APPLICATIONS OF LOGICAL REASONING VI SEMESTER ADDITIONAL COURSE (In lieu of Project) BA PHILOSOPHY (2011 Admission) UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION Calicut university P.O, Malappuram Kerala, India 673 635.
UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION STUDY MATERIAL Additional Course (In lieu of Project) BA PHILOSOPHY VI Semester APPLICATIONS OF LOGICAL REASINING Prepared & Scrutinized by: Layout: Computer Section, SDE Dr. V. Prabhakaran, Sreevisakh, Thekkegramma Road, Sastha Nagar, Chittur. Reserved Applications of Logical Reasoning 2
CONTENTS PAGE No MODULE I 4 MODULE II 11 MODULE III 14 MODULE IV 17 MODULE V 22 Applications of Logical Reasoning 3
Aims and objectives: 1. To develop the student s critical thinking and intellectual problem-solving capability. 2. To test the ability of the student to apply the theory and solve problems involving tests of reasoning. 3. To develop skill for linguistic analysis and the ability to detect errors in reasoning. MODULE I REDUCTION OF ORDINARY LANGUAGE SENTENCES INTO STANDARD FORM SENTENCES. The traditional logic recognized four forms of propositions(a, E, I, O). A proposition which is not expressed in one of these forms is to be reduced to one of these according to the meaning of the proposition. Categorical propositions are classified with regard to quality and quantity: From the point of view of quality categorical propositions are either affirmative or negative. An Affirmative proposition is one in which an agreement is affirmed between the Subject and Predicate, or in which the Predicate is asserted of the Subject, e.g. snow is white. A Negative proposition is one which the Predicate is denied of the Subject. It indicates a lack of agreement between the Subject and Predicate. E.g. The room is not cold. The quantity of a proposition is determined by the extension of the Subject. on the basis of quantity categorical propositions are either universal or particular. A universal proposition is one in which the Predicate refers to all the individual objects denoted by the Subject. (the subjects is taken in its full extension) E.g. All men 2 are rational. A particular propositions is one in which the Predicate belongs only to a part of the denotation of the subject. E.g. some metals are white. Particular propositions usually begin with some word or phrase showing that the subject is limited in extent. The logical sign of particular proposition is some, but other qualifying words or phrases, such as the greatest part, nearly all, several, a small number, a few, etc. also indicate particularity. Applications of Logical Reasoning 4
A.E.I.O. Combining quantity and quality we get four types of categorical propositions, Universal Affirmative,Universal Negative,Particular Affirmative,Particular Negative.. A.E.I.O. are used to symbolise them A and I from affirmo stand for affirmative propositions; E and O, the vowels from Nego for negative propositions. The four types of propositions are: Universal affirmative: It is a categorical proposition in which the predicate agrees with the whole subject, e.g. All men are rational. All S is P Universal negative proposition: It is a categorical proposition in which the predicate does not at all agree with any part of the subject. E.g. No men are perfect. No S is P particular affirmative proposition: is a categorical proposition in which the predicate agrees only with a part of the subject. E.g. some flowers are red. Some S is P particular negative proposition is a categorical proposition in which the P does not agree with a part of the S. e.g. some Indians are not religious. Some S is not P S = Subject; P = Predicate The purpose of reducing sentences to logical form is to make the meaning of the sentence clear in logical reasoning. If propositions are stated in their logical form, testing of inferences becomes easier. The points to be remembered in changing sentences to logical form are: 3 1.The meaning of the original proposition is to be preserve in the standard form proposition. 2.The proposition must contain all the three parts in he proper order, subject, copula and predicate. 3. A suitable copula must be used between the subject and the predicate. 4. The sign of negation must go with the copula, and not with the predicate. Applications of Logical Reasoning 5
