LOGIC AL REASONING • Understanding the structure of arguments: • argument forms, • structure of categorical propositions, • • • • • • • • • Mood and Figure Formal and Informal fallacies, Uses of language, Connotations and denotations of terms, Classical square of opposition. Evaluating and distinguishing deductive and inductive reasoning. Analogies. Venn diagram: Simple and multiple use for establishing validity of arguments. Indian Logic: Means of knowledge. • Pramanas: Pratyaksha (Perception), Anumana (Inference), Upamana (Comparison), Shabda (Verbal testimony), Arthapatti (Implication) and Anupalabddhi (Non-apprehension). • Structure and kinds of Anumana (inference),Vyapti (invariable relation), Hetvabhasas (fallacies of inference). LOGICAL REASONING • Understanding the structure of arguments: • argument forms, • structure of categorical propositions, • Mood and Figure, • Formal and Informal fallacies, • Uses of language, • Connotations and denotations of terms, • Classical square of opposition. LOGICAL REASONING • The word ‘Logic’ is derived from the Greek word ‘Logos’ which means ‘thoughts in language’ • In other words ‘Logic’ is the ‘science of thought’ expressed in words. • Logical Reasoning is the process of deriving a logical inference from a hypothesis through reasoning. • Understanding the structure of arguments: • argument forms, • structure of categorical propositions, • Mood and Figure, ARGUMENT • An argument is defined as a series of statements known as premises that are said to support a conclusion. • In simple language , argument is a point of view on a particular matter supported by certain evidences STRUCTURE OF ARGUMENTS • The central part of an argument include 1. Premise 2. Conclusion 1. PREMISE • A premise of an argument is a sort of thing that is put forward as a truth but which is not yet proven. • Although it is not proven true, it is assumed to be true. • A person’s argument can challenged through questioning the truth of their premises. • The more complex the premises, more scope to challenge it. • Also as arguments are main structure of the argument, there may be two or more premises in an argument. 2. CONCLUSION • Conclusion is a statement with which you want others to agree. • It is derived from the premises of the argument. • Best way for spotting a conclusion is that it probably will be a statement of necessity, saying what must or should happen. Premises • P1: All musicians can read music. • P2: Ram is a Musician Conclusion • Ram can read music • A simple argument must have a conclusion and at least two premises • Premises or Propositions are statements that directly support the conclusion • A conclusion is what an author or an individual wants me to believe , accept or do so. TYPES OF ARGUMENT • There are generally two types of arguments as follows 1. Deductive Argument 2. Inductive Argument DEDUCTIVE ARGUMENT • Deductive argument means a form of logic in which specific inferences are drawn from multiple premises (general statements). • It establishes the relationship between the proposition and conclusion. • When all the proposed statements are true, then the rules of deduction are applied and the result obtained is inevitably true. • Premise 1. All cats have long tail. • Premise 2. Tom is a cat. • Conclusion. Tom has long tail. • Statement I: All vegetables contain vitamins. • Statement II: Carrot is a vegetable. • Conclusion: So carrot contains vitamins. INDUCTIVE ARGUMENT • Inductive argument refers to the logical process, in which specific instances or situations are observed or analyzed to establish general principles. • In Inductive argument, there are certain possibilities that the conclusion drawn can be false, even if all the assumptions are true. • The argument vests on experience and observations that supports the apparent truth of the conclusion. • Statement I: Most vegetables contain vitamins. • Statement II: Carrot is a vegetable. • Conclusion: So carrot contains vitamins. • Premise 1: Most of the pulses contain proteins. • Premise 2: Moong dal is a type of pulse. • Conclusion: Moong dal contain proteins. • The above conclusion can be completely true or partially true as it is mentioned in the premise the Most of the pulses, but is not mentioned all the pulses. • Therefore Moong dal may have proteins or may not have proteins but it is not totally guaranteed. • So, this is an example of Induction reasoning. NOTE:- • Inference • Inference is the reading of the clues or making best guess. • Inference is similar to prediction but they are not same • A critical aspect of logical argument is that inferential statements are either true or false and nothing in between. PROPOSITION PROPOSITION • A proposition is a sentence that makes a statement and gives a relation between two or more terms. • In logical reasoning, any statement is termed as a proposition. • A proposition is assumed to be true and from which a conclusion can be drawn. • The statement, • ‘All cats are lemons’ • is assumed to be true as a proposition (or premise), • but actually we all know that cats and lemons are entirely different entities. • Proposition consists of the following four parts: • 1. Quantifier: All, no, and some. They