 Welcome back in the last lecture we have seen the semantics of propositional logic where we introduced a very simplistic method that is called as truth table method. So today what we are going to see is we are going to study this truth functional connectives that is not and are implies and double implies in greater detail and then once we construct the truth table we can come to know many things such as when two groups of statements in propositional logic are said to be consistent when a conclusion follows from the premises that means validity takes care of that particular thing and when a particular formula given well form formula is a valid formula or when a given propositional formula is a tautology contradiction or contingent sentence all these things one can come to know with this particular decision procedure method which is called as truth table method. So this is a decision procedure method because given a well form formula we can check whether that particular given well form formula is a valid formula contingent formula tautology etc. So this is the most simplistic method with which we begin with and then we will move on to some other kind of decision procedure methods such as semantic tab-lux method resolution refutation method if time permits we will go into the details of it in the next forthcoming lectures and then there are other methods such as given a well form formula one can reduce the formula into conjunctive and disjunctive normal forms and then one can talk about whether that particular given formula is a tautology or not. So once we find out that a particular formula is a tautology then we can say that it is a valid formula so logicians task is to identify what are the tautologies because all tautologies in propositional logics propositional logic are obviously valid formulas in all. So to begin with in this lecture what we will be doing is we will be talking about some of the basic definitions such as consistency validity logical consequence when two formulas are logically equivalent to each other etc using truth table method. So before that we have defined a truth functional connective in this sense a truth functional connective is a one in which the truth value of a compound sentence is solely determined by the truth value of its individual constituents suppose if you want to determine the truth value of PRQ so the truth value of PRQ the compound sentence is solely determined by whatever values PQ etc the propositional variables takes in that given well-formed formula. So there is no other thing which comes into picture in determining the truth value of that particular formula PRQ but in day to day practice we need to go beyond this truth functional connectives and all then we need to talk about some other things such as for example one example we already studied in some detail so that is I went to I became sick and I went to the doctor and the second one is I went to the doctor and I became sick I became sick these two statements are totally different one is there is a temporal order present in this particular kind of thing this example so went to the doctor and then followed by that I mean you became sick that is that is what is following in the second statement in the first statement you became sick and that is why you went to the doctor is these two statements mean two different things and so if the truth value of a compound statement or sentence is not solely determined by the truth value of its individual constituents then the usage of the truth functional connective we call it as non-truth functional usage of that particular kind of connective so there are many non-truth functional connectives which we come across not in this course but in advanced courses in logic especially in modal logic etc and all there are some operators which are considered to be intentional if the truth value of a compound statement is solely determined by the truth value of its constituents then the truth functional connective is considered to be extension if it is not determined solely by its individual constituents then it is called as intentional so there are many operators such as I believe that P I for example it is possible that P it is necessary that P etc all these things are some of them are considered to be modal operators which are considered to be intentional so the truth value of I believe that P is not solely determined by whatever value P takes in there are many things which I believe them to be true but it may not be actually true enough I believe that God exists to be true but actually we do not know whether God exists are not it may be the case that what exists may not be may the case that God does not exist and all so I believe that there are so many things that ghost exist and all that may be true to someone or may be others it may be false enough so the truth value of a of that particular kind of statement is not solely determined by whatever value that the individual components takes and all in that sense it is intentional which is beyond the scope of our study so we will be studying about only extensional operations operators such as not implies and are etc. Now these are called as truth functional connectives so this is the one which we came up with the negation is the simplistic one when the preposition is true then the negation of P is obviously false and P is false obviously not P is true and all it is raining that means the opposite of that one is it is not raining so in the same way you can define other connectives in this way I do not want to go into the details of all these things which we studied earlier the first to begin with the conjunction conjunction becomes true only when both the constituents are true whereas in all other cases it is going to be false enough distinction on the other hand it can be used in two different senses and all the one which is shown in the red is used for exclusive or the one which is used in the black color is meant for inclusive or in the case of inclusive or P or Q that is a disjunction it can be also called as a disjunction or you can call it as either P or Q etc and all etc that is going to be false only when both are both constituents are false in all other cases it is going to be true in the case of conditional a little bit little bit tricky so a conditional will become false only when when the antecedent that is P is true