Logic Standard XI PDF

STANDARD XI - Maharashtra State Bureau of Textbook Production and _ Curriculum Research, Pone. Hcfil!II¼ ( l{T ) t98.00 -- The coordination committee formed by GR...

STANDARD XI - Maharashtra State Bureau of Textbook Production and _ Curriculum Research, Pone. Hcfil!II¼ ( l{T ) t98.00 -- The coordination committee formed by GR No Abhyas - 2116 (Pra. Kra. 43/16) SD-4 Dated 25.4.2016 has given approval to prescrie this textbook in its meeting held on 20.06.2019 and it has been decided to implement it from the educational year 2019-20. LOGIC STANDARD XI 2019 Maharashtra State Bureau of Textbook Production and Curriculum Research, Pune. Download DIKSHA App on your smartphone. If you scan the Q.R.Code on this page of your textbook, you will be able to access full text. If you scan the Q.R.Code provided, you will be able to access audio-visual study material relevant to each lesson, provided as teaching and learning aids.  First Edition : 2019 © Maharashtra State Bureau of Textbook Production and Reprint : 2020 Curriculum Research, Pune - 411 004 The Maharashtra State Bureau of Textbook Production and Curriculum Research reserves all rights relating to the book. No part of this book should be reproduced without the written permission of the Director, Maharashtra State Bureau of Textbook Production and Curriculum Research, Pune. Logic Subject Committee Logic Study Group Smt. Dr. Smita Save Shri. Suresh Thombare Smt. Shraddha Chetan Pai Ms. Chhaya B. Kore Smt. Meeta Hemant Phadke Shri. Vasant Vikramji Lokhande Ms. Sandhya Vishwanath Marudkar Ms. Farzana Sirajoddin Shaikh Dr. Dilip Namdev Nagargoje. Smt. Pinki Hiten Gala Dr. Balaji Marotrao Narwade Shri. Dhanaraj Tukaram Lazade Dr. Sadanand M. Billur Ms. Janvi Shah Member - Secretary Co-ordinator Dr. Sadanand M. Billur Cover Special Officer, Kannada Shri. Yashavant Deshmukh Type Setting Shri. R.M. Ganachari Asst. Special Officer, Kannada Nihar Graphics, Mumbai Paper Production 70 GSM Creamwove Sachchitanand Aphale Print Order Chief Production Officer. Shri Liladhar Atram Printer Production Officer. Publisher Vivek Uttam Gosavi Controller Chief Co-ordinator Maharashtra State Textbook Bureau, Prabhadevi, Mumbai - 25 Smt. Prachi Ravindra Sathe NATIONAL ANTHEM  PREFACE Maharashtra State Bureau of Textbook Production & Curriculum Ressearch Pune, takes immense pleasure to introduce ‘Logic’ as a subject for Standrad XI. Logic is a science of reasoning. ‘Rationality’ is fundamental distinguishing characteristic of human being. This unique ability helps man to draw conclusions from the available information. Though ability to reason is an inbuilt feature of human beings, logicians have identified the rules of reasoning. Logic deals with these rules of reasoning. In logic one studies methods and principles of logic which enables one to distinguish between good and bad reasoning. Training in logic sharpens our ability to reason correctly and detect fallacies in reasoning if any. Logic, therefore is a fundamental discipline, useful to all branches of knowledge. With the introduction of logic at Standard XI students will be able to understand, argue and convince with considerable amount of maturity. Study of logic will enrich their ability of logical, analytical and critical thinking. The aim of this textbook is to explain basic principles of logic and their appllications. We have tried to make this textbook more interesting and activity based, which will facilitate easy understanding of the subject and create interest in the subject. The textbook is written keeping in mind needs of students from both urban and rural areas. Various activity based questions, exercises, puzzles given in the textbook will help students to understand the basic concepts of logic and master the methods of logic. Q.R. code is given on the first page of the textbook. You will like the information provided by it. The bureau of textbook is thankful to the Logic Subject Committee and Study Group, Scrutiny and Quality Reviewers and Artist for their dedication and co-operation in preparing this textbook. Hope, Students, Teachers and Parents will welcome this textbook. Pune (Dr. Sunil Magar) Date : 22nd May, 2019 (Sankashta Chathurthi) Director Indian Solar Date : 01 Jyeshtha 1941 Pune Maharashtra State Bureau of Date : 20 June 2019 Text Book Production and Indian Solar Date : 30 Jyestha 1941 Curriculum Research, Pune  For Teachers We are happy to introduce ‘Logic’ textbook for standard XI. As per the revised syllabus, two new topics are added in this textbook. These are : Origin and development of logic and Application of logic. Accordingly students will get brief information about historical development of Western as well as Indian logic. It will be interesting for students to know how logic has developed globally. Information about origin and development of Indian Logic will facilitate in enhancing pride in students mind about Indian contribution to the subject. Logic is a fundamental subject and basis of all the branches of knowledge. The chapter, Application of logic illustrates the importance of logic in day to day life as well as in the important fields like - Law, Science and Computer science. This chapter will enable students to understand the importance of logical thinking while taking decisions in personal as well as professional life. They will also realize how taking rational decisions at right time can lead to success and happiness in life. Importance of logic can be highlighted by informing students, how study of logic can help them to appear for various competitive exams, which test the reasoning ability of students. Logic as an independent subject is introduced as standard XI. At this stage students begin to think independently and express their thoughts and opinions. Logic being the science of reasoning, can help students in consistent and logical thinking. As teachers of logic it is our responsibility to train students to think rationally and reason correctly. As Standard XI is the first year of stydying logic, it is necessary for teachers to take in to account students age and level of understanding. Logic studies abstract concepts, so the important concepts in logic need to be explained step by step, in easy to understand language and by giving examples and various activities in such a way that, students can relate the subject to their experiences in life. Keeping this in mind the textbook is made activity based. Teachers are expected to make use of various examples, teaching aids and activities like debates, logical puzzles and giving examples of good arguments and fallacies from everyday experience. In this way teaching and learning can become interesting and enjoyable experience for both students and teachers.  Competency Statements Competency To acquire knowledge about the origin and development of logic. To understand the importance of logical thinking. To acquire knowledge about the fundamental concepts and principles of logic. To understand the types of argument and develop the ability to recognize the types of argument. To develop the ability of rational thinking. To understand the difference between sentence and proposition. To study the characteristics of propositions. To understand the types of propositions and to develop the ability to symbolize the propositions. To study the basic truth-tables. To study the method of truth-table. To develop the ability to apply the method of truth-table to decide whether a statement form is tautologous or not and to decide the validity of arguments. To study the method of deductive proof To develop the ability to prove the validity of deductive argument. by the method of direct deductive proof. To understand the need and importance of induction. To acquire knowledge about the types of inductive arguments and their use in our day to day life and science. To develop the ability of recognizing the types of inductive arguments. To enhance argumentation skills. To understand the different types of fallacies. To devleop the ability of recognizing the types of fallacies. To develop the ability to reason correctly and to detect errors in others argument. To understand the application of logic in day to day life, in the field of law, science and Computer science.  