5. Compound sentences must be split up into simple propositions. 6. The quantity and quality of the proposition must be decided and stated clearly. The following procedure is to be followed while reducing propositions to their logical form: Subject and predicate of the given proposition are to be identified. Subject is that about which the assertion is made. Predicate is that which is asserted of the subject. Having identified the subject and predicate, the quality of the proposition is to be known, affirmative or negative whether the predicate is affirmed or denied of the subject. Next, the quantity of the proposition is to be known. If the predicate is affirmed or denied of the entire denotation of the subject, the proposition is Universal. If the predicate is affirmed or denied of a part of the denotation of the subject, the proposition is Particular. Certain general rules are to be followed : *Sentences which have words like all, every, each,any, whoever, with the subject and words like always, necessarily with the predicate are to be reduced into A form. Every ticket-holder must be admitted. L.F: All ticket-holders are persons who must be admitted. Any man can do that L.F: All men are persons who can do that. Any criminal is punishable L.F: All criminals are punishable. Poets always love nature. L.F: All poets are lovers of nature. Virtues are absolutely desirable. L.F: All virtues are desirable. *Sentences with all, every, any etc containing the signof negation, not are to be reduced to O form. All that glitters is not gold 4 Applications of Logical Reasoning 6
L.F: Some things that glitters are not gold. Every disease is not fatal L.F: Some diseases are not fatal Any fruit is not sweet L.F: Some fruits are not sweet Not every good bowler is a good batsman. L.F: Some good bowlers are not good batsmen Every cloud does not bring rain L.F: Some clouds are not those which bring rain *Sentences with no, none, never, nothing, nobody, not one, not a single, are to be reduced to E proposition. Not a single student has passed L.F: No student is one who has passed. Nothing done in hurry is well done. L.F: No acts done ina hurry are acts which are well done. Not one was saved in the shipwreck L.F: No sailors are those who were saved in wreck Misers are never happy L.F: No misers are happy. No lazy man succeeds in life. No lazy men are successful. None can live for ever L.F: No persons are those who can live for ever *Sentences containing words as some, most, a few, mostly, generally, all but one, almost all, frequently, often, many, certain, nearly all, a small number, the majority, sometimes, nearly always are to be reduced to the particular( I or O). Girls are generally shy L.F: Some girls are shy Many a flower is born to blush unseen 5 Applications of Logical Reasoning 7
L.F: Some flowers are things born to blush unseen Most Hindus are vegetarians L.F; Some Hindus are vegetarians Many rules are easy L.F: Some rules are easy A few students do not work hard L.F: Some students are not those who work hard Most of the students are not hostellers L.F: Some students not hostellers *Sentences containing words as few, seldom, hardly., scarcely, are to be reduced to O if there is no sign of negation and to I if there is a sign of negation. Few men are reliable L.F: Some men are reliable Few men have not suffered disappointments in life L.F: Some men are those who have suffered disappointments in life Students seldom pass this examination in the first attempt L.F: Some students are not those who pass this examination in the first attempt 6 *Singular proposition is reduced to a Universal proposition when the singular subject is a definite individual or a collection of individuals. If the subject is an indefinite singular term,the singular proposition should be taken as a Particular proposition. Gandhiji is the Father of Indian Nation.- A The sky is blue.-a One minister is not rich L.F: Some ministers are not rich *Indefinite or indesignate propositions are treated as Universal when the predicate is an invariable and common attribute of the subject. Applications of Logical Reasoning 8
*Indefinite propositionsare treated as Particular when predicate is only an accidental quality. Glass is breakable L.F: All glasses are breakable. Material bodies gravitate L.F: All material bodies are things which gravitate. Lemons are not apples. L.F: No lemons are apples Catholics are Christians All Catholics are Christians Indians are poor L.F: Some Indians are poor Trains are not punctual L.F: Some trains are not punctual 7 *Sentences containing words as except, all but, save, called exceptive propositions are to be reduced to Universal if the exceptions are definitely specified. If the exceptions are indefinite, the exceptive sentences are reduced to Particular propositions. All students except Shyam have passed L.F: All students except Shyam are those who have passed No students except Shyam have failed L.F: No students except Shyamare those who have failed All students except two have passed L.F: Some students are those who have passed No students except two have passed L.F: Some students are not those who have passed All but James passed L.F: All persons other than James are persons who passed All but a few were saved L.F: Some persons are those who were saved Applications of Logical Reasoning 9