and the consequent Q here is false that means when P is T Q is false then P ? Q is false in all other cases it is going to be true so this is quite difficult for us to accept but it works for mathematical reasoning especially when the antecedent is false the consequent is false then also the conditional is going to be true and all for example if I say if Tajmal is in Andhra Pradesh then Uttar Pradesh is in Uttar Pradesh is in Pakistan for example if you say that thing both both statements are false so even then in that case the conditional total conditional the truth value of the conditional is going to be true because both P and Q are false so that is what is the semantics of material implication so it works perfectly all right for a mathematical reasoning for example simple examples can be 2 plus 2 is equal to 4 then 3 plus 2 is equal to 5 so in this case the conditional is going to be true if 2 plus 2 is equal to 5 then 3 plus 2 is equal to 6 if you say that thing both P and Q are false but it the conditional is going to be true so this is what we mean by material implication it is defined simply as P ? Q is nothing but not P or Q and the by implication which is especially used for invoking necessary and sufficient conditions or it can invoke some kind of equivalence relation so that is simply like this so when both when P is T Q is false the conditional P if and only if Q is false or if P is false Q is for Q is true then P if and only if Q is false in all other cases it is going to be true so this is what how we define the connectives and all this is the way the connectives behaves in behaves so what are we going to do with this truth tables so in what way it is they are going to be useful to us so truth tables are used especially to determine first whether the preposition is a logical truth or a logical false so that means it is a tautology or a contradiction so we will look into some examples and show that when a given formula with simple examples we show that a given formula is a tautology and given formula is a contradiction and it can also be used to determine whether a set of sentences are satisfiable that is whether the sentences can be simultaneously true or not that is also we can determine with the help of truth table we can also determine whether two prepositions are logically equivalent to each other for example if you say P implies Q it is logically equivalent to not PRQ so how do we know that these two are logically equivalent equivalent so there is a decision procedure method with which you can say that these two are logically identical that is what we are trying to do in the next few minutes using truth table we can find out when two given formulas are logically equivalent to each other we can also determine whether one preposition follows from another after the logic is all about what follows from what so that is validity is a concept which takes care of what follows from what so usually the conclusion follows from the premises so that means we can also determine the validity of a given argument using truth table method so we will consider few examples and then see how we can determine whether to begin with first we will talk about whether a given logical formula well form formula is a tautology contradiction or contingent kind of state so let us consider some examples for proving there are three kinds of statements which exist in propositional logic so they are tautologies one the formulas which are obviously true always true and there are some other formulas which are called as contradictions and there are some formulas which are sometimes true and sometimes false in all which are considered under the category of contingent statement the first foremost thing which we need to do is to identify what kind of formula it is whether it is a tautology or is a contradiction or is it a contingent kind of state so let us consider some example suppose if you have a formula like this so now we want to check whether this particular kind of formula is a tautology or contradiction or contingent statement using truth table method so now there are two variables here p and q so that is why 2 to the power of n possibilities entries will be possible in a truth table that means four entries are possible in a given truth table so now what we need to do here is this thing so this is considered to be a minor logical connective because it connects only two variables now this is considered to be a major logical connective so so under the main logical connective the idea is this that under the once you identify the major logical connective under this main logical connective if you get only t 6 and all it is all t's only then it is called as tautology formula which is always true whatever values you assign to p and q is always going to be true and all if that is a case it is a tautology and if you have only f's and all under the main logical connective then it is a contradiction and in the truth table under the main logical connective you have t's f or t f etc 1 t 1 f 1 t all f's and all then also it is called as contingent a formula so now how to check whether the given formula that we have taken is a tautology or not so for this we need to construct the truth table so for this you know you have to the variables that you need to write p and q so these are the only two variables that we have and then the first one which you have to take into consideration is this one because it is a sub formula of the main formula q ? p is the one which you take into consideration and then you write truth table for this one so now p is used can take only these values t t f f t f t f so there is a way of writing this particular kind of thing suppose if you have three variables so what you will do here is this t t t because three variables are there to the power of three entries are possible that means rows which are possible in a given truth table that means eight entries are possible first you write all these four T's and all followed by that four f's so this is what you do and then second one what you do is t t f f two T's and two f's two T's two f's and all and then two T's and f so this is enough four variables are there so now once you write first four T's four f's and two T's two f's two T's two f's and then you write alternative t f t f t f and t f and all so this is going to take care of all the possibilities that