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Logic

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1. 1. Logic Logic is the science of reasoning. Reasoning is a formal activity The notion of form has wider implication in logic. It pertains to the form of proposition as well as the form of argument. The notion of form refers to the norm/ rule/ laws that constitute expression. All expression follows a grammar. Our thoughts are formal – they are structured. The structurality of thoughts presupposes law. They are laws of thoughts. There are three laws of thoughts, they are, 1. Law of identity; 2. Law of contradiction; 3. Law of excluded- middle Law of identity: p is identical with itself. It asserts that if any statement is true then it is true. (If p stands for a true proposition then p is true only.) Law of contradiction asserts that ‘no statement can be both true and false’. (If p is a true statement then p cannot be false at the same time.) Law of excluded-middle asserts that ‘any statement is either true or false’. (If p is a true statement (¬p) its negation is false, both cannot be true together and both cannot be false together) Proposition is a logical sentence. The form of proposition is constituted of terms. A simple proposition is constituted of at least two terms; they are, the subject term and the predicate term. The subject term and predicate term refer to two different classes. They are related by a copula. Copula is a ‘to be’ verb. Example, All men are mortal. Here, in the above, proposition ‘men’ refers to the subject class term and the term ‘mortality’ represents predicate class. The copula ‘is’ relates the subject and the predicate terms. There are four type of categorical propositions used in Aristotelean logic. Their types are made with reference to the quality and quantity of the propositions. The categorical Propositions are: Logical form of Propositions 1. A: All men are mortal (Universal Affirmative) All s is p 2. E: No men are mortal (Universal Negative) No s is p 3. I: Some men are mortal (Particular Affirmative) Some s is p 4. O: Some men are not-mortal (Particular Negative) Some s is not-p 1
2. 2. Two Inferential Process of Deduction: 1. Immediate Deductive Inference 2. Mediate Deductive Inference Immediate Deductive Inference: Conclusion is deduced from one of the given propositions. Conversion and Obversion are deductive inferences. Conversion: The Rules of Conversion: 1. The conversion proceeds with interchanging the subject term and the predicate term, i.e. the subject term of the premises becomes the predicate term of the conclusion and the predicate term of the premise becomes the subject of the conclusion. For Example, No Hungarians are Cricketers (Convertend) No Cricketers are Hungarians (Converse) The given proposition is a premise is otherwise called as Convertend, where as the conclusion drawn from the premise is called Converse. 2. The quality of the premise (convertend) remains same. The quantity of the proposition may change. Table of Valid Conversion Convertend Converse A: All S is P (All students are smart) I: Some P is S (Some smart persons are students) E: No S is P (No student is tall) E: No P is S (No tall persons are students) I: Some S is P (Some students are poets) I: Some P is S (Some poets are students O: Some S is not-p (Conversion is not valid) 2
3. 3. Obversion: Rules of Obversion 1. Obversion is one of the immediate inferences. 2. To obvert a proposition, we change its quality and replace the predicate term by its complement. Instances: A: All poets are emotional (Obvertend) E: No poets are non-emotional. (Obverse) E: No singers are barbarians A: All singers are non-barbarians. I: Some politicians are statesmen O: Some politicians are not non-statesmen. O: Some teachers are not-cricketers I: Some teachers are non-cricketers. Syllogism A syllogism is a deductive argument in which conclusion is inferred from two premises. The standard syllogistic argument will have 3terms and 3 propositions. The term that occurs as the predicate term of the conclusion is called the ‘major term’. The term that occurs as subject term of the conclusion is ‘minor term’. The term, which does not appear in the conclusion but appears only in the premises, is called ‘middle term’. Major premises Minor premises Conclusion. 3
