 Welcome back in the last few lectures and the corresponding lectures we presented propositional logic and the predicate logic now we have come to an important topic of discussion that is we will be discussing the limitations of classical logic, classical logic we mean predicate logic and the propositional logic. So before that we have presented informal logic and fallacies and then Aristotle's logistic logic etc but mainly classical logic we mean propositional and predicate logic. So the question that we are trying to answer is this is the case that classical logic explains all kinds of reasoning or it is restricted to just mathematical reasoning, we say that it is restricted to mathematical reasoning classical logic as far as possible it tries to capture mathematical reasoning in a better so in this lecture I will be talking about some of the important limitations of classical logic in a very nutshell. So as we know classical logic we mean first order logic and logic is considered to be a systematic study of argumentation principles of valid reasoning whether deductive or inductive kind of reasoning mostly we focused our attention on deductive reasoning. So classical logic we mean first order logic that is what we have presented so far it is propositional and predicate logic and they are considered to be the good starting point for the study of reason and it is most appropriate for the mathematical reasoning and it is important property of this classical logic is that it is considered to be bivalent and then that means the sentence can be either simply true or false it has only two values and it is based on the material implication the material implication A in place B is defined as not A or B it is introduced by Bertrand Russell so all the classical logics are based on the material implication. So now classical logic is not appropriate for formalizing human reasoning we will be seeing some examples where it cannot be applied and where it does not make to apply make sense to apply this classical logic classical logic fails to provide satisfactory account of the following things one is conditionals conditional statements if all the conditional statements are simply expressed as if P then Q then there will be some kind of problems that we come across we will talk about paradox of material implication etc and then we will see that not all conditionals can be expressed in terms of simple if P then that makes use of the definition of material implication and also the arguments involving possibility necessity and logic of knowledge belief vagueness etc all present numerous problems for the classical logic that means we need to move beyond the classical logics and we need to talk about some other kinds of logics so non classical logics are basically developed to overcome these defects in the classical logic that means what essentially happens here is that classical logic is based on some fundamental laws of logic you have to withdraw at least one of these fundamental laws of logic so usually classical logic so based transitivity prints property of by violence that means it has only two values and it is considered to be monotonic in a sense that even addition of new premises will not lead to the withdrawal of the conclusion that you have drawn earlier and the problem here is that but most of the common sense reasoning that we usually employ in day-to-day discourse is considered to be non monotonic that means addition of new information you need to withdraw the conclusion that you have drawn earlier classical logic obviously will have will be having no help to represent intentional concepts like modality and time these are considered to be intentional but in this course we have we have focused our attention only on extension kind of concepts that means the truth value of a sentence is solely determined by truth value of its individual constituents then we say that the logical operator is considered to be extensive in nature if the truth value of a compound formula is not solely determined by the truth value of its individual constituents then it is called as intention and human knowledge is usually be incomplete and obviously it can be inconsistent we draw a lot of inferences based on incomplete information we do not wait for the perfect information to come by we base our decisions on incomplete and inconsistent kind of information and classical logics cannot express this incomplete and inconsistent kind of information so we are not charging this classical logic but basically classical logic is a starting point and it is used for capturing the mathematical reasoning and it also fails to explain the concepts which are related to vagueness and as we know vagueness is considered to be part and parcel of our life most of the time we will be using expressions English language is obviously considered to be vague which consists of vague predicates such as tall rich poor etc all these things the judgment about these things inferences based on these things will be purely based on our observations rather than the principles of logic etc now non classical logics have emerged as a response to this particular kind of defects in the classical logic there are number of logics which are which exist there these are modal logics tense logic is talks about time epistemic logic is not logic of knowledge and belief logistic logic logic of belief deontic logic talks about logic of obligation forbidden etc and all dynamic logics conditional logic intentional there are lots of names which are given to these logics and all so depending upon the usage to capture some of the interesting inferences will be you making use of this different kinds of logics and we have on the other hand there all extensions of classical logic it is it is not enough that only classical logic explain all kinds of situations and all but logic has to be applied to our day-to-day discourse as well it is just we not have to apply to devil's machines etc and all our computers etc but it has to apply to our day-to-day discourse so in that process we will be either extending the classical logics or we will be deviating from the classical logic deviating from the classical logic in a sense that we have three fundamental laws of logic law of identity law of excluded middle law of non-contradiction if at least you deviate from one of these things that means you drop one of this fundamental laws of logic then you are doing deviant kinds of logics so into statistic logic where law of excluded middle need not be a theorem in your in that logical system then it is into