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Categorical Propositions Theory Deduction
- 2. Categorical 3.1. The Theory of Deduction : Propositions ==================================================================== ===============================
- 3. Categorical 3.1. The Theory of Deduction : Propositions The theory of deduction aims to explain the relations of premises and conclusion in valid arguments. It also aims to provide techniques for the ==================================================================== evaluation of deductive arguments, that is, for discriminating between valid and =============================== invalid deductions.
- 4. Categorical 3.1. The Theory of Deduction : Propositions The theory of deduction aims to explain the relations of premises and conclusion in valid arguments. It also aims to provide techniques for the ==================================================================== evaluation of deductive arguments, that is, for discriminating between valid and =============================== invalid deductions. To discriminate valid and invalid deductions two theories have been developed. 1. Classical Logic or Aristotelian Logic, and 2. Modern Logic or Modern Symbolic Logic.
- 5. Categorical 3.1. The Theory of Deduction : Propositions The theory of deduction aims to explain the relations of premises and conclusion in valid arguments. It also aims to provide techniques for the ==================================================================== evaluation of deductive arguments, that is, for discriminating between valid and =============================== invalid deductions. To discriminate valid and invalid deductions two theories have been developed. 1. Classical Logic or Aristotelian Logic, and 2. Modern Logic or Modern Symbolic Logic. Aristotle (384-322 B.C) was one of the towering intellects of the ancient world. His great treaties on reasoning were gathered together after his death and came to be called Organon, meaning literally the instrument, the fundamental tool of knowledge.
- 6. 3.2. Classes and Categorical Propositions :
- 7. 3.2. Classes and Categorical Propositions : What is a class ? By a class we mean a collection of all objects that have some specified characteristic in common. Everyone can see immediately that two classes can be related in at least the following three ways:
- 8. 3.2. Classes and Categorical Propositions : What is a class ? By a class we mean a collection of all objects that have some specified characteristic in common. Everyone can see immediately that two classes can be related in at least the following three ways: 1. All of one class may be included in all of another class. Ex: The class of all dogs is wholly included in the class of all animals.
- 9. 3.2. Classes and Categorical Propositions : What is a class ? By a class we mean a collection of all objects that have some specified characteristic in common. Everyone can see immediately that two classes can be related in at least the following three ways: 1. All of one class may be included in all of another class. Ex: The class of all dogs is wholly included in the class of all animals. 2. Some, but not all, of the members of one class may be included in another class. Ex: The class of all chess players is partially included in the class of all females.
- 10. 3.2. Classes and Categorical Propositions : What is a class ? By a class we mean a collection of all objects that have some specified characteristic in common. Everyone can see immediately that two classes can be related in at least the following three ways: 1. All of one class may be included in all of another class. Ex: The class of all dogs is wholly included in the class of all animals. 2. Some, but not all, of the members of one class may be included in another class. Ex: The class of all chess players is partially included in the class of all females. 3. Two classes may have no members in common. Ex: The class of all triangles and the class of all circles may be said to be exclude one another.
- 11. In deductive argument we present propositions that state the relations between one category and some other category. The propositions with which such arguments are formulated are called “Categorical Propositions.” Like a proposition “Categorical Propositions” also contain Subjective term and Predicative term. Categorical Propositions are about quantity. So Categorical Propositions are quantitative Propositions.
- 12. In deductive argument we present propositions that state the relations between one category and some other category. The propositions with which such arguments are formulated are called “Categorical Propositions.” Like a proposition “Categorical Propositions” also contain Subjective term and Predicative term. Categorical Propositions are about quantity. So Categorical Propositions are quantitative Propositions. No sportspersons are vegetarians All Hockey players are sportspersons Therefore no Hockey players are vegetarians
- 13. In deductive argument we present propositions that state the relations between one category and some other category. The propositions with which such arguments are formulated are called “Categorical Propositions.” Like a proposition “Categorical Propositions” also contain Subjective term and Predicative term. Categorical Propositions are about quantity. So Categorical Propositions are quantitative Propositions. No sportspersons are vegetarians All Hockey players are sportspersons Therefore no Hockey players are vegetarians This above argument contain three Categorical propositions. We may dispute the truth of its premises but the relations of the classes expressed in those propositions yield an argument that is certainly valid. In this illustrative argument the three categorical propositions are about the class of all sportspersons, the class of the all vegetarians and the class of all hockey players.