4. 4. Mood & Figure of Syllogism The standard form of categorical propositions determines the mood of the syllogism For example, in an argument like; A: All men are sincere (Major Premise) I: Some men are hard working (Minor Premise) I: Therefore, Some hard working persons are sincere (Conclusion) The mood of the above argument is: A I I The different positions of the middle term determine the figure of the syllogism. Ist 2nd 3rd 4th For Example, A: All scholars are IITians I: Some scholars are scientists Therefore, Some scientists are IITians. In the above argument ‘scholar’ is the middle term. It appears in the subject place of major premise and also the subject place of the minor premise. Hence, it constitutes the 3rd figure. 4
5. 5. Rules and fallacies of Syllogism: An argument in syllogism becomes fallacious if and only iff it violates the rules of syllogism. Here forth we are stating about some of the rules of syllogism and some of the fallacies 1. In a syllogism there must be at least three terms. If an argument involves four terms then we cannot draw a valid conclusion. The fallacy is called fallacy of four terms Example: All men are mortal Some scholars are sincere. There is no term common in the above argument, which makes it impossible to draw a valid conclusion. 2. The middle term must be distributed at least once in the premises. If the middle term is not distributed in any of the premises then the arguments commits the fallacy of undistributed middle. Example: All students are scholars Some scholars are technocrats Therefore, Some technocrats are students As you know that A proposition which is universal affirmative prop., does not distribute its predicate term. So the predicate term ‘scholar’ is not distributed in the major premise and it is also not distributed in the minor premise. The minor premise is ‘I’ prop., which does not distribute any term. Therefore, the argument commits the fallacy of undistributed middle. 5
6. 6. 3. If both the premises are negative then no conclusion follows. It commits the fallacy of exclusive terms. Ex. No judges are sentimental No judges are singers Therefore, No singers are sentimental The argument is fallacious because in negative proposition whether it universal negative or particular negative proposition the terms (the subject term and the predicate term) exclude each other. Their exclusion implies exclusion of the relationship of middle terms. 4. If a term is distributed in conclusion it must be distributed in the respective premises. If this condition is not fulfilled them it leads to the fallacy of either Illicit Major or Illicit Minor. Illicit Major: All students are regular No hardworking persons are students. Therefore, No hardworking persons are regular As the E proposition distributes its predicate term, the term ‘regular’ in the conclusion is distributed. It is the major term and as major term it has appeared in the predicate place of the major premise, which is undistributed. It is because A’ proposition distributes only its subject term not the predicate term. Hence, the argument commits the fallacy of Illicit Major Illicit Minor: All students are singers All students are poets Therefore, All poets are singers The predicate term poet in the conclusion is distributed which is the minor term. As a minor term it must be distributed in the minor premise. The minor premise is A type of proposition which distributes only the subject term. The poet occurs as the predicate term in the minor premise and remains undistributed. Thus the argument commits the fallacy of illicit minor. 6
7. 7. In a syllogism if one of the premise is particular then the conclusion must be particular proposition In a syllogism if one of the premise is negative then conclusion would be negative proposition. Induction: Inductive Generalization Causality Casual Relations: 1. Necessary Condition 2. Sufficient Condition NC: the presence of oxygen is a necessary condition for combustion to occur. Et1 → Et2 We can legitimately infer cause from effect only in the sense necessary condition. And we can legitimately infer effect from cause only in the sense of sufficient condition. Postulates of Induction 1. Law of Causality 2. Law of Uniformity of Nature 3. Law of Conservation of Energy Induction by Simple Enumeration: The method of arriving at general or universal propositions from particular facts of experience is called ‘inductive generalization’. 7
8. 8. Mills’ Method for Understanding Causal Relation: • Method of Agreement • Method of Difference • Joint Method • Method of Concomitant Variation • Method of Residues Science Science replaced truth by authority. Simple view of Scientific Method Induction used in Scientific Prediction: • uniformity of nature • conservation of energy • causality Limits of Observation Is observation theory laden? The Problem of Induction • Problem of certainty (Hume & Russell) • Different generalization can be made looking at the past instances. (Goodman) Ex. ‘GRUE’ [Bule / Green] Ex. 1. All emeralds are blue (t/f) before 2. All emeralds are green (t) after Inductive Generalization based on large number of observation • Context of Discovery 8
