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School of Distance Education

Essential of Formal Logic Page 2

UNIVERSITY OF CALICUT

SCHOOL OF DISTANCE EDUCATION

STUDY MATERIAL

Core Course

B A Philosophy

III Semester

ESSENTIAL OF FORMAL LOGIC

Prepared by

Dr. Unnikrishnan. P, Assistant Professor, Department of Philosophy, Sreesankaracharya University of Sanskrit , Kalady.

Scrutinised by: Dr.V.Prabhakaran(CoOrdinator), Principal, EKNM Govt. College, Elerithattu, Kasargod.

Layout: Computer Section, SDE

© Reserved



CONTENTS PAGES

UNIT-1 INTRODUCTION 5

UNIT – II PROPOSITION 13

UNIT – III CATEGORICAL SYLLOGISM 22



UNIT-1

INTRODUCTION

The word “Logic” is derived from the Greek adjective “Logike” corresponding to the Greek noun “Logos”, which means either “thought” or “word” as expression of thought. Hence etymologically Logic is the science of thought as expressed in language. Thinking is the act of the mind by means of which knowledge is obtained. For the purposes of simpler a general description, however, the simpler word ‘reasoning may be substituted for ‘thought’, and it may be said that Logic is concerned with Reasoning as expressed in language and with certain subsidiary process. Reasoning means passing from something known to something unknown. The something known constitutes the data or materials of reasoning while the something unknown is the conclusion at which we arrive by reasoning. In other words reasoning is a kind of indirect knowledge based on some direct knowledge.

Logicians differ in their definitions of logic. Let us examine the definition given by Creighton for a comprehensive understanding of the nature and subject matter of logic. According to him Logic is the science which treats of the operations of the human mind in its search for truth. This definition states three facts about logic. First of all it states that logic is a science. Secondly it says that logic is about operations of the human mind. Lastly it maintains that logic is concerned with truth.

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Science is a systematic body of knowledge relating to a particular department of the world. As distinguished from popular knowledge, a science presents the following characteristics:

a) A science deals with a particular department of the world within which it confines its investigations while the ordinary man appears to be interested in the whole field of knowledge. That is science confines itself to one branch only. For example Botany deals with plants and Zoology with animals.

b) Science is a systematic and organized body of knowledge. That is scientific knowledge is systematic, unified, organized, and general while ordinary knowledge is a jumble of isolated and disconnected particular facts.

c) Science employs special means and appliances to render knowledge true and exact, while ordinary knowledge trusts to immethodical observations.

Thus logic is a science because it has the above mentioned three characteristics namely it confines itself to thinking; gives us systematic knowledge regarding correct thinking and knowledge given is correct and precise.

Secondly the definition says that logic deals with the operations of the human mind. The operations of the human mind with which logic is concerned are the three processes of thinking known as conception, judgment and reasoning. Conception or simple



apprehension is the function of the human mind by which an idea or a concept is formed in the mind. It is a process of forming a mental image of an object. For example when we see an elephant we form an idea of the elephant in our mind. Judgment is another function of the mind by which relation between things is established. It is a process of comparing concepts or ideas. By judgment the mind affirms or denies something of something else. Example: we have the idea of ‘man’ and the idea of ‘mortality’; when these two ideas are connected; we have the judgment “Man is mortal”. Reasoning or inference is the process of passing from certain known judgment to a new judgment. In other words Inference is a process by which one proposition is arrived at on the basis of some other proposition. By the act of reasoning the mind draws a new truth from certain given truths. Example: from the two known judgments,” All men are mortal” and “Socrates is a man” we draw the conclusion that “Socrates is mortal”. Thus these

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three processes of thinking namely conception, judgment and reasoning, when expressed in language are known as Term, Proposition and Argument.

In logic ‘thought’ means both the processes and the products of thinking. Logic is chiefly concerned with reasoning but reasoning presupposes concept and judgment. Thought consists of the great truths expounded by great minds. It is those well established truths which constitute the subject matter of logic.

Creighton’s phrase’ search for truth’ shows that truth is the goal or aim of logic. Truth may be either formal or material. Formal truth means agreement of thought among themselves. It consists in self consistency or freedom of thought from contradiction. For example, ‘Circular Square’ is contradiction in terms. It is formally false because it is self contradictory. But a ‘golden mountain’ though not real is formally true because the two ideas ‘golden’ and ‘mountain’ are not contradictory to each other. Material truth means the agreement of ideas with the corresponding objects in the world outside. We have truth when the ideas in our mind agree with the actual things to which they refer in the external world. If we have an idea about an object and that object exists in the outside world, our idea is materially true. Material truth is also known as objective truth as it refers to the objects in the external world. Formal truth has no reference to objects outside the mind.

Logic is a Normative Science

Sciences in general have been divided into positive and normative. A positive science deals with as they are, while a normative science deals with things as they should be. A positive science is also known as natural science as it studies the nature of things. It is also called descriptive science as it describes the facts of the world as they are. A normative science is one which set up a norm or a standard or an ideal to which the facts under study must conform. It tells us what a thing ought to be in order to agree with the deal before us. For instance Ethics is the



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science of good conduct. It is a normative science because it teaches us how our conduct ought to be in order to reach the ideal of Goodness. Similarly Aesthetics is the normative science of beauty. It examines the ideal of beauty and judges the value of things in the light of this standard of beauty. Aesthetics explains how things ought to be in order to be called beautiful. A normative science is also called regulative or evaluative science because it regulates or directs its materials towards a standard and estimates their value with reference to that standard. The difference between positive and normative science is that while a positive science deals with the things actual and real, a normative science is concerned with an ideal. The former tells us what a thing is while the latter teaches us what a thing ought to be. The laws of positive science are universal principles and cannot be changed. But the laws of normative science can be transgressed. Logic sets up truth as its ideal and teaches us how our thoughts ought to be in order to reach the goal of truth. It is not concerned with how or what we think in order to reach truth. Thus, logic is a normative or a regulative science of thought.

