 Welcome back in the last lecture we presented syntax of propositional logic where we have seen how to construct well formed formulas and not only that we constructed well formed formulas we also came to know how to read a given well formed formula with the help of parenthesis etc and we gave some kind of preferential ordering for the connectives not and implies are etc and all with that you know we came to know how to read a given well formed formula especially when the parenthesis are not given and all but you know usually we find that you know parenthesis are usually given and all in case if it is not given we need to follow those particular kinds of rules and all. So in this class we will be talking about semantics of propositional logic so what we will be doing simply is this that we will be talking about what we mean by the well formed formulas that we have formulated earlier for example if you have come up with a formula p implies q in place r for example and what we mean by that formula. So we go with a slogan that meaning means giving truth conditions to a given formula and all so following Fregé we will be going into the details of the semantics of propositional logic with an idea that meaning of a given formula means me giving truth conditions you know. In this lecture I will be focusing on one simplistic method which is called as truth table method which is very simple and very easy to use so it works for it works especially when the number of variables propositional variables are less that means two or three it works better of course it works for even more than three variables also but it is difficult for us to formulate difficult to for us to construct a truth table a computer can easily do it when the number of propositional variables are more than five and all six etc. So today we will be talking about the semantics of propositional logic so there are five connectives that we have discussed so far that is negation and r implies n double implication all so what do we mean by these connectives in all so the semantics will take care of this particular kind of thing and all so before going any further we need to talk about these connectives in a sense that these connectives are all truth functional and all so why these are truth functional it is because of this that the truth value of a compound statement that that is generated out of this propositional connectives can solely be determined by the truth value of its individual components in all for example if you have a formula p implies q the truth value of p implies q can can be solely determined by whatever truth values the propositional variables that occur in the formula that is p q etc takes suppose if p takes value t and q takes value f the truth value of the compound formula p implies q takes the value f so this means that there will be a rule telling which tells us exactly what the value of the compound must be for each combination of values for the components in all so that means if the components takes some values in all you can determine the truth value of a compound sentence in all you see it is in this sense these connectives that we are trying to discuss are considered to be truth functional connectives in all so if the truth value of a compound formula is not solely determined by the truth value of its constituents then it is not considered to be a truth functional connective in all so there are many non-truth functional the connectives which are used in the non-truth functional way but we are not going to talk about those particular kinds of things in all to start with the simplistic kind of connection that we are logical logical operator that we are trying to talk about is a negation so negation is simply represented as not p which simply results especially when you transform a given sentence to its negation in all for example if you say it is raining and the negation of that one is it is not raining in all you have to put just not before the verb or some kind of auxiliary verb in all men are mortal the negation of that one is all men are not mortal in all so it is as good as saying the same thing p does not hold in all suppose something as a property p metal metals expands upon heating and if it does not hold then you say that metals does not expand this metal does not expand upon heating in all that particular property does not hold for this particular kind of thing in all or if you want to say that it is not the case that p it is not the case that this is a this is a chalk piece or something like that so so in the same way if you want to represent p is it so then you have to represent it in terms of negation in all so the negation sometimes it is used with this particular symbol sometimes we use this particular kind of symbol it stands for negation only in some textbooks it is written in this sense so these two are the same symbols so it is only for our convenience we are using this particular kind of thing the next connective is or can be used in two different senses one is in the inclusive sense and other one is in the exclusive sense and understand until it is stated in classical logic that means the prepositional logic we treat it as inclusive or so what is inclusive or it is simply p or q and all it is raining or it is not raining or it can be both it is raining and it is not raining it can be both else like it is tea or coffee or both of them or it can be simply represented as p or q or p unless q is also represented as p or q and sometimes in the legal documents we write this kind of language where you write like this p and oblique or q and all it is used for some specific kind of purpose and all so we do not go into the details of this legal documents etc and all sometimes they use this kind of long language and all that means it means p either p or q or it can be both and all coffee or tea a coffee and tea either you can be satisfied with coffee I can be satisfied with tea or even if you serve both of them it will it will