9. 9. • Context of Justification Certainty of the conclusion is replaced by Probability. Probability is based on the consistency of the available data. Ex. Laws in science are not absolutely proven to be true, rather generalization which is high probability of being true. Justification for Induction: • ‘Invariable and unconditional’ causal connection Law of causation is established by empirical grounds – confronts a paradox. *causal relation is proved by experience *conclusion presupposes law of causation *problem of circularity of definition. Karl Popper’s Falsifiability thesis: “Empirical method is continuously to expose a theory to the possibility of being falsified” Formulation of Conjectures/ hypothesis Increasing the degree of Falsifiability • Any theory, which is shown to be false, is discarded or at the very least, modified. Science thus progresses by means of conjectures and refutation. Verisimilitude (Approximation of truth) : Truth content Vs Falsity content Hypothesis – corroborates with reality Corroboration – belief in the approximate truth of theory. 9
10. 10. • Context of Justification Certainty of the conclusion is replaced by Probability. Probability is based on the consistency of the available data. Ex. Laws in science are not absolutely proven to be true, rather generalization which is high probability of being true. Justification for Induction: • ‘Invariable and unconditional’ causal connection Law of causation is established by empirical grounds – confronts a paradox. *causal relation is proved by experience *conclusion presupposes law of causation *problem of circularity of definition. Karl Popper’s Falsifiability thesis: “Empirical method is continuously to expose a theory to the possibility of being falsified” Formulation of Conjectures/ hypothesis Increasing the degree of Falsifiability • Any theory, which is shown to be false, is discarded or at the very least, modified. Science thus progresses by means of conjectures and refutation. Verisimilitude (Approximation of truth) : Truth content Vs Falsity content Hypothesis – corroborates with reality Corroboration – belief in the approximate truth of theory. 9

Transcript

1. 1. Logic Logic is the science of reasoning. Reasoning is a formal activity The notion of form has wider implication in logic. It pertains to the form of proposition as well as the form of argument. The notion of form refers to the norm/ rule/ laws that constitute expression. All expression follows a grammar. Our thoughts are formal – they are structured. The structurality of thoughts presupposes law. They are laws of thoughts. There are three laws of thoughts, they are, 1. Law of identity; 2. Law of contradiction; 3. Law of excluded- middle Law of identity: p is identical with itself. It asserts that if any statement is true then it is true. (If p stands for a true proposition then p is true only.) Law of contradiction asserts that ‘no statement can be both true and false’. (If p is a true statement then p cannot be false at the same time.) Law of excluded-middle asserts that ‘any statement is either true or false’. (If p is a true statement (¬p) its negation is false, both cannot be true together and both cannot be false together) Proposition is a logical sentence. The form of proposition is constituted of terms. A simple proposition is constituted of at least two terms; they are, the subject term and the predicate term. The subject term and predicate term refer to two different classes. They are related by a copula. Copula is a ‘to be’ verb. Example, All men are mortal. Here, in the above, proposition ‘men’ refers to the subject class term and the term ‘mortality’ represents predicate class. The copula ‘is’ relates the subject and the predicate terms. There are four type of categorical propositions used in Aristotelean logic. Their types are made with reference to the quality and quantity of the propositions. The categorical Propositions are: Logical form of Propositions 1. A: All men are mortal (Universal Affirmative) All s is p 2. E: No men are mortal (Universal Negative) No s is p 3. I: Some men are mortal (Particular Affirmative) Some s is p 4. O: Some men are not-mortal (Particular Negative) Some s is not-p 1
