 Today I will discuss a unit that is rules of inference and unit number 7. In this unit we will discuss only two basic points. First one is rules of inference and the second one is some sole examples of rules of inference. Now this unit not only introduces to you the rules of inference but also the uses of the rules in case of formal proof of validity. It also provides us to go through some sole examples of the rules of inference in order to know how the rules of inference are applied to stage the validity of arguments. Now dear learners you will see that what is rules of inference. In the preceding chapter you have already come to know the formal proof of validity which discusses the concepts like strategy for deduction. Rules of inference rules of refreshment differences between rules of inference and the rules of refreshment. Then test of formal proof, general suggestion for formal deduction, etc. Therefore this unit emphasizes only on the application of the rules of inference. Now dear learners you will see there are nine rules of inference. These rules are considered self-evident and are therefore valid without any proof. On the basis of these rules we can determine the test of validity or invalidity of arguments. As we can derive a conclusion on the basis of self-evident rules, so the method of deriving a conclusion from the rules is called deductive method. Now dear learners you will see what are the rules of inference. You see dear learners first one is modus ponens that is P implies Q P therefore Q. This is the rules of modus ponens. So what it means? This rule means affirmation of the antecedent and on the basis consequences affirm. We can take an example of valid argument by applying these rules. Now dear learners you see we can take an example. You see dear learners if modu is intelligent he will be able to pass in the examination and modu is intelligent therefore he will be able to pass in the examination. And this argument can be symbolized as valid argument form of the rule of modus ponens. So you see dear learners how can we symbolize this type of argument. You see we can symbolize this example. So in this way P implies Q P implies Q means if modu is intelligent then he will be able to pass in the examination. And you see P means modu is intelligent and therefore Q indicates that he will be able to pass in the examination. Now dear learners you see there are also other valid argument forms of modus ponens. Now dear learners you will see there are also other valid argument forms of modus ponens. There negation P implies negation Q negation P therefore negation Q. You see the second rules of inference that is modus ponens. Now dear learners you see in the blackboard P implies Q negation Q therefore negation P. Now dear learners you see this argument means the denial of the consequence and on that basis the antecedent is denied. Means P implies Q negation Q therefore negation P. So dear learners we can take an example of valid argument form by applying this rule and this the argument is you see if he wins in the match he will give us a party. He will not give us a party therefore he does not win the match. So this is the example of modus ponens. But how can we symbolize this type of argument dear learners you see that the symbolic form of this argument is P implies Q P implies Q negation Q therefore negation P. This is the symbolic form of modus ponens. So this symbolic form is considered valid argument form of modus ponens. Now dear learners you see the third rule of inference is rules of inference is disjunctive syllogism. So you see dear learners P val Q negation P therefore Q. So this is the rule of disjunctive syllogism. This argument holds that if one of the options or disjunct is denied in the minor premise the other option or disjunct is accepted in the conclusion. So we can take real example of disjunctive syllogism dear learners you see in the blackboard. Either he will win or defeat in the match he will not win in the match. Therefore he will defeat in the match. So this is the real example of disjunctive syllogism. But how can we symbolize this real example you see dear learners how can you symbolize. So P val Q means either he will win or defeat in the match. Either he will win or defeat in the match if we if we take P and we take Q here. Then it means P val Q P means he will win or Q means defeat in the match. Then you see this means he will not win in the match. We means negation P. So and the last one that is conclusion he will defeat in the match means Q. So this is in this way we can symbolize the example P val Q negation P therefore negation Q. Now dear learners you see the fifth rule of inferences that is hypothetical syllogism. So how can you write this type of rule of hypothetical syllogism P implies Q Q implies R therefore P implies R. Dear learners you see this rule means that both the premises are conditional statements and they have one common statement. The common statement is known as antecedent in one premise and the other is consequent in another premise. Now we can take real example of hypothetical syllogism in order to better understand its logical form. Now you see dear learners we can take a real example of hypothetical syllogism if modu is intelligent he will be able to pass in the examination. If he passes in the examination he will be able to get a job therefore if modu is intelligent then he will be able to get a job. So this real example we can symbolize dear learners. So you see how can we symbolize this real example of hypothetical syllogism P implies Q Q implies R therefore P implies R. So this is a valid form of hypothetical syllogism dear learners. You see the next rule of inference that is conjunction that is you see dear learners P and Q therefore P and Q. You see dear learners P Q and therefore P dot Q. This rule means that the first two premises are joined together by the word AND. Next one you see simplification rules. So P dot Q therefore only P so this is the rule of simplification. This rule states that if the two premises are joined by the word AND or by the symbol dot the first variable or conjunct can be inferred as the conclusion P dot Q therefore P. So this is the rule of simplification. Now dear learners you see the seventh rule that is addition. So P so you see P therefore P fell Q. So this rule is known as addition. So this rule means that we can add variable or variables to the premises by the symbol fell. Now the next rule is absorption P implies Q therefore P implies back at begin P dot Q back at close. So according to this rule consequent carries antecedent consequent carries antecedent in the conclusion. So this is the rule of absorption dear learners. Now you see constructive dilemma this is another rule of rules of inference. How can we write the learners you see P implies Q R implies S P fell R therefore Q fell R. So what this rule means that this rule indicates that or this rule states that there are two conditional propositions and a major premise possesses two different antecedents. You see major premise indicates or major premise possesses two different antecedents here it is P or here it is R. So you see therefore both denotes different consequent we can take a concrete example to understand if a man is honest he must be respected and if he is a social worker he must get reward. So a man is either an honest man or a social worker therefore the conclusion is either he must be respected or he must get reward. So how can we symbolize this type of concrete example dear learners