 Welcome back in the last lecture we discussed in detail about one of the important methods which is considered as a decision procedure method which is called as a truth table method truth table method is considered to be the most simplistic method especially in this course introduction to logic it is simplistic in the sense that as long as the number of propositional variables are less in number that means two or three for example if you have two propositional variables you have four entries in your truth table and if you have three propositional variables like P, Q, R, etc representing some kind of propositions we have eight entries in the truth table but the problem is that it is very difficult for us to manage for example when you have more than five or six variables propositional variables. If you have six variables that means two to the power of n entries will be there in the truth table that means two to the power of six maybe 64 entries you need to inspect to find out whether group of statements are consistent or whether a particular kind of conclusion follows from the premises that means the validity etc for that you know you need to check all the 64 entries you know that means you need to inspect each and row each and every row of your truth table meticulously that means 64 rows are there and all the rows you need to inspect I mean those rows in which whether or not you have true premises in a false conclusion if you have a true premise in a false conclusion then the argument is obviously considered to be in value. So instead of inspecting the 64 rows which will be difficult for us and there are some other better methods one such method which we will be discussing in this class that is semantic tableaux method. So this is also called as analytic tableaux method or it is also called as tree method they are one of the same. So this tree method is a very very useful kind of method which is originated in the works of a famous logician Beth Beth in the year 1955 Beth lived from 1908 to 1964 is considered to be a Dutch philosopher and in the history of logic it seems that this method has originated in the works of Beth later Raymond Spoolian has formulated in his book first order logic his own trees and all which is little bit simpler than what Beth has proposed in his analytic tableaux method. So then simultaneously around the same year Hintika also developed independently the same kind of method it seems that somehow Hintika seems to propose this method around the same year but there is no evidence that whether or not Raymond Spoolian has borrowed something from Hintika. So there is a controversy or debate in what who has actually formulated this method first so that is not of interest to us but what is of interest to us is to understand this particular kind of method especially in understanding in understanding the validity of a given formulas are when two groups of statements are consistent to each other or one can even show whether a given well form formula is a tautology or a contradiction or a contingent statement using this tableaux method and you can also show when two logical formula two well form formulas are logically equivalent to each other using the same kind of method that is semantic tableaux method so it has its applications in automated theorem proving and it is also applications in the logic of programs etc and all which will not go into the details of these things but we will try to introduce what we what exactly this method is all about we will introduce this method and then we will try to show with some examples that given group of statements are consistent to whether or not they are consistent to each other or when a conclusion follows from the premises that means the question that is the validity of a given argument etc. So what is a semantic tree before we begin it is also a kind of constructive method so what we will be doing is given a well form formula we will be constructing a corresponding tree diagram for this particular kind of usually trees will have trunks and you have leaves etc branches etc so the same way we have we usually here in this case we have upside down kind of tree usually trees will be down and then up we will have branches etc and all but here we have some kind of upside down kind of tree which you will find it here for each and every formula you will be constructing a corresponding tree and then we will try to evaluate whether the fall following formula is a tautology etc. So a semantic tree is considered to be device for displaying all the valuations on which the formula or set of formulas are going to be true so one of the basic and important the essence of this method is this that you know we will be constructing some kind of counter example so the essence of this one is like this it consists of finding some kind of counter examples where what is considered to be a counter example suppose if an argument is considered to be invalid especially when your premises are true and the conclusion is false so if we can construct one kind of one particular kind of counter example in which your premises are true and the conclusion is false then obviously the argument is invalid so that means we have we are said to have constructed a counter example so the basic idea of this method is this that an inference is considered to be valid if and only if there exists there exists a counter example otherwise the inference is going to be valid if and only if there are no counter examples so that is there is no situation in which the premises hold and the conclusion is false so this also involves some kind of rule based construction which we are going to talk about in a while from now and using those rules we will construct trees and then we are going to show that if there are no counter examples then obviously the formula is going to be valid otherwise if there are any counter examples we