work for us so p except when q is also represented as p or q and all so if you want to represent the negation of p or q you have to say it in an English language as neither p nor q and all it is not the case that p or q and all so right now we are just trying to talk about what we mean by this connective set what exactly these connectives are then we will talk about the meaning of this connectives and all that means how these connectives behaves in the formal logic that is the preposition logic that we are trying to discuss in the conjunction that you come across like this p and q or p but q is translated into p and q and this is the list of things and all which can be translated into simply p and q and all p despite the fact that q p although q p though q p even though it is q etc all these things are translated into conjunction and all that is p and q p whereas q p for q etc p no sooner than q p still q p besides q all these things which you come across in day to day discourse so these are all translated into simple translation that is p and q p on the other hand q and all which is commonly used in the scientific discourse so is also translated into p and q this is translation kind of list and all one has to go through this thing in greater detail once we solve the examples then we will equip ourselves in a better way with this translation and all so this is also called as disjunction and all either p or q and all it is simply either p or q or p or q or both p or alternatively q and all these all simple things and all so now the conditional which is represented as p implies q in many textbooks this implication is used in this particular kind of sense so this is called as material implication so in many textbooks so you will find this particular kind of symbol but it is convenient for us so we are using these things in all for negation we use this for implication we use this one so p implies q by definition means it is not the case that p or q or if you apply demagons rule for this one then it becomes it is not possible that p and q so this is same as this one it is not p negation of this conjunction will become disjunction and negation of negation of q will become q and all so these are all one of the same now so whenever you come across material implication this is what we mean in prepositional logic we take into consideration this particular kind of definition for the implication that means implication can be expressed in terms of disjunction in implication is expressed in terms of our operator so all these things come under the category of p implies q and all for example in the circumstance that p then q in the event that p then q in the case of p then q etc assuming that p then q and the supposition that p then q etc and all all these things are translated into simply p implies q and all in p implies q p is usually considered as an antecedent hypothetical situation if that is a get satisfied and the q follows and all so how did better Russell invited it is the material implication is attributed to better Russell invited Russell came up with this idea that how this p materially implies implies q and all so it was that means somehow p it has to lead to q and all so what substitution one has to make here so that p materially implies q and all they were testing with so many different kinds of formulas here and ultimately what fitted well here is this particular kind of thing if you can substitute inside the missing preposition compound preposition that is not p or q then these two by disjunctive syllogism leads to q and all so it is because of this particular kind of thing so they thought so this is the definition of this one p materially implies q in no other way then this particular kind of thing if p is the case not p or q is the case then only p materially implies q and all it is a way of transfer so way of moving from p to q and all is it is in this sense is not p or q served as the definition of material implication so this is what we mean by this thing p if q so these are translated in a reverse way instead of p implies q we translated as q implies p and all so these are the things which we come across in day to day discourse in all p in the event that q p in case q p assuming that q etc all these things are translated into q implies p and all so we need to go into the details of these things a little bit later it will be boring if I go into the details of each and everything if I read out all these things and all so now by condition is not just we are trying to see what these conditionals what these connectives are all about then we tried we tried to talk about meaning of these connectives in all with the help of truth table. So p if and only if q and all this kind of by conditionals are usually used in natural sciences especially when you are trying to invoke necessary and sufficient conditions are when you are trying to bring in equality between two things when you want to say that p and q are logically identical to each other and you will say p if and only if qn so it is simply represented as p if and only if q then in some textbooks it is written in this sense now we are using this by conditional in some textbooks it is simply represented in this these two symbols are one of the same so all these things come under the category of whenever you find p if q otherwise not p just in case q p whether or not q p even if q all these things will be translated into p if and only if q and all you can go into the details of these things little bit late. So let us consider some of the examples how we can translate the phrases that occur in the English language into the appropriate language of prepositional logic the first one is neither p nor q it is exactly the negation of p or q so that is why we had just put negation before p or qn so now p unless q if you take it in the inclusive sense it is simply not q in place p and all if you want to say p except if q again it is used in inclusive sense it is written in