2. 2. Two Inferential Process of Deduction: 1. Immediate Deductive Inference 2. Mediate Deductive Inference Immediate Deductive Inference: Conclusion is deduced from one of the given propositions. Conversion and Obversion are deductive inferences. Conversion: The Rules of Conversion: 1. The conversion proceeds with interchanging the subject term and the predicate term, i.e. the subject term of the premises becomes the predicate term of the conclusion and the predicate term of the premise becomes the subject of the conclusion. For Example, No Hungarians are Cricketers (Convertend) No Cricketers are Hungarians (Converse) The given proposition is a premise is otherwise called as Convertend, where as the conclusion drawn from the premise is called Converse. 2. The quality of the premise (convertend) remains same. The quantity of the proposition may change. Table of Valid Conversion Convertend Converse A: All S is P (All students are smart) I: Some P is S (Some smart persons are students) E: No S is P (No student is tall) E: No P is S (No tall persons are students) I: Some S is P (Some students are poets) I: Some P is S (Some poets are students O: Some S is not-p (Conversion is not valid) 2
3. 3. Obversion: Rules of Obversion 1. Obversion is one of the immediate inferences. 2. To obvert a proposition, we change its quality and replace the predicate term by its complement. Instances: A: All poets are emotional (Obvertend) E: No poets are non-emotional. (Obverse) E: No singers are barbarians A: All singers are non-barbarians. I: Some politicians are statesmen O: Some politicians are not non-statesmen. O: Some teachers are not-cricketers I: Some teachers are non-cricketers. Syllogism A syllogism is a deductive argument in which conclusion is inferred from two premises. The standard syllogistic argument will have 3terms and 3 propositions. The term that occurs as the predicate term of the conclusion is called the ‘major term’. The term that occurs as subject term of the conclusion is ‘minor term’. The term, which does not appear in the conclusion but appears only in the premises, is called ‘middle term’. Major premises Minor premises Conclusion. 3
4. 4. Mood & Figure of Syllogism The standard form of categorical propositions determines the mood of the syllogism For example, in an argument like; A: All men are sincere (Major Premise) I: Some men are hard working (Minor Premise) I: Therefore, Some hard working persons are sincere (Conclusion) The mood of the above argument is: A I I The different positions of the middle term determine the figure of the syllogism. Ist 2nd 3rd 4th For Example, A: All scholars are IITians I: Some scholars are scientists Therefore, Some scientists are IITians. In the above argument ‘scholar’ is the middle term. It appears in the subject place of major premise and also the subject place of the minor premise. Hence, it constitutes the 3rd figure. 4
5. 5. Rules and fallacies of Syllogism: An argument in syllogism becomes fallacious if and only iff it violates the rules of syllogism. Here forth we are stating about some of the rules of syllogism and some of the fallacies 1. In a syllogism there must be at least three terms. If an argument involves four terms then we cannot draw a valid conclusion. The fallacy is called fallacy of four terms Example: All men are mortal Some scholars are sincere. There is no term common in the above argument, which makes it impossible to draw a valid conclusion. 2. The middle term must be distributed at least once in the premises. If the middle term is not distributed in any of the premises then the arguments commits the fallacy of undistributed middle. Example: All students are scholars Some scholars are technocrats Therefore, Some technocrats are students As you know that A proposition which is universal affirmative prop., does not distribute its predicate term. So the predicate term ‘scholar’ is not distributed in the major premise and it is also not distributed in the minor premise. The minor premise is ‘I’ prop., which does not distribute any term. Therefore, the argument commits the fallacy of undistributed middle. 5
6. 6. 3. If both the premises are negative then no conclusion follows. It commits the fallacy of exclusive terms. Ex. No judges are sentimental No judges are singers Therefore, No singers are sentimental The argument is fallacious because in negative proposition whether it universal negative or particular negative proposition the terms (the subject term and the predicate term) exclude each other. Their exclusion implies exclusion of the relationship of middle terms. 