 Welcome back in the last class we presented semantic tableaux method which is due to three logicians it is originated in the works of Beth and then later it was simplified by Raymond Smollion and we will find the same kind of work in the work of Hintika in his work modal sets. So there seems to be one of the same so semantic tableaux is a very interesting and important method with which you will come to know whether or not a given well-formed formula is valid when two sub statements are consistent to each other or when two statements are two well-formed formulas are logically equivalent to each other etc so in continuation to the last class so we discussed about some rules with which there are all four rules and beta rules and then we have also said that there is some kind of strategy which will be adapting in the process of using this particular method that is this that so whenever you come across a non-branching formula first you have to utilize this thing first and then use branching formulas. So now in continuation to the last class we will be talking about some definitions in the context of the semantic tableaux method. So in the semantic tableaux method so these are some of the definitions that will be following in a way indirectly we discussed all these things but in a more formal way we will be defining these terms the first and foremost important thing is what we call it as path of a tree. So what we are simply trying to do is given a well-formed formula we are trying to construct it upside down kind of tree and then we are trying to see whether or not a given well-formed formula is valid or consistent when two groups of formulas are consistent etc. So a path of a tree is a complete column of formulas from top to bottom of the tree so then that is considered to be the path of the tree for example if you have this particular kind of thing this is the root this is the upside down kind of tree a tree will be looking like this these are all branches and this is a trunk but in our case it will be like this so that it is an upside down kind of tree. So now suppose if you have a formula like this and then this and then this leads to this branch leads to three more branches like this and for example this leads to this. So now we have a given well-formed formula here and then this is considered to be the root that is where this your well-formed formula will be sitting and then it is reduced reduced into some kind of atomic variables PQs are which cannot be further reduced so that is why these are called as atomic propositions at the end of this branch this path you will end up with only atomic sentences. So now in this tree diagram this is considered to be one path and there is one more path it is going like this and there is one more path this is the third path like that you know this is considered to be a path of this particular kind of trick once you construct a tree for a given well-formed formula and that is going to be the path this is a complete column of all the formulas from top to bottom top is the root and the bottom is usually ends up with some kind of atomic sentences because atomic sentences you cannot apply any you cannot further apply any rules so that is why it remains at the end of the node as atomic sentences once you come across atomic sentences you will stop constructing the tree. So the other definition is the finished path that means a closed path a path which is considered to be finished if it is said to be closed especially if it is closed when a prepositional variables or its negation of the variables exist in the branch. So if x and not x exists the branch closes or the other way the branch remains it cannot be further extended is this that if it ends with some kind of atomic variables atomic sentences p q r s etc then no further rules can be applied on this p s q s r s etc. So that is why the path ends there itself so you will be working you will be using alpha and beta rules till to such an extent that you will end up with only atomic prepositions either that is a case or you may come across some kind of conflicting information that is it is considered to be a literal and its negation exists then usually you put a mark like this that means the branch closes here. So when the conflicting information exists the branch closes or when you are ending up with atomic prepositions there are no further rules which you can apply that means the tree cannot be extended further it stops there itself. So that means a tree is said to be closed or finished if all its paths are closed that means you will be checking all the formulas you will start all the well form formulas 1 2 3 etc and all you keep checking those formulas using alpha beta rules and constructed tree if all the formulas are checked and then you will end up with only atomic prepositions then there is no nothing else you can do I think you cannot extend the tree further it stops there itself because you have exhausted all the rules and other things and you ended up with atomic sentences and that is considered to be finished part or the other way of saying that it is a finished path is that when you have a conflicting information once you check all the well form formulas then also it is considered to be a finished part. So now an open path is a path that has not been ended with that mark X or a close path is a one path that has been ended with an X so this means this particular kind of thing. So suppose if you have some formula PRQ and Q implies for example so now you start constructing applying alpha and beta rules first you apply on this one so this is Q R and then you apply since this is checked so you put tick mark here and then so what you will be doing is you will be checking this formula and then this is PRQ and now each branch you need to write this information so now so all these branches are open so there is no conflicting information so this whenever you have a conflicting information Q and not Q