INDEX Chapter No. Title Page No. 1. Nature of Logic 1 2. Nature of Proposition 12 3. Decision Procedure 31 4. The Method of Deduction 46 5. Inductive Inference and its type 64 6. Fallacies 72 7. Application of Logic 84  Chapter 1 Nature of Logic.... bad reasoning as well as good reasoning is possible, and this fact is the foundation of the practi- cal side of logic. ---- CHARLES SANDERS PEIRCE DO YOU KNOW THAT.............. Logic is a branch of philosophy. Logic developed independently in India. Ability to reason is the unique characteristic of man. Logic will train you to reason correctly. You need not have formal training in logic to use the rules of logic & reason correctly. grammar, Panini (5th century BC) developed 1.1 ORIGIN AND DEVELOPMENT OF a form of logic which is similar to the modern LOGIC Boolean logic. Logic is traditionally classified as a branch The Buddhist and Jaina logic also comes of philosophy. Philosophy is fundamental to under the Indian logic. Jain logic developed all spheres of human enquiry, and logic is the and flourished from 6th century BCE to basis that strengthens philosophical thinking. 17th century CE. Buddhist logic flourished In philosophy one needs to think clearly to deal from about 500 CE up to 1300 CE. The main with the most fundamental questions related to philosophers responsible for the development our life and this universe. Use of principles of of Buddhist logic are Nagarjuna (c. 150-250 logic in thinking, reasoning and arguments is CE), Vasubandhu (400-800 CE), Dignaga (480- central to the practice of philosophy. 540 CE) and Dharmakirti (600-660 CE). The In ancient times Logic originated and tradition of Buddhist logic is still alive in the developed in India, Greece and China. The Tibetan Buddhist tradition, where logic is an beginning of modern logic as a systematic study important part of the education of monks. can be traced back to the Greek philosopher Mozi, “Master Mo”, a contemporary of Aristotle (384-322 B.C.). Aristotle is regarded Confucius who founded the Mohist School, was as the father of logic. The development of logic mainly responsible for the development of logic throughout the world is mainly influenced by in China. Unfortunately, due to the harsh rule of the Aristotelian logic, except in India and China Legalism in the Qin Dynasty, this line of study where it developed independently. in logic disappeared in China until Indian Logic Logic originated in ancient India and was introduced by Buddhists. continued to develop till early modern times. The Aristotelian logic is also known as Indian logic is represented by the Nyaya School traditional logic. Aristotle’s logic reached its of philosophy. The Nyaya Sutras of Akshapada peak point in the mid-fourteenth century. The Gautama (2nd century) constitute the core texts period between the fourteenth century and the of the Nyaya School. In Mahabharata (12.173.45) beginning of the nineteenth century was largely and Arthashastra of Koutilya (Chanakya) we find one of decline and neglect. Logic was revived in reference of the Anviksiki and Tarka schools of the mid-nineteenth century. logic in India. For his formulation of Sanskrit 1  At the beginning of a revolutionary period to the mathematics. Development in mathematics logic developed into a formal discipline. Logic along with the contribution of thinkers like is therefore classified as a formal science. Leibniz, Francis Bacon, Augustus De Morgan, The development of modern “symbolic” and Bertrand Russell, George Boole, Peirce, Venn, “mathematical” logic during this period is the Frege, Wittgenstein, Godel and Alfred Tarski most significant development in the history of has influenced the evolution of traditional logic logic. As a formal science logic is closely related in to today’s modern logic. Can you answer? 1. If you attend lectures then you will understand the subject You attend lectures Therefore......................... 2. Wherever there is smoke there is fire There is smoke coming out from the building Therefore.............. Solve the puzzles 1. A famous mathematician was walking on a street. He saw a beautiful girl on a bus stop and he asked her, ‘what is your name? The girl recognized him as a famous mathematician and replied that her name was hidden in the date 19/9/2001. Guess the girls name. 2. Manikchand was looking at the photo. Someone asked him, ‘Whose picture are you looking at? He replied: “I don’t have any brother or sister, but this man’s father is my father’s son. So whose picture was Manikchand looking at? 1.2 DEFINITION OF LOGIC The word logic is derived from the Greek word ‘Logos’. The word ‘logos’ means ‘thought’. We all can solve puzzles, give proofs So etymologically logic is often defined as, and deduce consequences as illustrated above. ‘The science of the laws of thought.’ There are This is possible because we are blessed with three types of sciences, 1) Natural sciences like the ability to reason. This is the unique ability physics, chemistry etc. 2) Social sciences like which differentiates man from other animals. history, geography, sociology etc. and 3) Formal This ability of ours is revealed when we infer, science like mathematics. Logic is a formal argue, debate or try to give proofs. We are born science. The etymological definition of logic, rational and may not require any formal training however, is not accurate, firstly because it is to reason. However our reasoning is not always too wide and may lead to misunderstanding that good / correct / valid. Sometimes our reasoning logicians study the process of thinking, which is good and sometimes it is bad. It is necessary is not correct. Thinking process is studied in that we always reason correctly and this is where psychology. Secondly the word ‘thought’ refers the role of logic is important because logic trains to many activities like remembering, imagining, us to reason correctly. day dreaming, reasoning etc. and logic is Reason has applications in all spheres of concerned with only one type of thinking i.e. human affairs. The study of logic, therefore, reasoning. has applications in many important fields Another very common and easy to like Mathematics, Philosophy, Science, Law, understand definition of logic is – ‘Logic is the Computer science, Education and also in our science of reasoning.’ But this definition also day to day life. Training in logic thus can help is too wide. This definition restricts the study one in all the endeavors of life. 2 of logic only to reasoning but logicians are not All artists are creative. interested in studying the process of reasoning Sunita is an artist. as is implied by this definition too. Logicians in Therefore, Sunita is creative. fact are concerned with the correctness of the completed process of reasoning. The propositions, ‘All artists are creative’ and ‘Sunita is an artist’ are premises and the The aim of logic is to train people to proposition ‘Therefore, Sunita is creative’ is the reason correctly and therefore the main task of conclusion which is established on the basis of logic is to distinguish between good reasoning evidence in the premises. and bad reasoning. This practical aspect of logic is accurately stated in I.M. Copi’s definition of Thus premise (premises) and conclusion logic. He defines logic as – ‘The study of the are the two basic constituent elements of an methods and principles used to distinguish argument. In every argument the conclusion good (correct) from bad (incorrect) is derived from the premises and an attempt is reasoning.’ This definition is widely accepted made to show that the conclusion is a logical by logicians. consequence of the premises. Reasoning is a kind of thinking in which 2) Valid argument : Every argument claims inference takes place i.e. a thinker passes to provide evidence for its conclusion. However, from the evidence to the conclusion. The term every argument is not valid. The validity of an ‘inference’ refers to the mental process by argument depends on the nature of relationship which one proposition is established on the basis between its premises and conclusion. If the of one or more propositions accepted as the premises provide ‘good’ evidence for the starting point of the process. An argument is a conclusion, the argument is valid otherwise it verbal representation of this process of inference is invalid. What is regarded as ‘good’ evidence, and logic is mainly concerned with arguments. however, depends upon the type of argument. (In this text we shall use the words reasoning, 3) Form of argument : The two important inference and argument as synonyms) aspects of any argument are – form and content. 1.3 SOME BASIC CONCEPTS OF Every argument is about something and that is LOGIC the subject matter or the content of the argument. In the same way every argument has some form. To get precise understanding of the nature Form means pattern or structure of the argument. of logic it is further necessary to understand For instance, pots may be of various shapes or certain technical terms used in logic viz. patterns. These different shapes are the forms of 1) Argument 2) Valid argument 3) Form of pots. These pots may be made up of any material argument. 