 Welcome back so far we discussed about syntax of predicate logic and partly we also discussed semantics of predicate logic where we discussed that a given well-formed formula it is true with respect to domain and it depends upon the domain that you have taken into consideration that is the same kind of well-formed formula can be true with respect to suppose let us say you take the natural numbers it might be true if you take the other numbers into consideration integers etc the same formula can be false as well so you can talk about truth value of a given well-formed formula only with respect to a model a model consist of a domain and an interpretation function I so what we will be talking today is one important decision procedure method which we already discussed in the case of prepositional logic so that is the semantic tableaux method so this method is due to first originated in the works of hintika and then later it was reformulated by Raymond's mullion etc. So these are the people who are responsible for this particular kind of method and using this method as is as in the case of prepositional logic we can find out whether a given formula is a valid formula that means all the tautologies are valid formulas just as in the case of prepositional logic and you can also talk about when two sentences in a predicate logic are consistent to each other or when a given formula satisfies within a domain so we will be talking about this particular kind of method in some greater detail with respect to the predicate logic and then we will consider some examples so that we can get familiarized we will realize ourselves with this particular kind of method semantic tableaux method it consists of some kind of semantic tableaux rules for the prepositional logic and in addition to those rules for the prepositional logic we have four additional rules for dealing with the quantifies what is extra in predicate logic is simply the quantifies so we have all the connectives and negation implies if and only if etc plus in addition to that in our language of predicate logic we have quantifies that is for all x and there exists some x in the case of prepositional logic we know that these are some these are the rules that we used in case of prepositional logic in the case of semantic tableaux for the predicate logic you have all the rules that are already there in the case of prepositional logic plus in addition to that we have some rules for quantifies so there are four such rules so I will be discussing these four rules in greater detail in a while from now but before that so there are something called alpha and beta rules in the context of prepositional logic so we have these connectives negation are and implies and if and only if now suppose if you have a formula like this thing not P and you simply write it as not P only and then P and Q so this is written as P and Q so we are constructing a tree diagram for this particular kind of formula so this looks like a trunk of a tree so whereas P or Q branches out it will be like the branches of the tree and only thing which is which you need to observe here is that this is an upside down kind of tree a tree will be like this trunk will be like this and then branches will be there but for our convenience we are taking the upside down kind of tree so now P or Q the tree structure for this one is this P Q and then we have other kind of connectives implication so we have not P or Q so this is a structure for this P implies Q and whatever is left is this P if and only if Q so it is like this both P Q or true and not P Q or true so these rules are based on the semantics of these prepositional logic we know that a conjunction is going to be a formula with the conjunction is going to be true when both the conjuncts are true that is why it is sitting at the trunk of the tree if any one of this thing is false that is going to be false so these are considered to be alpha rules so now beta rules are with respect to the negation of all these things suppose if you come across negation of negation of P you simply substituted with P and then negation of P and Q so negation of conjunction is a disjunction so it is not P not Q and then negation of P or Q so it is negation of disjunction is a conjunction so it will be not P not Q and then there are three other things which are left two things other things which are left not of P in plus Q so this is simply P and not Q and then P if and only if Q negation of that so this is going to be P and not Q and not Q and P I can even write it as not P Q and not P Q and not Q it does not matter what way you write it and all so these are the rules for the propositional logic so now we will we need some more rules to deal with the quantifiers that means the formulas that begin with the quantifiers we need to have a few more rules that I will be talking about in a while from now so the tableau method is based on an attempt to construct a counter example to a given formula the main idea of this main idea behind this method is that given a well-formed formula instead of checking for whether or not it is true and all what you look for is you look for a counter example where that formula is going to be false so if at least in one instance the formula is going to be false at least that means you have come up with a counter example so in the beginning of this course we discussed that an argument is considered to be invalid you have especially when it is possible for the premises to be true and the conclusion is false if it is impossible for the premises to be true and the conclusion to be false then that is called as in a valid argument it is considered to be a valid argument so in the semantic tableau method the main idea is that you try to look for a counter example because that is going to make it invalid so the tableau begins with the formula not alpha suppose if you are given a formula alpha which expresses some kind