 Welcome back, in the last few lectures we have presented basic concepts and then we also presented traditional logic which has remained as paradigm in logics for more than 2000 years that is Aristotelian logics. In fact Aristotelian logics needs to be presented while studying the predicate logics. In fact they are closer to predicate logics than the preposition logics that we are going to study right now. So what we will be doing now is will be studying the basic concepts of prepositional logic. So as the name suggests preposition logic is study of logic of prepositions and then in this lecture what we will be doing is we will be talking about what we mean by prepositional logic, basic introduction of prepositional logic and we try to cover the syntax of a prepositional logic. So in this the coming lecture forthcoming lectures what we will be covering is like this first we will start with the syntax and semantics of prepositional logic. So any language has its own syntax and semantics so in the same way language prepositional logic is also viewed as a language which has its own syntax and all just like our English language has its own syntax and semantics and then we will talk about how this syntax and semantics are related to each other. So just like that in prepositional logic is also a kind of a language so which has its own syntax and semantics and we will be talking about in this lecture we will be focusing on syntax part and all. So later we will be studying some of the translations of English language sentences into the language of prepositional logic that is what we will be studying from the next few lectures and then we will be studying three important kinds of statements which occur in prepositional logic. They are the statements which are always true which are considered to be tautologies and the statements which are considered to be always false they are considered to be contradictions and the statements which are neither true or false are considered to be contingent kind of propositions. So we will classify our group of statements into three categories and then we will only be interested in tautologies because it so happened that all the tautologies in our language that is the language of prepositional logic they are all turned out to be valid formulas. So ultimately our journey begins with constructing some kind of well formed formulas so that is what syntax will take care of it and then after constructing basic formulas etc. and all then we will talk about what we mean by these formulas and all that means we will start giving some kind of truth conditions to a given formula that means meaning of a given formula means giving truth conditions to a given well formed formula. So then we will talk about validity in the context of syntax and in the context of semantics and then we will be talking about some of the important decision procedure methods which tells us whether or not a given sentence is valid or when two groups of statements are consistent to each other or when how can we check whether a given formula is tautology or contingency or a contradiction etc all these things are common will come under the category of decision procedure methods. So there are few decision procedure methods to start with we will begin with the simplest possible method that is the truth table method and truth table method works as long as the number of variables are considered to be less in number that means two or three variables are there then it will be easy for us to handle and then if it would not if number of variables increases 5 to 6 and then there will be 2 to the power of n entries in the truth table so things will be difficult for us but maybe a computer can manage it in a better way but for us it will be difficult to handle. So then we will be talking about one interesting and important method which serves as core of this course and all so that is the semantic table x method or it is also called as tree method etc so then there is another method with which you can talk about validity etc and all so that mean in that method what we will be doing is we will reduce the complex well form formula into that well form formula consist of implication double implication etc and then we will reduce it to a formula which consists of only negation disjunction and conjunction so if you can reduce that complex formula into this particular normal forms which are usually we usually call then these things these formulas are called as conjunctive and disjunctive normal forms and then once you reduce the formula into conjunctive and disjunctive normal forms you can clearly you will easily come to know whether a given formula is a tautology or all these things will be studying in the forthcoming lectures and then we will also talk about one of the interesting methods that is resolution and repetition method. So these are all semantic kind of methods there are some other methods with which you can still talk about validity in the context of syntax so that is what we call it as syntactic consequence or syntactic validity etc so there we study the natural deduction method with which you can talk about some of the syntactic consequences etc which are there are some formulas which are obviously valid and all those things should talk we talk about in the context of syntax we present natural deduction method. So now what is propositional logic and why we need to study propositional logic so first of all it is a systematic study of logical propositions logical propositions sense that a proposition is a sentence which can be clearly spoken as either true or false in the basic concepts we have we have said that language can be used in three purposes one is logical usage another one is expressive usage and then there is other kind of way in which you can use the language for example language can also be used to express some kind of feelings etc and so we will be considering or we will be interested in only those statements which can be clearly spoken as either true or false suppose if I say that this is a chalk piece if it is referring to the fact etc and all then actually this is a chalk piece and that is the statement is considered to