you put this mark X mark that means is this branch cannot be further extended it closes here itself so this is considered to be a closed path so now we have exhausted all the rules and all so that means the final formulas that exist in your tree are going to be only atomic prepositions so this is one thing which will be so we are defining what we mean by closed path and open path etc and all so whenever you do not come across the mark X it is considered to be open path whenever it is whenever you come across mark X that means there is a conflicting information you mark it with X so these are some of the definitions so these are some of the definitions that we will be using so another definition is this that a formula occurs on a path if it is on the path and it is not merely a sub formula of some other formula on that path or second it is unchecked so if you once you check the formulas and all it goes it gets exhausted so it cannot be further used so that means either the formula should be unchecked or if you if it is checked and you should end up with only atomic prepositions and here is an important strategy which one will be using that strategy is that first you apply non branching rules before the branching rules one example could be for example if you have a formula P and Q and you have a formula P implies Q let us say P or Q implies R for example so now first you need to apply non branching rule so non branching rule can be like this so these are some of the rules alpha and beta rules the branching non branching rules are like this that means the formula does not lead to some kind of branch so this is a non branching rule or this is also another kind of non branching rule suppose if not of P implies Q is simply P and not Q so usually these two are considered to be a non branching rules another one is if you have a negation of negation of P and you will get so now here the strategy here is to use this non branching rules first either you can use this or that suppose if you open it open this first it leads to so many branches then it gives us excessive information and also better to use non branching rules first you expand this one so P and Q is simplifies to P Q so this is one simplification etc one simplification and then again we are not supposed to use this one because it leads to branch it is PR Q in place R so better exhaust this particular kind of thing so now this is P and not Q so this is a strategy one adopts in this particular kind of technique first you use non branching rules I mean those formulas that doesn't lead to branch are the ones which needs to be taken into consideration so now as you clearly see here so now you have Q and not Q and all this process here itself so now it doesn't matter whether or not you use this particular kind of formula PR Q implies R so the branch closes here itself so this is considered to be a proof for particular thing that these three statements are inconsistent to each other are inconsistent to each other so that's why all the branches closes it doesn't matter what formula is there in the third kind of thing for however suppose if you have used branching rules first here is the problem which comes to the same formula which we have taken into consideration so this is the correct way of applying this thing non branching rules first first we have used non branching rule so now instead of that we have used branching rule first then here is the problem the problem is like this now instead of expanding these two formulas instead of checking these formulas first you are trying to check this particular kind of formula so now this leads to P Q in place R so now we can use any other formula and all P and Q so now this is the one which we have used both are checked so now this needs to be expanded so now P and Q needs to be added on both sides so now you check these two formulas now you yourself will see that when you use branching rules first although you are you can use a non branching rules here this leads to the problem the problem is is that first of all the whatever you are trying to show will have more number of steps or if you are suppose if you are showing this if you are showing the validity of a given formula it involves more steps so now this ended in seven steps itself with seven still you could say that these three statements are inconsistent to each other but here they are already four five six and then you have seven here and there will be some more so now you need to write this one also P and not so now here it is P and not Q P and not so now here you have eight nine or something so instead of seven steps you have nine steps and all so that is the reason why so we will always be using non branching rules first when compared to this branching kind of rules in a semantic tableaux tree this is a very important strategy which comes through practice is a convention which logicians follow so always whenever you have a non branching formula better to open it up rather than a branching kind of formula because it involves more number of steps now so in fact this semantic tableaux method can also be used as some kind of proof method proof procedure kind of method a proof is a one which consists of finite number of steps and which ends in finite intervals of interval of time no proof can be considered to be an effective proof if it never ends in it goes on and on and on and on so a proof has to end in finite steps of course it should take a finite number of finite amount of time then it is considered to be an effective kind of proof but you know once you start with non branching rules of course you will get the answer and all by the number of steps will be more informational economy cannot be maintained if you use branching rules first over non branching rules that is the reason why we follow this particular kind of strategy that is always apply non branching rules before the branching rules. So now we defined all these things in the context