All metals except one are solid L.F: Some metals are solid. *Exclusive sentences containing words as alone, only, none but, none except no one else but, nothing but are reduced to E proposition. By contradicting the original subject and used as the 8 subject of the logical proposition. Exclusive sentences may also be changed into Universal propositions by inter-changing the original subject and predicate. None but citizens can hold property. L.F: No non-citizens are persons who cn hold property. Or All those who can hold property are citizens Only the wise are fit to rule L.F: No non-wise persons are fit to rule Or All persons fit to rule are wise Graduates alone can vote L.F: No non-graduates are voters Or All voters are graduates *Where the quantity of the proposition is not explicit, it is the meaning of the sentence that must be taken into account in reducing to the logical form. To be wise is to be happy L.F: All wise people are happy To err is human L.F: All cases of erring are human Blessed are the pure L.F: All pure persons are blessed There are bright colours L.F: Some colours are bright 9 Applications of Logical Reasoning 10
MODULE II Conversion of A, E,I, O propositions according to relations of opposition between categorical propositions as shown in the traditional square of opposition. IMMEDIATE INFERENCE Inference is a mental process of drawing something new from something known. Mediate inference consists in drawing a new proposition from two known propositions. The mediate inference asserts the agreement or disagreement of a subject and predicate after having compared each with a common element or middle term. The conclusion is thus reached mediately or indirectly. There are two kinds of immediate inferences: Opposition and Eduction. Traditional logic considers opposition of propositions as a system of inferences. Traditionally opposition deals with inferences within the four-fold scheme of propositions. Opposition of propositions is a scheme of inferences between two propositions which have the same subject and the same predicate, but differ either in quantity or in quality or both in quantity and quality. There are four kinds of logical opposition, contrary opposition, contradictory opposition, sub-contrary opposition, and sub-altern opposition. 1. Contrary Opposition or contrariety: is the relation between two universal propositions having the same S and P but differing in quality only. A and E E.g. All A is B--. No A is B. All misers are unhappy.----no miser is unhappy. 2. Contradictory opposition is the relation between two propositions having the same S and P but differing both in quality and quantity. I and O; I and E A and O E.g. All boys are clever-some boys are not clever. I and E Applications of Logical Reasoning 11 10 Some boys are clever- No boys are clever. 3. Subcontrary opposition or subcontrariety: is the relation between two particular propositions having the same S and P but differing in quality only. I and O
E.g. Some able men are honest. Some able men are not honest. 4. Subaltern opposition or subalternation: is the relation between two propositions having the same S and P but differing in quantity only. In subalternation the universal is called subalternant and the corresponding particular is called subalternate. A and I : E and O All men are mortal - Some men are mortal. No men are mortal- Some men are not mortal. As an immediate inference opposition consists in drawing out from the truth or falsity of a given proposition the truth or falsity of its logical opposite having the same subject and predicate but differing in quality only or in quantity only or in both. Laws of Oppositional Inference 1. Law of Contrary Opposition Between contraries if one is true the other s false, and if one is false the other is doubtful. Contrary propositions cannot both be true, but both may be false. All men are rational T No men are rational F No students are industrious F All students are industrious-undetermined 2. Law of Contradictory Opposition 11 If one of the contradictories is rue the other must be false; if one is false the other must be true. Both can neither be true or false at the same time. No men are perfect T/F Some men are perfect T/F 3. Law of Subcontrary Opposition Between subcontraries if one is false the other is necessarily true; but if one is true the other is doubtful. Both may be true; both cannot be false. Applications of Logical Reasoning 12
Some men are angels F Some men are not angles T Some students are honest T Some students are not undetermined Some fruits are sweet T Some fruits are not sweet T 4. Law of Subalternation Between subalterns if the universal is true the corresponding particular is also true; but if the universal is false the particular is doubtful. E.g. No gamblers are honest T Some gamblers are not honest T All students are clever F Some students are clever--undetermined If the particular proposition is true its corresponding universal is doubtful; but if the particular is false the universal must be false. Some politicians are not honest T 12 No politicians are honest--undetermined Some men are not rational F No men are rational F Applications of Logical Reasoning 13