be true. So only those sentences we will be taking into consideration and then these sentences are combined with the help of what we call it as some kind of truth functional connectives which are considered to be or and if not if and only if these are the things which we express it in the English language but in the formal language we express it in terms of its own symbols and all. So we will talk about it little bit later about these symbols with which you can represent these phrases or and if not and if and only if. So preposition logic is a systematic study of these things logic of prepositions or logic of study of the connectives etc all constant all come under the category of prepositional logic and we it only solely studies about these connectives only and there are no quantifiers etc and all in this kind of language quantifiers in a sense for all x there exists some x etc and all. This is the most simplistic kind of language which usually with which you can represent mathematical kind of reasoning and all. So one should note that there are different kinds of reasoning which we have said in the beginning of this course that you know reasoning can be divided into deductive and inductive kind of reasoning and deductive in the deductive reasoning prepositional language prepositional logic is one of the outcomes of this deductive reasoning. There are other kinds of deductive reasoning in which some of the fundamental principles of logic they are weakened and all and then we will enter into some other kinds of logics which are called as non classical logics. So it is also considered to be a formal language which is used to express sentential formulas or sentential language or it is also called as a sentential language etc. The most basic logical inference are about usually their expressions that are combinations of sentences involving not are and if and if and only if it is very difficult for us to talk about connecting two sentences without using these logical connectives. Suppose if you say this is a chalk piece and this is a duster and all if you want to combine these two sentences there should be some kind of connective which can we should connect these two sentences. So in that case either you say this is a chalk piece or this is a duster in that case it is P or Q if it is chalk piece and it is a duster it is called as P it is expressed as P and Q and all. So in order to join this atomic propositions which we call atomic propositions are the propositions which cannot be further reduced into any other kind of preposition and all just like P's Q's R's etc they are all called as atomic propositions and all. So these atomic propositions combine with these logical connectives and form complex formulas and all or compound formulas etc. So this is also considered to be truth functional calculus or it is also called as propositional calculus it is a study of those statements whose truth values are determined by the truth value of their component parts. One of the important point one needs to note is that a truth value of a given formula in the propositional logic is solely determined by whatever truth values that its individual constituents takes and all. Suppose in the formula P or Q the truth value of P or Q is solely determined by whatever values P and Q takes and all and depending upon how the logical connective that is in this case R behaves and all. So we will be talking about semantics under the context when we talk about semantics of propositional logic we will see how this these connectives behaves and all R and implies etc. So they behave in a certain way in the ordinary day to day language and as far as possible we are trying to convert we are trying to translate the English language statements into the appropriate language of propositional logic. So in the process of translation one might even be disappointed for example a simple example could be like this usually we say when we are sick we go to the doctor and all and this can be put into this format. So I became sick and I went to the doctor so that is what we usually do and all and that is represented as A and B. The same sentence can be represented as B and A also because A and B and B and A are identical to each other the logically equivalent to each other because they have only same truth values and all. Suppose if you say that I went to the doctor and I became sick so these two are different things you know nobody goes to the doctor to become sick enough. So I went to the doctor and I became sick is totally different from I became sick and I went to the doctor and all this seems to be some kind of other operators which are important to analyze these particular kinds of statements. So that means truth value of A and B is not solely determined by the truth value of A and B which is but it depends upon some other extra logic some other factors and all. So these kinds of things are called as intentional logics and all intentional logics are not part of our course and all but we will be talking about only extensional logic. That means the truth value of a given compound formula is solely determined by the truth value of its individual constituents. So to motivate ourselves we will ask this fundamental question where we use this propositional logic and all. So usually in general you know we present propositional logic as a formal language which tries to capture basically mostly mathematical reasoning or it can be used to capture logical reasoning and the argumentation that involved in day to day discourse but not all the all kinds of propositions can be captured in this propositional logic and all. For example if you say all birds flies tweet is a bird and tweet flies and you got an extra information that tweet is a penguin and penguins does not fly. So in that case you need to give up some of the conclusions that you have derived earlier that is all birds flies or maybe tweet is a bird that flies there are some other things which you have derived earlier we need to withdraw those conclusions which is not permitted in the language of propositional logic. So these are the cases which come under the category of defaults and default reasoning is not the