could construct a counter example then obviously the argument is considered to be invalid so each step of the construction is given in a count of some kind of tree leg structure which is also called as a tableau so usually this tableau closes when there is a conflicting information completing information in sense that suppose if you have a formula x and not x then obviously the branch close because you have a conflicting information suppose if you have some information like it is raining and simultaneously you say that it is not raining then that is a conflicting kind of information which one to believe you will be in some kind of dilemma so it is in you are in conflict so in that case the branch closes a literal and its negation appears in a branch then obviously the tree closes that particular branch closes so no counter examples can be constructed for a if the branch is not open so if the branch is closed that means it implies that there are no counter examples exist so now this tree method is based on some kind of rules so now what we will be doing in another 20 to 25 minutes is this that we will be talking about these rules and then we will construct this tree diagrams for various kinds of well form formulas then we are going to show that when a given well form formula is a tautology contradiction or contingent statement and second we will talk about when a given well form formula is a tautology and third where we will talk about when two groups of statements are inconsistent to each other and fourth we will talk about when two groups of statements are or two well form formulas are considered to be logically equivalent to each other so all these things which we will be trying to talk about in terms of this particular kind of semantic tableau method and then in the process we will also be trying to talk about some of the important strategies that will be following while adopting this particular kind of method so the rules are like this first to begin with we have repositional variables etc. PQRs etc so these are all prepositional variables and then we have these symbols one which is always true is represented in this way usually with the symbol t and then this is what you presented as bought one is always something which is always false is represented in this way and then we have some other things like parenthesis etc and all and apart from that you have this logical connectives negation are and implies and if and only if and then the other symbol which we will be using is this one that means the when the branch closes will put this mark cross mark so that means the branch closes so now what we are trying to do simply is that so we will be assigning some kind of truth values to these prepositional variables that means we are interpreting this formulas so now there are some kind of rules which we need to understand before applying this semantic tableau method so to begin with so there are something called root root and then nodes so you need to understand this thing little bit later we will talk about this thing little bit later so first we will talk about some kind of rules with which you know you can say whether a given formula is valid or not this is the construction tree construction rules so suppose if you come across a simple formula like this negation of P then you simply write it like this only the construction of this not P is same as this one so now if you have a formula like this PRQ the construction tree for this one is PRQ so now if you have a compound formula P and Q then the construction tree will be like this so usually a tree will be like this so these are all branches and this is considered to be the root and so this is the formula that we are trying to begin with and now these formula is reduced is reduced into some kind of atomic prepositions in all as you will see in all these rules the things which are there at the nodes are considered to be only atomic sentences as a PRQ or maybe negation of that one etc. So whenever you have a formula P and Q you just write it like this it is a trunk it is an upside down kind of tree usually the tree will be like this now you have to reverse it a little bit and then you will see these things so now P implies Q so the definition of P implies Q is not PRQ so that means it is not PRQ so the branch suggests that there is a disjunction so that is either not PRQ so this is exactly in alliance with the semantics that we have talked about in the last few classes so that means PRQ is going to be false only when both PRQ or false in all other cases it is going to be true in the same way P and Q is going to be true only when P Q's are true in all other cases it is going to be false so that is a semantics of preposition logic in the same way P implies Q is going to be false only when P is T and Q is false in all other cases it is going to be true so based on that kind of information we are just trying to come up with some kind of constructive method and then what we are trying to do is for simple formulas like this we are trying to construct trees tree diagrams usually a picture says thousand words so a given formula we are trying to construct trees like this so now this is called as a branch and this is called as a trunk of a tree etc. So now the only logical connective which is left now here is this one so we will write it here P if and only if Q so this is either P Q is the case or not P not Q you can write not P not Q here itself P and Q can shift it to the other side it does not make any big difference it is one of the same so these are the rules which we have for each and every logical connective for not this is a thing for or the tree appears to be like this for P and Q it appears to be like this so these are considered to be alpha rules so now in this alpha rules so there are some rules which are considered to be branching rules that means wherever you find a branch this