MODULE III Changing categorical propositions into converse, obverse, and contrapositive according to rules of eduction/immediate inference. Conversion, Obversion, Contraposition and Inversion are immediate inferences. Conversion Conversion is an immediate inference in which from a given proposition another proposition having the original predicat as the new subject, and the original subject as the new predicat but expressing the same meaning as that of the given proposition. The proposition to be converted is called the convertend and the converted proposition is called the converse. It is a process by which from a proposition of the form S-P a proposition of the form P-S is inferred. Conversion expresses the same idea by interchanging the subject and predicate. Rules of conversion: Term undistributed in the convertend not be distributed in the converse. Keep the same quality Interchange subject and predicate Two types of conversion, simple conversion and conversion by limitation Simple conversion---quantity and quality not changed Conversion of E and I No Hindus are Muslms- convertend No Muslims are Hindus-- Converse 13 Conversion by limitation quantity changed Conversion of A: All tigers are animals(convertend) Some animals are tigers(converse) O proposition has no converse. The subject undistributed in the convertend become distributed in the converse. This would go against the rule of distribution. Applications of Logical Reasoning 14
Obversion Obversion is an immediate inference in which from a given proposition a new proposition is inferred which has the original subject as subject and the contradictory of the original predicate as it s predicate. The quality of the proposition is also changed. The original proposition is called obvertand and the inferred proposition is called the obverse. obverse E No tigers are non-animals A All liers are non-honest O Some students are not non- I Some men are non-lazy Obvertend A--All tigers are animals E No liers are honest I Some students are sportsmen sportsmen O Some men are not lazy Contraposition Contraposition is an immediate inference in which from a given proposition another proposition having the contradictory of the given predicate as it s subject is inferred. There are two forms of contraposition, partial and full. When the predicate of the contrapositive is the same as the original subject, it is partial contraposition. When the predicate of the inferred proposition is also the contradictory of the original subject, it is full contraposition. To get the contrapositive, first obvert and then convert the obverse. Contraposition of A: All tigers are animals No tigers are non-animals (obverse) 14 No non-animals are tigers (partial contrapositive) All non-animals are non-tigers (full contrapositive) Contraposition of E: No liers are reliable All liers are non-reliable(obverse) Some non-reliable persons are liers (partial contrapositive) Some non-reliable persons are not non-liers (full contrapositive) Applications of Logical Reasoning 15
Contraposition of I: Some metals are heavy Some metals are not non-heavy (obverse) Some non-heavy things are not metals (no converse, hence no contrapositive) I proposition has no contrapositive Contraposition of O: Some politicians are not honest Some politicians are non-honest (obverse) Some non-honest persons are politicians (partial contrapositive) Some non-honest persons are not non-politicians (full contrapositive). 15 Applications of Logical Reasoning 16
MODULE IV Detecting fallacies according to the rules of categorical syllogism. CATEGORICAL SYLLOGISM Definition of Syllogism A Syllogism is a form of mediate deductive inference, in which the conclusion is drawn from two premises, taken jointly. It is a form of deductive inference and therefore the conclusion cannot be more general than the premises. It is a mediate form of inference because the conclusion is drawn from two premises, and not from one premise only as in the case of immediate inference. Eg : All men are mortal Structure of Syllogism All kings are men.`. All kings are mortal. A syllogism consists of three terms. The predicate of the conclusion is called the Major Term; subject of the conclusion is called the Minor Term; and that term which occurs in both the premises, but does not occur in the conclusion, is called the Middle Term. The Major and Minor terms are called Extremes, to distinguish them from the Middle term. The Middle Term occurs in both the premises, and is the common element between them. The conclusion seeks to establish a relation between the Extremes the major term and the minor term. The middle term performs the function of an intermediary. The middle term is thus middle in the sense that it is a mediating term, or a common standard of reference, with which two other terms are compared and is thus means by which we pass from premises to conclusion. The middle term having performed its function of bringing the extremes together drops out from the conclusion. Thus, we reach the conclusion in a Syllogism, not directly or immediately, but by means the Middle term. 16 The premise in which the major term occurs is called the Major Premise and the premise in which the minor term occurs is called the Minor Premise, For example, in the following Syllogism: All men are mortal All kings are men.`. All kings are mortal. Applications of Logical Reasoning 17
