2. Categorical
3.1. The Theory of Deduction :
Propositions
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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.
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
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
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
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).