considered to be a branching rule and wherever you do not find the branching kind of thing it is called as non-branching rules you know so why we are talking about branching and non-branching rules because so while adopting this particular kind of technique or method so there are some kind of strategies that one will be following so the one one of the important strategies is that always apply non-branching rules first so once you exhaust with the non-branching rules you enter into branching rules so now so these are this is non-branching rule it is not leading into any branch so non-branching rule so now all these things are branching kind of rules so given suppose if you are supposed to apply this method you have to ensure that first you apply the non-branching rule and then apply all these rules so now this list is called as alpha rules usually it is considered to be positive kind of rules and all so now we will be writing beta rules so beta rules are exactly the negations of these things so now if you come across a formula like this that is negation of negation of P it is it is not the case that it is not the case that it is raining that means it is raining so now if you have a formula like this you simply substituted with this particular kind of formula P so now using De Morgan's laws it is quite simple so now negation of P and PRQ if you push this negation inside then it will become negation of P and the negation of disjunction will become conjunction so that is why we need to write it in this format so now as you clearly see this is a formula and then once you apply these rules and all at the end you will find only atomic prepositions what is an atomic preposition an atomic preposition is a one which cannot be further reduced into any other kind of preposition PRQs can be reduced into P Q etc and all but P Qs are etc they are prepositional variables are the most simplistic kind of sentences which cannot be further reduced into other thing so that is why they are called as atomic sentences so now this is the rule for this one negation of P in the same way negation of P and Q using the De Morgan's law it leads to a branch it is negation you push it inside it becomes negation of P and negation of conjunction will become disjunction that is why disjunction will always have a branch so this is the form that we have so now negation of P implies Q is simply P and not Q why because P P ? Q is not PRQ and negation of not PRQ is not not P that is P and not Q so now negation of P implies Q is a branch again P and not Q and not not P and Q so these are the only things that we have these are the rules which will be applying for judging whether a given well-formed formula is a tautology whether it is a contradiction or contingent statement on the one hand or when two groups of statements are consistent to each other etc so now using these rules we will be trying to talk about whether or not given formula is valid or invalid etc. So now to start with we start with some simple examples so we want to see whether this particular kind of argument is valid or not we start with some simple examples in the beginning and then we will move on to some other things usually when you write P implies Q Q implies R P implies R then usually you say that it is a valid argument by virtue of transitive property obviously P implies Q Q implies R and obviously P implies R so now let us do not we do not talk about the valid argument obviously valid arguments which we know so instead of this what we do is we slightly change this thing P implies Q to instead of this thing R implies Q and then P implies R let us assume that this is the whether or not this P implies R follows from these two things are not so now these two are considered to be usually premises and this is usually called as conclusion. So now how do we know that P implies R follows from these two statements P implies Q and R implies Q whether it is it follows then it is valid otherwise it is invalid so how do we check that this particular kind of formula that is P implies R follows from these two things there are various methods one method which we have already discussed that is a truth table method and since there are three variables eight entries will be there in the truth table so that is also a little bit easy to do but so what we are trying to do is we are trying to see whether P implies R follows from these are not. So now the very essence of semantic tableaux method is that we are trying to construct a counter example if you fail to construct a counter example that means the original conclusion is going to be valid so what we will be doing is we will begin with the same thing we will store these things P implies Q and R implies Q so these are premises and we have a conclusion P implies R so now what we will be doing is we will be negating the conclusion so the idea here is that negation of the conclusion leads to the closure of branch that means it is unsatisfiable then obviously your negation of conclusion is going to be false that means all the branch closes it is going to be false that means X has to be a tautology if X is a tautology obviously the formula is going to be valid because all tautologies are considered to be valid formulas so now we are trying to check whether P implies R follows from these two are not this is denial of conclusion that means we denied the conclusion and all and we are trying to construct a counter example you fail in the process then obviously this is the actual conclusion which we that follows so now so now these are the compound formulas in all so now we will be applying alpha beta rules etc so now one of the important strategies is that we need to apply non-branching rules first so which formula you take into consideration is come under non-branching rules non-branching rules are this is the one and then of course this is another so these are the two