Logic as Science and as Art

Logicians differ in their opinion as to whether logic is a science or an art. Logicians like Mansel and Thompson accept logic only as a science while Aldrich and others consider logic only as an art. Mill and Whately and some others recognize Logic both as a science and an art.

A science is a systematic and exact body of knowledge about a particular branch of the universe. An art lays down rules for the attainment of certain ends. A science teaches us to know something while an art teaches us to do. That is theoretical knowledge is the aim of science but an art aims at practical utility. Proficiency in arts is acquired by practice while sciences are learned by hard study. A science is concerned with the matter or the things already existing in the world while an art is interested in the method of doing something. A science is based on fixed and unchangeable laws but art is ever changing the methods of producing new things.

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In spite of these differences art and science are neither opposed to each other nor exclusive of each other. They are not independent but interdependent. Arts depend upon sciences for their perfection and progress. The art of medicine for example depends on the sciences of anatomy, physiology, chemistry, etc. The great discoveries of sciences paved way for progress in the arts. Sciences in their turn are to some extent based on arts. For example the art of debate has given rise to the science of logic, the art of singing to the theory of music. The science of grammar has grown out of writing and speaking. Thus art and science or theory and practice go together and determine each other.

Logic is definitely a science because it possesses all the characteristics of a science. Though logic does not make us infallible reasoners it gives us certain practical rules to discover truth, to detect errors and avoid them in our reasoning. In this sense logic is an art and may be called the art of correct thinking. Logic does not give us any set of practical



rules to reason or think correctly. A study of logic is not necessary to reason correctly, yet whenever errors creep into our reasoning it is only logic which will guide us to detect and rectify the mistakes. Thus we can conclude that logic is primarily a science and only secondarily it is an art.

Uses of study of logic

1. We already know how to think. We cannot avoid reasoning until we studies logic. We study logic to improve our ability to use certain kinds of reasoning skills.

2. Logic does not teach us scientific, historical, or religious truths. Rather, it teaches us logical truths. It enables us to express our beliefs in logically correct arguments, to avoid many kinds of errors in reasoning, and to increase our knowledge by drawing logically correct inferences.

3. Logic has a central role in the humanities and sciences. It features prominently in Philosophy, History, Law, Art and Aesthetics, Comparative literature and literary criticism, and Mathematics and Computer science.

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4. The study of logic will help to improve our reading and listening skills by acquainting us with the formal structures of argument. Here we learn the consistency of essential concepts, definitions, propositions, and inferences. These are valuable techniques for every branch of learning.

5. We can better remember the content of information sources as a result of studying logic because logic teaches us to listen and read with a more definite purpose. We can check and improve our reasoning in drawing conclusions.

6. For these reasons logic has been an integral part of education since ancient times, beginning with the Greek philosophers.

7. To learn means to understand its concepts and put your pen to paper to solve logical problems. We must practice skills we are trying to acquire. As you study the lessons and meet new challenge, you will discover increasing confidence in your understanding of logic and an improvement in your reasoning abilities.

8. Logic is known as the “science of sciences”. There are mainly three reasons for this. Firstly, the suffix ‘logy’ attached to some of the sciences shows that logic is used in these sciences. Secondly, Logic is a regulative science. Hence it forms the foundation of other sciences. Lastly, Logic deals with the general rules of thought such as definition, division, classification, which must be observed in every science. Thus, in order to understand different sciences, knowledge of logic is indispensible



BASIC TERMS OF LOGIC

Inference

The term “inference” refers to the process by which one proposition is arrived at and affirmed on the basis of one or more other propositions accepted as the starting point of the process. To determine whether an inference correct, the logician

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examines those propositions that are the initial and end points of that process and the relationships between them.

Proposition

A proposition is either true or false, and in this they differ from question, command, and exclamation. Only propositions can be either asserted or denied; questions may be asked and commands given and exclamations uttered, but none of them can be affirmed, denied, or judged to be true or false. A proposition is a sentence, but every sentence is not a proposition. Only informative or indicative sentences may be said to be propositions.

It is customary to distinguish between sentence and the propositions they be used to assert. Two sentences, which are clearly two because they consist of different words differently arranged, may in the same context have the same meaning and may be used to assert the same proposition. For example,

Two different sentences: “Pranab won the election” and “The election was won by Pranab”, have exactly the same meaning. We use the term “proposition” to refer to what such sentences as these, declarative sentences, are typically used to assert.

While a sentence is always a sentence in a particular language proposition is not peculiar to any language. That is a proposition is language neutral entity. For instance, the sentence “It is raining” can be translated and expressed in various languages such as English, Malayalam, Hindi, German, etc. But all these sentences in different languages assert the same proposition.