of formula in the predicate logic and you start with the negation of that formula and then you start constructing the tree based on the rules which I am going to discuss in a while from now so the tableau begins with not alpha for some sentence alpha in a language L so now a counter example is now is considered to be another kind of structure a which means we must specify again the domain D an interpretation of each constant in that particular kind of language L and an interpretation of each predicate in that particular kind of language L usually once you construct the semantic tableau tree for a given formula suppose if you come across if you negate the formula and then you your branch is open at least some of the branches are open and that will serve as our counter example that means you have come up with an instance where your true premises and a false conclusion so from the open branch you can construct a counter example and then you can cook up a domain and then based on the open branches you can come up with a structure in which this formula is considered to be invalid now suppose these are some of the rules that we will be making use of with respect to quantifies suppose at some point in our consideration some point in our construction of your tree that means you are trying to construct a tree for the given predicate logical formula so we come to to an extent where let us say you have a formula there exists some x ? x so there exists some x is an existential quantified so this suppose if you find it on the node of the tableau so then we want to make it make this thing true so we need some element a in our domain D so some element a has to exist so that it will become an instance of that particular kind of thing such that ? of a has to be true so our rule should allow us to introduce ? a on the path provided that the parameter a has not yet appeared on the path so now there are four more rules that we will be using with respect to for all x and there exists some x etc so these are the four rules that we will be using so they are all sitting at the background and all alpha and beta rules and all so similarly with respect to quantifies suppose if you come across a formula like this in the given tree and all so now from this you need some kind of for an object a a parameter in your domain a has to belong to a domain D so now we should be in a position to say that it is p a there exists some x px is true when obviously one of the instances is also true so now this is where a is considered to be new so this rules says that for example if you come across two existential quantifiers like this q x etc once you eliminate this quantifier you use an individual letter a and then another time in another occasion if you remove this existential quantifier you do not use this parameter a but use b any other thing which is other than this a so whatever you use just below this one it should not figure out in your branch earlier so this is one of the important rules with respect to existential quantifier so each time you remove this existential quantifier you will you are to use a new parameter that means this a should not exist anywhere else anywhere else in the in the tree earlier parts of your tree diagram so now the other rule is this thing for all x px if it has to be true so this says that for any except px is going to be true so now so you can simply substitute as p a you can say for any arbitrary value a you can freely substitute any value for this particular kind of thing it is true for a it is true for b it is true for c etc and all it is as good as saying that all crores are black saying that particular crore a is black another crore which you have taken into consideration b that is also black etc so now these are the alpha rules with respect to the quantifiers now the other rule which we have is the beta rule it talks about the negation of this quantifies so now negation of for all x px so this is nothing but if you you push the negation inside and negation of universal quantifier will become existential quantifier and you push the negation inside then if you remove this particular kind of thing then use the same kind of rules you cannot simply apply rule for this one and saying that for not for all x px you cannot simply say that it is not px and all so this changes to there exists some x not px then we can eliminate this existential quantifier using the same kind of rules each time you eliminate the existential quantifier you have to use a new parameter so now the other rule is this thing not for all x px suppose if you come across this particular kind of formula in the tree in the construction of your tree you come across this one then it is nothing but sorry this is there exist some x px so this changes to for all x not of px so these are the rules that we require to construct tree diagrams for any given predicate logical formula so the only thing which you need to note is that when you are trying to eliminate this existential quantifier you need to move to you need to use a new parameter each time when you remove this existential quantifier you need to use a new parameter so this a bc etc has to exist in the domain there are the objects in a given domain so essentially what we are trying to see is that you are trying to construct a tree diagram based on we are looking for a counter example by negating the given well-formed formula and you are constructing a tree based on these particular kinds of rules so now this is what we have explained already suppose at some point the second rule says that is also considered in all four rules suppose at some point in our construction we come to an universal formula for all x ? x and the node of the tree then since for all x px is true ? x is true ? of a also has to be true for any element of your domain so there is never a problem with substitution of any term into this particular kind of thing unlike the case of there