non-branching rules that you are finding because there is no branch here only trunk of a tree so now this looks like this one not of P implies Q so instead of Q we have R so now this reduces to P and not R so this is three simplification three is this formula and we simplified it then this leads to this one so we applied beta rule here beta rule is talking about the negation of this formula so now this is the one so now once you apply this particular kind of rule you need to see whether there is any conflicting information in your branch in your tree so right now we do not have since we have checked this formula then we put this tick mark so that you know you will not use it again and again otherwise you will confuse and we will use it again so once is got exhausted then you put a check mark here that means you need not have to use this one again so now these are the formulas which are left so now we use this particular kind of thing P implies Q that means you apply this tree you are considering a tree for this one means you have to use this thing so it is not P or Q so you draw a diagram like this under this you put not be and Q you need to pause a second and you need to see whether there is any conflicting information in your in your branch so now this branch is going like this all the way till here it is a trunk and this is a branch now now this is going like this so one is going like this another one other branch is like this so now you have P here and you have not P here so that means there is a conflicting information so the branch closes here itself that means it stops there itself there is no question of any construction of any counter example possible here in this case since you have conflicting information there is a contradictory information it closes so now this branch is still open and all so now since this is a formula that we checked out already so that is why we have put tick mark here so now what is the formula which is left here is this one R implies Q so now again apply same rule alpha rule and it becomes not R and Q so now this is also over now we checked out checked all the formulas and all and then obviously at the end of all the branches and all you have only atomic propositions not Q etc. So now you have to inspect this particular kind of branch so now this branch is open and even this branch is also open that means even after the denial of the conclusion we could still construct a counter example that satisfies this formulas that means it is possible for the premises to be true and the conclusion to be false that is the reason why we have at least one or two open branches so now we can study this open branches and we can talk about this particular kind of thing so now not R means R is false that means not R is true that means R has to be false sorry and Q is true Q is T and then of course not R we have taken into consideration and P T so that means assigning these values R false Q T and P T is satisfies this particular kind of formula this particular kind of formula that means P implies Q R implies Q not P implies R this is an assignment which said which makes this formula true and all open branch means satisfiability and all so this is going to make this formulas true so this is one model which we have that means whenever you give R F Q T and P T that satisfies this particular kind of formula what is satisfying this formula that means if you have true premises and false conclusion that means this is a counter example for this one and another counter example is there here whenever Q takes value T that means Q T R false not R is true means R is false and then P T so this is another counter example and all so what is that we have achieved with this particular kind of thing denial of the conclusion does not lead to branch closure that means you already said to have constructed a counter example whenever you have said to have constructed a counter example then obviously the argument is invalid so it is possible that your premises are true but at your conclusion is false how the conclusion is going to be false it is going to be false in two different ways these two different ways are considered to be the two different open branches so especially when R is F Q is T P is T that is going to make this thing true and when Q is T R is equal to F P is equal to T then also your premises are true and the conclusion is false even if at least one branch is open by denying the conclusion then obviously the argument is considered to be invalid. So this is based on some kind of falsifiability kind of method instead of looking for things which are true will be looking for things which are false that means you are looking for always you are looking for a counter example so it is like you have 100 for example in a bag which consists of 100 tomatoes even if you have one rotten tomato and all you will say that not all tomatoes are considered to be good in this basket one tomato will spoil the entire thing so this is based on some kind of falsifiability a kind of method as long as it is not falsifiable the formula is going to be true it is like as long as you do not find a white crow it is going you are going to accept it as statement that you are going to accept the statement that all pros are going to are going to be black. So now this is the way to show that this argument is invalid so what about so the arguments which are valid will take up some kind of arguments which are obviously valid so this is the one P implies Q and P and Q this is obviously the modus ponens rule which is considered to be obviously valid so how do we show that this Q follows from these two things P implies Q and P etc you list out these premises as it is these are premises and then what you do here is you negate the conclusion so now we need to write here denial of conclusion so now once you deny the conclusion we need to check whether it leads to