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Argument

An inference is expressed in an argument. An argument is any group of propositions of which one is claimed to follow from the others, which are regarded as providing support or grounds for the truth of that one. An argument has a structure. The propositions of an argument are either a premise or the conclusion. The conclusion of an argument is that proposition that is affirmed on the basis of the other propositions of the argument, and these other propositions, which are affirmed as providing support or reasons for accepting the conclusion, are the premises of that argument. In other words the proposition or propositions which substantiate the conclusion are premises and that which is drawn on the basis of the premises is a conclusion.



For example in the following argument,

All men are mortal

Socrates is a man

Therefore, Socrates is mortal

The first two propositions‐ “All men are mortal” and “Socrates is a man” are the premises. The last proposition‐“ Socrates is mortal” is the conclusion.

Word, Name and Term

A word consists of a letter or a combination of letters which convey some meaning. A word may consist of one letter (example: ‘A’, ‘I’) or it may consist of more than one letter (example: ‘dog’, ‘alas’).A word always has some meaning. Dog is a word but ‘ogd’ is not a word, even though the letters are the same.

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A name is a word or group of words which can become the subject or predicate of a proposition. Every word cannot be called a name (for instance ‘of’, ‘before’).But all names are words.

A term is a name which serves either as the subject or predicate in a proposition. In the proposition “Gandhiji is the father of the Indian nation”, ‘Gandhiji ’and ‘the father of the Indian nation’ are terms because they are the subject and predicate of the proposition. Words and names can become terms only if they are used in a proposition. Names may have different meanings. But a term can have only one definite meaning in its proposition. Outside the proposition, a term loses its significance and is merely a name. For example, the name ‘blind’ may mean either a screen or absence of sight; but inside the proposition ‘Beethoven is blind’ the meaning of the term ‘blind’ is certain.

Deduction and Induction

There are two types of relations between premises and conclusion namely deductive and inductive. Thus there are two types of logic: deductive logic or deduction and inductive logic or induction. Consequently arguments are divided into deductive and inductive.

A deductive argument is one in which it is asserted that the conclusion is guaranteed to be true if the premises are true. Though every argument involves the claim that its premises provide some grounds for the truth of its conclusion, only a deductive argument involves the claim that its premises provide conclusive grounds for its conclusion. In a deductive argument the conclusion necessarily follows from the premises. The relationship between the premises and conclusion in deductive argument is of implication and entailment. For example consider the following argument:



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All men are mortal

Socrates is a man

Therefore Socrates is mortal

The conclusion of the argument “Socrates is mortal” is already inherited in the premises “All men are mortal” and “Socrates is a man”. The premises provide conclusive grounds for the conclusion. That is complete or sufficient and total evidences to support the conclusion. A deductive argument involves the claim that its premises provide all the evidence required for a conclusion. The deductive arguments are characterized as valid or invalid. A valid argument is one that has the property of being legitimately derived from premises by prescribed rules. The task of deductive logic is to clarify the nature of the relationship that holds between premises and conclusion in valid arguments and to propose techniques for discriminating valid from invalid arguments.

Inductive argument is one in which the premises provide only some grounds for the truth of the conclusion.The conclusion of an inductive argument is probable. Induction refers to the process of drawing conclusions from specific evidence. It is a process of reasoning in which conclusions are typically drawn from the observation of particular cases. The claim of induction is that we can experience directly only what is concrete and particular. Particulars are the individual units of perception or experience rather than the general or universal aspects. When we generalize about these particulars, we go beyond the immediate experience of those particulars. Consequently, a conclusion reached through the inductive process is never absolutely certain. In induction, reasoning proceeds to a conclusion that is not confined to the scope of the premises but is somehow additional to, or beyond, them. Inductive logical procedure is tightly linked to the concept of probability. The conclusion is only probable. There is the possibility of discovering new evidence. Inductive arguments are characterized as strong or weak. The claim is that their premises provide some evidence for their conclusion. One advantage of inductive reasoning is that it helps human beings to frame their expectations of the future on the basis of what they know about the past and present. The following argument is an example for inductive argument:

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Aristotle is human and mortal

Bacon is human and mortal

Castro is human and mortal

Descartes is human and mortal

Therefore all humans are mortal.



Truth and Validity

Truth and falsehood may be predicated of propositions, but never of arguments. Validity or invalidity can be predicated of deductive arguments, but never of propositions. There is a relation between validity or invalidity of an argument and truth or falsehood of its premises and conclusion. But this connection is not a simple one. It is important to note that an argument may be valid even when one or more of its premises is false. Some valid arguments contain only true premises as in the following example: All mammals have lungs. All whales are mammals. Therefore all whales have lungs. But a valid argument may contain exclusively false propositions as in the following example: All mammals have wings. All reptiles are mammals. Therefore all reptiles have wings.

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This argument is valid because, if its premises were true, its conclusion would have to be true also. An argument may contain true premises and conclusion nevertheless is invalid as in the following example: If I am President, then I am famous. I am not President. Therefore I am not famous. The invalidity of this argument can be proved by showing another argument of the same form to be invalid as in the following example: If Mahatma Gandhi is President of India, then Mahatma Gandhi is famous. Mahatma Gandhi is not President of India. Therefore Mahatma Gandhi is not famous.

The premises of this argument are true, and its conclusion is false. Such an argument cannot be valid, because it is impossible for the premises of a valid argument to be true and for its conclusion to be false. Arguments with true premises and false conclusions may be valid as in the following example:

All fishes are mammals. All whales are fishes. Therefore all whales are mammals.