one which we are trying to consider here. There are many other kinds of reasoning like counterfactual reasoning for example if you say if I become the prime minister I would wipe off all the corruption etc. So that means you have not become the prime minister that means the antecedent of that particular conditional is false and the semantics of propositional logic tells us that if the antecedent is false irrespective of whatever the consequent, irrespective of whatever truth value the consequent takes your conditional the truth value of the conditional is always true and all. In that case there is no distinction between if I had become the prime minister of India let us say pigs would fly and if you say if I had become the prime minister of India pigs would not fly it does not make any big difference in all because in both the cases the antecedent is false and irrespective of whether pigs flies and pigs would not fly it does not matter that makes the whole conditional sentence if I had become the prime minister of India pigs would fly is true in the same way equally if I had become the prime minister of India pigs would not fly will also be true. So these are the things which we do not want in day to day discourse because this goes against our intuition so in that case we need to maybe we need to extend this propositional logics and we need to talk about some other things in all so that is not what is of concern to us but there are some cases in which the propositional logics works better mostly the mathematical reasoning and all. And the another interesting point is that propositional logics are directly used in analyzing digital switching circuits and all we have a simple complex digital switching circuits which has on and off switches and all there are various kinds of gates that we use and all and are any NAND gate etc nor gate etc XOR gate etc all these things the underlying logics are nothing but Boolean logics I mean Boolean logics are usually which is also called as propositional logics because propositional logics is they are talking in the same way as Boole represented this kind of reasoning with the help of algebraic operations plus and multiplication and these plus in propositional logics stands for R the connective R and multiplication stands for the connective N. So there are more or less the principle of duality helps us in translating the Boolean algebra into the language of propositional logic which we will talk about it little bit later. So you are given a complex circuit and all so you will translate that complex switching circuit into an appropriate formula of a propositional logic again a complex formula and all you will forget about the switching circuits for a while and then you simplify that complex formula maybe it is P in plus Q in plus R or something like that PR Q or something like that big compound formula and then you will reduce or simplify it using demarcans rules all kinds of rules law of absorption law of distribution etc and all then you simplify the logical formula and all. So then again you will go back to the switching circuit and then you will directly study the simplified formula complex formula let us say it is reduced into just PR Q and all that means if you have a complex formula which involves 4 or 5 switches and all which somehow reduces to just PR Q and all that means you just require only 2 switches that means the complex digital circuit is exactly same as the simplified digital circuit that is the simplified formula that we got we arrived at. So in that sense proposition logics are directly used in analyzing only simple digital switching circuits where you have on and off switches so we can also study how this logical connectives behaves by using SNAND and OR gates etc. Exclusive OR gates etc that is a practical application of this propositional logic and one of the most important thing is that suppose if you want to talk to the machines etc and all if you talk in English and all it will not help us in any way it would not answer you anything. So the first thing one needs to do is to represent our knowledge that knowledge claims etc and all into some kind of formal language and all the most simplistic kind of formal language that we use is the propositional language of propositional logic which is also considered to be a formal language. Once you represent it in terms of some kind of formal language then the next thing is you know one can write one can prove some theorems in all and one can write programs one can write algorithms etc and all these things follows in all. So basically propositional logics are used as a knowledge representation tool in artificial intelligence and even in the computer science. So our concern is to capture the mathematical reasoning or reasoning that is that we use it in as far as possible in the day to day discourse as well. So what is this propositional logic? So when we study the history of logics into consideration in greater detail I will talk about how this propositional logics originated and how Aristotle logics are gradually logician have gradually had to ignore this Aristotle logics because it could not account for many things although it can still be used as one of the paradigm cases of logics in all because it dominated for more than 2000 years. So one can again fall back on Aristotle logics and you can talk about some of the important points in important things that we can borrow from Aristotle logics to the modern logic. What is this propositional logic? It is a branch of formal logic, formal logic in a sense that there is a distinction between content and form and all. Suppose if your argument involves the analysis of just forms only like all x or y all y are z and all x are z and all in that particular kind of argument or pattern. So what is important is the form all x in place y y in place z and x in place z. So only form matters and all it does not matter what you substitute for x y z it can be donkeys it can be cats it can be anything and all. So whatever you substitute into x and y and z suppose if it is in this particular kind of form x in place y y in place z then x has to be z n as far as you believe in the two value