- 14. In deductive argument we present propositions that state the relations between one category and some other category. The propositions with which such arguments are formulated are called “Categorical Propositions.” Like a proposition “Categorical Propositions” also contain Subjective term and Predicative term. Categorical Propositions are about quantity. So Categorical Propositions are quantitative Propositions. No sportspersons are vegetarians All Hockey players are sportspersons Therefore no Hockey players are vegetarians This above argument contain three Categorical propositions. We may dispute the truth of its premises but the relations of the classes expressed in those propositions yield an argument that is certainly valid. In this illustrative argument the three categorical propositions are about the class of all sportspersons, the class of the all vegetarians and the class of all hockey players. Ex: All humans are mortal Socrates is a Human Therefore Socrates is mortal
- 15. In deductive argument we present propositions that state the relations between one category and some other category. The propositions with which such arguments are formulated are called “Categorical Propositions.” Like a proposition “Categorical Propositions” also contain Subjective term and Predicative term. Categorical Propositions are about quantity. So Categorical Propositions are quantitative Propositions. No sportspersons are vegetarians All Hockey players are sportspersons Therefore no Hockey players are vegetarians This above argument contain three Categorical propositions. We may dispute the truth of its premises but the relations of the classes expressed in those propositions yield an argument that is certainly valid. In this illustrative argument the three categorical propositions are about the class of all sportspersons, the class of the all vegetarians and the class of all hockey players. Ex: All humans are mortal Socrates is a Human Therefore Socrates is mortal All H are M X is H Therefore X is M
- 16. In deductive argument we present propositions that state the relations between one category and some other category. The propositions with which such arguments are formulated are called “Categorical Propositions.” Like a proposition “Categorical Propositions” also contain Subjective term and Predicative term. Categorical Propositions are about quantity. So Categorical Propositions are quantitative Propositions. No sportspersons are vegetarians All Hockey players are sportspersons Therefore no Hockey players are vegetarians This above argument contain three Categorical propositions. We may dispute the truth of its premises but the relations of the classes expressed in those propositions yield an argument that is certainly valid. In this illustrative argument the three categorical propositions are about the class of all sportspersons, the class of the all vegetarians and the class of all hockey players. Ex: All humans are mortal Socrates is a Human Therefore Socrates is mortal All H are M X is H Therefore X is M H = The category of all humans
- 17. In deductive argument we present propositions that state the relations between one category and some other category. The propositions with which such arguments are formulated are called “Categorical Propositions.” Like a proposition “Categorical Propositions” also contain Subjective term and Predicative term. Categorical Propositions are about quantity. So Categorical Propositions are quantitative Propositions. No sportspersons are vegetarians All Hockey players are sportspersons Therefore no Hockey players are vegetarians This above argument contain three Categorical propositions. We may dispute the truth of its premises but the relations of the classes expressed in those propositions yield an argument that is certainly valid. In this illustrative argument the three categorical propositions are about the class of all sportspersons, the class of the all vegetarians and the class of all hockey players. Ex: All humans are mortal Socrates is a Human Therefore Socrates is mortal All H are M X is H Therefore X is M H = M = The category of all humans The category of all things that are mortal