An argument with false premises and true conclusion may be invalid as in the following example: All mammals have wings. All whales have wings. Therefore all whales are mammals.

13 There are invalid arguments whose premises and conclusions are all false, as in the following example: All mammals have wings. All whales have wings. Therefore all mammals are whales.

From the above examples it is clear that the truth or falsity of an argument’s conclusion does not by itself determine the validity or invalidity of that argument. And the fact that an argument is valid does not guarantee the truth of it conclusion. But the falsehood of its conclusion does guarantee that either the argument is invalid or one of its premises is false.

Thus a valid deductive argument is one in which the conclusion cannot possibly be false if all the premises are true. If it is possible for the premises of a deductive argument to be all true and its conclusion to be false, that argument is invalid. If a deductive argument is invalid, it may be constituted by any combination of true and/ or false premises and a true or false conclusion. If a deductive argument is valid, and its conclusion is true, it may have any combination of true and/ or false premises. But if the conclusion of a valid deductive argument is false, at least one of its premises must also be false. If a deductive argument is valid and at least one of its premises is false, the remaining premises and its conclusion may be true or false in any combination. But if all the premises of a valid deductive argument are true, its conclusion must also be true.

A valid deductive argument with true premises and conclusion is called a sound argument. Thus a deductive argument is sound if and only if it is valid and all of its premises are true.

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Deductive Logic (Formal Logic) and Inductive Logic (Material Logic)

Deductive and Formal Logic and Inductive and Material Logic are sometimes used as synonyms. It is more usual, however, to employ Deduction and Induction to mean the two main forms of reasoning, and in this sense, they are equivalent to formal and material reasoning respectively.

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UNIT – II

PROPOSITION

Just as a term is the verbal expression of a concept so also a proposition is the expression of a judgment. A proposition is unit of reasoning in logic. Both premises and conclusion of an argument are propositions. A proposition is a statement; what is typically asserted using a declarative sentence. So every sentence is not a proposition. Informative, indicative or factual sentences alone are propositions. Interrogative, exclamatory, and imperative sentences cannot become propositions. In a proposition we affirm or deny something of something else. It is a statement which shows a relation between two terms.

There are two kinds of propositions‐ Categorical and Conditional. A categorical proposition states the unconditional relation between the two terms. A conditional proposition on the other hand asserts a relation between the subject and the predicate on a condition. Conditional propositions are of two kinds‐ Hypothetical and Disjunctive.

Subject Term and Predicate Term.

A proposition consists of three parts‐ two terms and the sign of relation between them. The two terms are the subject term and the predicate term and the sign of relation is known as the copula. Thus a proposition contains two terms connected together by means of a copula. The term about which the affirmation or denial is made is called the subject. The term which is affirmed or denied of the subject is called the predicate. The connecting link is known as the ‘copula’.

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Connotation and Denotation of Terms.

Terms are employed for two different purposes; (1) to refer to things, to name and identify them (2) to describe, to represent the qualities or attributes belonging to things for which they stand. For example the term ‘man’ refers to different individual men and implies or suggests that the objects so named have certain qualities e.g. of animality and rationality.

Denotation

The function of indicating objects to which the term applies is called denoting. Denotation is the same as extension, i.e. the extent, or range of objects to which a term is applied. It is the sum total of objects to which the term can be applied.

Connotation

The function of suggesting qualities possessed by the objects is known as connoting. Connotation is the same as intension, the sum total of qualities or attributes which a term signifies.



PROPOSITIONS

Proposition: “A proposition is the expression in words of an act of judgement.” It is statement of something about something else. It is composed of two terms, a subject and a predicate, connected by a copula.

The logical proposition corresponds to the grammatical sentence. But not every sentence is a logical proposition. Sentences expresses not only judgements but also questions, commands, wishes, emotions. A logical proposition corresponds to an indicative or assertory sentence. But even indicative sentences frequently require to be rewritten in order to assume the logical form of a proposition. “It is very important to change the grammatical sentence to strict propositional form before attempting to treat it logically.” E.g. The sun shines, ‐ The sun is a body which shines.

Classification of Propositions

The most general division of proposition is that which classifies them as categorical and conditional.

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Categorical Proposition: is one in which the predicate is either affirmed or denied unconditionally of the subject. It is either a simple affirmation or a simple denial. e.g. A is B, this room is not cold.

Conditional Proposition: is one in which the predicate is affirmed or denied of the subject on the basis of a condition. It simply states the consequence that necessarily follows from a supposition and does not directly assert anything about particular matters. Conditional propositions are of two kinds: Hypothetical and disjunctive.

A hypothetical proposition is a conditional proposition in which the condition is introduced by the conjunction “if” or any other equivalent word. E.g. If (should) you work had you will succeed.

The part which expresses the condition introduced by “if” or its equivalent is the antecedent. It is the cause or ground from which the result follows. The clause which states the result that follows from the antecedent is the consequent. Only when these two are necessarily and intimately connected as cause and effect the hypothetical proposition is a correct logical one.

A disjunctive proposition is a conditional proposition which expresses two or more alternatives which are exclusive and exhaustive. Alternatives are exclusive when one excludes the others and all cannot go together, e.g. A line is either straight or curved. The alternatives are exhaustive when all the possibilities are mentioned.

Classification of Categorical Propositions on the Basis of Quantity and Quality

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

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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.