statistic logic if you drop the law of non-contradiction you are dealing with the para consistent logic and you are allowing for many values other than T and F you are talking about many value logics in one instance of many value logic is the fuzzy logic the father the founding father of fuzzy logic is of the view that doing classical logics is like coming to the party with formal dress shoes tie hat everything in all so that is what is considered to be doing formal logic and doing fuzzy logics or non-classical logics is like going to the party with simply T shirt jeans slippers some some kind of comfortable dress etc or short or something like that so that is what we prefer mostly so this is the way we nicely puts this analogy into the context and very distinguishes classical and non-classical logics the means non-classical logic for him is fuzzy logics so we will be talking about some of the basic principles of logic they are like this law of identity which is stated as P is P law of excluded middle either P is the case or not P is the case and law of non-contradiction it is not the case that something is simultaneously true and false that is law of non-contradiction a contradiction occurs when one statement excludes the possibility of another but it both are claimed to be true so truth is not considered to be self contradictory so these are the three fundamental laws which defines this classical logics now these are the three laws are considered to be foundation for mathematical and physical and rational thinking now one question that we can ask is this thing or all these laws complete it completely describe all kinds of phenomena or not now a variety of arguments can easily be produced to show that these laws are considered to be incomplete that means the applies to only static case when it come comes to dynamic case it may not apply they do not specify all kinds of reality for parts of reality can be shown to contradict one or more of Aristotle in loss here are critters find it BC is come up with this problem of change is of the view that you cannot step into the same water twice if you step into the water twice the water goes away and then the next time when you step into it is not the same water so everything changes this is what Buddha also talks about so here are critters pointed out that for a thing to change it must turn out to be something else for example if this is the desktop it looks like same even now tomorrow or day after tomorrow also it looks like to be the case like that but something is changing in this particular kind of desktop so if this tester changes means that it must turn into something else and then he asked how a thing could be something other than itself if this desktop changes to some other thing of course the desktop only with some other properties then how can whatever is changed the jester is same as the old desktop so it is in that contest is introduced this thing you cannot step into the same water twice so now if I restart loss are taken are taken to be all the fundamental laws of logic then logically they can be no change whatsoever because the change negates all these three laws for example if that means either change does not exist or it is totally illogical in a sense that it is now it is violating all the three fundamental laws of logic since all measurements detections thoughts perceptions are simply considered to be changes they are dynamic in nature whereas Aristotle and logics are simply applies to on the static case it follows that these operations logically cannot exist and then in that context logicians in they introduce the fourth fundamental laws of logic if you allow if you accept the fourth law then you need to deny the three laws of logic or if you accept the third law of logic then it is explicit implicitly you will be the fourth law will become implicit except so what happens is that when something changes and on a is not equal into a that is law of identity fails if that fails in law of excrual middle also changes and then of course it has its consequence on law of non contradiction in that sense how the three laws fails this is one problem with respect to this thing and the second problem is when you are referring to modal sentences you need to move away from the classical logics one example could be if it is necessary the case that two is the smallest prime number then two is the smallest number you need to have some kind of modal operators to make it distinct from the actual operators so we need to maintain the difference between something is actually true something is possibly true something is necessarily true this is what we all the time use in the day to day discourse you can say that it is possible that water exists on the mass obviously it is not necessary that it exists on or in the other hand you might you will be saying that it is necessary that 2 plus 2 is equal to 4 so what do you mean by saying that it is necessary that something is the case it is possible that something is the case and it is actually the case that is P is the case actually the P is the case is referring to some kind of factual sentences so that is what we are interested in in this particular kind of course. Now we have said that this is based on first order logics classical logic is based on paradox of first order logic is based on material implication if we accept material implication into consideration it leads to some kind of problems which we call it as paradox of material implication it concerns some logical consequences which are obviously considered to be valid principles of propositional logic but they contradict our universal linguistic intuitions all these things which I have presented here are considered to be an instances of paradox of material implication so from P you deduce Q implies P and not Q P implies Q etc all this list of implications which are okay for the classical logic but when you make use of these kinds of inferences in the day to day discourse then we generate some kind of counterintuitive results. One simple example that you can take into consideration let us assume that not P stands for there is no oil in my coffee if there is oil in my coffee that is represented as P and not P represent there is no oil in my coffee and Q means I like it so now you substitute it into the inferences that we have here not P implies Q implies P then this will become like this there is no oil in my coffee implies that something like there is a there is oil in my coffee then I like it so that it seems to be going against our intuition in the same