exists some x px we have some restriction once you use one particular kind of parameter you are not supposed to use the same thing when you are eliminating some kind of existential quantifier in the next time when you come across existential quantifier in your tree but there is no such kind of restriction in case of universal quantifiers they are readily available for it is it is true for all kinds of is etc so we always add the instance ? of a to a branch recursively when you can a number of times you can use this particular kind of thing as well as for all x ? x again to the path whenever you want to use again this thing you can reintroduce the same formula for all x ? x again into the path so now these are some of the truth conditions with respect to quantifiers it tells us when a quantifier quantifier is going to be valuation of a given quantifier is true or false so the valuation of a there exists some x ? is going to be true especially when valuation of a ? a x that has to be true that means this the formula ? a has to be true for at least one particular kind of a if at least one a satisfies this particular kind of thing then it is called as there exists some x ? otherwise it is going to be false in the same way for all x ? this has to be for all x ? it has to be represented in that way there is a mistake here so for all x ? has to be true especially when ? a has to be true for any kind of a that you are taking into consideration so in the second case for all a it has to that particular kind of ? a has to be true in the first case it has to be true in at least one kind of occasion so this is the only difference between these existential and universal quantify so these are the quantifier rules which we which we need it while constructing the semantic tableau streets example if you come across a formula for all x ? x then you will simply substitute it with ? of T and for all x ? x where T is considered to be a ground term for example if you come across in a not for all x ? x then you simply substituted it as not ? a where if you come across there exists some x ? x you simply substituted as ? of a where a is always considered to be a new parameter in the same way it is not the case that there exists some x ? that is simply replaced by not ? T so these are the things which we have explained already so now one important remark is that we are saying that each time when you are removing the existential quantifier we are using a new parameter so what do we mean by saying that a is new a is new means that this particular kind of a does not occur in the path that is being extended so suppose if it is used earlier then you are not supposed to use the same kind of literal a but you have to use some other kind of thing it can be a ? or B or some other thing just to make distinction you are using a different kind of parameter or we can insist that a not occur in the tab that is being extended so you have to ensure that nowhere else in that particular kind of tree diagram this particular kind of a occurs then you can use this particular kind of literal a so a little bit more about this tab looks of predicate logic as we look into it quickly all these things will be very clear once we talk about some kind of examples so now let us consider some kind of formal analysis of tab looks for the predicate logic let ? B set of sentences from L so a sentence in predicate logic that we have defined earlier a sentence in predicate logic is a one which does not have any free variables if it is free variables it is considered to be a formula in the predicate logic so if suppose ? is considered to be set of sentences from in the language of predicate logic it can be axioms it can be any other thing and all and let us consider a finite tabular from that sigma you constructed a finite tabular from sigma it is considered to be binary tree labeled with a formula La which satisfies the following kind of properties following definition all it goes like this all one node trees are labeled with a formula are finite tabular from ? obviously the path ends after path ends in finite steps and all so now the second thing is that if ? is considered to be finite tabular from sigma we constructed a kind of tree from sigma and ? is a path through this ? that means it is extended in a tree starting with the starting the main formula at the node and then you have started expanding that particular kind of formula in a tree and a is on that particular kind of path of the tree then the extension placing the components a1 and a2 on that particular kind of path is also considered to be finite table it is this tells us that what is considered to be a finite table on the third thing says that if ? is considered to be finite tabular from sigma that means it ends in some finite steps and all finite intervals of time so ? is a path through ? and B is on that particular kind of path then the extension of ? placing the components of b1 on the left hand side of the branch and b2 on the right hand side of the branch is also considered to be finite table from sigma essentially what we are trying to talk about is that you have some sigma which consists of some basic formulas etc and all which we know that they are all sentences in the predicate logic and together with that we have a given formula a we add to that particular kind of thing and then we are trying to construct a tree so now we are trailing that so when that construction after constructing the tree is it going to be a finite table or not so these are some of the things which tells us that this is going to be a finite table is not going to end forever and ever and all this is it will end at some point so ending means in a sense that once you end up with only atomic kind of prepositions then you have to close the tree or whenever you come across a formula X and its negation then also you will close the tree that means the path ends there itself that is