of truth table method for example a formula is valid especially when you have true premises and you do not find a false conclusion as long as you do not find a true premises in a false conclusion then obviously the formula is going to be valid so you have to inspect in the truth table all the rows and if there is any row in which you have premises are true and the conclusion is false then the argument is invalid and in the same way a formula is said to be satisfiable especially when you need to inspect at least one row in which your premises at least in one of the under the main logical connective at least you have one T if all if there are all T's and all then it is said to be unsatisfiable the group of statements are unsatisfiable or inconsistent. So now the same things which we will define in the context of semantic tableaux method so that is first if you want to determine whether a formula is valid one needs to construct a tree using all the alpha beta rules that we have been discussing so far first what we will be doing is you take the premises into consideration and the negation of the conclusion if all the paths close this and the formula is considered to be valid if it is not then it is going to be invalid some examples which we take into consideration and then we will see when a given formula I mean when the conclusion follows from the premises and all that means the validity some simple examples can be like this P implies Q implies R let us consider this one Q and from that you got Q implies R so these are considered to be premises and this is separated by an iPhone this is considered to be the conclusion. So now in the semantic tableaux method in order to show that Q implies R follows from P implies Q implies R and P what you will be doing is first you deny the conclusion so that means you need to write here denial of conclusion once you deny the conclusion now you will be constructing the tree diagram for this thing and then you need to see whether all the branches closes are not so now as usual we follow we apply non-branching rule first so that means you need to open up this one so this is Q and not R so this is 3 then simplifies it gets simplified and then you will get Q and not so now we check this formula so that is why we had put this mark so now we open up this thing this is an atomic sentence you need not have to do much so now we need to open up this one P implies Q implies R so now this is a branch because you have a formula X implies Y X implies Y the construction tree for this one is not X and Y so now this is not P and Q implies R is as it is so now you have P here and not P here this branch closes so now we need to expand this thing a little bit further so this will Q implies R will become again this rule use X implies Y is not X Y so now this becomes Q implies R becomes this one not Q R so now you have Q and not Q is a conflicting information this branch closes and you have R here and you have not R here results branch also closes so what is that we got simply this that negation of the conclusion leads to branch closure that means it is unsatisfiable that means leads to a contradiction I mean all the branches closes then negation of X is this one then X has to be T where this stands for board is always false the formulas which are always false and this T this is totally different from the formula that we use small T so this is this stands for truth of course this is also considered to be true but it is always true in all so this is the symbol is represented as top so that means X is a tautology X is a tautology that is what we have said that means the actual conclusion is the one which stands because we negated the conclusion and led to contradiction in all so that's why we have to retain the original conclusion that is Q implies R so negation of Q implies R leads to contradiction that means Q implies R has to be true Q implies R is the true conclusion of these two premises so it is in this way one can show a given form when a conclusion follows from the premises that means the validity for validity what you need to do simply is that you take the negation of the conclusion and see whether all the branches closes if all the branches closes then the negation of the conclusion is false that means the actual conclusion stands as it is so if the negation of the conclusion doesn't lead to a branch closure then that means there are at least some kind of interpretations which satisfies your truth premises are true and the conclusion is false that means you already constructed a counter model so that is the reason why this semantic tablox method one of the essence of this semantic tablox method is to look for some kind of counter models here we could not come up with any counter model that's why the argument is valid so the another way of showing a particular kind of argument is valid or invalid is simply this thing suppose if there are two formulas a and b and b is a logical consequence of a or this a implies a logically implies b as valid it is sufficient to show that a and not b is unsatisfiable so if you want to say that for example so this is the one which we want to see so now let us take formula a as p and q and then b as something like not p or not p not something like there is two formulas on the one hand you have this and you have this one so now if you want to show that this is a logical consequence of this one what you need to do is first you need to write like this not p or not q and then you construct a tree for this one and then this becomes not p or not q and in all the branches it becomes it the branch closes you know but that is not what we are trying to say so what we are trying to show is this particular kind of thing so p and q and not of not p or not q if this remains unsatisfiable then so this is a logical consequence of this one but actually that is not the case so now you expand this one it becomes p and q not not p is p not not q is q and