The term mortal is the major term, being the predicate of the conclusion; the term `kings` is minor term, because it is the subject of the conclusion; the term `men` which occurs in both the premises but is absent from the conclusion, is the middle term. The first premise `All men are mortal` is the major premise, because the major term `mortal` occurs in it; the second premise `All kings are men` is the minor premise, because the minor term `kings occurs in it. It may be pointed out that when a syllogism is given in strict logical form, the major premise is given first, and the minor premise comes next, and last of all comes the conclusion. The symbol M stands for the Middle term, S stands for the Minor term and P stands for the Major term. The above syllogism can be represented as,\ All M is P All S is M.`. All S is P General Rules of Categorical Syllogism and the Fallacies. I. Every syllogism must contain three, and only three terms and these terms must be used in the same sense throughout. There are two ways in which this rule is violated. If a syllogism consists of 4 terms instead of three, we commit the fallacy of 4 terms quartenioterminorum e.g. The book is on the table The table is on the floor.`.the book is on the floor 17 Here there are four terms, viz., The book, on the table, The table and on the floor. Hence no conclusion can follow. There is another way in which the above rule can be violated. If any term in a syllogism is used ambiguously in the two different premises, we commit a fallacy. If a term is use in two different meanings, it is practically equivalent to two terms and the syllogism commits the fallacy of equivocation. There are three forms of equivocation. They are: 1.Fallacy of ambiguous major 2.Fallacy of ambiguous minor 3.Fallacy of ambiguous middle Applications of Logical Reasoning 18
1.Fallacy of ambiguous major is a fallacy which occurs when a syllogism uses its major terms in one sense in the premise and in a different sense in the conclusion. e.g., Light is required for taking a photograph. Feather is not required for taking a photograph.`. Feather is not light. Light in the major premise is used in the sense of physical phenomenon ; in the conclusion it is used in the sense of not heavy 2.The fallacy of ambiguous minor occurs when in a syllogism the minor term means one thing in the minor premise and quite another in the conclusion. e.g., No boys are part of a book. All pages are boys..`.no pages are part of a book. In this syllogism, minor term pages mean boy servant in its premise and the side of a paper in the conclusion. Hence the fallacy of ambiguous minor. 3.The fallacy of ambiguous middle will be committed by a syllogism if it uses the middle term in one sense in the major premise and in another sense in the major premise and in another sense in the minor premise. e.g.,all criminal actions ought to be punished by law 18 prosecutions for theft are criminal actions.`.. prosecutions for theft ought to be punished by law The middle term criminal actions means crimes in the major premise and an action against a criminal in the minor premise. Hence the syllogism commits the fallacy of ambiguous middle. II. Every syllogism must contain 3 and only 3 propositions. Syllogism is a process of reasoning in which a conclusion is drawn from two given premises. Two propositions are given ad a third one is inferred. III.The middle term must be distributed at least once in the premises. The function of a middle term in a syllogism is to serve as the connecting link between the minor and major terms. In the major premise P is Applications of Logical Reasoning 19
compared with M and in the minor premise S is compared with the same M. thus the relation between S and P is established through the mediation of M. The violation of this rule leads to the fallacy of undistributed middle. e.g., All donkeys are mortal. All monkeys are mortal..`.all monkeys are donkeys. In this argument the middle term mortal is undistributed in both the premises as the predicate of an affirmative proposition. Hence the fallacy of undistributed middle occurs. IV. No term which is undistributed in the premises can be distributed in the conclusion. This rule guards us against inferring more in the conclusion than what is contained in the premises. In any syllogism, the conclusion cannot be more general than the premises. The violation of this rule would result in two fallacies illicit major and illicit minor. The fallacy of illicit major occurs when the major term which is not disturbed in the major premise is distributed in the conclusion. e.g. All men are selfish MAP No apes are men SEM 19.`.No apes are selfish SEP The major term selfish is undistributed in the major premise but distributed in the conclusion. Hence the fallacy of illicit major. The fallacy of illicit minor is one which occurs when the minor term is distributed in the conclusion without being distributed in the minor premise. e.g. All thugs are murderers MAP All thugs are Indians.`.All Indians are murderers SAP MAS Here the minor term Indians is distributed in the conclusion without being distributed in the minor premise. So it commits the fallacy of illicit minor. Applications of Logical Reasoning 20