Singular or individual propositions are these which have a singular or an individual name as subject. E.g. Mt. Everest is the highest peak in the world. Since it is impossible to limit a singular subject, individual propositions are to be regarded as universal.

Indefinite or indesignate propositions are those in which the quantity of the subject is not stated clearly but left vague. E.g. men are to be trusted; women are talkative.

Indesignate propositions are considered universal when P is an invariable and common attribute of the subject, e.g. glass is breakable = all glasses are breakable, Catholics are Xtians = All Catholics are Christians.

Indefinite propositions are treated as particular when P is only an accidental quality. E.g. Indians are poor: some Indians ………

trains are not punctual: some ……….. not ……….

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.

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.

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No S is P

A 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

A 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

A propositions are introduced by ‘All’ except in the case of singular propositions All means ‘every’, distributive not collective sense. E propositions begin with the negative sign ‘No’ or ‘None’, copula looks like an affirmative one.

The form of E proposition is No S is P, not All S is not P.

‘Some’ – popularly means only a small quantity, a minor portion of a group of objects, e.g. some men are rich. In logic ‘some’ means any indefinite quantity. ‘Some’ in logic does not suggest anything about others. ‘Some’ simply means ‘some at least’; it does not exclude the possibility of ‘all’. But in ordinary language some means ‘some only’ and refers to others by implication, e.g. some boys are honest suggesting that the others are not honest.

Formal logic recognizes only A, E, I and O propositions. If a sentence is not in the strict logical form it must be first reduced to one of these four logical forms before it can be considered in logic. In this way the meaning of the statements can be made clear and exact.

Distribution of Terms

A term is said to be distributed when it is used in its entire extent referring to all the objects denoted by the term. A term is undistributed when it refers only to a part of the class of things denoted by the term.



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Proposition ‘A’ distributes its subject only, e.g. All men are mortal. Here the S ‘men’ is taken in its full extent. But its P is undistributed because it does not refer to the whole class of mortails. The proposition means that all men are some mortal beings; besides men there are other mortal beings also.

‘E’ distributes both S and P. e.g. No men are angels. The proposition means that the entire class of men falls outside the entire class of angels.

‘I’ distributes neither S nor its P. e.g. Some birds are white. ‘Some’ shows that the subject is undistributed. The P also is undistributed because it refers only to some of the white objects. Hence the proposition means that some birds are some of the white things in the world.

‘O’ distributes its predicate only, e.g. Some birds are not white. The S is undistributed – ‘some’. But the P is distributed since it denotes all the white objects in the world. The proposition means that a part of the class ‘birds’ is excluded from the whole class of white objects.

Note: Singular propositions, where S and P are both singular names, distribute both S and P, e.g. Mt. Everest is the highest peak in the world.

In the case of propositions where S and P are equal in number (coextensive) both S and P are distributed. E.g. All men are rational animals.

A distributes S only SAPx

E distributes S and P SEP

I distributes Nil XSIPx

O distributes P only xSOP

Asebinop

ASEBINOP is the key word for distribution of terms in propositions. In A only subject distributed, in E both subject and predicate distributed, in I none of the terms distributed, in O predicate only distributed.

“Universals distribute S while particulars do not. Negatives distribute P while affirmatives do not”.

The German mathematician Euler represented the distribution of terms of the relation between S and P by means of certain diagrams known as Euler’s circles:

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Universal affirmative where S and P are both distributed. S is coextensive with P. Hence S circle fully coincides with P circle. Each circle stands for the class of objects denoted by the term.



‘A’ proposition is represented by placing the S circle inside the predicate circle (P). The small circle (S) falls within and coincides with a part of the bigger circle representing the predicate.

‘E’ is represented by two circles falling outside each other because in E, the class of objects denoted by S has no connection whatsoever with the class of objects denoted by P.

In ‘I’ a part of objects denoted by class S agrees only with a part of the class of objects denoted by class P. hence the two circles intersect each other and the shaded segment stands for I.

In ‘O’, a part of the S is completely excluded from the whole class of objects denoted by P. Hence a portion of the S circle falls outside the whole circle of the predicate. O refers to the shaded portion.

IMMEDIATE INFERENCE

Inference is a mental process of drawing something new from something known. It may be deductive or inductive. Deduction is the process by which our minds proceed from a more universal truth to a less universal truth. Induction is the process by which our minds proceed from particular facts to universal laws.

Deductive reasoning may be immediate or mediate. Immediate Inference consists in passing directly from a single proposition to a new proposition whose truth is implied in the former.

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.

Logical Opposition: is used to denote any relation either of exclusion or inclusion that exists between propositions having the same subject and predicate.

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Among the AEIO propositions all having the same S and P; certain relations of exclusions and inclusion exist. The truth of some of these propositions excludes the truth of others and also that the relation between certain of the propositions is such that one assertion necessarily involves that of another.

Square of Opposition

The ‘square of opposition’ illustrates the opposition of propositions:

Propositions are said to be opposed when they differ either in quantity only, or quality only , or both in quantity and quality. There are four types of opposition of propositions‐Contrary, Contradictory, Sub contrary and Sub altern oppositions.



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

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

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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?

2. Law of Contradictory Opposition

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.

Some men are angels F

10 Some men are not angles T Some students are honest T Some students are not…? 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?

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 No politicians are honest? Some men are not rational F No men are rational F



UNIT - III

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

All kings are men

.` . All kings are mortal.

Structure of Syllogism

A syllogism consists of three terms. The predicate of the conclusion is called the Major Term; that 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

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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.