a closure of a branch or not so now we need to apply is all for beta rules depending upon this thing there are no branching rule non branching rule which we can apply here so any rule which you can use so now we need to apply this one this is already an atomic sentence nothing needs to be done so now P implies Q is simply not P or Q so now you have P here and you are not here this branch closes and not Q and Q here is a conflicting information this branch also close so what did we get negation of the conclusion leads to the closure of a branch that means it is false usually represented as this one bought is a symbol let me use so that means X has to be obviously true since X is a X is true means X is a tautology X is a tautology means it is a valid formula all tautologies in propositional logics are obviously all valid formulas that is reason why logicians will be insisting on tautologies there are special kinds of statements which are obviously are always considered to be true they are like God given kind of truths which are always true and there are some other groups of statements a group of formulas which are obviously false it is like 2 plus 2 is equal to 5 which is obviously false and 2 plus 2 is equal to 4 is always true so now this is the way which you can show the validity of a given formula for validity what you need to do is you do have to deny the conclusion and then you need to see whether the branch closes or not all the branches close if at least one branch is open that means the open branch is considered to be it has to be analyzed in detail so that means open branch indicates us that there is a counter example what is a counter example you have two premises and a false conclusion so that is going to be the case if you can construct one counter example that means it is obviously going to be any invalid argument so what else one can do with the help of this particular semantic tableaux method so now we are trying to see whether a given group of statements are consistent to each other or not so for that what you will be doing is like this let us consider an example and we will try to see whether this particular kind of group of statements are consistent to each other are not b and b the first statement and the second statement is not k some statement which does not matter whatever you take into consideration and 3 not b b r b implies not g forget about what this c is b is etc and all they all correspond to some kind of statement it can be it is raining or it may be something like that some kind of statements which it expresses and so now given English language sentences we have converted into the language of propositional logic and now we are going to see whether these are consistent to each other or not that means they simultaneously they can be true or not that is what you are trying to explore so now here there is obviously there is no conclusion you need not have to look into the negation of the conclusion and see the branch closes etc and all so we will keep it as it is and all so now we are trying to find out a model in which trying to interpret is formal this propositional variables that means assigning some kind of values to it so that all these things will turn out to be true and all that means you need to inspect the open branches here so now so as usual we need to apply non-branching rules and all here there is no scope for any non-branching rule and all so you can open up any formula and then construct tree for these things so now let us consider the second one you can choose the first one also so this is not k RG so that is why we applied this particular kind of PRQ means either P is the case or Q is the case so now this is over you tick mark this one now the second formula you can take any formula into consideration so now we will open up the first one it is either C or B and not be and the same kind of information you need to pass it to the other branch also so C and then B and not so now this formula is also over so now each time you apply this alpha beta rules etc you need to see whether any branch closes or not so this branch remains open so now this you can write it in this way B and not be it can be written in that form of trunk and all that is B and not be so this is also B and not be so now you have a conflicting kind of information P and not be this branch closes and here also this branch closes now the branches that are open are these two so now whatever has remained the formula needs to be applied to this needs to be written just below this open branches so now what is left now this is the one so now this can be written as this is X and this is Y and now X implies Y is not X or Y that means not of not B and not G remains the same thing and now this branch is these two branches are already closed you do not have to worry much so now this is not B or B same information you put it in the all the open branches that you will see here so now this is not G now you further simplify it using demargin's laws etc or maybe you can apply any one of these rules there then not of this one you push the negation inside this becomes B and then not of negation of disjunction will become conjunction that is why we are writing one below the other so this is like a this formula and then this one and not G remains same and then this becomes B and not be not not be is B and not of disjunction is conjunction that is why we are writing just below this one and this is not B so now all the things are exhausted so now we need to see this particular kind of your B here and not be here is branch closes and C not K and all this branch remains open and this obviously closes because of conflicting information B and not B and then this branch also remains open open branches are the ones which you need to inspect so these are the branches that makes this formulas true that means each open branch will serve as a model some