logic that means your sentences are only true or false it cannot be neither true nor false all these things are ruled out. So then the formal logic means you are studying the you are interested in the forms of an argument but there are some other arguments which we have presented in the basic concepts while studying about the basic concepts they are like this for example if you say all atoms are invisible so all atoms all this room is made up of atoms for example let us say this room is made up of atoms are invisible so this room is invisible. So you will not be able to find out any flaw in this argument understand till you analyze the terms that you have used in your argument so in that case there is a shift in the meaning of the uses of the term atoms so in the first sentence it is used in a different sense the second sentence is used in a totally different sense but in propositional logics or the formal logics which we take into consideration it is presupposed that there is no shift in the meaning of the words or phrases that you have used in your logics. So it is a branch of formal logic where the basic units are sentences so all the sentences will take this simple letters and all which are called as sentential variables suppose if you want to represent so critisies wise it can be represented as a simple letter s or it can be represented as a simple letter w it is up to us to take our own letters because we have any number of letters which are available so you can take any other letter it can be represented as even all false. So it only deals with the constants that stands for entire natural language sentences and the ways these constants may be combined to form more complex expressions what are these constants and are implies if and only if etc. So we have some sentences which are out there and then we have some logical connectives and these sentences combined with this logical connectives and form a complex compound formula and all. So now sentential logic is also concerned with the way in which simple sentences are combined by means of sentential operators again negation conjunction disjunction etc. And they form the most complex kinds of sentences and all. And you have to note that sentential logics or prepositional logics or prepositional calculus all these things are one of the same in all. So they have no quantifiers in all so quantifiers in a sense for all x there exists some x etc. For example if you want to represent all men are mortal it simply prepositional language you represent it as m a letter m or something like that you represent it in terms of quantifiers in all you will say for all x x is a implies x is mortal in all. So we do not have quantifiers in all that sets some kind of limitation to the prepositional logic and if you want to make your language little bit richer in all then you need to add these quantifiers in all. But again prepositional logics on the one hand they are elegant very simple and it has wonderful features in all like there are wonderful features everyone every logician would be striving for that is consistency completeness etc and all a system is called as consistent especially when you are not able to derive you cannot derive both x and not x at the same time from the same set of assumptions etc. If you can derive both x and not x then your system is considered to be inconsistent suppose if you talk about it is raining and it is not raining given some kind of assumptions then this seems to be something wrong with your system not wrong is a system is called as inconsistent and logicians will hate this kind of inconsistency in fact even mathematicians as well inconsistency is treated as some kind of hell why because if you are given an inconsistent statement you can derive anything any kind of strange kind of preposition all without violating the truth value of your prepositional formulas in all. So that is one problem which arises so we look for only consistent kind of we look for a system to be consistent in and the second wonderful feature is completeness in all. So there are in our language that we are going to construct we will form some kind of well form formulas some well form formulas are considered to be valid some are invalid in all. So all the well form formulas which are considered to be valid so they are all can be provable in all. So you have to find a proof for this all the well form formulas and in the same way if all the proof all the formulas that you have proved that has to be at the end of the day it has to be true and so you prove lots of things and all but at the same time it is false and all you know you are not you are not solved your purpose is not solved your purpose and all at the end of the day that statement has to be proved has to be true and all that means it has to be tautology and all. So whatever is provable is true and whatever is true is provable and then your system is called as complete and one of the wonderful features of this preposition logic it is considered to be the minimal kind of representation of your knowledge these are consistent complete compatible all these wonderful features are there present in this prepositional logic. Once you add quantifiers etcetera and all then the complexities arises and we will talk about incompleteness etcetera in the context of Godel at the end of this thing while talking about predicate logics we will talk about Godel's incompleteness theorem which tells us that there are some formulas within the language of first order logic in the sense first order logic means it is preposition logic plus predicate logic in those languages in particular in the predicate logic there are some formulas which are obviously true but it cannot find proofs and all so that makes a system incompletely. So it has no quantifiers the preposition logics has no quantifiers and the other thing is that sentences that are generated from the other sentential connectives are considered to be compound sentences and all so you have some sentences like it is raining etcetera and all and you have other statement grass is wet and all if you combine these two sentences with it is raining implies grass is wet and all that will