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.



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

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Kinds of Syllogism

Syllogisms have been classified into Pure and Mixed. In a Pure Syllogism, all the constituent propositions are of the same type. If all of them are categorical, the syllogism is Pure Categorical; if all hypothetical, the syllogism is Pure Hypothetical; and lastly, if all of them are disjunctive, the syllogism is Pure Disjunctive. On the other hand, in a mixed syllogism the constituent propositions are of different types. Mixed syllogisms are of three kinds: Hypothetical, Disjunctive and Dilemma

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 quartenio‐terminorum e.g.

a. B is a friend of C

A is a friend of B

.’. A is a friend of C

Here there are four terms, viz., “My arm”, “that which touches the table,” “The table” and “that which touches 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:

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1. Fallacy of ambiguous major

2. Fallacy of ambiguous minor

3. Fallacy of ambiguous middle

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 essential to guide our steps.

Lead is not essential to guide our steps.

Lead is not light.

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 man is made of paper.

All pages are men.

No pages are made of paper.

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., Food is indispensable to life.

Plantain is a food.

Plantain is indispensable to life.

The middle term ‘food’ means ‘meals’ in the major premise and an ‘eatable’ 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 and a third one is inferred.



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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 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 preposition. 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

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.

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e.g. All thugs are murderers MAP

All thugs are Indians MAS

All Indians are murderers SAP



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.

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

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.

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I O I O

I I O O

X X X X

I and I can be cut off because there is no distributed terms in both the premises. This violate the third rule that the middle term must be distributed at least once.

O and O too is ruled out as these are two negative premises.

OI and IO taken together distribute only one term and this must be M to avoid the fallacy of undistributed middle. But the conclusion should be negative as one premise is negative. In the negative conclusion P would be distributed which must be distributed in the major premise also. But the only term distributed is M. So it commits the fallacy of illicit major. Thus from two particular premises, we cannot draw a conclusion.



VIII. If any one premise is particular the conclusion must be particular.

There are eight possible combinations of one particular premise and one universal premise, namely

A O A I E O E I

O A I A O E I E

Of these, the valid combinations are AO, OA, AI, IA and EI.

1. E & O and O & E are invalid as they commit the fallacy of 2 negative premises.

2. I and E is invalid as it commits the fallacy of illicit major.

The remaining valid combinations can yield only particular conclusions as shown below:

1. A & I and I & A taken together distribute only one term and that must be given to M to avoid undistributed middle. S and P are not distributed in the premises. Hence they cannot be distributed in the conclusion. So the conclusion can be only SIP.

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2. A & O and O & A taken together distribute two terms, of which one should go to M and the other to P because the conclusion is negative since one premise is negative. S is undistributed in the minor premise and so it cannot be distributed in the conclusion to avoid the fallacy of illicit minor. Hence the conclusion can only be SOP.

3. E & I taken together distribute two terms. One must be the middle term, the other the major term because in the negative conclusion P is distributed S is distributed in the minor premise and hence it cannot be distributed in the conclusion. So the conclusion can be only SOP.

IX. From a particular major and a negative minor we cannot get a conclusion.

The combination is I & E. Since one premise negative the conclusion is negative. If the conclusion is negative P will be distributed. If P is distributed in the conclusion it must be distributed in the major premise. But the major premise is an ‘I’ proposition and as such no term is distributed. Hence the syllogism commits the fallacy of illicit major. So from a particular major and the negative minor no conclusion is possible.

Figures and Moods

The Figures

According to the position of M in both the premises, there are four types of categorical syllogism known as the Figures. The figure of a syllogism means the form of the



syllogism as determined by the position of the middle term in the two premises. The figures of the syllogism are as follows:

1st Figure MP 2nd Figure PM 3rd Figure MP 4th Figure PM

SM SM MS MS

SP SP SP SP

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Special canons of the first figure

MP 1. Minor premise must be affirmative

SM 2. Major premise must be universal

SP

1. The minor premise must be affirmative.

Suppose the minor premise is negative. If one premise is negative the conclusion is negative, and it distributes ‘P’ which is the major term. Since ‘P’ is distributed in the conclusion it must be distributed in the major premise also, to avoid the fallacy of illicit major. In the major premise P is the predicate and only negative propositions distribute their predicates. So major premise must be negative to distribute P. By supposition minor premise is already negative. From 2 negative premises no conclusion is possible. Hence minor premise must be affirmative.

2. Major premise must be universal.

According to the first rule minor premise is affirmative. M is not distributed in it because the affirmative propositions do not distribute their predicates. But ‘M’ must be distributed at least once, to avoid the fallacy of undistributed middle. To distribute M in the major premise where it is the subject, major premise must be universal because particular propositions do not distribute their subjects.

Special canons of the 2nd figure

PM 1. One premise must be negative

SM 2. Major premise must be universal

SP

1. One premise must be negative

If both the premises are affirmative, M will not be distributed in both the premises because affirmative propositions do not distribute their predicates. M is predicate of both the premises. Hence one premise must be negative to avoid the fallacy of undistributed middle.



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2. Major must be universal

According to the first rule one premise is already negative. Since one premise is negative the conclusion will be negative and P will be distributed in the conclusion. If P is distributed in the conclusion it must be distributed in the major premise also to avoid the fallacy of illicit major. To distribute P in the major premise, where it is the subject the major premise must be universal because only universal propositions distribute their subjects. Hence the major premise must be universal.