you can talk about in a given formula these are the only values that p skews ours can take for example in the first case p q r etc takes only T's and all in the second case p takes value t q takes value t and r takes value f so like this you can fill the truth table and all but the problem here is that once the number of variables increases for example if you have n is equal to 5 then you have two to the power of five entries which are possible that is 32 entries you need to inspect to find out a given formula is a tautology or contradiction or contingent it is little bit difficult you know that is why we move on to some other kinds of methods which we can call it as indirect truth table method or there may be some other methods there are some other methods which we will talk about in the next few classes so they are all decision procedure methods but now coming back to this problem so first we need to consider q ?p this is sub formula so now this formula is going to be false only in this case in all other cases it becomes p and all so why because of this particular kind of thing this is p and q p ? q is defined as p or q t t f f t f so now this formula is going to be false only in this case in all other cases it is going to be true so that is a way we define the semantics of p ? q so this is the truth table of this one q ? p so what we are trying to do simply is this that given a well-formed formula we are trying to construct a truth table method truth table method is a very good construct constructive kind of method it is very easy to use especially when the number of variables are less than 4 and if it is more than 4 things will become very difficult it occupies the entire board and all so we will follow some other methods and all so now p ? q ? p now you need to look at these two rows and all you have to inspect a row in which p is t and q ? p is false and all so there is no row these are all rows and all there is no row in which p is t and the q ? t is false and all of course this is p is false q ? p is false obviously it is false and all so there is no row in which p is t and q ? p is false enough because that makes the whole conditional q ? p false enough so we do not have this particular kind of row so that means all these things are true so now what we got here is simply this thing under the main logical connective this is the main logical connective we got all t so that makes this particular kind of formula etotology so what happens if you put some kind of if you change it into some other kind of thing let us say this is p ? not q ? p so now what you need to do here is we need to draw some extra column and all so that is not so now observe this q so whenever it is t it becomes f whenever it is f it becomes it becomes t whenever it is t it becomes f and whenever it becomes f it becomes t that is the semantics of negation so what we have done we have changed this formula into some other thing and all when we are trying to see whether it is a tautology contradiction or contingent formula there are very simple things to do once you construct the truth table what you need to do is under the main logical connective you need to inspect whether a given formula is I mean you always get trees in all the rows of your truth table so now we need to consider not q ? p so now the truth table that is why I so now not q ? p you have to take into consideration this one you have to move from this to this so this becomes false only when not q is t and then p is false in all in all other cases it becomes t so that is a way we define the semantics so now we need to consider p ? not q ? p so that means you need to consider this particular kind of rows these two rows which you need to consider is there any row in which you have p is t and not q ? p t and not q ? p falls in all again in this case there is no row there are all the rows and all you have to inspect each and every row here is there any row in which you have antecedent this is the antecedent and this is whole thing is a consequent is there any row in which this particular thing is t and the right hand side that is not q ? p falls in all in this row this is not the case both are t is rolled out even in this case also it is t in this case also it is going to be t and in all the cases it becomes p so p ? not q is also going to be a tautology it is simply straight forward in all so this is usually an axiom in Hilbert Ackermann axiomatic system which we are going to talk about it little bit later suppose if you treat this particular thing is as an axiom whatever you substitute into this one that is uniformly substitute into this one is also going to be a going to be true and on for example here in this case not q is substituted for q and so this is also going to be true and suppose if you substitute not p for p and all so then this formula becomes not p implies not q implies not this is also going to be a theorem in all theorem is a one which is obviously true and on so uniform substitution we can retain the tautology hood and tautology hood so it is in this sense we just replace q by not q uniformly and it retained is tautology so this is what we are going to talk about a little bit later when once we study axiomatic systems so this is also an example of a tautology so let us consider some more examples and see whether a given formula is a tautology or contradiction or contingent kind of state suppose if you have a formula p and not p obviously this is a contradiction you are saying that it is raining and it is not raining so obviously it is a contradiction let us construct a truth table there is only one variable here propositional variable that is p on so that means it has only two entries in the truth table so once it is a case p and not p the only values that it can take is p can take value t or p can take value f and all and the corresponding negation is if p is t obviously it is f and if it is f then it is t so now you take p and not p so it is like 1 into 0 that is 0 that means f only even in this case also it becomes f so under the main logical connective you got only f's that means this is considered to be a contradiction now let us consider some examples for contingent kind of statements you can take other examples also p or q implies let us say not some formula whatever has come to my mind I have written here p or q implies not so now what you