4) True / False and Valid / Invalid. like clay, iron, bronze or silver. The material out of which it is made is the content of the pot. Now 1) Argument / Inference : An argument we may have pots of the same shape but made consists of proposition / statements. Every up of different material, we may have pots of the argument attempts to establish a proposition same material but of different forms or the pots by giving another proposition / propositions differing in both form and matter. In the same in its support. An argument may be defined way the arguments may differ in the content and as, ‘A group of propositions in which one have the same form, they may have the same proposition is established on the evidence content but different forms or they may differ of remaining propositions.’ The proposition both in the content and the form. For example – which is established is called the conclusion and the propositions which are stated in support of (1) All men are wise. the conclusion are called premises. For instance Rakesh is a man. in the given argument – Therefore, Rakesh is wise. 3 (2) All doctors are rich. states that a narrower class (men and doctors) Sunil is a doctor. is included in a wider class (wise and rich). The Therefore, Sunil is rich. second premise of both the arguments states that an individual (Rakesh and Sunil) is a member The content or the subject matter of the of the narrower class. In the conclusion of both above given arguments is different. The first the argument it is inferred that the individual argument is about men, wise and Rakesh. is, therefore, a member of the wider class. The The second is about doctors, rich and Sunil. following diagram clearly reveals how the form However, the form of both the arguments is of both the arguments is same. same. The first premise of both the arguments Argument – 1 Argument – 2 Wise Rich Men Doctors Men Rakesh Doctors Sunil Wise Rich Rakesh Sunil The form of the above arguments can also be expressed as follows --- All A is B X belongs to A Therefore, X belongs to B Can you give examples of....... 1. Two arguments having different forms and same content? 2. Two arguments having different forms and different content? Can you state the form of the following arguments? 1. All scientist are intelligent. 2. All men are rational. All intelligent are creative. Some rational beings are good. Therefore, all scientists are creative. Therefore, some men are good. 4 4) True / False and Valid / Invalid An arguemnt consists of propositions / statements. Proposition is either true or True / False and Valid / Invalid are false. The terms valid / invalid are not used important terms in logic. The terms valid / invalid for propositions in logic. A proposition is are used for arguments in logic. An argument considered to be true if whatever is stated in the is either valid or invalid and never true or proposition agrees with actual facts, if not it is false. Validity of an argument depends upon false. For example, ‘Washington is an American the evidence in the premises for the conclusion. city’ is a true proposition. And ‘London is an If the conclusion of an argument necessarily Indian city’ is a false proposition. follows from the evidence in the premises then the argument is valid otherwise it is invalid. 1.4 DEDUCTIVE AND INDUCTIVE ARGUMENTS / INFERENCES Can you find the difference in the evidence of following arguments? 1. If it rains then roads become wet. 2. All observed crows are black. It is raining. No observed crow is non-black. Therefore, roads are wet. Therefore, all crows are black. Arguments are classified into two types Another important feature of a deductive 1) Deductive arguments 2) Inductive arguments. argument is that, its conclusion is implicit in the This classification of argumments into deductive premises i.e. the conclusion does not go beyond and inductive is based on the nature of relationship the evidence in the premises. This means between premises and conclusion. Premises of we don’t arrive at any new information. By deductive arguments claim to provide sufficient deductive argument we can know what is implied evidence for the conclusion, whereas premises by the premises. Deductive arguments do not of inductive arguments provide some evidence give us any new information. For this inductive for the conclusion. arguments are useful. Thus, the certainty of deductive arguments comes at a cost. Deductive Argument / Inference – Every argument attempts to prove the conclusion. The In an invalid deductive argument, however, evidence needed to establish the conclusion is the claim that premises provide sufficient given in the premises. The evidence given in the evidence is not justified, therefore, the relation premises is not always sufficient. A deductive of implication does not hold between its premise argument claims to provide conclusive and conclusion. Even when the premises are true grounds i.e. sufficient evidence for its the conclusion may be false. For example, let us conclusion. If the claim that premises provide consider the following arguments. sufficient evidence is justified, the deductive argument is valid, if not it is invalid. (1) If Amit passes S.S.C. with good marks, he will get admission in college. In a valid deductive argument where the Amit passed S.S.C. with good marks. evidence is sufficient the relation between the Therefore, he well get admission in premises and the conclusion is of implication. college. Premises imply the conclusion means, if premises are true the conclusion is also true, it is (2) Meena will either go to college or study impossible for the conclusion to be false. Thus at home. the conclusion of a valid deductive argument is Meena did not go to college. always certain. Therefore, Meena is studying at home. 5 (3) If Anita gets the prize then she will become The form of all the above given deductive famous. arguments is as follows : Anita did not get the prize. All X is Y. Therefore, Anita will not become famous. All Y is Z. Therefore, All X is Z. (4) If it rains heavyly, the college will declare holiday. It is obvious that the form is valid and College has declared a holiday. therefore all the arguments being its substitution Therefore, it is raining heavily. instances are also valid. It is easy to accept that the first two arguments are valid because the All these arguments are deductive premises and conclusions of these arguments arguments as the conclusions of all the arguments are all true and conclusion necessarily follows don’t go beyond the evidence in the premises. from the premises. But one may find it difficult The first two arguments are valid as premises to accept that, the third and fourth argument is provide sufficient evidence. The premises imply valid as premises and conclusion of both the the conclusion. If premise are true, conclusion arguments are false. However they are also valid. cannot be false. The last two arguments, though Validity of deductive argument is conditional. deductive, are not valid because the claim that In case of a valid deductive argument if premises provide sufficient evidence is not premises are true the conclusion must be justified. Even when premises are true, the true. So if premises of the last two arguments conclusion may be false. So there is no relation of are assumed as true then the conclusions of implication, the conclusion does not necessarily both the arguments necessarily follow from the follow from the premises. premises and therefore both the arguments are The deductive arguments are formally valid. If conclusion necessarily follows from the valid. A formally valid argument is one whose premises then the deductive argument is valid. validity is completely determined by its form. Premises and conclusion of valid deductive In case of deductive arguments the content of argument may or may not be true. When the its premises and conclusion does not affect its deductive argument is valid and its premises validity. There is no need to judge the content and conclusion are true, such an argument is of the premises and conclusion, also there is no called sound argument. need to find out whether they are true or false to As deductive arguments are formally determine the validity. One only needs to check valid, the validity of deductive arguments can the form