way if suppose P stands for I will play the foot I will pay football tomorrow and secure is I will break my leg I break my leg today so for example Q implies P implies Q in that case I will pay football tomorrow from that what follows is this thing I will play football tomorrow if I break my leg today this is going to go see completely against our intuition it is like P implies Q implies P suppose P stands for I am alive Q stands for I am dead so I am alive from that Q implies P will come as an outcome as an instance of that one logical consequence of P I am alive to I am dead then I am alive and so all these things when you put it into the day to day discourse it presents numerous kind of problems and what needs to be done and all we need to move away from the classical logic either we need to talk about fixing this connective or you need to talk about you need to bring in relevance into picture and then talk about relevance logics etc it led to different kinds of logics in particular and there are some other kinds of examples like what about those sentences which are referring to future so these are called as future contingent sentences so you can only talk about truth value of the sentences which occur in the past are there are certain things which are factually true and what about the future sent the sentences which are referring to future for example if you say that I will be in my native place on December 25th and so I booked my ticket etc well and good then how do we judge the truth value of that particular kind of statement now so if I say that that sentence is true then I have to be in my native place and I have no choice that I can drop my plan etc and all that makes it necessarily true suppose if that is false and it pushes us to another extreme that it makes me impossible to go to my native place suppose if I cancel it now I can still plan for my trip on so-and-so data that sentence is false that I will be in my native place is false then it pushes us to into another extreme that is it is impossible for me to go to my native place so both extremes leads to some kind of position which is called as fatalism then what do we do with this future contingent sentences so how do we deal with this concern do we need to dismiss those things future contingent sentences are do we need to allow them in classical logics have no answer so we need to move to many value logics where a sentence is neither true nor false can be represented as 1 by 2 so I don't go into the details of this particular kind of thing another interesting important things are what we call it as paradoxes paradoxes are obviously valid arguments but they are considered to be counterintuitive so now let us consider a simple example why classical logic fails to explain this particular kind of phenomenon so consider a heap of grains of sand and let us assume that you have a heap of sand and all now we started removing one by one one grain after another now so take away one solitary grain from that particular kind of heap you still have a heap and nobody will be in a position to say that it is not a heap and still it retains his heapness heapness is considered to be a property a predicate which is considered to be a vague predicate so for one grain of sand is not enough to make that the transition from heap to non-heap now so in that sense if a pile of 10,000 grains of sand makes a heap then call this statement as H 10,000 then a pile of 9,999 grains also makes a heap so now a heap of sand is comprised of large collection of grains is the definition that we is a beginning with we began with any heap of sand minus one grain is still considered to be heap so these two premises are unquestionably true and all you just remove one grain it is not going to lose its character in the same way this problem can be extended to a different context where we can simply understand this thing a person with full of hair is considered not to be bald and if he removes one hair is not going to become that is not going to change much if a person with some let us say as one million kind of hair and all is not considered to be bald a person with one million minus one is also not considered to be bald so like that you will be using modus ponens n number of times and then at the end of the day you will say that even if you do not have any hair and all you still consider to be not bald enough that is little bit surprising for us and then it is counterintuitive to us so here the problem is that you have used modus ponens which is considered to be obviously valid principle of reasoning the premises are considered to be true the argument is valid so the conclusion is counterintuitive it is not at least we are not in a position to accept that it is true so this paradox is tricky for the philosophers because they must explain why one of the two premises are the conclusion is wrong even though they are they appear to be self evident so each time when you come to the next step it is an application of modus ponens rule which is obviously valid kind of influence so what is happening here is that you have a heap and you started removing that particular kind of thing one after another and all even if you remove all the grains and all then still you will have a property such as it still retains this heapness that is a little bit counterintuitive to us so now what to do with the classical logic and all do we dismiss this particular kind of predicates and then totally dismiss it that is deny the problem that is that it is legitimately set up and all or you hold that logic does not apply to vague expressions logic in a sense classical logic or accept that logic does not legitimately apply here but hold that this particular argument is still in value that means your premises are true and the conclusion is false somehow using degree theoretic account etc and all you can show that you have conclusions are probably true but sorry premises are probably true and the conclusion can still be false if you invoke the degree of truth and then accept both logic apply that means modus ponens etc all apply in such cases and the argument is considered to be valid but deny the premises you say that one of the premises is considered to be wrong like this is a number of solutions which are provided and which led to different kinds of logics which usually call it as many value logics I led to fuzzy logics as well the same example is used to motivate used as a motivation for doing fuzzy logics as well another important paradox which like to bring to your attention which has not found any solution