need to do here is first you write p and q these are the two variables that exist in this formula then you write p or q and here the main logical connective is this particular kind of while studying the syntax we found out we found what is going to be the major logical connective and what is going to be the minor logical connective a major logical connective is a one which connects as many propositional variables as possible and all that means this is the major logical connective because it connects one two three variables so that is why it is called as the major logical connective and whatever logical connective that occurs in the sub formula is considered to be the minor logical connective so you have p or q and the next one you are going to write is this one you have to consider not p also so this is the one which we need this brackets needs to be written clearly now we are trying to find out whether this is a tautology or contradiction or a contingent statement so now first we write this thing your two variables you write just like this t t ff alternative t send alternative f two t send two f in here alternative f and p or q is going to be false only in this case in all other case it becomes true and then you need to fill this not p and all not me so if this becomes t it becomes false it becomes false t and t so now we need to look into this particular kind of thing the arrow is important and all we should not move from not p implies p or q this is a totally different formula p or q is not p is different from not p is p or q it is never the one and one of the same until they are logically identical to each other if the by implication is there then you can go you can move in both directions directions so p or q implies not me so now you have to find out a row in which you have this as t and then this f then the whole conditional will become false enough so now this is t and this is f it becomes false again p or q and not p again this is a false now in this case both are t so that means it is t so now p or q is f and then this is t and by the semantics of conditional when whenever the antecedent is false the consequent is true the whole conditional is obviously going to be tn what is that we got is simply this thing so now in the final column of your truth table you have two f's and then two T's if it had been the case that all if you had got all f's and all then it is considered to be contradiction which you did not get it in the same way if I had got all T's and all under the main logical connective implication then it is considered to be a tautology which is not the case here so then that means it is considered to be a contingent of statement so using truth table you can classify the statements of prepositional logic into tautology contradictions and contingent this truth table method works perfectly fine especially when the number of variables are limited to three or four and all of course it works for many number of variables but it is very difficult for us to construct the truth table especially when n is greater than five because if n is greater than five for example six variables are there then you have two to four of six entries that means 32 entries which you need to inspect maybe 64 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 so this is 8 x 8 64 entries will be there it becomes difficult for a human being to see all these things and all perhaps it is easy but a computer can easily do this particular kind of thing so whatever is very difficult for us maybe the computer can easily calculate things so truth table can still be used as one of the important methods for judging whether a given formula is a tautology contradiction and contingent why are we doing all these things simply because we are trying to extract tautologies and all from a given well-formed formulas so we started with well-formed formulas and out of this well-formed formulas some are considered to be valid formulas means which are obviously tautologies some are contradictions which are obviously invalid and some are considered to be contingent kind of formulas if you can extract tautologies and all then what we we can have is all valid formulas because all tautologies are obviously valid formulas that is the reason why we are insisting on tautologies in particular so using truth table one can talk about whether a given well-formed formula is a tautology contradiction or a contingent state so now the second thing which you can do is this particular kind of you can also determine whether a given set of sentences are satisfiable or not so that means let us consider some simple example and we will see whether it is considered to be satisfiable or not now you have come across your translated the English language sentences and then you came across this particular kind of thing P or Q the first sentence Q implies something like not so now satisfiability means once you assign truth values to peace and queues ultimately these two should be simultaneously true and all these two becomes falls and all falls especially then it is called as unsatisfiability and all so how do we know by using truth table method these are some of the translations of English language sentence maybe it can be it is raining or it is not Q stands for grass is wet this stands for if it is grass is wet then implies it is not raining let us consider some examples like this then we will see whether it is satisfiable or not that means simultaneously it has to be true and all then only we can true especially then in that case we can say that these are satisfiable or the other way of saying that thing is consistent so now what you need to do is again you draw the truth table and then we need to see whether at least under some particular kind of in particular kind of row we generate these these so now you have P Q now this is the formula which we have here and then Q implies not P now you need to study this particular kind of P or Q and Q implies not so under the main logical connective that means and if at least there is one of one case one row in which you generate T then that is considered to be satisfiable if we get all the F's here then it is considered to be unsatisfiable so as usual you have two variables you can take T T F S and then alternative T alternative F is the only values that you can take so now we have not P so you write it for not so exactly