kind of each branch is considered to be satisfiable you know for example when you give when you assign value not K T that means K is false and C C as T and then not G as T means G is false under this particular kind of assignment of truth values these three statements are going to be satisfiable that means they are going to be true together in the same way another open branch that you will find it here in this tree diagram is this when you give valuation this is one kind of interpretation under which these formulas are going to be true together or satisfiable the other one is the one which you need to see not G is equal to T that means G is false the first one and then C is true if you assign some kind of true value T to C the only values that you can assign to C either T or F and G is not G we already have this information and all so you have this particular kind of information that also satisfies this particular kind of formula so these are the things interpretations which satisfies this formula that means these formulas are going to be consistent especially when you interpret in this particular kind of way so list out all the formulas one after another constructed tree diagram and you find any open branch and open branch corresponds to the satisfiability or consistency so that means the formulas can be true together especially when you assign some kind of values like this so that means these three formulas are simultaneously said to be consistent to each other so this is another way of showing that these formulas are going to be consistent suppose if you have used truth table method so now the number of variables are 1 2 3 4 and 4 variables are there that means there are 16 entries you need to inspect so that means each row it leads to true and all so all the rows under the final column whatever value that means the value that you are going to get is T those rows you need to inspect so instead of doing all these things it is the most simplistic kind of method is easy to use based on some simple kind of rules the muggins rules will help us so with the help of which we can easily see that you know here are these particular kinds of things which satisfies this formula that means under this particular kind of interpretation is going to be true all the formulas are going to be true so now we can also show that the given formulas are considered to be inconsistent to each other so obviously these three formulas are said to be consistent for example if you have formula P and Q and not P argue I have written deliberately chosen this example just to show that these two formulas are inconsistent to each other so now first you state all these things list out these things with some numbers and then apply non-branching rules first P and Q is a non-branching rule so that is why you apply this first and then you apply branching rules of course both are branching rules only when you are not very much so any rule which you can apply so this is not P and not of disjunction is conjunction so this will become not so now you will see here P and this negation is there so list out the premises for this list out these formulas and then construct a tree if all the branches closes then that is said to be unsatisfiable or it also called as inconsistent so P and Q is inconsistent with not P and Q that is obviously the case because P and Q is exactly not P or Q not of P or Q is a contradictory to P and Q obviously two statements which are in contradictory to each other are always considered inconsistent or unsatisfiable so this is the way in which you can show that given formula formulas are consistent or inconsistent to each other are satisfiable so now there are other things which you can do with the help of this semantic tableaux method so that is you can show whether given formulas are logically identical to each other and later we will see whether this is going to be statement is going to be tautology or not that also one can do with the help of semantic tableaux method so now we are trying to see whether so these are the two formulas now we are trying to see whether they are logically identical to each other or not P implies Q implies R we put one bracket here so now this one can do it in several ways and all again using truth table method if the truth values of this one exactly matches with the truth table truth values of this one under the main logical connective then these two are said to be logically identical to each other for example P and Q and Q and P Q and P and Q and P so you have two variables that is why you write two T's and two F alternative T and alternative F and this formula is going to be false going to be true only in this case in all other cases it becomes false in all exactly Q and P also the same thing so this formula Q and P is going to be true when both P Q are true in all other cases obviously it becomes false so now according to the truth table I mean this exactly matches with the other one for example when P and Q takes value T even Q and P also takes a value T so this matches with this this matches with this and this matches with this one in that sense P and Q and Q and P are logically identical to each other so that is one way of showing it the other way of showing it is using some other kind of thing so that is this one suppose X is a formula and Y is another formula so when do we say that these two are logically identical to each other these two are logically identical to each other especially when X if and only if Y is considered to be a tautology if you can show that X if and only if Y is a tautology then X and Y are said to be identical to each other that is logically equivalent to each other so now what we are trying to see here is this thing so now we are trying to see whether this formula is logically equivalent to this or not so now for that what we will be doing is will be putting this particular kind of