of the argument. If the form is valid the be determined or proved by using the rules and argument is also valid. For example – methods developed by logicians. (1) All men are animals. Inductive Argument / inference --- All animals are mortals. Inductive argument is an argument which Therefore, all men are mortals. provides some evidence for the conclusion. (2) All crows are birds. The conclusion of an inductive argument goes All birds have wings. beyond the evidence in the premises. There is a Therefore, all crows have wings. guess, prediction or something new is asserted in the conclusion for which the evidence given in (3) All singers are actors. the premises is not sufficient. As the evidence in All actors are leaders. the premises is not sufficient, the premises of an Therefore, all singers are leaders. inductive argument don’t imply the conclusion. (4) All cats are rats. This means even when the premises are true the All rats are lazy. conclusion may be false. The conclusion of an Therefore, all cats are lazy. inductive argument is always probable. Whether 6 the argument is good (valid) or bad (invalid), the the form of the argument, but is decided by its possibility of its conclusion being false always content. Inductive arguments are materially remains. valid. A materially valid inference is one whose validity is completely determined by its Technically the terms ‘valid’ and ‘invalid’ content. To decide whether the given inductive cannot be used for inductive arguments. Only argument is good or bad, one has to consider deductive arguments are either valid or invalid. the content / the subject matter of the argument. Inductive arguments can be judged as better or The form of the first and second argument is the worse. More the possibility of the conclusion same but the first one is bad whereas the second being true, better the argument. The addition one is good. of new premises may alter the strength of an inductive argument, but a deductive argument, The amount of evidence in the premises if valid, cannot be made more valid or invalid by determines whether the argument is good. If the the addition of any premises. We shall use the evidence in the premises makes it reasonable terms ‘good’ or ‘bad’ for inductive arguments. to accept the conclusion, then, the argument For example, consider the following arguments. is good otherwise it is bad. From the above given arguments, the first arguments is a bad (1) Whenever cat crossed my way in the past, one because the conclusion is based on the something bad happened on that superstition, there is no connection between day. a cat crossing the way and good or bad events Today morning a cat crossed my way. happening in our life. In the other two arguments, Therefore, I am sure that something bad is though, the conclusions may turn out to be going to happen today. false, the evidence on the basis of which the (2) Every morning I have seen the sun rising conclusions are derived is scientific. Hence the in the east. last two arguments are good. It is early morning now. Though the content decides whether So, I am sure I will find sun rising in the an inductive argument is good, this does not east. mean that the premises and conclusion of (3) The doctor told me that, Suresh is suffering good inductive arguments are true and of bad from cancer and he will not survive for inductive arguments are false. In case of the more than three months. first argument, even if premises are true and the After two months I got the news that conclusion turns out to be true, still the argument Suresh is no more. is bad. Similarly in case of the last argument So, Suresh must have died due to cancer. even if conclusion turns out to be false when the premises are true, the argument is good because All the above given arguments are inductive the inference is based on the doctor’s verdict. arguments as conclusions of all the arguments go beyond the evidence in the premises. The Like deductive arguments, whether the premises don’t imply the conclusion. Even given inductive argument is good or bad cannot if premises are true the conclusions of all the be determined by the methods and rules of logic. arguments are probable. The conclusion is In case of common man’s inductive arguments, probable does not mean that the argument is bad. as given above, one can easily decide whether In the above given arguments the first one is bad they are good or bad. However, in case of the where as the other two are good. inductive arguments, in various sciences, by judging the evidence in the premises only the Like deductive arguments the validity of experts in the field can decide whether it is good or inductive arguments i.e. whether the inductive bad. Unlike deductive arguments, the Inductive argument is good or bad, is not determined by arguments, provide us with new information and 7 thus may expand our knowledge about the world. an argument may be valid when one or more So, while deductive arguments are used mostly or even all its premises and conclusion are in mathematics, most other fields of research false and an argument may be invalid with all make extensive use of inductive arguments. its premises and conclusion true. The truth or falsity of an argument's conclusion does not Truth and Validity of arguments – The by itself determine the validity or invalidity of relation between validity or invalidity of the that argument. And the fact that an argument argument and truth or falsity of its premises and is valid does not guarantee the truth of its conclusion is not simple. As discussed earlier, conclusion. DEDUCTIVE ARGUMENT INDUCTIVE ARGUMENT 1. Premises claim to provide sufficient 1. Premises provide some evidence for the evidence for the conclusion. conclusion. 2. In valid deductive argument premises 2. Premises do not imply the conclusion. imply the conclusion. 3. In valid deductive argument if premises 3. Even when premises are true conclusion are true, conclusion must be true. may be false. 4. Conclusion of valid deductive argument 4. Conclusion is always probable. is always certain. 5. Conclusion does not go beyond the 5. Conclusion goes beyond the evidence in evidence in the premises. the premises. 6. Arguments are formally valid. 6. Arguments are materially valid. 7. Validity can be determined by rules 7. Correctness of arguments can be decided and methods of logic. by an appeal to experience and not by rules and methods of logic. 8. Deductive arguments cannot expand 8. With inductive arguments we can discover our knowledge of the world, by deduction something new and expand our knowledge we can only know what is implied by the of the world. premises. Summary In past logic developed independently in India, Greece and China. Modern logic is evolved from Aristotelian or traditional logic. Logic is study of methods and principles used to distinguish between good and bad reasoning. Arguments, Valid argument, Form of argument, True / False, Valid / Invalid are some important concepts in logic. The two important types of arguments are – Deductive and Inductive arguments. Deductive arguments claim to provide sufficient evidence for the conclusion. Inductive arguments provide some evidence for the conclusion. 8  Exercises Q. 1. Fill in the blanks with suitable words 15. Conclusion of.............. inference does not given in the brackets. go beyond the evidence in the premises. 1............... is regarded as the father of logic. (Deductive / Inductive) (Aristotle / De Morgan) Q. 2. State whether following statements are 2. The development of logic throughout the true of false. world is mainly influenced by the.............. 1. Logic is a branch of Psychology. logic. (Aristotelian / Indian) 2. Philosophy is fundamental to all spheres 3. The Nyaya Sutra of.............. constitute the of human enquiry. core texts of the Nyaya School. (Gautama / Nagarjun) 3. The Jaina logic is represented by the Nyaya School of philosophy. 4. The proposition which is established in the argument is called the.............. 4. Mozi, "Master Mo" was mainly responsible (Conclusion / Statement) for the development of logic in China. 5. The proposition which is stated in support 5. Etymologically logic is often defined as of the conclusion is called.............. the science of the laws of thought. (Premise / Conclusion) 6. Form means pattern or structure of the 6............... means pattern or structure of the argument. argument. (Content / Form) 7. Argument is either true or false. 7............... is either valid or invalid. 