Classical Square of Opposition MCQ Quiz - Objective Question with Answer for Classical Square of Opposition - Download Free PDF

The correct answer is - (A) and (D) Only

Key Points

• Contradictory Propositions
  • Two propositions are said to be contradictory if they cannot both be true at the same time and they cannot both be false at the same time.
  • Statement (A): "All judges are lawyers" implies that every single judge is a lawyer.
  • Statement (D): "Some judges are not lawyers" implies that there exists at least one judge who is not a lawyer.
  • Since statement (A) claims that every judge is a lawyer, it directly contradicts statement (D), which claims that there is at least one judge who is not a lawyer.

Additional Information

• Logical Relationships
  • Contradiction: Two statements are contradictory if one being true means the other must be false.
  • Contrary: Two statements are contrary if they cannot both be true, but they can both be false.
  • Subcontrary: Two statements are subcontrary if they cannot both be false, but they can both be true.
  • Subalternation: This refers to the logical relationship between a universal proposition and its corresponding particular proposition (e.g., "All S are P" and "Some S are P").
• Examples
  • Contradictory Example: "All cats are animals" vs. "Some cats are not animals" (cannot both be true or both be false).
  • Contrary Example: "All cats are black" vs. "No cats are black" (cannot both be true, but can both be false).
  • Subcontrary Example: "Some cats are black" vs. "Some cats are not black" (cannot both be false, but can both be true).
  • Subalternation Example: "All birds can fly" (universal) implies "Some birds can fly" (particular).

The correct answer is - A - II, B - III, C - I, D - IV

Key Points

• Square of Opposition
  • The Square of Opposition is a diagram representing the different ways in which each of the four propositions of the traditional categorical logic (A, E, I, O) relate to each other.
  • These relationships are defined by the truth values of the propositions.
• Specific Matchings
  • A - II: If 'I' is given as true, 'E' is false, and 'A' is undetermined.
  • B - III: If 'I' is given as false, 'A' is false, and 'E' is true.
  • C - I: If 'O' is given as true, both 'E' and 'I' are undetermined.
  • D - IV: If 'O' is given as false, 'E' is false.

Additional Information

• Propositional Forms
  • 'A' Proposition: Universal Affirmative (All S are P)
  • 'E' Proposition: Universal Negative (No S are P)
  • 'I' Proposition: Particular Affirmative (Some S are P)
  • 'O' Proposition: Particular Negative (Some S are not P)
• Logical Relationships
  • Contradiction: Opposite truth values (A and O, E and I)
  • Contrariety: Cannot both be true (A and E)
  • Subcontrariety: Cannot both be false (I and O)
  • Subalternation: Truth of universal implies truth of particular (A to I, E to O)

The correct answer is - God

Key Points

• God
  • The term God exemplifies an extension that is empty because it refers to a concept or entity whose existence cannot be empirically verified.
  • In philosophy and logic, an empty extension means that the term does not have any actual instances or members in the real world.
  • Other terms like Pen, Book, and Bag refer to physical objects that can be empirically verified and therefore do not have empty extensions.

Additional Information

• Extension and Intension
  • Extension refers to the set of all objects or entities that a term denotes.
  • Intension refers to the set of attributes or properties that a term connotes.
  • For example, the term Cat has an extension that includes all individual cats, while its intension includes properties like being a mammal, having fur, etc.
• Philosophical Context
  • In philosophical discussions, terms with empty extensions often pertain to abstract concepts, hypothetical entities, or mythological beings.
  • Examples include terms like Unicorn or Dragon, which do not have real-world instances.

The correct answer is - Corresponding

Key Points

• Corresponding
  • When two propositions have the same subject and predicate terms, and they agree in quality but differ in quantity, they are known as corresponding.
  • Quality refers to whether the proposition is affirmative or negative.
  • Quantity refers to whether the proposition is universal or particular.
  • For example:
    • "All S are P" (universal affirmative) and "Some S are P" (particular affirmative) are corresponding propositions.
    • "No S are P" (universal negative) and "Some S are not P" (particular negative) are also corresponding propositions.

Additional Information

• Contraries
  • Contrary propositions cannot both be true but can both be false.
  • For example, "All S are P" and "No S are P" are contraries.
• Sub-Contraries
  • Sub-contrary propositions cannot both be false but can both be true.
  • For example, "Some S are P" and "Some S are not P" are sub-contraries.
• Contradictory
  • Contradictory propositions cannot both be true and cannot both be false; one must be true and the other false.
  • For example, "All S are P" and "Some S are not P" are contradictories.