the sense not q or not q in place p or q if you want to say explicit in terms of exclusive operator that means one excludes the other one other possibility and all for example in the parties usually you will say when you are organizing some kind of function in the halls for example you will put only you will put a restriction in such a way that either you have to take soup or salad and all for example that means if you take soup you are not supposed to take salad if you take salad you are not supposed to take soup and all the one excludes the other possibility in the same way you put ice cream or some kind of sweet and all if you take ice cream you are not supposed to take sweet if you take sweet you are not supposed to take ice cream and all but if it is used in inclusive sense and all which is little bit flexible and all you can take either ice cream or even sweet suppose if the ice cream is finished you can even take sweet also so inclusive or has that fix flexibility and all that means p or q and even both also that means p and q also so p the sixth translation is like this p if q in whichever case r and all so it is represented as q implies p and p implies r and all p rather than q or p instead of q or p without q is simply translated as p and not q and all so this is only if a sake of translation and all which we are trying to mention all first the translation part is important and all then we can talk about you have to convert the given English language sentence into the appropriate language of prepositional logic for example if you are given this thing a man is either mentally deficient or mentally healthy and all so the first one is represented as mentally deficient is represented as D and mentally healthy is represented as H then it is simply a disjunction all so it is D or H in the same way either we accept quantum mechanics or we study objects larger than atomic size it is represented as M or O a successful man is either intellectually creative or mentally dynamic it is represented as CRD and sometimes you can even represented as CRD or it is not the case that both of them and all not it is not the case of C and D exclusive r the definition of exclusive r is this thing p r q but not both of them so we sometimes we use it in exclusive r sometimes we use it in the sense of inclusive sense we use this particular kind of operator r and all so p unless q can be represented in inclusions it can be represented in exclusions and all so this creates some kind of confusion and all so the same kind of connective can be used in both the senses in one sense it is like this this is the inclusive r and all and this is exclusive so both are represented in this way only usually we represent exclusive r in this way we do not write anything here it is p r q r both the things and all this is what we mean by this thing either p r q or even both of them and all either you have ice cream or sweet jalebi or something like that or it can be both and all ice cream and jalebi also you can have any one of this things in all but if you put restriction on this particular kind of thing suppose if you if you take ice cream you are not supposed to take jalebi then you have to write it in this particular kind of p r q r but not both of them so definitely you have to rule out one possibility and all so if you take this one you are not supposed to take the other one and all it is either p r q but definitely it is not both of them both of them you are not supposed to take it is in this sense exclusive r and all either ice cream or jalebi but you are not supposed to take both the things ice cream and jalebi followed by it followed by ice cream you are not supposed to take jalebi so it is what we mean by exclusive r and all so this is what is expressed here so these things can be represented in either in an exclusive sense or an inclusive sense and all so what we are trying to say here is that the same thing can be expressed in both the things inclusive or exclusive r but in classical logic or propositional logic if unless and until it is stated clearly it is usually taken by default as inclusive r only that means the first definition is the one which we will be taking into consideration either p r q or p n q n or both of them so that is the definition we usually take into consideration for the operator r and all unless and until it is stated explicitly that it is an exclusive r we do not mean by it as exclusive r and all for example these things can be interpreted in even both inclusive and exclusive r as well sometimes it is difficult to find out whether it is used in exclusive sense or inclusive sense and all but by default it is used as inclusive sense only you must pass this course or make up the credit hours in some other way and all it seems to be the case that if you use common sense and all it seems to be the case that one excludes the other possibility and all that means it is CRH or it cannot be both of them and all that is in one sense we can use this thing or you can use it in inclusive sense in such a way that CRH or even both C and H and all in both the senses we can use this particular kind of thing what I am trying to say simply is this unless and until it is stated explicitly we mean the connective that we have used that is r means it is inclusive it is used in inclusive sense and all so the conjunction has a list of these things p and q p but q p although q p in another less q whenever you have come across that kind of thing you simply substituted as p and q not only p but also q is also written as p and q that means p and q have to be there and all so that is what we mean by not only p but also q p at q is also translated as p and q etc p more over q p however q all these things translated into simple conjunction and all p and q and all a few examples of translation we consider and then we will move on to what we mean be this connectives and all so this class is all about what we mean by this correct is how these