4. If a term is distributed in conclusion it must be distributed in the respective premises. If this condition is not fulfilled them it leads to the fallacy of either Illicit Major or Illicit Minor. Illicit Major: All students are regular No hardworking persons are students. Therefore, No hardworking persons are regular As the E proposition distributes its predicate term, the term ‘regular’ in the conclusion is distributed. It is the major term and as major term it has appeared in the predicate place of the major premise, which is undistributed. It is because A’ proposition distributes only its subject term not the predicate term. Hence, the argument commits the fallacy of Illicit Major Illicit Minor: All students are singers All students are poets Therefore, All poets are singers The predicate term poet in the conclusion is distributed which is the minor term. As a minor term it must be distributed in the minor premise. The minor premise is A type of proposition which distributes only the subject term. The poet occurs as the predicate term in the minor premise and remains undistributed. Thus the argument commits the fallacy of illicit minor. 6
7. 7. In a syllogism if one of the premise is particular then the conclusion must be particular proposition In a syllogism if one of the premise is negative then conclusion would be negative proposition. Induction: Inductive Generalization Causality Casual Relations: 1. Necessary Condition 2. Sufficient Condition NC: the presence of oxygen is a necessary condition for combustion to occur. Et1 → Et2 We can legitimately infer cause from effect only in the sense necessary condition. And we can legitimately infer effect from cause only in the sense of sufficient condition. Postulates of Induction 1. Law of Causality 2. Law of Uniformity of Nature 3. Law of Conservation of Energy Induction by Simple Enumeration: The method of arriving at general or universal propositions from particular facts of experience is called ‘inductive generalization’. 7
8. 8. Mills’ Method for Understanding Causal Relation: • Method of Agreement • Method of Difference • Joint Method • Method of Concomitant Variation • Method of Residues Science Science replaced truth by authority. Simple view of Scientific Method Induction used in Scientific Prediction: • uniformity of nature • conservation of energy • causality Limits of Observation Is observation theory laden? The Problem of Induction • Problem of certainty (Hume & Russell) • Different generalization can be made looking at the past instances. (Goodman) Ex. ‘GRUE’ [Bule / Green] Ex. 1. All emeralds are blue (t/f) before 2. All emeralds are green (t) after Inductive Generalization based on large number of observation • Context of Discovery 8
9. 9. • Context of Justification Certainty of the conclusion is replaced by Probability. Probability is based on the consistency of the available data. Ex. Laws in science are not absolutely proven to be true, rather generalization which is high probability of being true. Justification for Induction: • ‘Invariable and unconditional’ causal connection Law of causation is established by empirical grounds – confronts a paradox. *causal relation is proved by experience *conclusion presupposes law of causation *problem of circularity of definition. Karl Popper’s Falsifiability thesis: “Empirical method is continuously to expose a theory to the possibility of being falsified” Formulation of Conjectures/ hypothesis Increasing the degree of Falsifiability • Any theory, which is shown to be false, is discarded or at the very least, modified. Science thus progresses by means of conjectures and refutation. Verisimilitude (Approximation of truth) : Truth content Vs Falsity content Hypothesis – corroborates with reality Corroboration – belief in the approximate truth of theory. 9
10. 10. • Context of Justification Certainty of the conclusion is replaced by Probability. Probability is based on the consistency of the available data. Ex. Laws in science are not absolutely proven to be true, rather generalization which is high probability of being true. Justification for Induction: • ‘Invariable and unconditional’ causal connection Law of causation is established by empirical grounds – confronts a paradox. *causal relation is proved by experience *conclusion presupposes law of causation *problem of circularity of definition. Karl Popper’s Falsifiability thesis: “Empirical method is continuously to expose a theory to the possibility of being falsified” Formulation of Conjectures/ hypothesis Increasing the degree of Falsifiability • Any theory, which is shown to be false, is discarded or at the very least, modified. Science thus progresses by means of conjectures and refutation. Verisimilitude (Approximation of truth) : Truth content Vs Falsity content Hypothesis – corroborates with reality Corroboration – belief in the approximate truth of theory. 9

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