become a compound kind of sentence. So this out of these connectives and are implies if and only if they are all binary connectives because it connects at least two sentences and all whereas the negation is considered to be an unary connective and all. So suppose if you say this example Mars is a planet which has no satellites suppose if you say Mars is a planet and Mars has satellites and all so it is a conjunction of two statements and all if at least one of the statements is false then it makes the whole conjunct falls and all that is a way that conjunction behaves and all suppose if you say Mars is a planet which has no satellite that makes maybe making the sentence true and all the second sentence in particular at least one of the conjunct is false then obviously it makes the whole conjunct falls and all even though Mars is a planet is true but the second one Mars has satellites etc that has to be that is falls and all it makes the whole sentence falls and all. So we will talk about truth and falsities little bit later so these are some of the pertinent or fundamental questions that we will be asking ourselves which motivates us to consider or study the preposition logics which are considered to be the minimal kind of logics with which you can represent basically mathematical reason. So these are some of the questions the first question is what does it mean to say that one sentence logically follow from others certainly from others that is what is validity will take care of this particular kind of question and all. So logic is after all all about what follows from what you have few sentences and why after all why we need to use tools of logic and all you have few sentences and then we want to move from these two sentences to another sentence and all. So there are some kinds of techniques which we use either it is the way you move from premises to the conclusion if it requires that requires some kind of necessity some kind of certainty or if you do not want any new information to be there in the premises do not want any new information to be there in the conclusion which is not there in the premises etc then we use deductive reasoning and all in the same way if you want we have to few premises to begin with and you want to conclude something then conclude something and then your argument can only be strong or we can all we cannot be 100% certain etc and all in that context you use inductive reasoning natural sciences usually follow this particular kind of reasoning and all. So this is what we have studied in the basic concepts and all. So one of the fundamental questions any logician would be interested to ask is what follows from what so that validity the concept of validity will take care of this particular kind of thing you know we know that particular argument is valid for example it is raining and grass is wet and all from that you can derive it is raining enough A and B you can derive A. So how do we know that this A and B A can be derived and all is there any logical procedure with which you can tell us you can tell whether it is raining is derived from it is raining and grass is wet. So if a sentence does follow logically from the others what are the methods of proof etc which are necessary to establish this particular kind of fact. So in that context we use two table method we use semantic tablox method we use some other interesting and important methods and all and then after finding out all these things we will ask ourselves is there any gap between what is what we are proving using some axiomatic system that means you start with fundamental axioms and then you use some kind of transformation rules some kind of natural principles that you use in the logic modus ponens modus toluens etc and then you prove something that means the last step of your proof is considered to be a theorem. So is there any gap between what we can prove and what is true about let us say natural numbers etc that means whatever is probable is not true or whatever is true is not probable and there is a gap enough. So if that is the case then the system is called as incomplete but in the case of propositional logics this problem will not arise there are no gaps so propositional logics is considered to be complete consistent etc. And the other question that computer scientist must be interested in is what is the connection between logic and computability. So we will not talk much about this particular kind of thing but we will try to cover the first three things. So let us start this thing with some kind of motivating example the one which we use it in day to day discourse we have said in the beginning of this lecture that propositional logic also covers some of the important things related to arguments related to day to day discourse. So it is an interesting example there was some robbery in which lots of boost was stolen and we are trying to find out who is the convict in all who is considered to be guilty here. So these are the conditions which are given the robbers left in a truck after stealing everything in all the bank they might have stolen something goods etc they left in a truck and all and our information has this particular kind of things and all. So we have seen only three robbers so running away from stealing everything etc and all. So nobody else could have been involved other than instead of naming them it is A, B, C and all. So that is the first thing which we know at least some information that we know and we also know that somehow the history of this culprits and all guilty people are robbers and all some information we have may be so C never commits a crime without A's participation and all that means whenever C does some kind of thing A will always be there and all that is what we are sure of that means C will not commit any robbery etc and all unless A's participation is there so they will be coordinating with each other nicely and all so C and A will always be there in that particular kind of site wherever the robbery has taken place that is one thing which we know and the third thing which we know is this B does not know how to drive that means after let us say he has stolen lot of money and all but he himself