in the classical logic that is the liars paradox the liars paradox is considered to be an ancient conredom it was there right from the period of Greeks it was originally cast in the form of a fable the fable goes like this in the ancient times all the inhabitants of great work capable of making true statements epimunities is considered to be one great great belongs to that particular kind of country who lived in great and made the following statements all he said in one context all creatines are less epimunities is also considered to be a comes to be belonging to the creatine now the question is easy a liar if he is a liar he can you tell the truths so now if you say is that all creatines are liars now the question is is epimunities lying or not so now slightly different kind of a problem which emerges with respect to self-referential kind of sentences such as this thing this sentence is false if you say that particular kind of thing now if you ask yourself is this sentence true or false so now in the case of classical logic we have said that a sentence has to be either true or false it only takes two values now what about these particular kinds of sentences that this sentence is false is neither true nor false so how do we incorporate these particular kinds of sentence in all do we dismiss this particular kind of sense and then or be silent about these things are you incorporate or incorporate these things and then talk about extending your logics and all these things presents problem to the classical logic so I don't want to go into the details of Lias paradox lies paradox just I will end it in this thing by saying that Lias paradox says to consider the statement this statement is false if this statement is false is true and whatever it says is whatever indeed stating is true then obviously the statement is considered to be false which would turn out to mean that it is actually true but this would mean that it is actually false so the problem here is that when it is true is a problem it leads to contradiction it falls also it leads to contradiction now so that leads to a tricky kind of situation paradoxical kind of situation again there are different attempts which are made to resolve this particular kind of paradox is if you maintain the distinction between object and meta language maybe you can come make come out of this particular kind of paradox for example truth of a particular kind of sentence you can talk about in the higher language and all you cannot talk about truths of particular kind of sentence within the object language if we maintain a distinction between object and meta language is a way to resolve this particular kind of problem there are some other attempts in the form of by taking into consideration three value logics such as you represent this sentence is false is neither true nor false neither true nor false is represented as something called as paradoxical or nonsense or something like that it takes a value one by two which is different from one and zero so there are other kinds of paradoxes which sets limit to the logic that is one particular kind of paradox which is important in the context of set three that is the Russell's paradox so now consider a set of all sets which are not members of itself so that is represented by R such that X as that X doesn't belong to X now the question is case one is that if R is a member of itself then it is one of those sets that is not a member of itself so R is not a member of itself in the case to if R is not a member of itself then it is there is one there is one set one of the sets in R and hence it is not a it is a member of itself it is not a member of itself it has to be a member of itself that means R belongs to R if and only if R does not belong to R so whether sets of all sets belongs to itself or not is a question which is posed by Russell and now in a letter to Fregge he writes like this a scientist can hardly meet with anything more undesirable than to have a foundation to give a give away just as the work is finished he has finished a grand book and all and then after that this result has come to him as a surprise it shakes the foundations of the logic itself so I was put because set theory logic is rested on set theory set theory is considered to be shaky in this sense that it led to this famous paradoxes that is one paradox is the Russell's paradox so now he's of the view that is positioned by a letter from Bertrand Russell when the work was almost nearly through the press then this result has come and shaken the foundations now so so this is the famous paradox which Russell's paradox which is expressed in terms of barbers paradox barbers paradox goes like this suppose there is a town in which there is just only one male barber was there and every man in this and the town keeps himself cleanly shaven that means he had to go to the barber to shave themselves some by shaving themselves some by attending to the barber and all it seems reasonable to imagine that barber obeys the following rule he sets it as in the notice board and he says like this he shaves all all and only those men in the town who do not shaves themselves only those people he shaves so that do not shave themselves they will go to the barber for shaving you know so now the question is does the barber shave himself so the problem here is that if the barber does not shave himself then he must abide by the rule and he has to shave himself if he shaves himself he violates whatever he has said shaves are only those men in the town who doesn't shave themselves and if it does himself then according to the rule the rule which is in the red color he will not shave himself in both cases there will be problem there is no way in which he can shave himself will be growing his beard like anything so these are some of the counterintuitive results that we come across especially in the context of classical logic there are some other important paradoxes such as omnipotence paradox with this I will end this lecture so suppose if you if you say that God can make a rock which he cannot lift then God is not omnipotent so usually we say that omnipotent means he is capable of doing anything omnipresent means he's present everywhere so now the question is can God create a stone which he cannot lift so if God indeed can make a rock that he cannot live then there is then obviously God is not considered to be omnipotent if God cannot make a rock then there is something which he could not do that means he cannot lift then God is obviously not considered to be omnipotent so either God