specify a quantity. ‘All’ and ‘no’ are universal quantifiers and ‘some’ is a particular quantifier. • 2. Subject (S): About which something is being said. • 3. Predicate (P): Something that affirms or denies about the subject. • 4. Copula: Relation between subject and predicate. • "Jane is friendly." • Quantifier + Subject + Copula + Predicate CLASSIFICATION OF PROPOSITIONS CLASSIFICATION OF PROPOSITIONS • Propositions are basically of two types, namely 1. universal and 2. particular. UNIVERSAL PROPOSITION • Universal proposition is further divided into two parts. • 1. Universal positive or affirmative (A) • 2. Universal negative (E) 1. UNIVERSAL POSITIVE OR AFFIRMATIVE (A): • Form: All S is P • Example: ‘All cats are animals’. • It is basically about inclusion 2. UNIVERSAL NEGATIVE (E): • It denotes exclusion. • Form: No S is P. • Example: ‘No fish are birds’ would be a universal negative. PARTICULAR PROPOSITION: • A particular proposition can also be divided into two parts. 1. Particular positive (I) 2. Particular negative (O) 1. PARTICULAR POSITIVE (I): • It denotes ‘partial inclusion’. • Form: Some S is P. • Example: Some men are foolish. 2. PARTICULAR NEGATIVE (O): • It denotes partial exclusion. • Form: Some S is not P or not every S is P. • Example: ‘Some birds are not carnivores’. • ‘All cats are animals’. • ‘No fish are birds’ • Some men are foolish. • ‘Some birds are not carnivores’. PARTS OF CATEGORICAL PROPOSITION • To find a logical deduction from two statements there should be three terms present . • Those three terms are • 1. Major Term • 2. Minor Term • 3. Middle Term • 1. All mammals are creatures that have hair • 2. All dogs are mammals. • 3. Therefore, all dogs are creatures that have hair • 1. Major Term (P) • It is the predicate of the conclusion • 2. Minor Term (S) • It is the subject of the conclusion • All mammals are creatures that have hair • All dogs are mammals. • Therefore, all dogs are creatures that have hair • 3. Middle Term (M) • It is the term common to both the statements. • The middle term should not appear in the conclusion No geese are felines. Some birds are geese. Therefore, Some birds are not felines. • Premise 1. All cats have long tail. • Premise 2. Tom is a cat. • Conclusion. Tom has long tail. • A • E • I • O MOOD • The mood of a syllogism can be simply understood as a statement of categorical propositions. • It comprises of (A, E, I, or O) categorical propositions which are listed in the order, in which they appear in standard form. • The ‘mood’ of a syllogism is determined by the ‘quantity’ and ‘quality’ of the three propositions. Example • All rocks are hard things. • No rocks are liquid. • Some liquid things are not hard. • Mood: AEI • 1. No states with coastlines are states that are landlocked. • 2. Some U.S. states are states that are landlocked. • 3. Therefore, some U.S. states are not states with coastlines. • 1. No states with coastlines are states that are landlocked. • • 2. Some U.S. states are states that are landlocked. • 3. Therefore, some U.S. states are not states with coastlines. • EIO FIGURE • There are three terms in a syllogism: a major, a minor, and a middle term (M). • The major and the minor are called the extremes and they form the predicate and the subject of the conclusion. • The middle term is what joins the two premises. • These three terms can be combined in different ways to form figures. • The ‘figure’ of a syllogism is determined by the position of ‘middle term’. • There is mainly four figure of syllogism. Figure 1 Figure 2 Figure 3 Figure 4 Major Premise M–P P -M M - P P - M Minor Premise S - M S–M M–S M - S MOOD AND FIGURE COMBINATION • The mood and figure together describe enough logical feature of an argument to tell if it is valid by applying some simple rules. • There are 256 combinations of different forms of mood and figure. • Out of these 256 combinations, only 24 were found to be valid. • You just need to remember these combinations only. • They are as follows: Figure Mood Figure I AAA, AAI, EAE, EAO, EIO and AII Figure II AEE, AEO, EAE, EAO, EIO and AOO Figure III AAI, AII, IAI, EAO, EIO and OAO Figure IV AAI, IAI, AEE, AEO, EAO, and EIO • All mammals are animals with hearts. • All dogs are mammals. • All dogs are animals with hearts. • All animals with hearts are mammals. • All dogs are mammals. • All dogs are animals with hearts. • All mammals are animals with hearts. • All mammals are dogs. • All dogs are animals with hearts. • All animals with hearts are mammals. • All mammals are dogs. • All dogs are animals with hearts. • All dogs are mammals. • All cats are mammals. • Therefore, all cats are dogs • P1: All ships are buses • P2: All cars are ships • Conclusion : All cars are buses No geese are felines. Some birds are geese. Therefore, Some birds are not felines. • All cats are felines • No dogs are cats • Therefore , No dogs are felines No islands are men Some islands are continents Some continents are not men Some paintings are works by Renoir All works by Renoir are masterpieces Therefore, Some masterpieces are paintings. All ducks are birds All griffins are ducks Some griffins are birds Some P are not M All S are M Therefore , Some S are not P