Special canons of the 3rd figure

MP 1. Minor premise must be affirmative

MS 2. Conclusion must be particular

SP

1. Minor premise must be affirmative

Suppose the minor premise is negative. Then the conclusion also will be negative distributing P. If P is distributed in the conclusion it must be distributed in the major premise also. P is the predicate of the major premise. Only negative propositions distribute their predicates. Hence major premise also must be negative. But by supposition minor premise is already negative. From 2 negative premises no conclusion is possible. Hence minor premise must be affirmative.

2. Conclusion must be particular

In the affirmative minor premise S which is predicate is undistributed. Hence it cannot be distributed in the conclusion. Only particular propositions do not distributed their subjects. So the conclusion must be particular to avoid the fallacy of illicit minor.

Special canons of the 4th figure

PM 1. If one premise is negative the major premise must be universal

MS 2. If major premise is affirmative, the minor premise must be universal

SP 3. If he minor premise is affirmative, the conclusion must be particular

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1. If one premise is negative the conclusion will be negative distributing P. P must be distributed in the major premise also to avoid the fallacy of illicit major. P is the subject in the major premise. Only universal propositions distribute their subjects. Hence the major premise must be universal.



2. If the major premise is affirmative M is undistributed because the affirmative propositions do not distribute their predicates. But M must be distributed at least once in the minor premise where M is the subject. Only universal propositions distribute their subjects. Hence minor premise must be universal to avoid the fallacy of undistributed middle.

3. If the minor premise is affirmative S which is the predicate is undistributed. Hence it cannot be distributed in the conclusion. Only particular propositions do not distribute their subjects. So the conclusion must be particular to avoid the fallacy of illicit minor.

Moods of a Syllogism

A mood is a form of syllogism determined by the quality and quantity of the three constituent propositions. If all the three propositions of a syllogism are A propositions, the mood of the syllogism is A, A, A. But any 3 of the categorical propositions A, E, I, O will not make a valid syllogism. There are certain conditions to be satisfied for a combination of 3 propositions to become a valid mood in any figure. Therefore it is enough if you determine the proper way of grouping together propositions, taking but 2 at a time. Thus we get the following combinations:

1. A A A A 2. E E E E

A E I O A E I O

3. I I I I 4. O O O O

A E I O A E I O

All the 16 sets do not satisfy all the general rules of a syllogism. Thus EE, EO and OE and O, O are rejected because both are negative premises. II, IO and OI are eliminated because both are particular premises. EI is also rejected because it leads to the fallacy of illicit major. There are only 8 pairs which fulfil all the general rules of a syllogism.

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The valid moods of the four figures

I. In the 1st figure the rules are:

a. Minor premise must be affirmative

b. Major premise must be universal

Applying these rules to the 8 combinations, we find that A, E and A, O are invalid by the violation of the 1st rule and IA and OA are invalid by violation of the 2nd rule. Thus we have four valid moods namely. AAA / EAE / AII / EIO.

BARBARA, CELARENT, DARII, FERIO



II. In the second figure the rules are:

a. One premise must be negative

b. Major premise must be universal

Applying these 2 rules to the 8 combinations we find that AA, AI and IA are invalid by the violation of the first rule and OA is invalid by the violation of the second rule. Hence there are only 4 valid moods in the second figure namely EAE, AEE, EIO, AOO.

CESARE, CANESTRES, FESTINO, BAROCO

III. In the 3rd figure the rules are:

a. Minor premise must be affirmative

b. Conclusion must be particular

According to the 1st rule AE, and AO must be excluded; thus we have six valid moods, viz., AAI, IAI, AII, EAO, OAO, EIO.

DARAPTI, DISAMIS, DATISI, FELAPTON, BOCARDO, FERISON.

IV. The valid moods of 4th figure are AAI, AEE, IAI, EAO and EIO.

BRAMANTIP, CAMENES, DIMARIS, FESAPO, FRESISON.

Thus there are 19 valid moods in the four figures.

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MIXED SYLLOGISMS

Mixed syllogisms are those in which the conditional and categorical propositions are combined in the same argument. There are three kinds of mixed syllogisms:

1. Hypothetical syllogism

2. Disjunctive syllogism, and

3. Dilemma

Hypothetical Syllogism

A hypothetical syllogism is a mixed syllogism in which the major premise is a hypothetical proposition, the minor and the conclusion are categorical propositions. It is also called a conjunctive syllogism.

e.g. If a man takes poison, he will die If A is B, C is D

This man takes poison A is B



This man will die C is D

In the Hypothetical major premise, there are two parts:

1. Antecedent: It is the part which express the supposition or condition, introduced by if or any equivalent of ‘if’. The antecedent is a condition or cause from which the result follows.

2. Consequent: It is the clause which states the result that follows from the antecedent.

Rules of Valid Hypothetical Syllogism

1. Affirm the antecedent in the minor premise then affirm the consequent in the conclusion.

The antecedent is the cause and the consequent in the effect. There may be several antecedents for the same consequent. E.g. several antecedents like poison, accident, disease, etc. produce the same consequent ‘death’. Hence when the cause exists, the effect or the consequent must also exist.

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2. Deny the consequent in the minor premise and then deny the antecedent in the conclusion.