even if P becomes T then it becomes false then it is false T and T so the negation of P is takes the values FFT so now you fill these things P or Q is going to be false only in this case when both are false both constituents are false in all other cases it is T so now Q implies not P that means you need to take this one Q implies not P the directions are important so this is going to be false when antecedent is true and the consequent is false and then in this case it becomes T now this is also T now this is also P now what we need to consider is these two things now only under this particular kind of thing it is false in all other cases it becomes the compound formula is going to be T so now what is that we have achieved with this particular kind of construction of the truth table we are trying to see whether these two statements that we have written on the board are jointly consistent or jointly becomes true so there are three different cases in which it becomes true so these are the three rows which you need to inspect only in this row it becomes false it does not matter but at least in three different situations the formula P or Q and Q or not Q implies not P is simultaneously true that means P or Q and Q implies not P are consistent to each other or if at least one T is there in this one under the major logical connective it is also considered to be satisfiable so we will talk about formal definition of satisfiability little bit later but as far as truth table is concerned when you take two formulas P or Q in the combined formula under this major logical connective if you come across at least one T then it is considered to be satisfiable suppose if you come across all f's and all all f's etc then it is considered to be unsatisfiable some examples we can take into consideration you can understand it in a better way so this formula is considered to be satisfiable or you can even call it as simultaneously you can call it as P or Q is consistent with Q implies not P. So now you take some other formulas and then you can see whether these are consistent to each other or not not P so the way I have written itself shows that these two are contradictory to each other of course using truth table again you can find out whether are not these are unsatisfiable satisfiable etc so now this is P or Q and not P and not Q so I will quickly go into the details of this one TT F and F is the one which you write it first and then alternative T's and F's it will be boring if I go into the details of all these things so now quickly PRQ is false only in this case in all other cases it becomes T now not P is exactly the opposite of this particular kind of column so now this is FFT now not Q is exactly opposite of this one FT so now this is not the end and all end so now we need to take the conjunction of this particular kind of thing so now we need to write not P and not Q so this is the first second third fourth fifth and six and this is seventh one so this is the way we need to write so now we need to take the conjunction of these two things so this is going to become true only in this case in all other cases it becomes false because at least in one of the conjuncts is false in these cases so now this is F F etc so now all these cases it becomes this thing now you need to consider this and this now in all these cases at least one of the conjunct is false in all so that makes the whole conjunct also so now what is that we got is simply this thing under no particular kind of interpretation interpretation means assigning some kind of values or what are we doing assigning values to P and Q so under no particular kind of assignment of the truth values to a given proportional variables we got these in all these in particular in the final column of your truth table that means you got all F so it is in this sense these two formulas are inconsistent to each other or it is also considered to be unsatisfiable so this is the way we can determine whether a given formula is satisfiable and consistency using the truth table method we can also determine whether two formulas are logically equivalent to each other so now again we go into the details of this one we take the same example into consideration so two formulas are said to be logically identical especially when the truth table matches so now what we need to see here is need to observe the truth values of this one are they logically equal that is what is the question that we are trying to answer so what we are saying is when the truth table matches exactly then they are considered to be logically equivalent so now we need to observe P and Q and this one so now if you have value F here you got T's and you have F here you have T and you have F you have T here and then whenever you have T you have F here that means the truth table does not match but what else we can say about it when the truth values of two particular given formulas are exactly opposite to each other and they are said to be contradictory to each other so that means here if you have F you will find T here and whenever you find T here you will find F here so in it is in this sense these two formulas are logically contradictory to each other because they have exactly opposite truth values in the truth their truth tables whenever PRQ is T now you will see here F whenever you have F here you have T here that means exactly opposite truth values it has these two formulas have exactly opposite truth values and also these two are said to be logically contradictory to each other so we can also talk about logical equivalence with the help of truth table method that means especially when the truth values of two particular well formed formulas matches then they are considered to be logically equivalent so now the fourth thing that we can determine with the help of truth table method is the very important thing which is considered to be validity of a given argument so let us consider some simple examples and we will see whether a particular formula follows from this thing or not so randomly we take some formula into consideration and then we will see whether particular thing follows from that or not let us say you have formula P implies Q and then you have formula not P and then this is the conclusion which is separated and then let us say you write not so now using truth table whether or not this particular thing follows