negation of disjunction is conjunction so now this branch is open it satisfies this particular kind of thing that means this is not a once again p and not q has to be unsatisfiable for showing that this particular kind of argument is valid otherwise it is considered to be invalid so that is another way of showing that whether or not a given formula is valid or invalid so what we are trying to do is a logically implies be that means B is a logical consequence of a so a is considered to be premises and B is considered to be conclusion suppose if you can come across at least two premises and a false conclusion and obviously the argument is invalid so another thing which you can do with the help of for the semantic tableaux method is you can show whether two sets of formulas are two given formulas are consistent to each other or not for that what you need to do is you list out all the formulas and then constructed tree when at least one branch is open then it is considered to be consistent if all the branches closes then it is considered to be inconsistent and there is another way of seeing that a given formula is a tautology a formula is considered to be tautology that means always true if and only if not a is unsatisfiable so that is this particular kind of thing so what we are trying to say here is this thing so you take any formula P implies Q implies this is what is the formula which is given to us so now according to the definition if you want to show that this particular formula is a tautology what you need to show is this thing not of X is unsatisfiable unsatisfiable in a sense that if take the negation of this one all branches should close so now that is what we are trying to see not of X is this one P implies Q implies P so this is denial of original well-formed formula the given well-formed formula so now this has to be unsatisfiable that means all the branches should close after applying alpha beta rules so now this is one formula this is another formula X and Y so not of X implies Y is X and not so now you construct a tree for this one this becomes Q implies P so now this further reduces to Q and not so now you have P here and not P here conflicting information the branch closes here so that means negation of the formula leads to the branch closure all the branches closes here is only one branch here is apart this. So this branch closes so that means it is considered to be unsatisfiable at least one branch is open in the construction of your tree of a given formula then it is said to be satisfiable but in all the branches only one branch here that causes here so that means not of X is said to be unsatisfiable if not of X is considered to be unsatisfiable then obviously the given formula is considered to be tautology and you need to note that all tautologies are obviously valid formulas, so this is the relation between satisfiability and validity or means the tautology, so something is considered to be a tautology only when the negation of the formula is considered to be unsatisfiable, so if the negation of X is considered to be satisfiable then it is not considered to be a tautology first of all it may be contingent statement or it can be even contradiction as well, so we cannot say that a given formula is considered to be valid, so based on the information that negation of the given formula is satisfied, so we have to ensure that not A is unsatisfiable then only you can say that A is a kind of valid formula or it is all valid formulas are obviously tautologies, so the other thing which you can do with the help of semantic tableaux method is this thing contingency, so we have defined statements into statements of propositional logic into three categories tautologies which are always true, contradictions which are always false and contingent statement which can be sometimes true sometimes false, so for this we need to construct two different trees one is to test whether test for the consistency another one is to test for the validity, so if the formula is consistent but not valid then it is said to be contingent, for example if you have a formula like this formula like this thing for example P implies Q R not P or something like that, so now this is the formula that we have, so now what you do is you negate this formula and then see whether all the branches close that means not of X is unsatisfiable that is what we are trying to show, so now this is if you expand this thing use alpha beta rules and all beta rule which you need to apply and this becomes Q R not P, so now this is not Q and not P, so this closes that means it is considered to be a tautology and all, so now let us take another example where the branches does not close, so now this example could be P or Q implies R implies something like yes some formula you take into consideration whatever comes to my mind that I am writing it on this thing, so now we want to see whether it is a contingent or tautology or contradiction, so for this again you deny the original formula and start constructing a tree, so now you write the same thing here brackets needs to be written clearly, so this is one formula is another formula, so now this can be written as P implies Q implies R not S because not of X implies Y is X and not Y, so now you further expand this thing then this becomes P R Q and R, so this is the one which this further simplifies to not P not Q and you will observe that all the branches remains open that means negation of the given formula does not lead to branch closer that means not X is not unsatisfiable, so that means definitely it is not a tautology this particular formula is not a tautology, so since if at least one branch is remains open, so then it is considered to be a kind of contingent kind of state, so we need to ensure that this is not unsatisfiable if it is once it is not unsatisfiable then we can clearly say that it is can be called as a contingent kind of statement, so for contingency what you need to do