considered to be contradictory path so now the fourth rule a fourth rule tells us that tau is considered to be finite table from sigma that means you constructed a tree from sigma and Phi is a path through tau and let us say another element another thing C on that particular kind of path Phi then the extension placing the component C1 and C on Phi is also considered to be finite tableaux from sigma it tells us that only we are just trying to construct a finite tableau based on you given any formula we have only finite kind of tableau because it the tableau ends after once you have an atomic formula at the end of the somewhere else in the branches tableau will end there itself so there is no way in which you can extend the tree so all the tableaux end with atomic prepositions so now tableaux for particularly logic if tau is considered to be a finite tableau and Phi is considered to be path through tau and D on Phi then the extension placing the component D1 on Phi is also considered to be finite tableaux in the same way if tau is considered to be finite tableaux from sigma and Phi is a sentence from sigma then the extension of Phi where Phi is placed on each path of each path Phi through particular kind of formula tau is also considered to be finite tableaux from sigma so like this if tau 0 tau 1 tau n etc is a sequence of finite tableaux from sigma such that for each n greater than 0 tau of n n plus 1 the next one is constructed from tau n by the application of all these rules which we have mentioned so far so then tau is considered to be tau is equivalent to union of n tau that is also considered to be tableaux from sigma is also considered to be a finite tableaux so we are not said anything great about this thing except that tableaux ends in finite steps so for any given formula we have a finite tableaux so now there are certain things which you can talk about with respect to this tableaux tableaux constructions so let us consider that tau be a tableaux that means you constructed tree diagram for a given formula and Phi is a path through tau that means sometimes it will have branches sometimes it might be only trunk and all and etc and all depending upon the formula that you are taking into consideration so now the path Phi Phi is considered to be contradictory if for some sentence Phi any sentence that you are taking into consideration Phi both Phi and 0 Phi appears in the same path of your tree Phi and Psi 0 Phi occurs in the same branch then it will close so then the path is considered to be contradictory path so Phi tau is considered to be contradictory if every path on tau is considered to be contradictory so whenever suppose if you have a branch and then you have two paths in particular one is going to the left hand side and the right hand side in both the paths you come across a literal and it is negation then obviously that is contradictory and then entire path is considered to be contradictory so if tau is considered to be proof of alpha from Sigma and tau is a contradictory tableaux from Sigma with this root node label not alpha if there is a proof tau of alpha from Sigma we say that it is considered to be alpha provable from Sigma and it is denoted by alpha is proved from Sigma and the other way of saying I mean when do we say that Sigma is considered to be inconsistent if there is a proof of a contradiction from Sigma that means you come across a and not a in a given branch so we can also use tableaux to show that Alpha is true in Sigma so whenever Sigma is true Alpha also has to be true in that sense Alpha is a logical consequence of Sigma that means what essentially we are trying to do is that given Sigma we are adding not Alpha to it and then we are trying to see when the branch closes are not in the branch closes then not of Alpha is considered to be a contradiction then obviously Alpha has to be true that means you cannot deny the formula and all because denial of the formula leads to the contradiction so what essentially we are trying to do is we are adding not Alpha to Sigma and then we are trying to show that it is unsatisfiable so if all the branches closes and all if it becomes unsatisfiable then the formula is going to be original formula is going to be valid otherwise it is going to be invalid because at least there will be one particular kind of counter example so now this is the thing which we will make use of in constructing proofs of some kind of formulas and all so now let us consider some examples so that we can consider we can understand this particular kind of method in a better way so let us try to say that you take some formulas into consideration and then we will see whether this is considered to be probable in the predicate logic or not so for all x Px implies PA so now this is the formula which is given to us x is the formula in the predicate logic so in the semantic tablox method what one essentially does is that you take the negation of the formula and show that this is considered to be unsatisfiable if negation of x is unsatisfiable that means all the branches closes then that implies x is considered to be a valid formula or tautology something so that is what essentially we are trying to do so now we negate the formula and then we need to use this alpha and beta rules that we have mentioned here so this is in this format x and y so this can be like this for all x Px and not PA so how did you get this to this one not of P implies Q is simply P and not somewhere else we have used this particular kind of rule negation of P implies Q is P and not Q so now so what is what essentially we are trying to say is that whether it is probable in your whether it is considered to be valid formula or not that is what we are trying to check so now you have something called for all x Px at the node of somewhere else in this kind of formula whenever you come across this particular kind of formula for all x Px means it in its instance is