- 18. In deductive argument we present propositions that state the relations between one category and some other category. The propositions with which such arguments are formulated are called “Categorical Propositions.” Like a proposition “Categorical Propositions” also contain Subjective term and Predicative term. Categorical Propositions are about quantity. So Categorical Propositions are quantitative Propositions. No sportspersons are vegetarians All Hockey players are sportspersons Therefore no Hockey players are vegetarians This above argument contain three Categorical propositions. We may dispute the truth of its premises but the relations of the classes expressed in those propositions yield an argument that is certainly valid. In this illustrative argument the three categorical propositions are about the class of all sportspersons, the class of the all vegetarians and the class of all hockey players. Ex: All humans are mortal Socrates is a Human Therefore Socrates is mortal All H are M x is H Therefore X is M H = Humans M = Mortals The category of all humans The category of all things that are mortal
- 23. 3.4. Quantity, Quality and Distribution : Quality :
- 24. 3.4. Quantity, Quality and Distribution : Quality : If we talk about the quality in Categorical proposition, it means that we are talking about the affirmative or negative aspect of that proposition. We can explain it in four ways:
- 25. 3.4. Quantity, Quality and Distribution : Quality : If we talk about the quality in Categorical proposition, it means that we are talking about the affirmative or negative aspect of that proposition. We can explain it in four ways: All S is P – it is an affirmative proposition No S is P – it is a negative proposition Some S is P – it is an affirmative proposition Some S is not P – it is a negative proposition
- 26. 3.4. Quantity, Quality and Distribution : Quality : If we talk about the quality in Categorical proposition, it means that we are talking about the affirmative or negative aspect of that proposition. We can explain it in four ways: All S is P – it is an affirmative proposition No S is P – it is a negative proposition Some S is P – it is an affirmative proposition Some S is not P – it is a negative proposition Quantity : Every standard-form of categorical proposition has some class as its subject. 1. Universal and 2. particular. We can also explain it in four ways:
- 27. 3.4. Quantity, Quality and Distribution : Quality : If we talk about the quality in Categorical proposition, it means that we are talking about the affirmative or negative aspect of that proposition. We can explain it in four ways: All S is P – it is an affirmative proposition No S is P – it is a negative proposition Some S is P – it is an affirmative proposition Some S is not P – it is a negative proposition Quantity : Every standard-form of categorical proposition has some class as its subject. 1. Universal and 2. particular. We can also explain it in four ways: All S is P – it is Universal affirmative proposition No S is P – it is Universal negative proposition Some S is P – it is Particular affirmative proposition Some S is not P – it is Particular negative proposition
- 28. 3.4. Quantity, Quality and Distribution : Quality : If we talk about the quality in Categorical proposition, it means that we are talking about the affirmative or negative aspect of that proposition. We can explain it in four ways: All S is P – it is an affirmative proposition No S is P – it is a negative proposition Some S is P – it is an affirmative proposition Some S is not P – it is a negative proposition Quantity : Every standard-form of categorical proposition has some class as its subject. 1. Universal and 2. particular. We can also explain it in four ways: All S is P – it is Universal affirmative proposition No S is P – it is Universal negative proposition Some S is P – it is Particular affirmative proposition Some S is not P – it is Particular negative proposition Distribution : All S is No S is Some S Some S P – A Proposition Ex: All members of Parliament are citizens P – E Proposition Ex: No sports persons are vegetarians is P – I Proposition Ex: Some solders are cowards is not P – O Proposition Ex: Some students are not regular
- 29. 3.3. Kinds of Categorical Propositions :
- 30. 3.3. Kinds of Categorical Propositions : Aristotle‟s Logic is called „Categorical Logic‟ because it deals with the categorical statements. Categorical statements are statements about categories of objects. Syllogism is an argument which consist of two premises and one conclusion. Categorical syllogisms are syllogism composed of categorical statements.