cannot drive and all that means only maybe A and C knows driving and all but in this case maybe might have to depend upon A and C and all he himself cannot drive so that means suppose if he is alone and all then he cannot you will be caught and all so now using this information and all how do we know that whether or not A is innocent or A is guilty and all so in this case thus the letters that you are seeing here A, B, C etc and all so they all stands for A means A is guilty B means B is guilty C means C is guilty and all of course you can say you might say that you know I will take I will represent guilty as not A and all you can do it in that way as well but in general we treat whosoever is involved in a robbery is found to be guilty on so A, B, C means A is guilty B is guilty etc. suppose if you say not A not guilty means is innocent and all that is opposite of that so we are already use some kind of connectives and all. So now you can say that this puzzle or problem can be solved you can use some kind of mental reasoning etc and all you do some kind of exercise and then you can come up with your answer and all that seems to be very difficult and all so it will be very difficult if you do not have some kind of symbolic form and all. So this is what has taken place in the history of mathematics in particular ancient in the ancient days in the period of Egyptians so they do not have any knowledge of the unknown numbers and all they were all struggling about this particular important thing what adds to one-fifth of its number leads to 17 so they struggle for more than 200 years because they do not have this concept of unknown number you know so these days if you ask even six class student will be easily able to answer this particular kind of thing you know the problem there was is that there were no symbols that are available to them they did not they did not think in that direction so only after 200 300 years after their contribution to the mathematics and people could solve this particular kind of problems and all. So what I am why I am saying this particular kind of thing is this because this fact that if you do not have some kind of symbolic form and all things will become very messy you might say that there are only three sentences I can do it I can mentally calculate I can mentally know what is going to be the answer and all if the number of propositions increases and all our information is big enough then the best way to represent it is in terms of some kind of symbolic form. So this motivates us to study the propositional logics and all if you want to know whether A is innocent and A is guilty etc and all we have to represent it in terms of appropriate language that language could be like this so depending upon the sentences we translate the sentences that occur in this puzzle into the appropriate language of propositional logic and then we will see when these three sentences are simultaneously happen to be true and all when these three sentences satisfies and all that means when these three sentences taken together is going to be true and all so that gives us some kind of answer for this particular kind of thing and all. So this no one no one else could have been involved other than A and B and C means either A has to be there B has to be there or C has to be there and all there is no other kind of thing D is not there. So A, R, B or C that is the one with which you can represent the first sentence and the second sentence is represented as C never commits a crime without A's participation that means C is important for A and all presence is important for A's presence is important for C and all so this can be translated into C in plus A and all so it is like a sufficient kind of condition and all. So the third one is that B does not know how to drive that means B requires help of either A or C and all so in the first case it is like this B in place either B has to be with A and all so that you can run away or B has to be with C and all has to coordinate with C so that you can run away from that side and all after stealing everything. So now once you represent this thing into this particular kind of formulas and all then we will come to know when these three formulas are jointly true and all so when under what context they are all consistent to each other etc and all we will solve this problem little bit later but I will postpone this problem to for a while and then once we talk about either truth tables are some kind of semantic method or some other kind of method we will come back to this particular kind of problem but our basic thing basic idea that I am trying to give is that when you have some problems like this once you represent it in terms of some kind of symbolic form and all and use principles of propositional logic then you can solve this particular kind of problem. So if you are little bit impatient etc and all the answer for this thing is that A is considered to be guilty and all so how do we know that A is guilty some methods with which you can show that A is guilty follows from this particular kind of information all so one way of doing it is these are the three propositions that we have P1 P2 P3 etc for example for the time being we consider in that P1 and P2 and P3 are all leads to some kind of tautology statement which is obviously true if you can show that that is a tautology then instead of showing that this is a T and all what we have to show is this particular kind of thing and all. Let us say the first sentence is represented as P1 and the second sentence is represented as P2 and P3 this is the information that we have and this all should lead to suppose if A is guilty means guilty means it should be like this and A is innocent means it is not A innocent and guilty are opposite to each other. So if you can show that this whole thing P1 and P2 and P3 all these sentences implies that A means guilty if we can show that this is a tautology that means under all interpretations of P1 P2 P3 whatever values that you assign to the variables that occurs here it is always true then you can say that A is considered to be guilty. So if you want to show that A is guilty or to show that that