also true that means it has to be true for a formula A else A has to be having this particular kind of property P suppose if you say all are happy or something like that and but if you take a particular kind of individual that individual also has to be happy has to feel happy so that means for all x Px means one instance of that one is PA so now you come across PA and not PA in the path of this particular kind of tree so this particular kind of thing closes because it is a contradiction not PA and PA leads to contradiction so that means what essentially we have shown is simply this that not of x is unsatisfiable because we take the negation of this particular kind of given formula it leads to the closure of branches that means it is unsatisfiable satisfiable that means x has to be a valid kind of formula so that means we have derived this particular kind of thing for all x Px implies PA is a theorem in your in the language of predicate logic so let us consider some more examples so that for example if you say this particular kind of is a famous example that we have been trying to talk about right from the beginning of this course that is this thing all men are mortal socrates is man socrates is mortal this is a famous example which is given in all the introductory logic courses in all so now you represent it as H humans and then this as mortal as M because if you represent it with the same letter M will be no distinction between these two predicates so so now the first sentence will become like this if X is man of course we are representing it as HS men if X is a human being then X has to be more now socrates is mortal that means M socrates is a man means is this H X this has to be written with the individual letter s let us say s stands for socrates is represent some kind of specific kind of objects in the domain so what is our domain domain consist of people are human beings etc so you don't I don't have to take into consideration animals trees plants etc into that one that is not in this particular kind of domain so now this is HS and then socrates is mortal so now this is considered to be this is this predicates are written the capital letters whereas the individual constants that you come across this particular kind of formula written in terms of small letters so now this is the argument that we have so now we want to see whether this particular kind of argument is valid or invalid we know that this is a valid argument so now we are trying to establish it with the help of the predicate logic so what essentially we are doing in the second part of this lecture is that we are trying to consider some examples like this and then we are trying to see we are trying to make use of the semantic tableaux method which serves as some kind of decision procedure method it tells us when how the if the argument is valid tells us it gives us some kind of in a proof for this particular kind of thing that this conclusion follows from this premises and so now let us take this argument in detail HX implies MX and second is this thing H yes and this is going to be your conclusion yes so in the semantic tableaux method what you will do is we will start with the negation of the conclusion so what essentially we are trying to do is Sigma is this particular kind of thing so these two premises constitute Sigma so now what you are doing is you are adding not Alpha to it because where Alpha is considered to be the conclusion so now if Sigma union not Alpha is unsatisfiable then so how did we come to this unsatisfiability here we landed into this particular kind of problem especially when you are taking the negation of the conclusion that means negation of Alpha is unsatisfiable that means Alpha has to be valid that means Alpha has to be true so now you take the negation of the conclusion like this and then you try to construct a tree for this one so they are all atomic formulas you do not have to worry much about it so now HX implies MX happens for all X you know so that means this has to be true for even this particular kind of thing for all human beings if X is a human being then X has to be mortal that means all human beings have to die some day or other so that happens even for the Socrates are even for any thing which you substitute it into this one if it all is a human being then it has to die so now this S stands for Socrates so one instance of this one is this so we have used this particular kind of route for all XPX if you come across this particular kind of formula in your tree then you can simply represent it as a this is this happens for any so now HS implies MS is a thing which we have thing this is called a universal instantiation of this one instance of this one is this now this can be represented as not HS and MS so this is simply P implies Q nothing but not P and Q so it is in that sense you need to write like this so now you have HS here and not HS here and this branch closes now there is another path like this HS not MS etc and all and then MS here and not MS here and this also closes so that means negation of this particular kind of thing leads to contradiction so that means it has to be MS rather than not MS so this is a proof which is based on something called as Reduxio add absurdum kind of method so this is a that means the original conclusion is MS rather than not MS so in this way we can guarantee that so could this is mortal necessarily follows from all men are mortal and so could this is man so like this you can translate many formulas into the language of predicate logic then you can talk about the validity or truth the tautology of a given formula let us consider some more examples since we have sufficient time so we can consider some more examples let us say a formula like this for all x Px implies Qx and not Qx so this is closed by this particular kind of thing so this is one particular kind of formula and from this whether or not you will be able to