sign and then we are trying to see whether this formula is going to be tautology or not so now this is the formula which is there now so now the idea of semantic tableaux method is simple that you know you start with the counter example first that means you negate the conclusion negation of X is this one not of the entire thing which you need to write so P in place R is the first formula and then the second formula is P in place R if the negation of this formula leads to the closure of branch and all that means negation of X is false that means X has to be T if X is T and obviously this is considered to be valid and all so now if X if and only if Y is a tautology then obviously these two are said to be logically identical to each other so now this you treat it as X and this as Y so now we need to apply negation of P implies Q so now we need to apply this particular kind of beta so this is this is X P and Q implies R the first one that is the X part and then not of P implies Q implies R a little bit big and all here that is the first one X and not Y and then not of P and Q implies R that is not X and Y Y is same as brackets needs to be put properly so now we need to further simplify this thing then it will be like this we need to expand this branch and all so once you apply branching rule here so that is this becomes not P and Q and then this becomes R so now this is over now this is the thing so now you further expand this thing it becomes not P not Q and this is as it is so now you apply rule again here not of P implies Q implies R so this is so here what you need to do here is better to use non branching rule first so here we apply first rule for this one so this is P and not of Q implies R so now we apply rule for this one first so always the strategy is that first you open up a formula which leads to non branching kind of formula and all so first you explain this one rather this one so now this becomes this so now this further reduces to Q and not because not of Q implies R is Q not so now you apply this one here so this is not of P and Q and then R so now you see here R and not R is that this branch closes so now this further expands to not P and not Q so now you have P here and all the way down here you have not P this branch closes and you have not Q here and Q here this branch also closes so the left hand side all the branches closes so now you have to see the right hand side of this so now first you open a formula which leads to some kind of non branching non branching rule is the one which need to apply first so this is X and this is why so that is why not of this one leads to non branching kind of rule so that is P and Q and then not R so this is the formula which we have used here this is the one not of P implies Q is P and not Q so not Q instead of not Q you have R here so now this is over so now we apply branching rules it does not matter so now this is not P and Q implies R so now P and Q this P and Q can be written as P and not Q P and Q one after another this is the rule which we use P and Q means you can write P Q as a trunk this reflects the trunk so now you have P here and you are not P here this closes now you expand it further this becomes not Q and R now you are not R here and R here this closes and Q here this is hidden here Q here and not Q this also closes so now none of the branches remains open so that leads to negation of the given well-formed formula leads to the closure of the branch that means what we have showed is this particular kind of thing not of X is false that means X is a tautology so that means these two P and Q implies R and P implies Q implies R are said to be logically identical to each other so we have established that actually this is the thing which we need to use the mistake here so by implies so in this way you can show that two given logical formulas well-formed formulas are said to be logically equivalent to each other or the other way of showing it is that how do we show that a given well-formed formula is a tautology especially when you deny the well-formed formula that means negate the well-formed formula if all the branches close this then that means negation of the given formula X is false there is a contradiction that means X has to be tautology so these are some of the things one can do with the help of semantic tableaux method so you can show that a given formula is a tautology you can show that two groups of statements are consistent to each other are satisfiable or you can even show that two given logical formulas are consistent equivalent to each other etc all these things are considered to be some kind of decision procedure methods so the advantage of this semantic tableaux method is that it conducts a direct search for models the models in a sense that whenever you find an open branch that is considered to be a model that means under these particular kind of assignments a given formula is going to be true for example P and Q is going to be true especially when both are both P and Q are going to be true in all other cases is going to be false so that means when you assign truth values P T QT and that will serve as a come some kind of model all the open paths of the tree that you are seeing in this all this example corresponding to satisfiability of conjunctions of formulas at the node suppose if all the branches closes then it means unsatisfiable so now traditional approaches such as constructing truth table etc it can take 2 to power of n steps for any given n for example n stands for some number of variables that exist in a given formula if n is too big in all that means large the truth table method is difficult for us to handle because the number of repositional variables are large if it is more than 6 it becomes 64 entries are the entries you need to inspect to find out whether a given well-formed formula is valid or not so these are some of the definitions that