- 31. 3.3. Kinds of Categorical Propositions : Aristotle‟s Logic is called „Categorical Logic‟ because it deals with the categorical statements. Categorical statements are statements about categories of objects. Syllogism is an argument which consist of two premises and one conclusion. Categorical syllogisms are syllogism composed of categorical statements. Aristotle introduced four types of statements 1. Universal Affirmative – called as „A‟ Proposition: In this the whole of one class is include or contained in another class. Ex: All Humans are Mortal Humans are Mortal All Whales are Mammals Whales are mammals All layers are decent people Lawyers are decent people
- 32. 3.3. Kinds of Categorical Propositions : Aristotle‟s Logic is called „Categorical Logic‟ because it deals with the categorical statements. Categorical statements are statements about categories of objects. Syllogism is an argument which consist of two premises and one conclusion. Categorical syllogisms are syllogism composed of categorical statements. Aristotle introduced four types of statements 1. Universal Affirmative – called as „A‟ Proposition: In this the whole of one class is include or contained in another class. Ex: All Humans are Mortal Humans are Mortal Human All Whales are Mammals Whales are mammals All layers are decent people Lawyers are decent people Mortals
- 33. 3.3. Kinds of Categorical Propositions : Aristotle‟s Logic is called „Categorical Logic‟ because it deals with the categorical statements. Categorical statements are statements about categories of objects. Syllogism is an argument which consist of two premises and one conclusion. Categorical syllogisms are syllogism composed of categorical statements. Aristotle introduced four types of statements 1. Universal Affirmative – called as „A‟ Proposition: In this the whole of one class is include or contained in another class. Ex: All Humans are Mortal Humans are Mortal Human All Whales are Mammals Whales are mammals All layers are decent people Lawyers are decent people Mortals 2. Universal Negative – called as „E‟ Proposition Ex: No snakes are reptiles No bachelor are married
- 34. 3.3. Kinds of Categorical Propositions : Aristotle‟s Logic is called „Categorical Logic‟ because it deals with the categorical statements. Categorical statements are statements about categories of objects. Syllogism is an argument which consist of two premises and one conclusion. Categorical syllogisms are syllogism composed of categorical statements. Aristotle introduced four types of statements 1. Universal Affirmative – called as „A‟ Proposition: In this the whole of one class is include or contained in another class. Ex: All Humans are Mortal Humans are Mortal Human All Whales are Mammals Whales are mammals All layers are decent people Lawyers are decent people Mortals 2. Universal Negative – called as „E‟ Proposition Ex: No snakes are reptiles No bachelor are married In this Universal Negative proposition one important thing we have to understand very clearly. No snakes are reptiles means “all snakes are not reptiles” and “No bachelors are married” means “All bachelors are not married.” The word „not‟ applies only to the predicate term but not to the subject term.
- 35. 3.3. Kinds of Categorical Propositions : Aristotle‟s Logic is called „Categorical Logic‟ because it deals with the categorical statements. Categorical statements are statements about categories of objects. Syllogism is an argument which consist of two premises and one conclusion. Categorical syllogisms are syllogism composed of categorical statements. Aristotle introduced four types of statements 1. Universal Affirmative – called as „A‟ Proposition: In this the whole of one class is include or contained in another class. Ex: All Humans are Mortal Humans are Mortal Human All Whales are Mammals Whales are mammals All layers are decent people Lawyers are decent people Mortals 2. Universal Negative – called as „E‟ Proposition Ex: No snakes are reptiles No bachelor are married In this Universal Negative proposition one important thing we have to understand very clearly. No snakes are reptiles means “all snakes are not reptiles” and “No bachelors are married” means “All bachelors are not married.” The word „not‟ applies only to the predicate term but not to the subject term. 3. Particular affirmative – called as „I‟ Proposition Ex: Some dogs have long hair Some people earn Rs. 200 in a day Some girls taller than boys Here some means at least one person. We can imagine it in three ways: 1. at least one dog has long hair 2. There is a dog that has long hair 3. there exists a long-haired dog
- 36. 3.3. Kinds of Categorical Propositions : Aristotle‟s Logic is called „Categorical Logic‟ because it deals with the categorical statements. Categorical statements are statements about categories of objects. Syllogism is an argument which consist of two premises and one conclusion. Categorical syllogisms are syllogism composed of categorical statements. Aristotle introduced four types of statements 1. Universal Affirmative – called as „A‟ Proposition: In this the whole of one class is include or contained in another class. Ex: All Humans are Mortal Humans are Mortal Human All Whales are Mammals Whales are mammals All layers are decent people Lawyers are decent people Mortals 2. Universal Negative – called as „E‟ Proposition Ex: No snakes are reptiles No bachelor are married In this Universal Negative proposition one important thing we have to understand very clearly. No snakes are reptiles means “all snakes are not reptiles” and “No bachelors are married” means “All bachelors are not married.” The word „not‟ applies only to the predicate term but not to the subject term. 3. Particular affirmative – called as „I‟ Proposition Ex: Some dogs have long hair Some people earn Rs. 200 in a day Some girls taller than boys Here some means at least one person. We can imagine it in three ways: 1. at least one dog has long hair 2. There is a dog that has long hair 3. there exists a long-haired dog 4. Particular Negative – called as „O‟ Proposition Ex: some dogs do not have four legs
- 37. 3.5. The Traditional square of oppositions :
- 38. 3.5. The Traditional square of oppositions : In the system of Aristotelian Logic, the square of opposition is a diagram representing the different ways in which each of the four propositions of the system is logically related ('opposed') to each of the others. The system is also useful in the analysis of Syllogistic Logic, serving to identify the allowed logical conversions from one type to another.