is you have to construct two different trees one is to test for the consistency and the other one is to test the validity, so the one which we have mentioned it here is on the left hand side of the board is we checked for the validity of a given formula, so that is one test which we are trying to do definitely it is considered to be an invalid kind of formula, so now for contingency there is another thing which we need to follow so that is now here we showed that it is not X is not unsatisfiable that means it is this formula is not a tautology, but and obviously it is invalid, so this is not enough because there it can be a contradiction it can be even contingent statement, so now what you need to do here is instead of negating this formula you need to check this formula for the consistency for consistency what you will be doing is you do not take negation into consideration just you leave the formula as it is and then you start constructing a tree for this one, so now this becomes like this is X this is why so not X and Y, so now this also changes, so now this changes to this is not PQ, now this further simplifies to this one, so and this is the whole thing which we need to take into consideration negation of this one and yes, so now negation of PRQ implies R is this thing PRQ and not R, so now PRQ simplifies to this one, so now all the formulas are open I mean all the branches are open that means it is considered to be consistent kind of formula, so this formula is definitely not valid it is not a tautology, but definitely it is not a contradiction also because at least all the branches are open enough, so now for example in the process of checking the consistency you came across all the close branches and all then this formula is going to be a contradiction, so the idea here is that for contingency you need to construct two different trees, first you need to check the validity of a given formula that tells us whether a given formula is a tautology or not, but it does not tell us about whether it is a contradiction or contingent kind of state, for contingency and contradiction you need to go for another test that is the test for the consistency, for consistency you do not deny the conclusion, but you just leave the formula as it is and you construct the tree as in the case that is explained on the left hand side of the board. So suppose imagine a situation where in which all the branches closes that means the formula is said to be inconsistent, so if it is inconsistent obviously that formula is going to be a contradiction, but we did not come across that particular kind of situation, at least one branch is open that is considered to be satisfiable. So this formula the one which we have written on the board PRQ implies R implies S is considered to be a contingent kind of formula, it is not a contradiction because in the process of checking the consistency the branch does not closes all the branches does not close, so it is not a contradiction and for the tautology we already checked it in the beginning that you know negation of the formula leads to the closure of branch all the branches that means not X is unsatisfiable that also we did not get it, so it is not a tautology and not a contradiction, so it is considered to be a contingent kind of formula. So here that is the way in which you can test you can check the contingency of the given formula, you need to construct two different trees, so now here are some of the interesting and important theorems which will be using it further when we talk about something on meta logic that means theorems we are discussing about some important theorems later, so one of the interesting theorems is this thing we are not going into the details of proving this theorems but we will just highlighting one of the important theorems which will be making use of it later in another context I will be explaining these theorems in greater detail. So a completed semantic tableau for a given formula A is said to be closed if and only if A is said to be unsatisfiable, so you take the negation of the formula not X and then it leads to the closure of all the branches then obviously not X is considered to be unsatisfiable, if not X is going to be unsatisfiable then obviously X has to be a tautology, so that is one thing which we have observed it already and the second most important thing is the soundness theorem in the context of semantic tableau method, so in the context of semantic tableau method soundness is like this, if the tableau is closed then obviously A is said to be unsatisfiable that means you have a formula X and you construct a tableau for that and if all the branches closes then A is said to be unsatisfiable. So usually is the case that soundness relates there are few things which is important in propositional logic that is in the we have not discussed in detail about something about truth theory we will be talking about it in the next few classes, so if something is considered to be true then if it is also provable or something is provable and it is also true then it is something is provable and it is true it is called as soundness and then if something is already true then it has to find a proof and that is considered to be completeness in all whatever is provable is true and whatever it is true has to be provable and interestingly in the propositional logic in both the cases it happens all the true formulas are all provable and all the provable formulas obviously at the end of the day has to be true you proved lots of things but at the end of the day it is false and it does not make any sense to us that means all provable formulas are true and all true formulas are provable and propositional logic in this sense is considered to be complete completeness says that if a well-formed formula is unsatisfiable then any tableau for A is obviously closed you can show that all the branches closes and one of the important corollaries of these two theorems