The correct answer is - Option 4

Key Points

• Stipulative Definition as a Proposal
  • A stipulative definition is a proposal or a request to use the definiendum (term being defined) to mean what is meant by the definiens (defining phrase).
  • This is reflected in statement B which says, "A stipulative definition is a proposal or a request or a resolution to use the definiendum to mean what is meant by definiens."
• Example of Stipulative Definition
  • Naming a large number, such as 10100, as "googol" is an example of a stipulative definition.
  • This is captured in statement D which states, "Naming the number 10100 as 'googol' is a stipulation."

Additional Information

• True or False Nature
  • A stipulative definition is neither true nor false because it does not aim to describe the existing usage of a term but rather to introduce a new usage.
  • Hence, statement A is incorrect.
• Directive vs. Informative
  • A stipulative definition is directive as it dictates how a term should be used rather than providing information about an existing term.
  • Thus, statement C is incorrect.
• Purpose of Stipulative Definitions
  • Stipulative definitions are often used in technical fields, legislative documents, and during the introduction of new concepts or terms.
  • They help avoid ambiguity and ensure that all parties understand a term in the same way.

Categorical proposition :

• In logic, the categorical proposition is also known as a categorical statement that asserts or denies that all or some of the members of one category are included in another.
• It is an important branch of deductive reasoning.
• The categorical proposition can be classified into four under two heads, based on their quality and quantity, and distribution of terms.
  • Affirmative proposition (affirmo): A type and I type
  • Negative proposition (nego): E type and O type

Quantity of categorical proposition:

• It is the number of members of the subject class that are used in the proposition. 
• It could be the universal or the particular categorical proposition.
• It will consider universal if all members of the subject class refer to it.

Quality of categorical proposition:

• It is described as whether the proposition affirms or denies the inclusion of a subject within the class of the predicate. 
• It could be affirmative or negative.

Validity:

• An argument is valid if and only if in every case where all the premises are true, the conclusion is true. Otherwise, the argument is invalid.

Mood:

• The mood of a categorical syllogism is a series of three letters corresponding to the type of proposition the major premise, the minor premise, and the conclusion is (A, E, I, or O).
• The mood will help us to determine when such syllogisms are valid or invalid.

​Figure:

• The figure of a categorical syllogism is a number that corresponds to the placement of the two middle terms. 

Therefore, the quantity and quality of a categorical proposition decide the mode of the proposition.

The standard form of categorical proposition having the same subject term and predicate term but may differ from each other in quality or quantity or both. Such differing has been called opposition. The term opposition is used when there is no apparent disagreement between the propositions.

Types of Opposition:

1. Contradictories: The standard form of a categorical proposition that has the same subject and predicate term but differs from each other in both quantity and quality. Two propositions are contradictories if one is denial or negation of the other if they can’t be true or can’t be both false.

• Example:
  1. All trees are plants.
  2. Some trees are not plants.

2. Contraries: Two propositions are said to be contraries if they can’t both be true, and the truth of one entails of the truth of the other. i.e. both can’t be true and both can’t be false. If either of these propositions is true, then the other must be false.

• Example:
  1. All artists are dreamers.
  2. No artist is dreamer

3. Sub-contraries: If a particular proposition having the same subject and predicate terms but differing in quality, one affirming the other denying. Two propositions are said to be sub-contraries if they can’t both be false although both may be true.

• Example:
  1. Some cars are vehicles.
  2. Some cars are not vehicles.

4. Subalternation: It is the opposition between a universal proposition and its corresponding particular proposition. In the corresponding proposition, the universal proposition is called superaltern and the particular proposition is called subaltern. These propositions have the same subject and predicate term and agree in quality. Both are affirming or both denying but differ in quantity. On universal and the other particular.

• Example:
  1. No hens are birds
  2. Some hens are not birds

5. The square of opposition: There are four ways in which propositions may be opposed as contradictories, contraries, sub-contraries, subalterns and superaltern. These are represented using a diagram called the square of opposition.

Hence, from the given points it is clear that subaltern is when the subject and predicate of both the premises are the same but they differ only in quantity.

The correct solution is "The predicate is distributed".

Key Points

• The A statement distributes the subject term only.
• The E statement distributes both the subject term and predicate term.
• The I statement distributes no terms (neither the subject nor the predicate)
• The O statement distributes the predicate term only.