or not but initially speaking so we have some kind of rules which is like this if P implies Q is the case P is the case then Q follows so this is called as modus ponens rule it is obviously truth preserving kind of rule so obviously always valid in the same way there is another kind of rule which we commonly use that is this thing so here this is antecedent and this is consequent so now this is antecedent and consequent if you deny the antecedent you have to deny the consequent you need to deny the antecedent as well so this is called as modus ponens rule so now we can clearly see here instead of denying the consequent here we denied the antecedent part and then we are denying the consequent it might work in day to day discourse but in the first in the classical logic or the preposition logic that we are trying to talk about this is considered to be an invalid argument so we want to see why it is an invalid argument using a particular decision method procedure method which we have been talking about that is truth table so now what are the variables that exist here P and Q that means four entries are possible because there are two variables here so you write down the same thing TTF and F and alternative T is F etc. And the formulas that we have are here this P implies Q and you are not P and so P implies Q this becomes false only in this case in all other cases it becomes T so this is a semantics of implication that we have and not P is exactly opposite of this one F F and TT so now you need to write not Q as well not Q is this F T F now what you need to do here is this thing so now you can write like this P implies Q and not P and then separated by that you have not Q now so the conjunction of these two is the one which you need to write here that is P implies Q and not P so that is this becomes false becomes false this becomes T and F this becomes false and this is T so now you need to observe rows in such a way that is there any row in which the left hand side here left hand side is this particular kind of thing P implies Q and not P do you have any row in which this whole thing is T and the conclusion is false so now you need to inspect the rows what are the rows that you need to inspect these rows these are the two rows which you need to inspect so this is the first one so this is on the left hand side left hand side and this is on the right hand side so now is there any row in which you have T's on the left hand side and T on the right hand side and F on the left hand side and all so now so there are different ways to do this thing so now here is an instance where one second P implies Q and not P this becomes F F and F F T and T T and T this becomes T so this is the one which we need to look into it so now so when an argument is considered to be an invalid argument so invalid argument is an argument in which you have true premises and a false conclusion that is that is what we have been telling right from the beginning of this course and the basic concepts we have clearly said that an argument is invalid especially when it is impossible argument is valid in particular especially when it is impossible for the premises to be true and the conclusion to be false so now these are considered to be premises and this is considered to be conclusion here in this case is it possible that your premises are true and the conclusion is false so now observe this particular kind of row so your premises are true I mean both taken together are true and your conclusion is false not Q is considered to be the conclusion here so that is false that means we can have a counter example in which your premises are true and the conclusion is false that makes this argument invalid so that is why this argument is invalid so did we know that this is invalid we constructed a truth table method and then we are inspecting the left hand side in the right hand side and left hand side is usually considered premises if you have true premises you can only have true conclusion you cannot have false conclusion if you can come across true premises and a false conclusion then the argument is invalid so it is in that sense this particular kind of argument is considered to be an invalid kind of argument you can consider some other examples and you can show whether a given argument is valid or invalid let us consider some more examples and we will see whether with the help of truth table that particular kind of argument is valid or invalid P ? Q ? Q let us say P ? R then from that and you have derived not so now since we have three variables the truth table will be relatively little big so you have P ? Q ? Q and P ? R and then you have not are so now you need to inspect a row in which suppose if you can come across all these things true and not R is false then it leads to a particular kind of thing that this formula is considered to be invalid so it involves three variables so the truth table will be relatively bigger so instead of that one can try to solve this particular kind of problem using some different kinds of methods which we will be talking about in the next class so now we are trying to see whether not or follows from these two or not so simple thing is here P ? Q and ? Q from this you will get not and not P and P ? R so this is the one which we need to inspect from not P and P ? R whether or not not are follows so this reduces to two variables now instead of talking about Q you are talking about this thing not P P ? R not R so if you can say that this is considered to be valid if you can say that the premises are true in the conclusion is false then it is invalid otherwise it is going to be valid enough so now this reduces to this particular kind of thing so now quickly we can draw a truth table for this one not P and P ? R ? P ? R etc and then not R so there are three variables here and again the two variables in fact so P R and not so T T F F this is T F T F then not R is F T F is what is not P ? R this becomes false only in this case in all other cases it becomes T so now not P not P and not P ? R so these are the things so which we need to take into consideration not P is this F F T not P ? P ? R so now this is false now this is also false T and T true and T and T is going to be true so now not R so now we are trying to see whether not R follows from these things are not so now we have reduced the formula into two variables in