can make a rock and he cannot if you make sir I cannot lift and there is some problem or God cannot make a rock I mean there is something which you cannot do that is he cannot lift in all in both cases there is a problem therefore you can say that God is not omnipotent so in this lecture what we have seen is this that we have presented some kind of limitations to the classical logic we should not be under the impression that all kinds of reasoning we are trying to cover in terms of first order logic there are lots of things which are considered to be which come under the category of common sense reasoning which cannot be captured in terms of first order logic so that means we need to extend the first order logics or we need to deviate from the first order logics and talk about deviant logics by dropping one of the fundamental principles of logic such as law of identity law of excluded middle law of non-contradiction etc if you drop the law of non-contradiction you are doing para consistent logic if you are dropping law of excluded middle you are talking about many value logics or fuzzy logics etc so as far as possible what we understood from this course is that this first order logics in particular is a starting point it is a basic beginning point for doing all other kinds of logics so it obeys all the nice properties and all it has wonderful properties such as soundness consistency completeness etc now so the moment you drop some of this fundamental laws of logic then it will be at the cost of this nice features such as consistency completeness soundness etc now so this first order logics basically tries to capture mathematical reasoning this mathematical reasoning is based on mostly is based on the parent the material implication so what we have done in this class is that in the course we discussed about first order logic in this lecture we presented some of the problems are challenges to the first order logic so so welcome back and then we will be talking about introduction to logic I am the course instructor for this course my name is Ravi Shankar so I will be dealing with I am the course instructor for this course introduction to logic so as you all might be wondering why this logic course is taught in the humanities department so if you take the history into consideration logic has began the discipline of philosophy and then it has moved to mathematics and then now it has taken shelter in the department of computer science so I will be talking about what I am going to discuss in this particular kind of course and then what are the topics that I am going to cover in this particular kind of course before I begin so I will start with an important quote so it says like this he who knows not and knows not that he knows not he is considered to be a fool you need to shun him and he who knows not and knows not that he knows not is considered to be a simple person and we need to teach him and a person who knows that know that he knows that he does not know is considered to be is falling asleep so you need to awake him and then finally we have people like those who knows that he knows that he knows that something is the case is considered to be wise and we need to follow him so this is a famous Arabian a proverb and all it tells us that there are four kinds of people which exist in the world so the fourth one is the one which we will be following they have they are considered to be having some kind of wisdom mostly teachers will be having this kind of thing and there are other context the other quotation which I would like to bring to your attention is this to attain knowledge things every day that is what we have been doing all the time will be accumulating lot of knowledge etc day by day but if you go to the saints etc and all the first thing that they will tell you is to empty your mind so to attain wisdom you have to delete things every day so what do you mean by saying that deleting things every day and all we might have accumulated knowledge out of our prejudices biases etc and all we need to give up those things which we have accepted out of our prejudices biases are some of the things which might be just some kind of opinion etc this is the famous quotation by law and zoo so now coming back to this course what is that we are trying to do in this interesting and exciting course so first we will be starting with some of the basic concepts of logic where we will be introducing what we mean by an argument and what kind of arguments exist and if you once you identify that these are the arguments and what kind of argument it is so in that context we introduce inductive and deductive kind of arguments as two different kinds of arguments that you come across in logic then we will discuss about some of the important properties of these particular kinds of arguments and then we say we will move on to another interesting and very exciting topic that is considered with fallacies so fallacies are considered mistakes in the argumentation and both deductive and inductive arguments can be fallacious and all these fallacies can be used as some kind of strategies in all they are used as some kind of persuasive strategies which we commonly come across we usually come across in day to day discourse even you see that many politicians etc in order to who the custom who the voters they will be making use of these fallacies in all they will be making some empty promises etc and all they play with the emotions etc and all they will be making use of many fallacies. So then we will move on to traditional logics which are due to Aristotle which has dominated for more than 2000 years then we will be taking up the theory of syllogisms which more or less serves as some kind of predicate logically but Aristotle has no broad house as kind of formal equipment but at it he has discussed some of the important inferences of this categorical propositions how two categorical proposition leads to another kind of categorical proposition so that is where we introduce theory of syllogisms so theory of syllogisms have some kind of limitation then we move on to the propositional logic where we discuss about logic of propositions so a proposition is a sentence which can be simply spoken to be as a true or false and then we discuss the proposition logic is all about logic of the five connectives that we are trying to use that is and or implies if and only if and negation it is basically discussing about the properties of these connectives since this is the starting point of our representation of our knowledge this is the minimal tools that