Distribution

The square of opposition is a chart that was introduced within classical (categorical) logic to represent the logical relationships holding between certain propositions in virtue of their form.

Key Points

 The square of opposition:

• Universals on top vs particulars on the bottom
• Affirmatives on left vs negatives on right
• Contradictories (diagonals):

They always have the opposite truth values--you will always be able to determine the truth value of contradictories.

• Contraries:
  • Two propositions are said to be contraries if they can't both be true, and the truth of one entails the truth of the other. i.e. Two statements are contrary to one another if they are both universals but differ in quality.
  • Contraries cannot at the same time both be true, but can, at the same time, be false.
  • If either of these propositions is true, then the other must be false.
  • Contraries cannot both be true at the same time 
• Sub-contraries
  • ​The relation between two particular propositions having the same subject and predicate but differing in quality is subcontrary opposition
• Subalternation
  • It is a relation between the particular statement and the universal statement of the same quality (affirmative or negative) such that the particular is implied by the universal,
• Opposition:
  • It occurs when two standard-form categorical propositions refer to the same subject and predicate classes but differ in quality, quantity, or both.

Hence we conclude that the correct answer is one is denial or negation of the other 

When the subject and predicate of both the premises are the same but they differ in quality as well as quantity, This differing is called opposition. We shall establish various types of relations among the four classes of categorical propositions. This study is restricted to the categorical proposition.

Contradictories: This relation holds good for four pairs of propositions, which differ in quality and quantity.

The relation between A, O, E and I are always contradictory.

Accordingly, if it is true that ‘All rabbits are herbivorous’ then it is false that ‘some rabbits are not herbivorous and if it is false that ‘All rabbits are herbivorous’, then it is true that ‘some rabbits are not herbivorous’.

Hence, All the students cleared their examination” and “Few students did not clear their examination is an instance of Contradictories.

Contrary: This relation holds good ‘only’ between universal propositions, which differ in quality.                                                   

• For example, If “All bats are mammals” is a premise then its conclusion will be “No bats are mammals”.

Superaltern: When a particular conclusion is deduced from a universal proposition without affecting the quality, then superaltern relation holds good between the universal premise and particular conclusion. In this case, the quality of the proposition is irrelevant.

• For example, If “All metals are hard” is a premise then its conclusion will be “Some metals are hard”.

                        If “No fruits are bitter” is a premise then its conclusion will be “Some fruits are not bitter.”

Subaltern: When the premise and the conclusion in superaltern are reversed we obtain subalterns. In other words, when we deduce universal conclusions from a particular premise, the process results in subaltern.

• For example, If “Some planets are small” is a premise then its conclusion will be “All planets are small”

                       If “Some comets are not dense” is a premise then its conclusion will be “No comets are dense”

The standard form of categorical proposition having the same subject term and predicate term but may differ from each other in quality or quantity or both. Such differing has been called opposition. The term opposition is used when there is no apparent disagreement between the propositions.

Types of Opposition:

1. Contradictories: The standard form of a categorical proposition that has the same subject and predicate term but differs from each other in both quantity and quality. Two propositions are contradictories if one is denial or negation of the other if they can’t be true or can’t be both false.

Example:

1. All republics are grateful.
2. Some republics are not grateful.

2. Contraries: Two propositions are said to be contraries if they can’t both be true, and the truth of one entails the truth of the other. i.e. both can’t be true and both can’t be false. If either of these propositions is true, then the other must be false.

Example:

1. All artists are dreamers.
2. No artist is a dreamer

3. Sub-contraries: If a particular proposition having the same subject and predicate terms but differing in quality, one affirming the other denying. Two propositions are said to be sub-contraries if they can’t both be false although both may be true.

Example:

1. Some cars are vehicles.
2. Some cars are not vehicles.

4. Subalternation: It is the opposition between a universal proposition and its corresponding particular proposition. In the corresponding proposition, the universal proposition is called the superaltern and the particular proposition is called subaltern. These propositions have the same subject and predicate terms and agree on quality. Both are affirming or both denying but differ in quantity. On universal and the other particular.

Example:

1. No hens are birds
2. Some hens are not birds

5. The square of opposition: There are four ways in which propositions may be opposed as contradictories, contraries, sub-contraries, subalterns, and superalterns. These are represented using a diagram called the square of opposition. 

Therefore, according to the relations given in the image, If two propositions are so related that both can be true simultaneously but cannot be false simultaneously are called - Subcontraries.