deduce this thing they it does not exist they does not exist x such that Px so now we are trying to see whether this can be derived from this particular kind of thing or not so in the semantic tableaux method what you will do here is is that you just take this as it is and then you add the negation of the conclusion so this is what you are trying to do that means you say conclusion on so now what you will do in the second step is you negate this particular kind of thing and add to your sigma so now you will start constructing a tableau for this one and then we will see whether the tableau closes are not not not of there exists some x Px it is not not of P whenever you come across a formula not not P you simply represent it as P only so the same way you have this particular kind of thing there exists some x Px not of not of there exists some x Px is simply this one so how did we get this one to double negation you need to give the justification to the right hand side otherwise it does not make any sense to talk about this particular kind of thing we need to say how did we come to this particular kind of formula so that is why we need to write justification on the right hand side followed by this particular kind of formula so now we need to talk about one instance of this particular kind of formula so Px implies Qx and not Qx is true for all x and all I mean it has to be true for some kind of a also so before that there is one strategy that we need to follow always eliminate the existential quantifiers first rather than dealing with the universal quantifiers first you handle with the existential quantifiers eliminate those existential quantifiers first and then move on to the universal quantifiers so now there exists some x Px if you if you come across this particular kind of thing there exists some x Px in the tablux rule we replace it with we can replace it with PA where this a should not occur anywhere is in the branch which is above above this particular kind of formula nowhere else that a has to exist that means a has to be new so now whenever you have a formula there exists some x Px you can simply represent it with PA so now that is what we are trying to write so now this is a tablux rule you can say existential instantiation or something now so this Px implies Qx and not Qx is true for any kind of a now that means it has to be true for a also so that means it is PA implies Qa and not Qa so now if you further expand this particular kind of thing so this is one universal instantiation so one instance of this one is this so now this is going to be not PA and Qa Qa and not Qa you can write it as like this because P and Q can be simply written as PQ it is like a trunk of your tree whenever you have P or Q it is a branch PQ so now you have PA here and not PA here it closes and all it is like cutting your own tree and all you are sitting on a tree and you are cutting your own tree because there is a contradiction PA and not PA you cannot go further so that is why it closes here and already you have Qa and not Qa it closes so that means so negation of this conclusion leads to the contradiction that means the original conclusion holds that is what is the original thing which we have to deduce that is there now it is not the case that there exists some x Px so that has to be true so this is the way to show that this particular thing follows from the given formula Px implies Qx and not Qx so now let us try to consider some one more example here what we will do is a given predicate logic sentence we will transform it into the language of predicate logic and then we are trying to see whether that particular kind of formula well-formed formula is valid or not so here is the statement which you come across in the natural language and that is like this so there is an object which is allowed by which is allowed by every mathematician every mathematician we do not know what kind of object it is let us assume that it can be an abstract object it can be anything which is allowed by every mathematician this is for the sake of for example we take into consideration this thing so therefore every mathematician mathematician loves at least one object at least one this is abstract object at least one abstract object mathematician does not require any real entity entities to exist in the world they can even consider an abstract object and then they can still mathematics and still talk about the mathematics around surrounding that particular kind of abstract objects so now here is so it looks like that the conclusion seems to be following from the premises and all we have to establish with the help of some kind of decision procedure method so the first one states that there is an object there is an abstract object which is allowed by every mathematician and then that that implies that means every mathematician loves at least one particular kind of abstract object should follow from whether or not this follows from this are not is the one which you are trying to see so now we need to translate these things into the language of predicate logic using quantifies so the first one can be translated in this sense suppose if x is considered to be an abstract object abstract object is represented as a x and for all y if x is a mathematician then there is one more predicate that is there here allowed by every mathematician so that means L y x that means for all the mathematicians there is at least some kind of object x and that that mathematician all the mathematician whatever is considered here mathematician loves this x so there is an order which we need to follow suppose if you write x and y that means x loves y and y x means y loves x there is the order that we follow in the predicate logic so this is a translation of the first letter so there is an object means there is there should be at least one object that should exist now that means there exists some x and whatever is there here