science that is the method is called as resolution refutation method we discussed this particular kind of method in the context of propositional logic but lack of time we will not be able to deal with the resolution refutation method so in the next class we will be dealing with some of the important theorems of first order logic there we discuss about some of the important theorems such as completeness compact next and the celebrated result in the first order logic that is the Godel's incompleteness 0.