with which you can represent our knowledge then propositional logics are not sufficient enough they are not rich enough especially you know we do not have relations predicates etc and all quantifies etc and all which are missing in the propositional logics in order to make the language richer then we will be introducing two more quantifiers two quantifiers that is for all x and there exists some x you augment the propositional logic with these two quantifiers then we will be talking about the predicate logic and in the context of propositional and predicate logic we will be talking about some of the important decision procedure methods with which you can judge whether a given well-formed formula in this propositional predicate logic is considered to be valid when we say that two statements are considered to be consistent to each other or this was some of the important logical properties that we will be discussing with respect to this decision procedure methods so at least four or five decision procedure methods that we have used in this course to start with the most simplistic kind of method is the truth table method which we have used in two different senses direct and indirect kind of truth table and then what is what occupies the central position for this course is the decision procedure that occupies the central position for this course is the semantic tableaux method so we will be making use of the semantic tableaux method in both in the context of predicate logic and propositional logic and we will talk about validity consistency etc. So what is the learning outcome of this course so why should I do this particular kind of course so we will be able to learn to distinguish good from bad arguments that occupies the first part of this course so ultimately logic is all about it's a study of argumentation as well so we need to identify what is considered to be a good argument or effective argument compared to bad argument etc and make the process of making argumentation effective so in that context we introduced one important model of argumentation which is due to Stephen tool mean so tool mean has introduced a very nice model of argumentation we discuss it in greater detail about that particular kind of thing and one should be in a position to represent one should be able to represent various kinds of knowledge claims within the language of first order logic given an English language sentence we should be able to translate it into the language of propositional logic or predicate logic depending upon this phrases exist for example if the sentence begins with for all x etc and all some none etc and all will be using predicate logic and if you if it is enough that you express it in terms of prepositional just simple prepositions and all prepositional logic would suffice and one should be in a position one should be we will be able to learn more about some of the decision procedure methods to start with two tables semantic tablox method and one of the proof procedure methods such as natural deduction method and resolution defutation method etc these are the decision procedure methods that we will be using in this particular kind of course but what occupies the central position is the semantic tablox method and you will be one of the unique feature of this course is that we studied the basic principles of logic by making use of by solving some kind of logical puzzles so these logical puzzles are due to famous logician Raymond Smollion so we have used different kinds of puzzles in this course nights and naves puzzles etc tiger lady and tiger etc interesting puzzles that one can make use of to make this basic concepts clear to the student and also to learn some kind of underlying techniques of some of the important decision procedure methods which we have discussed just now so one important question that come that might come to your mind is this that what is considered to be logic I mean why it is studied in the humanities discipline etc so logic is usually considered as study of the principles of valid demonstration are principles of valid inference is not enough that is something follows from something but it has to be valid and it has to be sound as well so logic is considered to be branch of philosophy and the word logos and derives from the word logic derives from the word logos which means word thought idea argument account reason principle etc we make use of reason to be the most important thing out of these things logic also concerns with the structure of statements and arguments in formal system of inference and in the natural language so it also deals with topics such as validity fallacies paradoxes etc that means reasoning using probability and arguments involving causality this is mostly taken care by inductive kind of logics this is not what occupies the our attention all we will be dealing with the validity fallacies paradoxes etc mostly in this course we will be focusing our attention on deductive reason so what the subject matter of logic logic is used as a formal language with his own syntax and semantics and we discuss the relation between these two things syntax and semantics syntax will be taken care by probability and semantics is taken care by logical consequence so whatever is probable is true and whatever is true is full and system is considered to be complete it deals with the principles of valid reasoning that is what we have discussed earlier and in the good olden ancient days it was considered to be part and parcel of the discipline philosophy and it is still widely studied in the area of philosophy there are many problems philosophical problems which are raised in the Greek period they are still considered to be problems in the contemporary literature on logic they occupy the attention of logicians in the contemporary literature of logic so it has to it is part of mathematics in a sense that there is shift from ever since the fall of Aristotle in syllogistic logics it moved to mathematics in particular where the attempt was made to reduce mathematics to logic the program is called as logicism and this is recently it has become one of the important and most essential subjects to learn in the area of computer science so logic is studied in all these disciplines even tilt even now also it is a kind of interdisciplinary kind of subject so the three views which are dominant in the logic or logicism formalism and enthusiasm I am not going to the details of it the course these things will become explicit so there are different kinds of logic which we can talk about first is formal