particular formula which is there in the brackets is should be a tautology and all if it is not a tautology then A is not said to be guilty and all. So that is one way of proving it there are there are varieties of ways with which you can show that A is guilty follows from this particular kind of information. So one important method that we will be using a little bit later is the semantic tableaux method. So this is the motivating example which leads us to study the prepositional logics there are lots of puzzles we will be talking about some other puzzles little bit later which are cooked up by a famous logician Raymond Smollion he has constructed wonderful puzzles which are which come under the category of knights and the use puzzles. So these puzzles goes like this so you have to imagine an island in which there are only two kinds of inhabitants. So let us say type A and type B so type A always talks truths type B always tells lies for example if you ask type A who always tells to these two percent is equal to four they will say true yes it is and if you if you ask the same question to the news which are of different types which are a different type they will answer is two plus two is equal to four they will say no suppose in the same way if you ask them is two plus two is equal to five news will answer yes. So news always lies and knights always tells truths you know it is not the case that news all news tells truths you know so a liar cannot tell truths you know that is the condition that is there now. So now Raymond Smollion constructed lots of puzzles in which your a stranger goes to this particular kind of island and he asks some kind of questions you know he wants to find out what type he is whether he is an A whether he is an A or he is a knight you know for example if a stranger goes to an island and the stranger says the inhabitants says let us say A says I am not a name and B says I am a knight so based on that particular kind of information so what is that you are going to infer and all. So these are some of the puzzles again which are represented by the language represented in the language of propositional logic and then once you represent in terms of propositional logic and things will become simpler and all we can solve it with using basically semantic tab looks method especially these knights and his puzzles you know so basically what is that I am trying to say is simply this that there are some kind of puzzles which you can solve with the help of the propositional logic and then there are some instances like the one which we have seen earlier and there are some other things like how the complex digital circuits can be simplified into simplified digital circuits by using the principles of propositional logic except. So far I have discussed only the background of this propositional logic I have not discussed the origin of propositional logics in a different context I will talk about in the history of logic I will cover how this propositional logics have originated and all basically it began with Bool in the mathematical analysis of logic and then Frege Russell Whitehead advanced it further and then there are some developments Godel etc all these things which will be trying to cover basically from 1890s to 1930s so the developments that we will be covering under propositional logics. So now every language has its own syntax and its own semantics in all the languages that we are trying to talk about they are considered to be formal languages in all. So what do you mean by a formal language why in what sense it is distinct from the natural languages like English, Hindi, Urdu whatever it is. A formal language consists of a set of symbols those symbols can be there the ones which you usually choose whatever symbols you can take into consideration but you fix those symbols in the beginning of constructing your language and all. So you have some symbols in this case p's use r's etc and all and then you have some rules which tells us how these symbols are constructed and then you can form some kind of grammatically correct strings of symbols in that particular kind of language. For example in English language we have some alphabets to begin with a to z and then this a to z some different alphabets combined together and form some kind of words let us say cat, mat etc. So these words combined together in a certain way and form some kind of grammatically meaningful sentence for example if you say cat is on the mat it makes sense for us to talk about these kinds of things which is constituted out of alphabets a to z and then cat is another word which we know maybe concept mat is another concept cat is on the mat is considered to be a grammatically correct sentence or you can even say mat is on the cat is also usually we do not say that but you will say you can say that also but if you say mat cat on is and all and any child can easily recognize that it is not a grammatically well formed kind of sentence you know. So how do we know that is a grammatically correct sentence grammatically incorrect sentence we have to learn some kind of grammatical rules with which you can judge whether it is grammatically correct or not in the same way in the language of the formal language in particular we have some symbols and we know how these symbols are combined together and form some kind of a string so this string is defined in this particular kind of sense a string or a word just like you know cat is a string which consists of C A T and on in the English language we have a string called P Q R implies X etc and all a string or a word in a formal language is any finite sequence of the symbols in the language especially so we include in an empty string also as consisting which consists of no symbols at all that is also considered to be a string and all which which will not have any symbol and all examples of strings or there may be thousands of things which you we have what we have are simply like this we have some propositional variables P Q's R's etc and all and we have logical connectives or implies negation if and only if etc and all and out of this logical connectives and this thing and then we have parenthesis etc which