connectives behaves in all in what sense they are considered to be truth functional connectives in all that is what we are interested in but before that we are trying to translate some of the English language sentences appropriately into the language of propositional logic suppose if you want to express this particular kind of thing it is not necessary to give up Newtonian mechanics even though we accept quantum mechanics in all so this is in this one p although q etc it appears to be like p although q and all third one so this is simply translated as not m because it is saying that it is not necessary to give up it is necessary to give up means it is written as m but here it is saying that it is not necessary that means not m and all and the next one is accepting the quantum mechanics that is t represented as a letter t and all these are simple translations in all one needs to make lot of translations so that so once you the idea here is that once you translate English language sentence appropriately into the language of propositional logic then we can talk about many beautiful logical properties in all when you can say that two groups of statements are consistent to each other or when conclusion follows from the premises or when we can say that a given sentence is a tautology that means always true and all or when a given formula is always false that is a contradiction etc all these things you can talk about only when only once you translate this English language sentences in appropriately into the language of propositional logic. So the last one we last example we take into consideration anybody anybody anybody may be electrically charged under proper conditions but not everybody seems to be seems to have very strong magnetic properties in all if you represent the first sentence as C second sentence as P then the second sentence not every body seems to have very strong magnetic feels is represented as not P so it is a conjunction of these two statements that is C and not P it is represented the whole sentence gets translated as C and not these are some of the examples which we can take into consideration so it is raining but I am happy again it is translated as R and H although it is raining I am happy again same as R and H it is raining yet I am happy is also translated as R and H it is raining and I am happy so it is R and H so these are some of the simple examples you know you come to know how to translate a given English language sentence appropriately into the language of propositional logic and all this whatever I have presented here will serve as some kind of translation guide and all one needs to go into the details of each and everything whenever some kind of example arises and all you can look back this translation guide and see how to translate a given English language sentence appropriately into the language of propositional logic. So if you want to express this thing if P then Q it is simply translated as the one which is mentioned in the red color that is P implies Q in case of P Q is also translated as P implies Q provided that P is also translated as P implies Q only here is a little bit difference in all if you want to express sufficient condition P is a sufficient condition for Q is translated as P implies Q whereas if you want to say P is a necessary condition for Q it is represented as Q implies P and all so it is simply like this so usually conditionals are used for sufficient and necessary conditions so P is sufficient for Q this is translated as P implies Q and all so P is necessary necessary for Q for example if you want to say that oxygen is necessary for life and all but oxygen itself is not sufficient for us to survive and all we need fats carbohydrates and lots of other things in all so P is necessary condition for Q means it is represented as Q implies P so this is the only difference if both the cases happens in all that means P implies Q and Q implies P P is a sufficient and necessary condition and all then we write it as this one both it is both necessary and sufficient so this is what is important in natural sciences especially when you want to say that something is a sufficient necessary condition for something and all even in mathematics also we use this necessary and sufficient kind of conditions in all so something is sufficient for Q means P implies Q something is necessary for Q P is necessary for Q means Q is P Q implies P and all so that is the reason why in the 6 one Q is a necessary condition for P means it is the reverse one that is P implies Q suppose if you want to say P is necessary for Q and all it has to be Q implies P and all so this is the thing which we will be using it in the examples that follows so all these things are simply again represented as simple P implies Q and all Q if P that is the kind of necessary condition and all it is represented as simply P implies Q and all P only if Q is also represented as P implies Q P only when Q or P only in case of Q etc all these things are represented as P implies Q and all so what do we need to substitute this P is Q is with some kind of simple sentences atomic sentences that we commonly see in day-to-day discourse then we will see the importance of these translations and all one simple example of conditional sentence is like this if Tajmal is in Agra then Tajmal is in India it seems to be making some sense and all if the Tajmal is in Agra and Tajmal is in Singapore the first sentence is true the second sentence is false and what happens if you have this particular kind of thing and all P is true Q is false and all what will happen to the whole conditional that is P implies Q that is what we are going to talk about in a while from now that means the semantics of material implication that is false and Tajmal is in India that is true even then this antecedent is false the consequent is true the conditional is going to be true the whole conditional is going to be true if the Tajmal is in AP and the Tajmal is in Kerala