V. From two negative premises, no conclusion is possible. We cannot draw any conclusion from two negative premises. For, the major premise being negative, the major term does not agree with M. in the minor premise also, the minor term has no relation with M. Thus there is no mediating link between S and P. In the absence of a common link between S and P, no relation can be established between them. The violation of this rule commits the fallacy of two negative premises. e.g. No monkeys are rational No men are monkeys. No conclusion is possible VI. If one premise is negative, the conclusion must be negative and if the conclusion is negative one premise must be negative. If one premise is negative the other premise must be negative. In the negative premise M does not agree with the other term. In the affirmative premise M agrees with the other term. Hence in mediating between the two terms, M can establish only a relation of disagreement between S and P in the conclusion. In other words the conclusion must be negative. VII. Two particular premises yield no valid conclusion. This is proved by examining the four possible combinations of two particular premises. I O I O I I O O X X X X VIII. If any one premise is particular the conclusion must be particular. 20 Applications of Logical Reasoning 21
MODULE V Deriving the logical conclusion from two given premises. Logic deals with the question of validity of arguments. Logic is the science of the valid forms of reasoning. The study of logic contributes towards forming a critical habit of mind which has it s own value. The task of logic is to clarify the nature of the relationship which holds between premises and conclusion in a valid argument. To infer means to recognize what is implied in the premises. If we recognize that the conclusion is implied in the premises, then we can say that the inference is valid. Every scientist aims to arrive at correct conclusions on the basis of certain evidence. He has to see that his reasoning is in accordance with valid argument patterns. Such knowledge is provided by logic. Logic provides us the tools for the analysis of arguments. Knowledge of logic is helpful for the formation of critical habit of mind and for detecting fallacies in thinking. 21 By practicing various logical exercises our intellect become sharpened. Logic cultivates and develops our reasoning power. It trains and disciplines the mind. Intellectual discipline is of great importance to man. Deriving valid conclusions from the premises: Examples 1.All systematic knowledge of a particular subject is Science Logic is a systematic knowledge of a particular subject.`.. Answer- Logic is a Science 2All logicians are those who know how to reason well Some not-good reasoners are logicians.`.. Answer-Some not-good reasoners are those who know how to reason well 3.All repeaters are women students All who have passed are repeaters.`.. Answer-All who have passed are women students 4.,All youths are inexperienced All students are youths Applications of Logical Reasoning 22
.`.. Answer-All students are inexperienced 5. No lazy men are great Gopal is lazy 22.`.. Answer- Gopal is not lazy 6. All successful students are clever Some mischievous students are successful.`.. Answer Some mischievous students are clever 7. No tale bearers are reliable Some men are tale-bearers.`.. Answer Some men are not reliable. 8. All misers are unhappy X is a miser.`.. Answer X is unhappy 9. All men are mortal X is a man.`.. Answer X is mortal 10. All wicked people are unhappy Some people are wicked.`.. Answer Some people are unhappy 11. All statesmen are far-sighted Applications of Logical Reasoning 23
23 All politicians are statesmen.`.. Answer All politicians are far-sighted 12. All statesmen are far-sighted Some politicians are not far-sighted.`. Answer Some politicians are not statesmen 13. All clever people are enterprising All publishers are clever.`. Answer All publishers are enterprising 14. No men are angels Some rational beings are men.`. Answer Some rational beings are not angels 15. All greedy men are discontented All misers are greedy.`. Answer All misers are discontented 16. All discontented persons are unhappy All misers are discontented.`. 24 Answer All misers are unhappy 17. All unhappy men are selfish All misers are unhappy.`. Applications of Logical Reasoning 24
Answer All misers are selfish 18. No mortals are perfect All men are mortal.`. Answer No men are perfect 19. No men are perfect All politicians are men.`. Answer---No politicians are perfect 20. No politicians are perfect All ministers are politicians.`. Answer No ministers are perfect 21. All men are mortal Socrates is a man.`. Answer Socrates is mortal 22. All organisms are mortal All men are organisms 25.`. Answer All men are mortal 23.All composite things are mortal All organisms are composite.`. Answer All organisms are mortal %%%%%% Applications of Logical Reasoning 25