is the one which we have written so now this is the formula which is represented by premise and then the conclusion is every mathematician that means you need to start with the quantifier for all y if y is a mathematician which is written in this sense y is mathematician that means there exists some x such that A x that means x is an upstate object and that mathematician has to love that particular kind of object x so for all y if y is a mathematician then there exists some x such that x is an upstate object and that x has to be allowed by the mathematician that means y has to love x so this is the conclusion once you change once you represent it in terms of the language of predicate logic then you do not have to worry about what is the content of this argument and all because we will be handling only the symbols that you see here so now in the semantic tablox method as usual we start with the negation of the conclusion that means so this is the first thing there exists some x and for all y mx implies L y x etc and then negation of the whole thing negation of for all y my implies there exists some x A x and L x y so that means you are denying the whole formula so now once you deny this suppose if this conclusion indeed follows from the premises then if you take the negation of the conclusion then the branch should close and all that means for example if you take sigma into consideration sigma as your premises union not of alpha should lead to contradiction so now this is the second one now we need to use different rules you know so now the our strategy is always this that first you eliminate this existential quantify that means you need to find out an instance of this one so now if you eliminate this existential quantify that means wherever you find x you need to replace it with some kind of a parameter ABC whatever you feel like replacing it so now the first one existential instantiation will become this you taking replace x with a and this becomes like this for all y remains the same mx x is replaced by a and then L y remains the same and then x is replaced by a so this is what is existential instantiation of one existential instantial one instance of this one is this so now next time when you remove this existential quantify you should ensure that you are not using this particular kind of thing parameter a so you have to use another different kind of parameter so now this can be represented as this thing so this is for example if you have a formula x and y in the tree you can simply write x and y it is like a trunk and all so that means three simplification or it is a conjunction rule so it will be like this for all y ma implies L y a so this is the fifth step so now we have this particular kind of formula we need to simplify this one then so this will become like this so it is not for all y m y implies there exists some x so we have this particular kind of formula not for all y x if suppose if that is there like this then it will become there exists some x not x so the same way it is like not for all y that means there exists some y and then you push this negation inside and then this will become the entire thing m y whatever is there m y implies there exists some x a x and this formula L x y so now this is coming like this so this is considered with a six step so this is a simplification of this particular kind of thing now one instance of this particular kind of thing is like this just getting out so now this formula can be written in this sense if you replace y with this thing it will become B you are not supposed to use a here it will become B so now this is same there exists some x a x and L x and then you replaced y with B so this is one instance of six so now eight we have m B so for so this is double negation of seven implication negation and implication will become this negation of this particular kind of thing is like this negation of P implies Q is nothing but P and not Q so this is first one is MB and the second one is not of there exists some x a x and L x B so this is a simplification and then not of for all x this one will become negation of existential quantify will become for all x not of a x and L x B so now one instance of this one because it happens for all x and all it has to be true for this one also that means not of a a and not of L a B so there has to be true for all kinds of anything which you take into consideration for x it has to be true that means for a also it has to be true and all in that sense we have written like this 10 and universal instantiation so now we are getting closer to our proof 12th one is MB L B a let us come from this one M a L y a it happens for all the thing and all fifth one M y M y L y a is getting over so now 13th step if you expand it and all so it will become not MB and L B a and then this branch closes because not MB is here and MB is here and then this branch it can be further expanded to not a a and then not L B a this and this closes in all so ultimately what has happened is if you take the formula if you take this is premise and this is the conclusion if you negate the conclusion it leads to closure of all the branches and all so with this we will end this lecture and all so what what we discussed in this lecture is simply is that we discussed about semantic tablox method which serves as some kind of decision procedure method with which you can we can find out whether or not a given well-formed formula is valid we did not talk about the consistency satisfiability etc and all more or less you know in the case of consistency if you have if you are given two sentences in the predicate logic and you constructed tree using the same kind of rules and all if at least one branch is open and that is considered to be these two formulas are considered to be consistent so in the next lecture we will be talking about some more examples so that we get familiarize ourselves with this particular kind of method.