both are false and again the conditional is going to be true and all so how do we know that the first conditional is true and the last conditional is also true whereas the other one second one if the Tajmal is in Agra then the Tajmal is in Singapore that is false and all so how do we know that the given conditional is false given conditional is true so we need to have some kind of semantics for this material implication all which we will be talking about a little bit later sometimes this kind of connect operator unless all these things which I am trying to talk about they are all truth functional connective so I did not enter into the truth functional kind of thing yet but we are just trying to see how to translate a given language English language sentence appropriately into the language of preposition logic especially when you come across unless implies all these things is one of the most confusing kind of English words to be translated in English is unless so this word expresses a dependency between two prepositions that means it is a binary kind of operator sometimes it can be represented as R sometimes it can be represented as implies etc. But one which is not always what is not straight forward is that whether or not to represent it as if then or sometimes it is represented as R etc. In strong sense it is equivalent to if and only if not and in the weak sense it is translated as if not and if not P then Q and all. So suppose if you want to represent this particular kind of sentence it depends upon whether you are trying to make it strong kind of statement or trying to make a weak statement in all suppose if you want to say this particular kind of thing the library will remain open till 11 o'clock from Monday to Saturday unless it is some kind of Sunday or some kind of public holiday or some kind of strike happens or students close it for some other reason etc. So it is simply represented as a conditional if P then Q that means O stands for the library will remain open till 11 am if and only if not and all this is used in a strong sense and all if and only if it is not the case that SR P and all that is it is not the case that either it opens from Monday either it is Sunday that is yes or it is a public holiday means P and all it is used in a strong sense and all or it simply represented as O implies not S or P and all only one kind of conditional will apply by conditional will not apply there if not P then Q and all. So that is the way we translate it into a particular kind of thing and all unless is always always create some kind of problem to us. So now this is what we have already discussed and all whenever you want to invoke some kind of necessary and sufficient conditions we use conditionals and all either P in plus Q or Q in plus P some simple examples are like this being a bachelor is sufficient for being a male and being a male is a necessary and necessary condition for being a bachelor and all. So being a male is considered to be a necessary condition for being a bachelor and all. So these are all some of the translations which are which needs to be you need to go into the details of this thing. So Q provided that P Q is necessary for P P is sufficient for Q all these things are translated into P in plus Q and all. So all these things past 20 minutes which I am trying to talk about will come under the list of a kind of we are trying to provide a kind of translation guide with the help of which we can safely translate some of the sentences in English into the language of prepositional logic. So these things also we have covered and all so let us consider some simple examples then we will move on to the semantics of prepositional logic only those who do exercises will pass logic and all that means you must do exercises to pass the examination and all it is a kind of suppose if you represent it like only those who do exercises is E and who will pass the examination is P usually if it is a sufficient condition you will represent it as E implies P and all but it is a necessary condition that means it clearly stating that if you do not solve the exercises you are there is no question of passing the exam and all. So usually we simply write it as E implies P and all but it only satisfies the sufficient condition here it expresses the necessary condition all one is necessary for the other it is like oxygen is necessary for our existence and all. So instead of translating this the first sentence into E implies P we translated as the reverse one that is P implies E and all. So this is a necessary condition that is why P implies E and all that means in the last case we have seen if P is a sufficient condition of Q then you translated as P implies Q P is a necessary condition for Q means it is Q implies P and all. So instead of Q we have here P and instead of P here you have E and all that is why it is translated as P implies E. So in the same way you would not pass the course unless you do the exercises can also be translated as you would not pass the course unless and until you do the exercise E implies it is also translated as P implies E and all. So the last one if you do the exercises you will pass the course provided that you are diligent and intelligent and all. So it is like E and D and I the conjunction of all these statements implies P and all the fourth one is the one which is used it can be used in strong sense or it can be used in weak sense and all. So again unless creates some kind of problem in the translation and all. So this is the way to translate the things and all some more examples will be like this Ravi and Priya go to the movie while Sita goes to the work and all. So Ravi and Priyat is represented as R and P and while Sita goes is also represented as a conjunction operator that is why it is R and P and S. The second one in order for Ravi to go to the movie it is necessary that Sita goes to the school and all. So this is R implies S and all Ravi goes to the movie if Sita stays at home and