- 39. 3.5. The Traditional square of oppositions : In the system of Aristotelian Logic, the square of opposition is a diagram representing the different ways in which each of the four propositions of the system is logically related ('opposed') to each of the others. The system is also useful in the analysis of Syllogistic Logic, serving to identify the allowed logical conversions from one type to another.
- 40. 1. Contrary Propositions : Universal statements are contraries: Ex: All Poets are dreamers (A), No poets are dreamers (E). Both cannot be true together, although one may be true and the other false, and also both may be false . In this case both (A&E) Propositions are having the same subject and predicate terms but differing in quality (one is affirming and the other denying)
- 41. 1. Contrary Propositions : Universal statements are contraries: Ex: All Poets are dreamers (A), No poets are dreamers (E). Both cannot be true together, although one may be true and the other false, and also both may be false . In this case both (A&E) Propositions are having the same subject and predicate terms but differing in quality (one is affirming and the other denying) 2. Contradictory Propositions : Two propositions are contradictories if one is the denial or negation of the other. Two categorical propositions that have the same subject and predicate terms but differ from each other in both quantity and quality. Ex: „All Judges are lawyers‟ (A) and „Some Judges are not lawyers‟ (O). Ex: „No politicians are idealists‟ (E) and „Some politicians are idealists‟ (I).
- 42. 1. Contrary Propositions : Universal statements are contraries: Ex: All Poets are dreamers (A), No poets are dreamers (E). Both cannot be true together, although one may be true and the other false, and also both may be false . In this case both (A&E) Propositions are having the same subject and predicate terms but differing in quality (one is affirming and the other denying) 2. Contradictory Propositions : Two propositions are contradictories if one is the denial or negation of the other. Two categorical propositions that have the same subject and predicate terms but differ from each other in both quantity and quality. Ex: „All Judges are lawyers‟ (A) and „Some Judges are not lawyers‟ (O). Ex: „No politicians are idealists‟ (E) and „Some politicians are idealists‟ (I). 3. Sub-contrary Propositions Two propositions are said to be Sub-contrary if they cannot both be false, although they may both true. In this two particular categorical propositions (I&O) having the same subject and predicate terms but differ in quantity, Ex: „Some diamonds are precious‟ (I) and „Some diamonds are not precious‟ (O).
- 43. 1. Contrary Propositions : Universal statements are contraries: Ex: All Poets are dreamers (A), No poets are dreamers (E). Both cannot be true together, although one may be true and the other false, and also both may be false . In this case both (A&E) Propositions are having the same subject and predicate terms but differing in quality (one is affirming and the other denying) 2. Contradictory Propositions : Two propositions are contradictories if one is the denial or negation of the other. Two categorical propositions that have the same subject and predicate terms but differ from each other in both quantity and quality. Ex: „All Judges are lawyers‟ (A) and „Some Judges are not lawyers‟ (O). Ex: „No politicians are idealists‟ (E) and „Some politicians are idealists‟ (I). 3. Sub-contrary Propositions Two propositions are said to be Sub-contrary if they cannot both be false, although they may both true. In this two particular categorical propositions (I&O) having the same subject and predicate terms but differ in quantity, Ex: „Some diamonds are precious‟ (I) and „Some diamonds are not precious‟ (O). 4. Subaltern Propositions When two propositions are have the same subject and predicate terms, and agree in quantity (both affirming and denying) but differ in quantity (one is particular and another one is Universal). It is also called „corresponding propositions.‟ Ex: „All Spiders are eight-legged creatures‟ (A) and „Some spiders are eight-legged creatures‟ (I). Ex: „No whales are fishes‟ (E) and „Some whales are not fishes‟ (O).