Additional Information:

In term logic (a branch of philosophical logic), the square of opposition is a diagram representing the relations between the four basic categorical propositions. The doctrine of the square of opposition originated with Aristotle in the fourth century BC and has occurred in logic texts ever since.

Key Points

So, the correct match is A-III, B ‐ I, C ‐ II, and D ‐ IV.

Important Points

• A categorical proposition is a simple proposition containing two terms, subject (S) and predicate (P), in which the predicate is either asserted or denied of the subject.
• Every categorical proposition can be reduced to one of four logical forms, named A, E, I, and O.
• The 'A' proposition, the universal affirmative usually translated as 'every S is a P'.
• In the proposition, the universal negative is usually 'no S is P'.
• The 'I' proposition, the particular affirmative usually translated as 'some S are P'.
• In the 'O' proposition, the particular negative is usually translated as 'some S are not P'.

The propositions are placed in the four corners of a square, and the relations are represented as lines drawn between them, hence the name 'The Square of Opposition'. Therefore, the following cases can be made:

• If A is true, then E is false, I is true, O is false;
• If E is true, then A is false, I is false, O is true;
• If I is true, then E is false, A and O are undeterminate;
• If O is true, then A is false, E and I are undeterminate;
• If A is false, then O is true, E and I are undeterminate;
• If E is false, then I is true, A and O are undeterminate;
• If I is false, then A is false, E is true, O is true;
• If O is false, then A is true, E is false, and I is true.

The square of opposition is a chart that was introduced within classical (categorical) logic to represent the logical relationships holding between certain propositions in virtue of their form.

Key Points The four corners of this chart represent the four basic forms of propositions recognized in classical logic:

• A propositions, or universal affirmatives take the form: All S are P.
• E propositions, or universal negations take the form: No S are P.
• I propositions, or particular affirmatives take the form: Some S are P.
• O propositions, or particular negations take the form: Some S are not P.

Therefore the correct match is  A-II, B ‐ I, C ‐ IV, D ‐ III

Categorical proposition 

• To be in standard form a categorical syllogism meets the following strict qualifications:
• It is an argument with two premises and one conclusion.
• All three statements are categorical propositions.
• It contains exactly three different terms.
• Each term is used exactly twice.
• The following notes apply to standard form categorical syllogisms:
• Major term (P) = Predicate of conclusion
• Minor term (S) = Subject of conclusion
• Middle term (M) = Term that occurs in both premises.​

The figure depends on the arrangement of the middle terms in the proposition

For example,

Thus, the middle term decides the figure of a categorical proposition.

Copula:

• The part of a proposition, be it one word or more, which connects or couples the subject and the predicate.
• The copula is often defined as that which expresses the relation between the subject term and the predicate term of a proposition.
• But this is not sufficiently accurate for the purposes of exact logic.
• Passing over the objection that it applies only to categorical propositions as if conditional and copulative propositions had no copula.
• Contrary to logical tradition, it may be admitted that a copula often does fulfill the function mentioned; but it is only an accidental one, and its essential function is quite different. 

Categorical proposition :

• In logic, the categorical proposition is also known as a categorical statement that asserts or denies that all or some of the members of one category are included in another.
• It is an important branch of deductive reasoning.
• The categorical proposition can be classified into four under two heads, based on their quality and quantity, and distribution of terms.
  • Affirmative proposition (affirmo): A type and I type
  • Negative proposition (nego): E type and O type

​

Quality of categorical proposition:

• It is the number of members of the subject class that are used in the proposition. 
• It could be the universal or the particular categorical proposition.
• It will consider universal if all members of the subject class refer to it.

Quantity of categorical proposition:

• It is described as whether the proposition affirms or denies the inclusion of a subject within the class of the predicate. 
• It could be affirmative or negative.

The correct answer is B and D only.Key Points

Statements B and D are so related that they both cannot be false, although they may both be true.

• B. "Some animals are birds." There are indeed some animals that are birds.
• D. "Some animals are not birds." There are also animals that are not birds.
• Both these statements reflect the reality of the animal kingdom, where some animals are birds and some animals are not birds.
• They both cannot be false at the same time in the real world because even if one animal is a bird and one animal is not a bird, both statements become true.
• However, they can both be true simultaneously.

Additional Information

• On the other hand, statements A and C are direct contradictions of each other.
• If all animals are birds (A), then it cannot be true that no animals are birds (C), and vice versa.

Therefore, these cannot both be true or both be false at the same time.