all is some kind of sufficient condition. So that is why it is a necessary condition and all Sita must stay in the house so that Ravi can go to the house. So it is simply the first one is represented as R second one is represented as S since it is a necessary condition that one must be there in the house. So it is S implies R and all. Now we fail in the exam unless he studies in all. So either he has to study F is the case or S is the case and all. It is used again in inclusive or exclusive sense and all. So here you will see clearly here sometimes unless it is translated as if P then Q and some other cases it is translated as F or S and all that is why unless the phrase unless presents lot of problems and all in the process of translation. The last example we will have picnic unless it rains and all if it does not if it does not rain we will have will go for a picnic and all. So it is simply it can be used in both the senses and all it can be used as inclusive sense PRQ and both and all or it can be in this case at least it is used in a exclusive sense and all. So it is one excluding the other possibility and all. So that means if it rains you will not go to the picnic and all if you go to the picnic it does not rain and all. So one excludes the other possibility and all so it is simply represented as PRQ and all some more examples which we will take into consideration. So now before going any further and all so we need to see how these connectives actually behave and all. So let us try to talk about the semantics of propositional logic and all so how these connectives behave and all. So to start with we have negation then we are trying to express it how these connectives behave and all. So to start with we have simple negation and all suppose if you have a sentence P and then the negation of this one for example if this becomes T and all and the negation of P will become false. Suppose if you say this is a tester the negation of that one is it is not a tester and all. So that is represented by not P and all. So this is the way this connective behaves and all. If the sentence is false and all then the negation of that one will automatically be true. So this is the most simplistic kind of thing and all then coming back to this one R. So this is the truth table for this thing. So all these connectives that I am trying to mention on the board they are all truth functional connectives and all that means truth value of a compound sentence is solely determined by the truth value of its individual constituents and all. So here you have P and you have Q and then P R Q and this P R Q can be used in two different senses and all. So this is inclusive R which is defined as P R Q but not both the things and all and this one P R Q or but not it can be both and all but not both the things. So this is the definition of this one. So now there are two variables in our propositional variables P R Q that means there are 2 to the power of n entries in the truth table and all. So what we are trying to do simply is is that we are trying to see how these five operators like negation R and implies if and only if behaves in logic and all. So for that we are trying to discuss truth table we are trying to construct truth tables with which we can say something about this the behavior of this connectives in all. So suppose if you have only two variables in all then you have 2 to the power of n entries possible in a given truth table that means 2 square 4 entries are possible in the truth table and all. Suppose there are three entries in all you have three variables in all then you have eight entries possible in the truth table and all. So in eight rows we need to inspect and all. So as the number of propositional variables increases in all truth table will become bigger and bigger and all and then if the number becomes more than 5 4 and all there will be 32 entries in the truth table and all because 2 to the power of 5 is 32. So let us talk about simple thing in all. So first you will write alternative T's and F's and then so two T's and two F's and then you write alternative F's and all like this and then this connective is going to be false only in this case that means both the constituents individual constraints P and Q are false then only it will become false in all other cases it becomes true and all. So this is the way this behaves in all this connective behaves in all P or Q is going to be false only in this case in both a both constituents are false considered means P or Q in all other cases if P becomes F Q becomes T then also it becomes T or whenever P is T Q is F then also it becomes T etc. So now this is what is used in the inclusive sense in the exclusive sense one excludes the other possibility in all that means it cannot be both true and all so that means this becomes false and the same way it cannot be both false also so if that means one is false the other one has to be true and all or if this is Q is tree and P has to be false and all. So now this automatically anyhow this becomes false and all and this may become T and all so this is the only difference that you will find it in exclusive R and all it is like ice cream or G levy I mean that possibility needs to be ruled out and all it cannot be both and all so that is why it becomes false in all other case it remains the same and all so now coming back to the connective and all again let us assume that there are only two variables P and Q so why we are talking about this truth tables in all because a well-formed formula should get its meaning and all so the meaning means providing meaning means providing truth conditions for a given formula and all so this is the compound formula and all we are trying to provide meaning of this formula and all so meaning how it gets meaning and all whenever you provide truth conditions for the individual constituents that exist and all once you provide once you assign some kind of truth values to this individual constituents and all then