8. The classification of arguments into (Proposition / Argument) deductive and inductive is based on the 8. A deductive argument claims to provide nature of relationship between premises.............. evidence for its conclusion. and conclusion. (Some / Sufficient) 9. When the deductive argument is valid and 9. In Inductive argument premises provide its premises and conclusion are true, such.............. evidence for the Conclusion. an argument is called sound argument. (Some / Sufficient) 10. A formally valid argument is one whose 10. In case of a valid.............. argument if validity is completely determined by its premises are true the conclusion must be content. true. (Deductive / Inductive) 11. Conclusion of inductive is always certain. 11. A materially valid inference is one whose validity is completely determined by its 12. Conclusion of inductive argument goes.............. (Content / Form) beyond the evidence in the premises. 12. Conclusion of valid deductive argument is 13. Even when premises are true conclusion always.............. (Certain / Probable) of valid deductive argument may be false. 13. Validity of.............. arguments can be 14. The truth or falsity of an argument's determined by rules and methods of logic. conclusion does not by itself determine the (Deductive / Inductive) validity or invalidity of that argument. 14. Correctness of.............. arguments is 15. Deductive arguments cannot expand our determined by an appeal to experience. knowledge of the world. (Deductive / Inductive) 9 Q. 3. Match the columns. Q. 6. Explain the following. (A) (B) 1. Truth and validity. 1. Nyaya 1. Sufficient evidence 2. Form of argument. 2. Aristotle 2. Mozi ‘Master Mo’ 3. Distinction between form and content. 3. Mohist school 3. Some evidence 4. Distinction between formal and material 4. Nagarjun 4. Valid or Invalid validity. 5. Argument 5. Greek logician 5. Distinction between deductive and inductive argument. 6. Statement 6. Buddhist philosoper Q. 7. Answer the following questions. 7. Deductive 7. Akshapad Gautama 1. Explain in brief origin and development of argument logic. 8. Inductive 8. True or false 2. Write short note on Indian Logic. argument 3. Define logic and explain the terms - Q. 4. Give logical terms for the following : Argument, Premise and Conclusion. 1. The study of methods and principles used 4. Explain the difference between terms - to distinguish good from bad resoning. Reasoning, Inference and Argument. 2. A proposition that is stated in support of 5. Explain with illustration nature of the conclusion in an argument. Deductive argument. 3. The proposition that is established in the 6. Explain with illustration nature of argument. Inductive argument. 4. An argument that claims to provide Q. 8. State whether the following arguments sufficient evidence for its conclusion. are deductive or Inductive. 5. An argument in which premises provide 1. Either it is a bank holiday or the bank is some evidence for the conclusion. open. It is not a bank holiday. Therefore the bank is open. 6. An argument whose validity is completely determined by its form. 2. There are no good players in our college team. So the team will not win the match. 7. An argument whose validity is completely determined by its content. 3. Whenever I went to my sister’s house she cooked biryani for me. As I am visiting Q. 5. Give reasons for the following : my sister today, I am sure my sister will 1. Etymological definition of logic is not make biryani. accurate. 4. My aunty is a doctor, so she is a female 2. Deductive arguments cannot expand our doctor. knowledge of the world. 5. If Mohan takes admission for science then 3. Conclusion of valid deductive argument is he will take computer science. Mohan has always certain. taken admission for science. So he must have opted for computer science. 4. Conclusion of an inductive argument is always probable. 6. Meena is smart. Seema is smart, Neena is smart. These are all girls. Therefore all girls are smart. 10 7. Sunil is hardworking, intelligent and 12. India has taken loan from the world bank, smart. Therefore Sunil is smart. so India is sure to develop economically. 8. Nikita is not happy with her job, so I am 13. If and only if a student is sick during sure she will leave the job. examination, he is allowed to appear 9. Mukesh is an actor and Mukesh is for re-examination. Ashok is allowed to handsome. Therefore Mukesh is handsome appear for re-examination. So Ashok must actor. have been sick during examination. 10. If I go to college then I will attend lecture. 14. Suresh is taller than Naresh. Naresh is If I attend lecture then I will understand taller than Ramesh. Therefore Suresh is logic and if I understand then I will pass taller than Ramesh. with good marks. Therefore if I go to 15. Hardly any man lives for more than college then I will pass with good marks. hundred years. Mr. Joshi is ninety nine 11. Amit and Sumit are in same class, they year old. So he will die next year. both play cricket and go to same tuition class. Amit is a good singer. Therefore Sumit is also a good singer. v v v 11  Chapter 2 Nature of Proposition Logic studies the preservation of truth and propositions are the bearers of truth and falsity. Identify the following arguments. EXAMPLE 1. EXAMPLE 2. All men are mortal. All actors are handsome. All artists are men. Prasad is an actor. Therefore, all artists are mortal. Therefore, --------------- We have seen earlier that one of the 2.1 PROPOSITION (STATEMENT) functions of logic is to study arguments. AND SENTENCE However, to study the arguments, it is essential Definition of proposition – to understand the statements that constitute an argument. A proposition is defined as a sentence, which is either true or false. We begin by examining propositions, the building blocks of every argument. An Activity : 1 argument consists of premises and conclusion. These premises and conclusion are in the Make a list of true or false form of propositions or statements. Hence, a propositions. proposition is a basic unit of logic. From the definition of proposition we can Find the premises and the conclusion conclude that all propositions are sentences but from the following : all sentences are not propositions. Only those sentences which are either true or false will be propositions. Hence, the class of proposition is EXAMPLE 1 narrow, whereas the class of sentences is wider. All monuments are beautiful. This leads to a question that, which sentence can be true or false? To answer this question we shall Taj Mahal is a monument. have to consider various kinds of sentences. Therefore, Taj Mahal is beautiful. Activity : 2 EXAMPLE 2 Make a list of sentences you know and state its kind. All Mangoes are fruits Kinds of Sentences : All fruits grows on trees. (1) Interrogative Sentence : This kind of Therefore, All mangoes grow on trees. sentence contains a question. Example : What is your name? 12 (2) Exclamatory Sentence : It is a kind of Grammatically given examples are Inerrogative sentence which expresses some kind of and exclamatory sentences respectively but feelings. logically they are propositions. Example : Oh! God Activity : 3 (3) Imperative Sentence : This sentence Make a list of Assertive / Declarative / expresses a command or an order. Informative sentences. Example : Get Out. PICTURE: 1. (4) Optative Sentence : This sentence expresses a wish, desires, urges. Example : May God bless us all. (5) Assertive Sentence : It is a sentence which asserts something about an individual. This sentences can make positive or negative assertion. (It refers to identifiable particular individual possessing definite, particular property.) PICTURE: 2 The word “Individual” stands not only for persons but for anything like city, country, animal, or anything to which attributes can be significantly predicated and the property/ attribute may be an adjective, noun, or even a verb. Example : Sanika visits her grandmother during the holidays. (Positive assertion) Example : The tiger is not a domestic animal. (Negative assertion) Activity : 4 These kind of sentences can be either Observe and describe these pictures and true or false. Hence, they are considered as make a list of assertive propositions. statements or propositions in logic. They are (positive assertion and negative also called as declarative sentences. They are assertion) informative sentences because they provide us A proposition is expressed in the form of a with information. So declarative sentences can sentence. But it is not the same as sentence. The make logical propositions. So, we can conclude same proposition may be expressed by different