you see P implies Q dot R implies S and P fell R therefore Q fell S. So this is a valid form of constructive dilemma dear learners. Now you see another rule of rules of inference is that is destructive dilemma dear learners you see. So again P implies Q dot R implies S dot negation Q fell negation S therefore negation P fell negation R. So that is destructive dilemma in the previous one you have you have already the idea of constructive dilemma here the next one the opposite one that is destructive dilemma. Now dear learners you take a concrete example then you will be able to know. So we can take a concrete example in order to understand its logical form if a man is honest he must be respected and if he is a social worker he must get reward. A man is either not respected or does not get reward therefore the conclusion is either he is not honest or not a social worker. So this is all about a concrete example of destructive dilemma dear learners. So now the next step is how can we symbolize this type of concrete example now dear learners you see how can we symbolize so in this way we can symbolize P implies Q dot R implies S dot negation Q fell negation S. Therefore negation P fell negation R. So this is a valid form of destructive dilemma. So dear learners this is all about the rules of inference. So how can we symbolize the concrete example the idea is given to you in the previous slide. Now you see how can we determine the validity of arguments applying the rules of inference. So you see dear learners therefore it is given the sole example of sole examples of rules of inference. Now you see dear learners there are some sole examples of formal proof of validity given in this section in order to understand how the rules of inference are applied to test the validity of arguments. Now dear learners you see this is an example so we have to apply the rules in order to find out the conclusion. Now dear learners you see first step is P implies Q second one is Q implies S third one is P and the conclusion is S. Now you see how can we find out the conclusion S applying the rules of inference. Now dear learners you see first and second by hypothetical syllogism first and second if we use the rules of hypothetical syllogism then we find out the conclusion that is S. So you see first we have to apply first and second by hypothetical syllogism first and second we apply a hypothetical syllogism then we find P implies S. Then fourth and third this is fourth and third we have to apply by modus ponens then we find the conclusion S. So in this way we have to find out the conclusion or we can test the validity of arguments by applying the rules of inference. So this is an solved this is a solved example of rules of inference dear learners. Now you see another one that is first one is P value Q P value Q implies R second one is S value P third step is negation S and conclusion R. So here also we have to see what are the rules we have to apply here in order to deduce the conclusion R. So dear learners you see so the first step we find P how can we find P we have to apply second and third by this junctive syllogism here you see second and third we have to apply this junctive syllogism then we have to find P. Next you see how can we find P value Q here also you see four addition this is four P value Q. So this is the we apply addition rules now you see the next step are the conclusion are how can you find out how can you deduce one and five modus ponens one that is one and another five. So that is R we apply here modus ponens rules one and five then we find out the conclusion R. So this is another solved examples of rules of inference dear learners. So dear learners you see here P implies Q R implies S P value R fourth step Q value S implies T and the conclusion T. Now you see how can we find out or how can you deduce the conclusion T out of these four steps now you see one and two that is constructive dilemma. So we have to apply this then we have to find Q L S next one four and five we apply modus ponens rules then we find out the conclusion T. So this is another example of rules of inference dear learners. Now you see first we have to another way another way of deducing conclusion out of the premises means there are some real examples given to you. You have to symbolize the argument after that you have to prove how can we deduce the conclusion out of the premises. So you see out of the premises so you see there is a real example given to you that is if rain comes in time farmers will be happy. So how can we symbolize that is R or F R means rain or F means farmer here. So R implies F we can symbolize this example if rain comes in time farmers will be happy so we can symbolize R implies F. Next one if farmers are happy there will be good crops how can you symbolize the again farmers F and that is C. So F implies C the rain comes in time so we can symbolize here giving R rains come in time so the conclusion will be there will be good crops so that it will be good crops that is conclusion C. So how can we deduce the learners the conclusion C out of these three steps. Now the learners you see we apply here one two hypothetical syllogism by applying one two hypothetical syllogism we find R implies C. Now you see how can we deduce the conclusion then C we apply four and three rules modus ponens rules here four and three means that is we find out the conclusion C applying modus ponens rules in four and three steps. So this is the sole example of rules of inference the learners. So this is all about the rules of inference how can we apply the rules of inference in order to deduce the conclusion the learners. Now you see what are the basic points in this unit rules of inference. Now the learners you see in the blackboard rules of inference are considered self-evident and are therefore valid without any proof. On a basis of these rules we can determine or test the validity or invalidity of arguments that is why these rules are known as deductive method. Second one modus ponens means what affirmation of the antecedent and on that basis consequent is affirm. Modus ponens means the denial of the consequent and on that basis the antecedent is denied. Another one another basic point is this argument holds that if one of the options or disjunctions denied in the minor premise the other option or disjunction accepted in the conclusion. Now you see the learners another basic point that is hypothetical syllogism means that both the premises are conditional statements and they have one common statement. The common statement is known as antecedent in one premise and other is consequent in another premise. The learners this one is disjunctive syllogism this one is disjunctive syllogism so the learners so this is all about the basic points of rules of inference. Now the learners you see what are the books you have to consult or you have to read in order to know the unit in a very comprehensive way. So the learners you can take the book that is sanda sakravarti logic informal symbolic and inductive. So this book you can take another one are being copied that is book is symbolic logic and copy copy that is introduction to logic and RC Munshi handbook of logic and Samki Sourcing book modern logic. And you can also took another very important book that is krishna join it takes book of logic. I think the learners you have understood by going through this unit and for understanding in order to know more you have to consult with this book what are the books I have already given to you. So the learners thank you.