the behavior of the connective is such that whenever you have this thing T F T F so whenever both the conjuncts are true and all is going to become T and in all of the cases it becomes false in so this is exactly in correspondence with what we have here one analogy that we have seen here R stands for this is Boole's representation of usually algebraic interpretation so this is a logical interpretation that we are giving the same thing which we are trying to talk about this is a connective R which means plus in Boolean algebra and N stands for multiplication and all so usually represented in this way but you can represent it is star or sometimes it can be represented as even dot also so this stands for multiplication so now these P's T stands for one for example value one a digital number or F stands for zero then usually you will see one plus zero is equal to one one into zero automatically zero and all so it is zero into zero that is zero only so it is in this sense also you can understand this particular kind of thing and all so now this is how the connective and behaves in all you have to note that this is the way it behaves in the prepositional logic and all but in actual day-to-day practice you might be surprised with this particular kind of thing that suppose if P and Q is there this is same as Q and P and all suppose if you represent P as I became sick let us say became sick and all and then usually when you become sick in all you went to see the doctor in all and see the doctor so now P and Q is same as Q and P and all so now the sentence will become compound sentence will become like this I went to see the doctor and I became sick that is one thing nobody goes to the doctor to become sick and all that means Q and P and P and Q is I became sick and I went to the doctor that makes sense to us but the second one does not seem to be making sense in all these two are totally two different things in all I went to the doctor and I became sick and I became sick and I went to the doctor so P and Q and Q and P are totally different especially when you use it in the day-to-day discourse and all but in classical logic P and Q is same as Q and P because the truth table of P and Q matches with the truth table of Q and P that means these two are logically identical to each other it is by saying the computability law holds and all so that is why P and Q is Q and P but in actual day-to-day discourse it is not just the meaning of a compound sentence is solely determined by the meaning of its constituents and all but there is something more and all so those kinds of connectives are called as non-truth functional connectives in all which we do not talk about it at this moment and all so how this the other one behaves in all P implies Q so we have said that P implies Q by definition is not P so better Russell invited for trying to understand how P materially implies Q and all what kind of substitution you make it here so that P materially implies Q and all so they substituted this one and these two P not PRQ descent is a little piece to Q and all so that is why this the substitution word the formula that is they used here is served as the definition of P implies Q and all so not P or Q is the definition of this one we follow this particular kind of definition then again if you have two propositional variables exist in your truth table then you have four entries possible in all that means four rows which are possible first you write it two Ts and two F's and then alternative F's and all 21 F and even F and all and then this becomes false only in this case in all other cases it becomes P so this is a way this connective behaves and the only connective that is left here is P implies Q P implies Q Q implies P so now this is what we mean by P implies Q so that is both P implies Q it is a sufficient condition and Q implies P is the case so now again you write alternative Ts and alternative F's T F T F and all so now first you write it for P implies Q is the same as this one you write the same thing T F T T so now Q implies P now you have to move from this site so it becomes false only in this case when Q is T and P is F this becomes false in all other cases it becomes T and all so this is the way the connective behaves in all so because of that are by that by using the definition we are trying to write the truth value of a given compound formula and all so now this is a conjunction of these two things and all it is both P implies Q and Q implies P so now this is an end connective so whenever both are true this is T and now this becomes F F and T is F that is 0 into 1 is 0 only so now this is also becomes F and this becomes T so that means P if and only if Q becomes T when both are true or when both are false in all that case it becomes T so in all other cases it becomes false in all in this case it becomes false so this is the way this the connectives behaves in all based on this thing we can talk about based on this truth table which is formulated by at least two kinds of people Wittgenstein two philosophers have formulated a mathematicians and logicians one we can attribute it to Wittgenstein and the one is attributed to Emily post so these are the two logicians which are responsible for this truth table method in all it is a very intuitive kind of method it is a kind of constructive method with which you know you can talk about whether a given formula is tautology or given a formula is contradiction or a kind of contingent kind of statement in all so we will go into the details of this little bit later so this is what we mean by semantics of propositional logic in all meaning of a formula means providing truth conditions for it we have used one particular kind of method that is truth table method and with which we form we have given meaning to a given formula in all the providing truth conditions for a given formula in all so what we will be doing here is which we have done it already it is we need to know something about