tells us which are punctuation marks just like in the case of English language we have comma full stop etc and all we have parenthesis and all which tells us that you know how to read a given well-formed formula when a formula is considered to be a well-formed form so P R P Q and then closed by right parenthesis in the first case implies P Q R they are all strings and all but not all strings are considered under the category of well-formed formula you know for example if you say cat C A D cat makes some kind of sense to us we know the concept of cat we know that it is seems to be a correct string you know it can be T A C also or A C T or something like that so you can even talk about this particular kinds of things you know but that is not a meaningful kind of word which we know in the same way in the formal language if you say P R P Q and closed by some kind of parenthesis then that is not a meaningful kind of well-formed string and all so there are thousands of strings which we can form and all using symbols and logical connectives etc then then the question arises is what is considered to be a meaningful kind of string and all or in this case it is well-formed formula so these are the things that we are trying to begin with we have prepositional variables which can be infinite and all because there are so many sentences which we can express it in terms of formal language so we our symbols are also exhaustive and all although it is finite but you can even take it as infinite also P 1 P 2 P n and all these things and if it is exhausted you can use Q 1 Q 2 something we have n number of natural numbers you can use that so we often use usually P Q R etc and all to represent the prepositional variables suppose if you say it is raining and it is not raining it is represented as R and not all so symbols for the prepositional we have prepositional connectives negation or and implies and if and only if we need to talk about how this what we mean by these connectives and all when we talk about semantics we talk about this so we do not we are not worried about the meaning of this connectives at this moment in the syntax what we will be interested in is we just know how this formulas are generated and all it is just like start trying to build a building and all suppose if you are trying to build a building and what you need is a raw material that is bricks concrete and cement etc and all you just mix it and all and then later you know fine kind of restructuring etc that you that involves involves some kind of with that you can say that it is analogous to semantics. So in addition to the symbols and logical connectives we have parenthesis left hand left parenthesis you have a comma or pull stuff or your right parenthesis in all parenthesis tells us how to read a given formula. So now with this I think I will close this lecture and all we have we need to know we have generated lots of strings etc and all so now we need to talk about what we mean by a well formed formula and all so how do we know that a particular string is considered to be the well formed formula there should be some kind of definition with which you can talk about decide about when you can say that a given formula is a well formed formula and all. So this is the definition goes like this every preposition variables PQRs etc and all these things are considered to be a well formed formula whatever variable that occurs is considered to be a well formed an empty string is also considered to be a well formed formula which is nothing is there that is also considered to be a well formed formula and if a is a well formed formula if assuming that there is a atomic sentence is obviously well formed formula if you add negation in the left hand side that is also considered to be a well formed formula. In the same way if A and B are two atomic prepositions and which are considered already considered to be well formed formulas that means their preposition variables as is the case of first rule tells us and then not A is a well formed formula A and B is a well formed formula which is in put in left and right parenthesis A or B is also considered to be well formed formula A implies B is also considered to be a well formed formula and all. So the fourth rule tells us that nothing is a well formed formula which does not follow these three rules and all. So for instance finally we will see that for example if you have particular kind of sentence like P1 implies not something like P2 not implies P R and all for example if you have this whole complex sentence like this so this is a string and all definitely this is called as a string because it is generated out of your symbols and logical characters but this is not a not considered to be a well formed formula because of this thing this is not constructed out of the four rules that we have said only three rules appears here but the fourth rule is implicit in this one that is nothing is a well formed formula which is does not follow these things and all. So it is none of it is not in this particular kind of format so it is not considered to be well formed formula. So in this class what we have discussed is simply like this we presented the basic idea of propositional logic that means what we are trying to do in the propositional logics and then we started with the syntax and then we talked about what we mean by a meaningful string and all mean not meaningful string when you say that a given well formed formula when do you say that a given formula is considered to be a well formed formula that means you have generated thousands of strings and all not all strings are all of importance to us just like at C A T is one word which we use we do not use T A C etc. in English. So in the same way maybe we have some kind of meaningful so we have some kind of strings which are important to us so these rules the definition gives us some kind of idea of whether or not a given formula is a well formed formula or not. Once we have well formed formulas then you can talk about whether or not this given well formed formula is a valid or it is a tautology all these things you can talk about it little bit later. So with this we will end this lecture.