logic homilogic means it is a study of influence with purely formal content where that content can be made explicit we are not worried about the content of the argument where we are only worried about the form if P then Q and P that is why Q follows but there are certain kinds of arguments which require the analysis of content such as this room is made up of atoms atoms are invisible that means this room is invisible so these kinds of argument requires you have to analyze the content of the argument and all so there we have used a shift in the meaning of the usage of the word atoms in the premises that is why that kind of argument is called as a fall fallacy and that fallacy is called as informal fallacy there are certain kinds of argument which requires the analysis of content but mostly we will be dealing with the formal logics where what matters to us is only form of an argument an informal logic is considered to be study of natural language arguments that occupies our first part of our course study of fallacies is an especially important branch of informal logic so that is the reason why we look into the informal logic in the beginning of this course especially for example the dialogues of Plato a wonderful is a wonderful example of informal logic and there is another way which you can refer logics with the name symbolic logic symbolic logic is a study of symbolic abstractions that capture the formal feature of logical inference symbolic logic is often divided into two branches that is what we are done in this course we are going to do in this course propositional logic and the predicate logic so what occupies the second part of this course is the propositional and the predicate logic and there is another thing which is important that is mathematical logic it is an extension of symbolic logic into other areas such as in particular to the study of modal theory proof theory set theory and recursion theory we are not going to study all these things but we will be focusing our attention partly on modal theory and partly on proof theory in the context of proof theory we introduce Russell whitehead axiomatic system and then Hilbert Ackerman axiomatic system and then we also talk about some of the important proof procedure methods such as natural deduction and how do we deduce theorems from the given axiomatic system there are things which we will be studying so if you see the content of this course it is a mixture of all these things formal logic in form of formal logic to a certain extent we are taken into consider study logics also considered to be formal logics partly we have taken into consideration that and in the informal logic we will be studying various kinds of fallacies and in the symbolic logic that is the core of this particular kind of course that is propositional and predicate logic and as far as possible we introduce the concepts of propositional and predicate logic with the help of some kind of puzzles the solving some kind of puzzles puzzles we get familiarize ourselves with some of the important decision procedure methods such as semantic tablux method and in the mathematical logic we focused our attention on proof theory it is a mixture of all these things and nature and scope of this logic you are it should not be under the impression that all kinds of reasoning that we are trying to cover in this particular kind of course there are other kinds of reasoning which we employ in day-to-day discourse that is common sense reasoning which is considered to be non-monotonic for example if you say all birds flies Tweety is a bird and Tweety flies and if somebody comes to you and tells you that Tweety of course it's a bird but it will not it comes under the category of penguins and penguins doesn't fly so now what you will what you are going to do so are you are you supposed to withdraw the conclusion that you have drawn that Tweety flies are what exactly you are going to do at this stage so accepting the new information will lead to the withdrawal of the conclusion that you have drawn earlier so this is not what is permitted in the classical logics because classical logics are considered to be monotonic they are deductive and monotonic in nature so this course doesn't study all kinds of reasoning although it tries to capture some of the things of course logic has to apply to day-to-day discourse to certain extent in solving puzzles etc and all we make use of the basic principles of logic but when it actually comes to the day-to-day discourse we will talk about some of the limitations of this first order logic especially when it is referring to vague predicates when it is referring to sentences such as this sentence is false whether it is true or false that kind of questions liar sentences for example etc all these things present some kind of challenges to the classical logic in the same way when you use material implication the day-to-day discourse you have counterintuitive results so in that sense we are not going to talk about all kinds of reasoning but we are restricting ourselves to just two valued logic that is predicate and preposition logic takes care of that one which obeys the fundamental laws of logic that is law of identity law of excluded middle and law of non-contradiction and of course the monotonicity as far as possible we are trying to capture the mathematical reasoning this is the most minimal kind of ways to represent our knowledge claims it's used as a language for the for representing the knowledge so now what is important here is is that just we need to consider a brief history of logic is not considered with a complete kind of all the complete details are not there in this one but it is very difficult to go through a brief history of logic it started with there are many things which might be missing in this list but one of the most important things that that usually find it in the history of logic are these things I will go through it quickly to start with we have stoics they are before Aristotle there they seems to be the major proponents of this prepositional logic and they also proposed some rules of inference which needs to be studied in greater detail some research is still going on in this direction to what extent they have come up with the rules etc. So then we have an important work by Aristotle in his works Argonen which consists of set of books and all like prior analytics supposed posterior analytics etc where he has introduced in one of these things he introduced theory of syllogisms you know in a way more or less he has introduced quantifiers because