which will be explaining it little bit later just we are just highlighting what we mean by these theorems and all a well-formed formula a is considered to be satisfiable formula if and only if any tableau for a is open if at least one branch is open it is considered to be satisfiable and in the same way corollary 2 is this that a well-formed formula a is said to be a valid formula if and only if the tableau for not a consider tableau for not a any so happens that all the branches closes that is the case then it is considered to be a valid formula. So soundness in the context of tableau method is simply like this if alpha is tableau provable then obviously alpha is considered to be a tautology what is tableau provable so you have a given formula x and you negate the formula and it leads to all the closure of all the branches that means not x is false that means x has to be true. So if alpha is tableau provable then obviously alpha has to be that given formula has to be a tautology if something is provable and it is true then it is called as a sound soundness. So the tableau method is also considered to be consistent in a sense that while proving certain theorems using tableau method it never happens that you come across a you prove both alpha and not alpha. So either it is to be the case that you have to prove only alpha or it has to be the case that you have to prove only not alpha. So now the soundness of tableau method is like this if alpha is provable in the natural reduction system which we will be talking about a little bit later then alpha is also considered to be tautology natural reduction method is the one which in which you will find this proofs of some given formulas and all the same thing is you can prove it with the help of semantic tableau method as well. So if something is provable then it has to be true and something is true it has to be provable then the system is considered to be complete. So far we discussed about semantic tableau method and then we talked about when given formula is valid consistent satisfiable and all these things. So now where we will apply this semantic tableau method. So one of the important things which we will be making use of this semantic tableau method is solving some kind of puzzles. So here are some of the interesting puzzles which are cooked up by Raymond Smollion and Raymond Smollion has come up with the various books all his books are quite interesting it includes lots of puzzles. One of the interesting books are like this the title of the book is what is the name of the book that is the digital is considered with the title of the book and the other book is Lady or Tiger and lots of other books where he discussed all these puzzles and he tries to solve these things using the principles basic principles of logic. A few puzzles which we will take up in this class and then we will end this lecture and all. So here is an interesting puzzle which is called as Nights and Naves puzzles this puzzles can be solved by using semantic tableau method it can be solved by using truth table method. So the puzzle is puzzle goes like this a story behind the puzzle is like this. So on some island there are two inhibitors one always speaks truths they are considered to be nights and there are some other kinds of inhibitors where they always lies for example if you ask particular kind of inhibitor is 2 plus 2 is equal to 4 then if he is a night you will tell the answer is yes if he is a name you will tell that is 2 plus 2 is equal to 4 if he is a name you will answer that no that means he always lies whatever truths are there you will always say it is not the case. So now that is a particular kind of island that means cleverly designed in such a way that they speak only true and false everything is crystal clear black and white either something a sentence is either true or false so that is an island where you will be going. So now you went to that particular island is all an imaginary kind of situation so all stories but a lot of things can be done with the help of this particular kind of things. So now you meet two islanders let us call them A and B now you hear the first one saying that at least one of us is a new. So now can you tell whether the islanders are nights or names based on whatever information that the person is trying to give. So what is happening here is that you are a stranger you visited that particular kind of island and then you are trying to question this you are trying to ask some questions so that you will get definitely SR no kind of answers and with the help of those answers which you are trying to judge whether their nights are news. So for solving this kind of nights and news problems the first thing which you need to note is some kind of notation. So what we are discussing is nights and news puzzles using either truth table or semantic tableau some simple problems which will be considering and then we will move on to some kind of difficult kind of problems. So now so this is an island so there are only two kinds of inhabitants A and B. So now suppose if A is a night then you represented it like this only so A is night in the same way suppose if you write simply letter B that means B is night suppose if you write like this not A that means A is a name and not B means B is a name. So now so there are some particular kind of problems the problem here is this thing so now you are a stranger you know you visited this island now you are trying to know what they are. So you ask some questions and all this is what they tell you hear the first one says at least one of us is a name the first one is a so now what is the thing A says at least one of us is a name this is the information that A gives so now you are a stranger you went to this particular kind of island now you are trying to decide whatever answers that they give you are trying to decide what type they are you do not know whether A is a night or B is a night and all but based on