that all sentences are not propositions. Only sentences. those sentences which can be either true or false can be propositions. Example : (1) This is a fish (English) Sometimes declarative sentence may be (2) Das ist ein fisch (German) in the form of a question or an exclamation e g. (1) Do you feel you can fool your friend. (3) ¶h ‘N>br h¡& (Hindi) (2) Thief......... Thief (4) hm ‘mgm Amho. (Marathi) (5) kore wa sakana desu. (Japanese) 13  Here a sentence in English, Marathi, If a proposition does not represents a Hindi, German, Japanese may differ as sentence fact and if the claim is not justified then, the but they express the same proposition. proposition is false. Anything that is known through sense Example : Mumbai is capital of India. organs has physical existence. A proposition (truth value of this proposition is false) refers to the meaning or content expressed in the form of a sentence. Therefore, it does not have (2) A proposition has only one truth value. a physical existence. It is expressed through the A proposition cannot be true and false medium of a sentence. together. On the other hand a sentence has a physical existence. A sentence when spoken, is in the E. g. Chalk is white. (This proposition form of sound waves. When written, it is a sign cannot be both true and false.) or a symbol on a surface. e.g. In five different (3) The truth value of a proposition is sentences given above. The meaning expressed definite : in these sentences is the proposition which does not have a physical existence because one A proposition has unique truth value. If a cannot see it, touch it but one can understand it if proposition is true, it is always true. If it is false, and only if the language in which it is expressed it is always false. In other words truth value of a is known. proposition does not change. The following are the main characteristics Example : The earth is a flat disc. of proposition : Though, the truth value of the above (1) Every proposition has a truth value : proposition appears to have changed but in The truth or falsity of proposition are reality it not so. This proposition was believed called truth values. The truth value of a true to be true due to ignorance (lack of scientific proposition is true and that of a false proposition knowledge) but it is proved to be false today. is false. Thus, all propositions are sentences but Now the question arises, “what determines all sentences are not propositions. Only those the truth value of a proposition? sentences which are either true or false are The answer is “The Fact”. propositions. If a proposition represents a fact or facts, it is true. It means a proposition is true when the assertion in a proposition {that which is said in a proposition} agrees with the facts. Example : Butter melts in heat. Activity : 5 Look at the pictures carefully and construct the propositions describing the pictures. 14 There are important differences between the proposition and sentence. Yet they are interconnected. Proposition (Statement) Sentence (1) It is sentence which is either true or (1) It is a meaningful group of words in a false. grammatical order. (2) A proposition is conveyed through (2) A sentence is a vehical through which a a sentence. statement is expressed. (3) Only declarative sentences are proposition. (3) The sentences which expresses feeling, wish etc are sentences only. (4) Every proposition has a truth value (4) Sentence does not have a truth value. i.e it is either true or false. It is neither true nor false. (5) A proposition does not have physical (5) A sentence has a physical existence. existence. (6) Example : Taj Mahal is white. (6) Example : How are you? 2.2. Classification of proposition : Compound proposition : Classification of proposition can be Example : done on the basics of whether the statement (1) (Delhi is the capital of India) and (it is a contains another statement as it’s component crowded city). propositions. Some propositions do not contain another proposition as a component, while others (2) (Peacock generally lives in jungle) or do. The former are called simple proposition and (bushes.) the later are compound propositions. (3) If (polygon has six sides) then (it is a Simple Proposition : hexagon.) It is a basic unit in logic. Simple (4) If (turmeric reduces my arthritis pain) proposition is defined as a proposition that then (I will eat turmeric everyday). does not contain any other proposition / propositions as it’s component. (5) (Anil is eligible to drive) if and only if (he is eighteen years old). Example : (6) It is false that (Mumbai is the capital of (1) Delhi is the capital of India. England). (2) Peacock generally live in jungle. Proposition that comes as a part of a proposition is called as a component (3) Polygon has six sides. proposition. The propositions in a compound (4) Turmeric reduces my arthritis pain. statements are called its components. (5) Anil is eligible to drive. Activity : 7 (6) Mumbai is the capital of England Identify Component proposition from the above example. Activity : 6 Thus, a compound proposition is defined Make a list of simple propositions. as a proposition which contains another proposition / propositions as its component. 15  Activity : 8 (4) Class membership proposition : Construct compound propositions using A class membership proposition asserts statements constructed by you in Activity 6. that an individual is a member of a class. Thus, it shows that the subject term belongs to the class Kinds of simple proposition : indicated by predicate. So, here predicate term is general. There are four kinds of simple proposition. These are : Example : (1) Subjectless proposition : (1) Rani Lakshmi bai was a great warrior. The simplest kind of proposition is the (2) Bhagat Singh was a freedom fighter. subjectless proposition. Kinds of compound proposition : Example : Compound proposition are further (1) Bomb! classified into two kinds – (2) Fire! (1) Truth – functional compound proposition Subjectless propositions make an (2) Non Truth – functional compound assertion. They give information. Therefore proposition. they are propositions. However the subject of the assertion is not clear. They are primitive (1) Truth functional compound proposition: propositions. In a compound proposition there are two or (2) Subject – Predicate proposition : more component propositions that are connected by some expression like ‘and’, ‘or’ etc. These A subject – predicate proposition states component propositions are either true or false. that an individual possesses a quality or attribute. The component proposition as a whole also has A subject predicate proposition is that which has some truth value. a subject , a predicate and a verb. An individual is a singular term. Therefore, the subject of this Example : Sameer is intelligent and kind of proposition is a singular term. Sameer is smart. Example : Ashok is intelligent. In this proposition there are two propositions (3) Relational Proposition : (1) Sameer is intelligent. A relational proposition states a relation between two subjects. The subjects between (2) Sameer is smart. which a relation is stated are called terms of relation. Example : Ram is taller than Shyam. The above proposition expresses a relation between two subjects namely Ram and Shyam. 16  Now when there are two component propositions, we get four possibilities as given below : Sameer is intelligent And He is smart TRUE TRUE TRUE TRUE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE The truth value of compound propositions Thus Non – truth functional compound (which is stated in the middle column) changes as proposition is defined as a compound per the truth value of its component proposition. proposition whose truth value is not determined by the truth value of its component In the above example when both the proposition / propositions. components are true one can say that the compound proposition is true. Otherwise under Types of truth functional compound other possibilities it is false. proposition : Thus the truth functional compound On the basis of the connectives which proposition is defined as a compound combine the components in truth functional proposition whose truth value is determined propositions, we get five types of truth functional by the truth value of its component proposition compound proposition. / propositions. (1) Negative proposition (2) Non – truth functional compound proposition : Example : This book is not interesting. There are some compound propositions (2) Conjunctive proposition whose truth value is not determined by the truth Example : This book is interesting and value of its component proposition / propositions. informative. Such compound propositions are called (3) Disjunctive proposition Non – truth functional compound proposition Example : Either this book is interesting Example : I believe that Soul exist. or informative. Here the component proposition “Soul (4) Material Implicative or conditional exist” may be either true or false. proposition – Whatever may be the truth value of the Example : if this book is interesting then component proposition, the truth value of the people will buy the book. compound proposition does not get affected. (5) Material Equivalent or Bi – conditional If the proposition, ‘I believe that Soul proposition – exit” is say true, then whether the component proposition “Soul exist” is true or false. The Example : People will buy this book if truth value of the compound proposition will and only if it is interesting. remain true. Hence, It is a Non Truth functional compound proposition. 