categorical propositions starts with all some none etc and all they're all quantifies only more or less but formal interpretation is missing in that particular kind of thing and then he also talked about modal propositions etc and although we did not deal with these things in much more greater detail these are things which are already there and then followed by that we jump to the medieval period 1565 Cardano he has come up with probability theory probabilistic logics etc uncertainty and 1646 Leibniz has come up with a grand program that is research for general decision process to check the validity of a given formula this is considered by the origin of the computer computers etc usually treated as the origin of the computers in 1847 this is considered be the most important work in the first order logic this is a starting point where the logic has taken the shape of mathematics in particular logic there is a turn there is a mathematical turn in logic so that is this thing George bull has come up with algebraic interpretation of syllogisms and he has also come up with the prepositional logic what he has done is given the algebraic interpretation of syllogism and one of the most important works in first order logic is this thing got to go to love freggies this is spelling mistake here frege first order logic is considered to be the father of first order logic it tries to reduce mathematics to the branch of logic and 1889 you have pianos nine axioms for the natural numbers which are considered to be the important thing and in the 20th century I mean in the 19th century mid 19th century the Mon the monumental work is due to Wett and Russell invited principia Mathematica and Hilbert's program has emerged after that one he provided decision procedures for mathematical theories and he also presented 23 challenging open problems that still considered to be open problems and Wittgenstein is said to be attributed to the development of the truth tables and troops based on the truth tables and the celebrated result of Godel sin Godel's complete as theorem in one of the lectures we will be discussing about that related to first order logic and Herbrand has come up with an important theorem which is called as deduction theorem which will be dealing within one of these lectures a proof procedure for it is considered as a proof procedure for first order logic based on proportionalization and could Godel's very important result in 1931 path breaking results which sets limit to the program of Russell Whitehead and then Hilbert Ackerman that is the incompleteness theorem in the context of a consistency of pianos axioms in 1936 Benson has come up with a method of natural reduction he showed a proof for the consistency of pianos axioms in the set theory so we it is not the case that we will be dealing with all these topics in all but as far as possible we will be dealing with some of the important and interesting topics out of this so in 1936 church and touring has come up with undecidability of first order logic and after 1950s most of the work is done in the area of computer science and the mathematics in particular of course the issues are philosophical in nature only so philosophers mathematicians computer centers in all work together mostly logicians work together in on these particular kinds of problems so 1954 Davis Patnam has come up with the first machine-generated proof automated kind of proof and then 1955 Beth and Hintika has come up with the semantic tableaux method and semantic tableaux method is considered to be a occupying the central position for this particular kind of course and Neville and Simon Simon has come up with first machine-generated proof in the context of logical calculus 1957 kanger and pravis come up with some interesting method which is lazy substitution by free and dummy variables and profits has come up with first provers for first order logic and then 1958 good Godel has come up with a method of proving consistency of axioms with the type 3 and these are some of the important developments after 1959 and after 1963 of course we will be making use of Robinson's unification resolution refutation method in one of these lectures so in the context of prepositional logic and then after that after 1960s 70s and all there was a turn towards non-classical logic there was a lot of this is a dissatisfaction with respect to classical logic fails to explain many interesting phenomena so there is a shift towards fuzzy default and modal logics so these are the things some of if you take the history into consideration and we will try to do justice to some of the important parts of important content that arises out of this history and we will not be dealing with all these works and all but just to present the continuity and all just to know the subject matter of logic I have presented a brief history or timeline of logic there are some of the important developments there are many things which might be missing in so as far as the course is concerned I will be using this particular kind of textbooks Patrick Hurley concedes introduction to logic there are many many good books on introduction to logic Mendelssohn introduction to mathematical logic Sean Hedman the first course in mathematical logic and then we will be making use of the original work by Russell invited at his principia Mathematica we will be taking we will making use of a portion of it then we will be talking about various kinds of proofs based on is axiomatic system then what is interesting in this course is paradoxes and the puzzles for the puzzles we refer to Raymond Smollion's book there are lots of books which are written by Raymond Smollion what is the name of the book logical librarians etc these are the books which will be referring to it so these are the online differences which will be following so this is considered to be an interesting and exciting course in the sense that it is it has all the flavors and all philosophical logical and mathematical flavors philosophical in the sense that will be dealing with some of the philosophical issues such as lies paradoxes paradoxes etc and all and then it has mathematical flavor that you know it deals with the foundations of mathematics are the problems related to the foundations of mathematics and it also deals with computational favor in the sense that many methods that we will be making use of decision procedure methods which we are trying to make use of will have some kind of implications for the computer science it is in that sense it is considered to be an interesting and exciting course.