all including P and R so now not R is this one so whenever you have T you have F here F T F whenever you have F is T so now what you need to do here is you need to observe these two rows is there any row in which your true premises and a false conclusion so now clearly you have this particular kind of a row in which your premises are true that means not P and P ? R are true and the left right hand side is false enough so now this row is sufficient enough to show that this particular kind of argument is invalid you need not have to inspect any other row because invalidity requires that at least one counter example if you can come across with one counter example in which your premises are true and the conclusion is false then obviously the argument is invalid so what is that we have done here first we have reduced P ? Q and ? Q into this particular kind of thing so this is a logical consequence of this one from this you can derive not P is a logical consequence of these two by using modus tolerance true so now we have removed the one particular kind of variable Q is not required here and then we constructed truth table for these things and we have seen that the premises are true and the conclusion is false and hence the argument is obviously an invalid argument. So what we have seen here is this that we can determine whether a given formula is a tautology a contingent statement or contradiction or you can even say when two groups of statements are satisfiable to each other or you can even say when two groups of statements are well-formed formulas are logically equivalent to each other they are equivalent to each other especially when the truth values matches and you can also talk about whether or not a given well-formed formula or a given conclusion follows from the premises again you can construct the truth table method truth table method works for works better for this thing but when the number of variables increases it presents some kind of difficulties see it is the one which we have said so far one can have tautologies especially when under the main logical connect you have only t's and a statement is considered to be contradiction especially under the main logical connect you have only f's and a statement is considered to be contingent if and only if it is true on some assignments that means you assign some kind of values p q's as t and f ultimately under the main logical connect you have two t's two f's or maybe one f one t and three f's etc and all etc at least one t should be there in under the major logical connect it is in that case it is considered to be contingent. So one can determine whether a given formula is a tautology or not using this particular kind of thing so there are some particular kind of statements which occur in the natural language and which are presented in natural language that is English then we first what we need to do is we need to translate the given English language statement into the appropriate language of propositional logic then we can talk about whether a particular formula is contingent or tautology or it is considered to be contradiction and all for example if you have this particular kind of thing if the neuron is alive and fires then it has a given minimum number of excited excitatory fibers a and f implies n and if you want to say if you the neuron is alive it has given number of excitatory fibers wherever it fires a implies f and n so now if you want to say that these one and two are logically equivalent to each other so then what you need to do here is this so the first formula in this one is like this so before that I will talk about what we mean by saying that these two are logically equivalent to each other the first formula is a and f implies n and the second formula is a implies f implies n so now these are the two formulas which are which we got by translating these two statements in statements so now if you want to show that this is going to be tautology then that means if one first statement is considered to be x the second statement is considered to be y and it obviously becomes x if and only if y so if you want to say that these two are logically equivalent to each other what you need to do here is you have a formula x and you have a formula y so x and y are considered to be logically equivalent especially when x if and only if y is a tautology if you can say that x if and only if y you can show that x if and only if y is a tautology then obviously then x and y are said to be logically equivalent so these two statements whether or not logically equivalent to each other for that what we need to check is first you need to translate the English language sentence into appropriately into the language of prepositional logic and then let us assume that the first formula is x that is a and f implies n and the second formula is a implies f implies n then x if and only if y if you can show that that is a tautology then you have already shown that x and y are said to be logically identical to each other. So in this class what we have discussed is simply this that we started with the truth table method which is considered with the simplistic method which works fine for a number of when the number of variables prepositional variables are less so with the help of truth table method one can talk about consistency one can talk about whether or not a particular formula whether or not a conclusion follows from the premises that is the logical validity or you can also talk about whether or not to given well form formulas are logically equivalent to each other by showing that x if and only if y is considered to be tautology. So in the next class we will be talking about a particular different kind of method which works perfectly alright even if the number of variables are more than four so that particular method is called as semantic tablex method. So you have to note that we have been covering semantic methods so far so then we will move on to this semantic tablex method and we will talk about the essential features of semantic tablex method and we can talk about the same thing like whether or not given well form formulas are consistent to each other contingent etc all these things whether a given formula is valid etc all these things we can know with the help of semantic tablex method in the next class we will we are going to talk about semantic tablex method.