the information that they give you you will be drawing some kind of conclusion. So these are some kind of reasoning problems which you can solve it with the various number of numerous methods and all but since we have studied semantic tableaux method and truth table method in greater detail so we will be talking about this particular kind of method. So now A says X some X that means you will be saying one of us is an A or some other kind of thing so this can be represented as a by kind by implication and this is represented as A if in only if X so now the first problem that we will be solving is this particular kind of thing you went to an island and then you came across two inhabitants instead of both are talking the first person A is saying that at least one of us is a name so from that what you can judge about A and B so now this at least one of us is a new can be translated as it is usually translated as inclusive so that is P or Q suppose if they have said exactly one of us is a name then it is considered to be an exclusive R that is either A has to be the case of B has to be the case but not both of them that gives us information about exactly one of us is a name but usually at least one of us is a name is usually translated as inclusive R that is PR Q so now this formula reduces to this particular kind of thing so now what it says at least one of us is a new so that means A is saying that is a if and only if a this one it says that at least one of us is new that means either A has to be new or B has to be a name so now we need to construct a semantic tableaux for this particular kind of thing then we can come to know what is A and what is B so what is what is happening here you went to a strange island you ask them they do not tell about anything but they will say this particular kind of thing at least one of us is a name from that you need to judge what they are so now this is the information that A is trying to give at least one of us is a name so now you need to see when this formula is going to be satisfiable that is going to give us the answer for this particular kind of thing whether A is an A or B is an 8 etc all this information is hidden in this one so now you construct a semantic tableaux for this one for example if you have a formula x implies y so this is either x y is a case x and y is a case or not x and not so this is a tree for this particular kind of thing so now you draw semantic tableaux for this one this is a and not a or B not B and then the second one is not a not of not a or not be so now you further expand it and this becomes not a and not be and here so not of not a is a not of negation of disjunction is conjunction that is why you write it just below this one and now this becomes B so now we need to see whether there is any conflicting information in the branch so now you have a here and not a here this branch closes and this branch remains open now you have a here and not a here this branch closes so now when this formula is going to be satisfiable especially when you need to inspect the open branch the open branch is only this one in this open branch the information that we have is a and not be so you have a here and not be here that satisfies this particular kind of formula when a is tree and B is false and this makes the whole formula true and this is the one which we are looking for and according to our original interpretation if you write only a and all that means a is a night and then if you write not be then B has to be a new so the solution of this one is this thing a is a night and B is a new so this is the solution for this particular kind of problem so when somebody tells her at least one of us is a name then this has to be the solution suppose if you had said that exactly one of us is a name and all that means one is ruling out the other possibility and all so then you need to write in a different way so this will become like this so not only this is the case and all exactly one of the things are nights or and it is not the case that both are nails and not a and not be so this also you need to take into consideration that means one excludes the other possibility then you need to draw the semantic tableaux for this one and then you can see whether or not the open branch is the one which you need to inspect then you can see the corresponding answer for this particular kind of problem so now let us consider some more examples of this sort and we will see what can be done with this particular kind of thing so now let us consider that this is the one which will be considering so this is the one which can be solved with the help of truth table method also so this is like this you have P Q and not P or Q is stands for at least one of us is a name and then P implies not P or not Q is the one which you need to see because P is saying this particular kind of statement so now at least it should be by implication and that is the correct one but you have to inspect a row in which this formula is satisfiable that means the second row is the one which satisfies this particular kind of formula that means P has to be T and Q has to be F that satisfies this particular kind of formula so using truth table method also you can solve this particular kind of problem so let us consider some more examples and then we will see what is the situation so these are some of the notations that we will be following in solving this particular kind of puzzles so you meet two kinds of people A and B suppose if A says I am an A but B isn't let us consider that particular kind of thing so A is saying A is saying this particular kind of thing I am a name but B is not I am a name but P is not that means B is a night so now from this information what is the one which you are trying to get so now again you construct a semantic tableau method using semantic tableau method you construct a tree for this one first it is A not B and then this is not of not A and B so now this is this