17  (3) It prevents confusion of vague and 2.3 Symbolization of proposition : ambiguous words. Need, uses and importance of Symbols are kind of short – forms. In a symbolization. natural language a proposition or an inference Symbolization is necessary because has a much longer expression. When we use arguments are expressed in language. The use symbols the expression becomes much more of symbols is not misleading but it helps us to shorter. reason correctly. For symbolizing of truth functional There are certain defects of natural compound propositions. We need certain language as follows. symbols. They are – (1) use of ambiguous words and vague words. (1) Propositional Constant (2) use of misleading idioms. (2) Propositional Variable (3) confusing metaphorical style. (3) Propositional Connective or Operator The symbolic language is free from the (4) Brackets above mentioned defects. (1) Propositional Constant : Logic is concerned with arguments. Propositional constant is defined as Arguments contain propositions or statements a symbol, which stands for a specific (or as their premises and conclusion. Arguments particular) proposition as a whole. They are may be valid or invalid. To determine the called constants because they have definite validity of the arguments we have to use certain meaning. The capital letters from A to Z (English logical procedures. These procedures can not alphabet) are used as propositional constants. be applied directly to the propositions with We are free to use any propositional constants ordinary language. Logicians have developed for symbolizing of a proposition. specialized techniques to bring out the form of the proposition. It is done by symbolizing Example : Yogasanas act as bridges to propositions. unite the body with the mind. Deductive Logic is concerned with the The above proposition can be symbolized form of an argument and not with the content of as “A” or by any other capital letter which will argument. It is form of a proposition. This can be stand for the whole proposition. done by symbolization. When an argument contains more number Use of symbols is convenient and of propositions as components we have to advantageous, for better understanding of observe following conditions or restrictions. arguments and drawing of inference from it. (1) The same propositional constant is to be Significance of symbolization in Logic – used for symbolizing a proposition if it occurs again in the same argument (or in (1) It helps to focus on what is important in the same compound propositions) an argument and to ignore unnecessary details, thus helps to decide it’s validity (2) The same propositional constant can not easily. be used for different propositions in the same argument. (or in the same compound (2) It helps to understand the logical structure proposition) of propositions and arguments more clearly. 18  Example : Santosh will take salad or for any specific proposition. It only marks or sandwich. indicates the place of proposition. Santosh will not take salad. For Example : The expression “if p then q” indicates that “p” stands for any proposition Therefore, Santosh will take sandwich. and “q” stands for any other proposition and In the above example for the proposition these two different propositions are connected “Santosh will take salad.” we will choose by the expression “if………….then”. the propositional constant “S” and for the A propositional variable is a symbol used proposition “Santosh will take sandwich” we to substitute a proposition. cannot use the same propositional constant “S”. (as per restriction no.2) so we will have to use When an argument form contains more different propositional constant, like “D” number of propositions as components we have to observe following conditions or restrictions. Example : (1) The same propositional variable is to be The first proposition (premise) is substituted by the same proposition if it Santosh will take salad or sandwich occurs again in the same argument (or in the same compound proposition) The symbolization of this proposition will be (2) The same propositional variable can not be substituted by different propositions in the S or D same argument. (or in the same compound The second proposition (premise) is proposition) Santosh will not take salad. In an argument of the following form, for The symbolization of this proposition will instance by substituting any proposition for “p” be and any other proposition for “q” we will get innumerable arguments. Not S Example : if p then q The third proposition (conclusion) is Not q Santosh will take sandwich Therefore Not p The symbolization of this proposition will be Example No : 1 Therefore D If a figure is a square then it has four sides. Thus the argument may now will be The figure does not have four sides. symbolized thus: Therefore the figure is not a square. S or D Example No : 2 Not S Therefore D If you have a password then you can log on to the network. (2) Propositional Variable : You can not log on to the network. Propositional variable is defined as a symbol which stand for any proposition Therefore, you do not have a password. whatsoever. Small latter p, q, r, s …….. We can substitute any proposition for a (English alphabet) are used as propositional propositional variable, it is therefore said to be a variable. Propositional variable does not stand place marker / place holder or dummy letter. 19  Activity 9. Read the following arguments forms carefully and construct arguments form it. (1) Either p or q (2) If p then q (3) If p then q Not p p If q then r Therefore q Therefore q Therefore If p then r Propositional Connective symbol Name of the symbol (1) Not ~ Tilde / Curl (2) And · Dot (3) Either ………. Or Ú Wedge (4) If ……….. then  Horse - shoe (5) If and only if then º Tripple Bar (3) Propositional connective – (Truth – Example : (from English Language) Functional logical operator) – Why we need commas because Propositional connective is defined as “I like cooked vegetables, fruits and dogs.” an expression which operates on proposition is not same as or propositions or they connects two “I like cooked vegetables fruits propositions in a truth functional compound and dogs.” proposition. There are five expressions which connect component or components in a truth In mathematics, to avoid ambiguity and to functional compound proposition. The name make meaning clear, punctuation marks appear of the symbols for five connectives, are given in the form of brackets. below. These symbols are also called logical Example : 6 + 7  8 constants / operators. It could be 6 + (7  8) or (6 + 7)  8 The propositional connective “not” So, in Logic some punctuation marks operates on one proposition only. are equally essential, to clear the complicated Therefore it is known as Monadic operator. propositions. In symbolic Logic parentheses, brackets and braces are used as a punctuation On the other hand, the last four connectives marks. or operators namely [and, either…….or, if….. then……, if and only if…… then…..] connects (1) Parentheses : It is a symbol ‘( )’ that is put two propositions. Therefore, they are known as around a word or a phrase or a sentence. Binary or Dyadic operators. Example : (p · q)  r Importance of Bracket (s) in symbolization – (2) Box Brackets : It is used to enclose words or figures. In logic it is used to group In language punctuation is requires to expressions that include parentheses. make complicated statements clear. Example : [ (p · q) Ú (q · p) ] º r Punctuation are a mark such as full stop, comma or question mark, exclamation mark, semicolon, inverted comma etc. which are used in writing to separate sentences and their elements and to clarify their meaning. 