‘Deny’ means give the contradictory of the original and not its contrary. When the consequent is said to be absent (denied) we can rightly infer that its cause or antecedent would also be absent. If a man is not dead, we can say that he has not taken poison. But we cannot infer that the antecedent poison must be present when death is present. Poison is not the only antecedent of death. Hence if we deny the consequent in the minor, we can deny the antecedent in the conclusion. There are two kinds of Hypothetical syllogisms:

1. Modus ponens or Constructive Hypothetical Syllogism. It is one which the minor premise affirms the antecedent and the conclusion affirms the consequent.

i. If a man is industrious, he will be successful.

X is industrious

He will be successful

ii. If a man is not virtuous, he will not go to heaven.

Some men are not virtuous.

Some men cannot go to heaven.

In the second example the antecedent is negative. But it is affirmed in the minor, since the negative quality is repeated.

2. Modus Tollens or Destructive Hypothetical Syllogism



It is one in which the minor premise denies the consequent, and the conclusion denies the antecedent.

i. If he is a thief, he will hide the goods.

He has not hidden the goods.

He is not a thief.

ii. If a man is a gentleman, he will not offend others.

This man offends others

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This man is not a gentleman.

Fallacies

There are two fallacies in relation with these two rules.

a. The fallacy of denying the antecedent and

b. the fallacy of affirming the consequent.

e.g. 1. If a country is highly industrialised, it is prosperous.

This country is not highly industrialised.

This country is not prosperous.

2. All industrialised countries are prosperous MAP

This country is not industrialised SEM

This country is not prosperous SEP

This fallacy corresponds to the fallacy of illicit major.

II. If a man is deaf, he talks loud.

This man talks loud.

He is deaf.

This fallacy corresponds to the fallacy of undistributed middle.

All deaf person are those who talk loud PAM

This man is one who talks loud SAM

This man is deaf SAP



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Disjunctive Syllogism

A disjunctive syllogism is mixed syllogism whose major premise is a disjunctive proposition and whose minor premise and conclusion are categorical propositions.

e.g. 1. Signal light shows either red or green.

It is red.

It is not green.

2. He is either mad of drunk.

He is not mad

He is drunk

Rules

1. The possibilities or alternative in the major premise must be exclusive of each other.

The alternatives must be contradictory to one another. One alternative should not co‐exit with others. The presence of one implies the absence of the other.

2. The alternative must be exhaustive.

No alternative should be omitted. The alternatives taken together must cover the whole subject without omitting any part of it.

3. Affirm one in the minor and then deny the other in the conclusion. Or, deny one alternative in the minor and then affirm the other in the conclusion.

This rule holds good only when the alternatives are contradictories and exclusive of each other because both cannot be true at the same time.

There are two kinds of disjunctive syllogism:

1. Modus Ponendo Tollens

2. Modus Tollendo Ponens

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In the first, one of the alternatives is affirmed in the minor and then the other is denied in the conclusion.

e.g. Students are either hostellers or day‐scholars.

X is a day‐scholar.



He is not a hosteller.

In the second the minor premise denies or rejects one alternative, while the conclusion affirms or accepts the other.

He is either a saint or a sinner.

He is not a saint.

He is a sinner.

Violation of the rules of a disjunctive syllogism commits the fallacy of improper disjunction.

Errors creep into disjunctive syllogism because it is not easy to obtain a perfect disjunctive major premise. Sometimes the alternatives are not exclusive, sometimes they are not exhaustive, sometimes they are neither.

Dilemma

A dilemma is a mixed syllogism in which the major premise consists of two hypothetical propositions, the minor premise is a disjunctive proposition and the conclusion is a categorical or a disjunctive proposition. A dilemma is constructive when disjunctive minor premise affirms the antecedents of the major, and destructive, when it denies the consequents. A dilemma is simple when the conclusion is a categorical proposition and complex, when the conclusion is a disjunctive proposition. Thus there are four forms of dilemma.

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1. Simple Constructive Dilemma

In this form, the disjunctive minor affirms the two antecedents of the major, and the categorical conclusion affirms the consequent of the major premise.

e.g. If a man acts according to his conscience, he will be criticised, and if he follows the opinions of others, he will be criticised.

Either the man acts according to the opinions of others or follows the opinion of others.

In any case he will be criticised.

In symbols: If A is B, C is D, and if E if F, C is D

Either A is B or E is F

C is D



2. Simple Destructive Dilemma e.g. If a man is moral he is honourable, and if he is moral, he is honoured. He is neither honourable nor honoured. He is not moral. In symbols: If A is B, C is D, and if A is B, E is F.

Either C is not D or E is not F. A is not B.

3. The Complex Constructive Dilemma

e.g. If the books agree with the Koran, they are superfluous, and if the books do not agree with the Koran they are dangerous.

Either the books agree with the Koran or they do not. Either the books are superfluous or they are dangerous.

19 In symbols: If A is B, C is D, and if E is F, G is H.

Either A is B or E is F. Either C is D or G is H.

4. Complex Destructive Dilemma

e.g. If he were clever, he would see his mistakes, if he were sincere, he would acknowledge it.

Either he does not see his mistake or he does not acknowledge it. Either he is not clever or he is not sincere. In symbols: If A is B, C is D, and if E is F, G is H.

Either C is not D, or G is not H. Either A is not B, or E is not F.

Rules and Conditions

1. Affirm the antecedents in the minor, and then affirm the consequents in the conclusion. Or deny the consequents in the minor and the antecedents in the conclusion.

2. The alternatives in the

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