branch closes here itself because not A and B because A and not A it closes here itself and now this can be expanded to this this is A not not A is A and then this is not so now this branch also closes again this is the information that we have so A is a name and B is also considered to be name and all so what if you says this particular kind of thing that I am a name but B is not the case so then it has to be the thing that both of them are names suppose this in the second problem suppose if A says if I am a night so is B then can we determine what are this thing again the same thing which will be using so with this I think will end this class so the other one is like this if A says the second problem if it is like this A says means A if only if this thing if I am a night that means A is a night then so is B that means B is also night and all so this is the formula we can translate that statement into this particular kind of formula A implies A implies single implication B so now again you construct semantic tableau method using semantic tableau method you construct a tree for this one and this becomes a plus B and then not a exactly in the negation of this one so now this further expands to not A and B because A and not A this branch closes and then this branch remains open now this is A and not B so this branch closes now open branches are the ones which you need to inspect that means you have A here that means A is a night and B is a night that means suppose if you went to an island a strange island and you ask them a particular kind of inhabitant replies by saying that if I am a night then so is B from that information if that has to be translated into appropriate language of propositional logic then this is the formula with which it can be translated into and then you constructed a tree and then you are observing the open branches and then the open branches corresponds to the answer that is A has to be night and B has to be a night to satisfy this particular kind of statement is statement so in this way suppose if you are making some kind of sample statistical survey that how many number of people are nights how many number of people are news etc then you need to translate the given formula in appropriately into the language of propositional logic and then you constructed semantic tablox method and then you can see whether or not whether they are nights or naves. So let us consider one last example and then here we have three inhabitants ABC one of them each of whom it can be a night it can be never it can be it cannot be both and all and other important thing which you need to note is that a layer cannot tell truths and all if that is the case then it leads to a big problem which is called as last paradox so that thing which we will try to avoid a layer cannot tell truths a lawyer always lies so here is the information that we have you asked A and B and then is talking about a particular kind of thing is saying B is a name. So now this particular kind of thing which I will describe it and then maybe you can solve it in your free time so now a says that B is a name that is the first sentence that we have in this problem and now B says A and C are of same type A and C are of same type who is saying this thing B is saying that A and C are of same type either it should be this particular kind of thing or you can even take into consideration B as not A and not C so this satisfies that both are nights that means they are of the same type and the other one satisfies this thing not A and not C you can take that also into consideration but not both of them so they are of same type and all so this satisfies this particular kind of formula now we are trying to determine what is CN quickly we can draw a semantic tab box method for this one A and not B and then this is not A and B not of not B is B is the first formula that we checked it so now the second formula is this thing B and A and C and not B not of A and C same information which you write it here that is B A and C and not B not of A and C sometimes the branch closes even before itself you need not have to go do anything in all so this B and not B here is branch closes here itself and now this is not A and not C since A and not A is a this all branch also closes so now this A and C simplifies to this one since A and not A is a this branch also closes so now B and not B so what is the problem here one second so there is some problem with the representation of this one so we will talk about it in the next class so there is the translation seems doesn't seem to be correct at all so we need to translate it properly so then we will get the answer so in this class what we have done is we have talked about some of the definitions of semantic tab box method some of the definitions in the context of semantic tab box method and then we have applied the semantic tab box method in solving some of the interesting puzzles that is these are the puzzles which are cooped up by Raymond's mullion these are all called as knights and nails puzzles in knights and nails puzzles what we will be doing is you are a stranger you went to the island and then you ask them what type of inevitability are you so suppose if they reply that based on the reply that you get from them and you are trying to convert those information into appropriately into the language of propositional logic and then we are judging what kind of what kind of type he is suppose if they say both are nails both are knights etc and all and based on that information you translated into the language of propositional logic and then we are trying to handle it properly with the help of propositional logic in the next class we will be seeing some more examples in the context of semantic tab box method and what we will be doing is we will be translating some of the English language sentences appropriately into the language of propositional logic and then we will see whether the argument follows or not.