20 (3) Braces : It is used to group expressions Symbolization : that include box brackets. Example: { } Example : Sadanand is not a mathematician. Example : ~{[(p · q) Ú ( q · p) ] º p } Step 1: The above example consists of one Truth functional compound propositions proposition and one propositional operator. On the basis of five propositional Underline the proposition and put a propositional connectives, there are five types of Truth operator in the box. functional compound propositions. They are as follows – So we will get following expression: (1) Negative proposition Example : (2) Conjunctive proposition Sadanand is not a mathematician. (3) Disjunctive proposition ~S (4) Material Implicative or Conditional Thus the form of negative proposition is proposition ‘~ p’. This is read as ‘Not p’. (5) Material Equivalent or Biconditional Always Remember : proposition ~ Sign to be written before the letter or (1) Negative proposition : on the left hand side of the letter. When any proposition is negated or ~P  P~  denied we get negative proposition. Negation is commonly expressed in English language by the Truth value for negation – word “Not”. But a proposition can be negated Negation is also known as contradictory with the help of words like it is not the case function. that, it is not true that, it is false that, none, never. A negative statement is true when its component proposition is false and vice versa. Example : Basic truth table for negation : (1) Sadanand is not a mathematician. (2) It is false that Ajit is taller than Rajesh. ~ P (3) It is not true that Urmila is a magician. F T (4) It is not the case that Ajay is a singer. T F In Logic we use symbols for propositional (2) Conjunctive Proposition – (conjunction) connectives as well as propositions. For the connective “Negation” or the word “not” the When two propositions are joined together symbol “ ~ “ is used. by truth – functional connective “and” it is called a conjunctive proposition. This symbol is called as “Tilde” or “Curl” The components of conjunctive proposition are called as Conjuncts. Using the symbol “~” for negation and the propositional variable “p” for any proposition The word “and” is called dyadic whatsoever, we get the form of negative connective or binary operator, as it connects two propositions as follows : propositions. ~p 21  Example : Be good and you will be happy. (7) Chocolates are neither nutritious nor good for teeth. The above example consists of two propositions – (8) Mr. Patil is a politician and Sai baba is a saint. (1) Be good Symbolic form of conjunctive (2) You will be happy. proposition will be as follows : These are connected by the word “and”. Example : Be good and you will be happy. Often we use word such as but, though, To symbolize a propositional operator although, while, yet, also, still, nevertheless, “And” we can use symbol ( · ) however, moreover, further, as well as, neither……. nor, in the conjunctive sense. Symbolic from of conjunctive proposition is as follows: Example : ‘p · q’ (1) The lion is called king of the forest and it has a majestic appearance. Example : Sugandha is a mother and a grandmother. (2) I want to go to the party, but I am tired. Above proposition consists of two parts (3) Gauri is playing, while Varsha is studying. (components) (4) The couch was shouting, yet the players (1) Sugandha is a mother. remained noisy. (2) Sugandha is a grandmother. (5) Hemangi kept working even though she was tired. These two parts or components of a conjunctive proposition are called as Conjuncts (6) It’s a small house still it is spacious. in the language of Logic. Sugandha is a mother and a grandmother. (First conjunct) (Second conjunct) Thus symbolization of above proposition is M·G Thus the form of conjunctive proposition is ‘p · q’. It is read as ‘ p and q’. Truth Value : Basic truth table for conjunction : A conjunctive proposition is a kind of truth p · q functional compound proposition. Hence, the truth value of a conjunctive proposition depends T T T on its components i.e. conjuncts. T F F A conjunctive proposition is true only F F T when both the conjuncts are true otherwise it is false. F F F 22 (3) Disjunctive proposition – (Disjunction) Example : When two propositions are joined together Either he is rich or poor. by truth functional connective …. ‘either ……. …… or’, it is called a disjunctive proposition. In the above example there are two The word “either …… Or” is called dyadic propositions. connective or binary operator, which connects (1) he is rich two statements. The components of disjunctive proposition are called as “Disjuncts”. (2) he is poor Example : These two propositions are connected by truth – functional connective or logical (1) Either I will go to Prague or Vienna. connective” Either…………..Or”. (2) Either she is weak or coward. Form of the disjunctive proposition is (3) The car is either blue or red. “p Ú q”. This is read as “p or q”. Symbolization : Either he is rich or he is poor. Proposition Logical connective or operator Proposition (First disjunct) (Second disjunct) R Ú P Therefore, symbolization of the above proposition will be: RÚP Form of disjunctive proposition is ‘p Ú q’. This is read as ‘p or q’. Truth Value : Disjunctive proposition may be used in the inclusive (weak) sense or exclusive (strong) A disjunctive proposition is a kind of truth sense. functional compound proposition. Hence, the truth value of a disjunctive proposition depends (1) The Inclusive or weak sense of “OR” – on its components i.e. disjuncts. When both the disjuncts can be true, the A disjunctive proposition is false, only word or is said to be used in inclusive sense. when both the disjuncts are false otherwise it Rajvi is either a mother or an actress. is true. In the above proposition there are two Basic truth table for disjunction : disjuncts. p Ú q (1) Rajvi is a mother. T T T (2) Rajvi is an actress. T T F Both these disjuncts can be true together F T T because a person can be both a mother and an actress. F F F 23  In other words the statement can be (4) Material Implicative or Conditional interpreted as “either p or q or both”. Means “p” proposition – alone can be true, “q” alone can be true and both When two propositions are joined together can be true together but cannot be false. by truth functional connective if …… then (2) Exclusive or strong sense of OR : ……. it is called an implicative proposition. When both the disjuncts cannot be true Example : together, the word “Or” is said to be used in (1) If you want a good pet then you should exclusive sense. get a dog. Example : Either it is a sparrow or a crow. (2) If my car is out of fuel then it will not run. In the above proposition there are two (3) If a figure is a rhombus then it is not a disjuncts. rectangle. (1) A bird is a sparrow. 4] If you do all the exercises in the book, you (2) A bird is a crow. will get full marks in the exam. Both these disjuncts cannot be true 5] If it is a molecule then it is made up of together. If one is true, other is necessarily atoms. (exclusively) false. (sometimes ‘ , ’ (coma) is used instead of In other words this can be interpreted as, word “then”) either “p” is true or “q” is true but both cannot Words indicating implicative proposition be true together. i.e. if a bird is a sparrow then it – The expression like “if ………then”, “in cannot be a crow or vice versa. case”, “had it”, “unless” (if not) indicate that In logic, disjunctive proposition is used the proposition is a conditional proposition. in the inclusive sense only. If it rains then the trains will run late. Proposition 1 Proposition 2 If ……….. then ………. is logical operator. The symbolic expression : RT Thus, the form of the implicative proposition is “ p  q. This is read as “if p then q” or “p implies q”. An implicative proposition is also called The proposition that states the condition as conditional proposition because they state is called as antecedent and the proposition that the condition and its consequences. states result is called as consequent. Example : If she is tall then she can become a model. Condition Result antecedent consequent 24 Truth value : The expression “if and only if ” indicates that the statements is a biconditional statement. An implicative proposition is false only when its antecedent is true and its consequent Example : is false. Otherwise it is always true. Birds fly if and only if sky is clear. Basic truth table for material implication :