will be fun we are just discussed in a very informal way about this particular kind of method there are some definitions which will be following while constructing the truth this method semantic tableau method so the first definition is about the path a path of a tree in any stage of construction wherever in the left hand side you will see tree diagram for this particular kind of thing so a path of a tree is a complete column of formulas from top to the bottom of the tree and all for example in this case so this is considered to be one path and this is one path and is going like this and ending like this this is path number one and there is one more path exactly the same thing and is going like this and it ends here another path to that means it starts all the way from the route and ends with some kind of atomic prepositions so that is a risk consider to be the path is already we have discussed in somewhere other but you know we are discussing in explicitly what we mean by path of a tree so this is what is considered to be part definition of a finished part that means the path is no way in which you can progress further that means then there is a conflicting information there is no way in which you can go for why we are not able to go further because whenever you have an inconsistent information you can derive anything so that is the reason why we want to avoid this conflicting kind of information so this is the way in which one can show that suppose if you have a conflicting information like this P and not to begin with you have a conflicting information like this it is raining and it is not raining so now you just state these things like this on it P and not pink so now fourth one not P or this is one simplification and one simplification you will get this one so now you can safely add another kind of proposition without disturbing the truth value of this one if suppose you assume that not P is already true then whatever you add after this one Q is always going to be true enough because of the semantics is like this that and this is obviously true irrespective of whether it is Q is T Q is false this is going to be true only that means P R Q and P R Q so obviously so this becomes false only in this case in all other cases it becomes in all other cases is going to be true so whatever you add after this one formula you retain the same thing in all it is also going to be true so that is why you can safely add any kind of strange kind of propositions suppose if we P stands for it is not raining you can safely add another kind of proposition that fix flies and all that is a strange thing about this particular kind of thing so now what we are trying to show is that whenever you have conflicting information you can derive anything so this is what is called as law of addition this is a truth preserving kind of law which I which you commonly seen in logic one of the important laws of logic so that means you have P and you can easily safely add P R Q without disturbing the truth value of this one so now quickly what you can see is three and four sorry two and four distinctive syllogism will lead to Q so that is the reason why we are not going further and then whenever you have a conflicting information we stop but in the same way so we have proved Q is the case Q is any kind of strange kind of proposition which which is which comes as a consequence of this completing kind of information exactly in the same way you follow some other kind of steps you can even prove not Q also or maybe some other kind of preposition so that means if you start with the contradictions inconsistencies etc. You can derive anything so that is the reason why whenever you have come across a closed branch now you will stop there itself so a path is considered to be finished if it is if the only unchecked formula it contains are only propositional variables and that means there is no way in which you can go beyond it or the negation of the propositional variables so that no more rules can apply on this one no more alpha beta rules can apply on that one so that means a tree is said to be closed or finished if all of its paths are closed and an open path is a path that has no that has been marked with X and a close path the mark with tick mark and the close path is marked with some kind of cross okay. In this class we just introduced semantic tableaux method as one of the important decision procedure method and we have seen with some examples that how the semantic tableaux method can be used to decide you know whether a given well-formed formula is a tautology suppose if you can show that given well-formed formula is a tautology obviously the formula is going to be valid or you can even show that when two groups of statements are consistent to each other that is what you can show and then you can also show with the help of semantic tableaux method is a constructive method that when two given logical formulas are propositional well-formed formulas are going to be logically equivalent to each other so what we will be doing in the next class is that we will be applying the semantic tableaux method particularly in solving some of the important logical puzzles as well as you know once we translate the English language sentences in appropriately into the language of propositional logic then we can see whether a conclusion follows from the premises are not again by using the semantic tableaux method the semantic tableaux method has an edge over the truth tableaux method especially in the sense that when the number of variables are more than 4 or 5 semantic tableaux method is easy to use so that is why it has an edge over the truth tableaux method so it depends upon our convenience which method will be using in the next class we will be seeing some of the important logical puzzles interesting logical puzzles that can be solved with the help of semantic tableaux method.