interpretation in all interpretation or valuation of a language it is an assignment of meanings to its various symbols or its well-formed formulas in all that means if you have a formula like any formula to take into consideration let us say p implies q r r so how it gets meaning in all by assigning some kind of truth values to this one this formula in all so how can we assign truth values to this formula in all so there are some individual components p q r and all so it takes some kind of values in all either it can take p can take value t q can take value t or r can take f there are many possibilities like this so whenever you have three variables in all in the case of truth table first you write all four t's and four f's in all and then you write two t's two f's two t's and two f's in all and then you write alternative f's in all t f t f t f in all so these are the values that p's q's are can only taken so we will be assigning some kind of values to the individual kind of components in all and then with the help of this we will be evaluating the truth value of the whole compound formula in all that means interpretation or valuation of a language is an as a kind of assignment of meaning means giving truth conditions to a given symbol or given well-formed formula in all so in formal terms it is a valuation v is a function from prepositional symbols that means p q's r's etc in all that occurs in the formula and then this p q r's etc can only take values only two values in all either it can be true or it can be false in all so that means valuation function takes a prepositional variable and assign some kind of value and that value is going to be a kind of binary value that is either 0 or 1 it cannot take any other value and all if you assign some kind of truth values and the few evaluate the truth kind of truth value of a given formula that will always be turned out to be either 1 or 0 1 stands for t and 0 stands for false in all so this is the way in which we try to talk about the same thing that we have listed out on the board in a totally little bit different way and all so v stands for a valuation function and there are two symbols which we will be using it here so that is also very important in all the symbols t means the formula which is always true and all which is written as top that means they are all tautologies in all like 2 plus 2 is equal to 4 etc and all on the one hand use another symbol the reverse of that one which is represented as board so that means sentence which is always false in so there are contradictions etc. So if you give valuation to those formulas which are always true that is obviously true and all two formulas will the valuation of truth formulas will always be true a bachelor is unmarried the truth value of that one is always true and all so in the same way if your truth value of a contradiction is obviously false and all so that is what is the case and all now the second one the valuation function of a and b so that is going to be true and all that means if you are saying various values to a and b the valuation of a and b becomes t only when the valuation of a and valuation of b is going to be true and all otherwise it is going to be false and all suppose if you observe this truth table method so the valuation of p and q so that is going to be true only in this case in all other cases it is false and all that is what essentially says in all in this third case valuation of a or b that is going to be false so now we need to observe this particular kind of thing and all so the valuation of p or q that is going to be false when both are false and all in all other case it is going to be true and all so that is going to decide the truth value of a or b in the same way well fourth one valuation of a implies b that is going to be false only when in this case when p takes value t q takes value of and this p implies q becomes false and in the fourth fifth case valuation of a if and only if b that is going to be true especially when the valuation of a and valuation of b is one of the same and otherwise it is going to be false and all so in this case you need to take the same thing the valuation of a that means f here and valuation of b that is also same and all in that case it becomes t and all sorry in this case so in this case and this one so these are the cases which decides whether these are true or true or false in all other cases it becomes false and that is what essentially it says and the last one is straight forward and very simple that is valuation of not a it becomes t especially when the valuation of a becomes false and valuation of not a becomes false only when the valuation of a that means giving value to this particular formula a it is already true and it becomes t and all so what we have studied in this class is simply this that we started with the translation of given English language sentence appropriately into the propositional logic then after studying in detail about the translation we moved on to how these connectives behaves in all how the truth functional connectives such as negation and r etc behaves and we express it in terms of truth table and then we express the same thing in terms of some kind of formal language with the help of this thing that how when we assign some kind of values to that thing interpreting the formulas and all so when the formula is going to be true and the formula is going to be false etc other things which we have provided especially in the case of valuation extended to propositional logical formulas in all in the next class we will be talking about some of the important logical properties such as when do you say that a given a well formed formula is a tautology that means always true when you say that a given form formula is always false when it is contingent or when two groups of statements are consistent to each other or when some kind of conclusion follows from the given premises in all all these things you will be studying in the next class.