Logic
Category : 10th Class
Logic
Logic
The term “logic” came from the Greek word logos, which is sometimes translated as “sentence” “discourse”, “reason”, “rule”, and “ratio”
Briefly speaking, we might define logic as the study of the principles of correct reasoning or the study of the principles of reasoning, especially of the structure of propositions as distinguished from their content and of method and validity in deductive reasoning.
Proposition: In logic, any categorical statement is termed as the proposition.
The standard form of proposition is consists of four parts
Example: 1
All \[\to \] books \[\to \] are \[\to \] pens
\[\downarrow \] \[\downarrow \] \[\downarrow \] \[\downarrow \]
Quantifier \[\to \] subject \[\to \] copula \[\to \] predicate
Example: 2
Some \[\to \] dog \[\to \] are \[\to \] cats
\[\downarrow \] \[\downarrow \] \[\downarrow \] \[\downarrow \]
Quantifier \[\to \] subject \[\to \] copula \[\to \] predicate
Quantifier: The words which specify the quantity like ‘all’, ‘no’ and some.
Subject (denoted by ‘S’): The subject is that about which something is said.
Predicate (denoted by ‘P’): It is the part of the proposition denoting that which is affirmed or denied about the subject.
Copula: It is the part of the proposition which denotes the relation between the subject and the predicate.
Note: We should keep in mind that the quantifiers (‘all’, ‘no’, ‘some’) are implied in all categorical propositions even though they are not explicitly stated. If these quantifiers are not explicitly used in the formulation of the propositions it is aid to be in nonstandard form.
Quality: The kind of affirmation made by the proposition
When we put the quality and quantity of categorical proposition together we see that there are four and only four possible arrangements:
Distribution: The quantity associated with either the subject or predicate term of a categorical proposition.
FourFold Classification of Propositions
A proposition, which begins with a universal quantifier, is said to have universal quantity. And a proposition, which begins with a particular quantifier, is said to have particular quantity. Besides, a proposition which asserts a quality to a class is called affirmative quality and a proposition which denies a quality to a class is called negative quality. Also, a term is distributed in a proposition if it refers to ail members of the set of objects denoted by that term. Otherwise, it is said to be undistributed.
Based on the above facts, propositions can be classified into four types
Example 1: All frogs are amphibians.
This is proposition A since we cannot say ‘All amphibians are frogs’.
Example 1: No book is pen.
Example: Some rats are snakes. Here, the subject term ‘rats’ is used not for all but only for some rats and similarly the predicate term ‘snakes’ is affirmed for a part of subject class. So, both are undistributed.
These facts can be summarized as follows:

Statement Form 
Quantity 
Quality 
Distributed 
(A) 
All S is P. 
Universal 
Affirmative 
S only 
(E) 
No S is P. 
Universal 
Negative 
Both S and P 
(I) 
Some S is P. 
Particular 
Affirmative 
Neither S nor P 
(O) 
Some S is not P 
Particular 
Negative 
P only 
Logical Deduction
The phenomenon of deriving a conclusion from a single proposition or a set of given propositions, is known as logical deduction. The given propositions are also referred to as the premises.
In this method, conclusion is deduced from one of the given propositions, by any of the three ways conversion, obvers ion and contraposition.
The given proposition is called converted, whereas the conclusion drawn from it is called its converse.
Table of Valid Conversions

Converted 
Converse 
(A) 
A: All A is B Example. All cats are dogs 
I: Some B is A Some dogs are cats. 
(E) 
E: No A is B. Example. No rat is fish. 
E: No B is A No fish is rat. 
(I) 
I: Some A is B Example. Some birds are cats. 
I: Some B is A. Some cats are birds 
(O) 
O: Some A is not B. 
No valid conversion 
Note: that in a conversion, the quality remains the same and the quantity may change.
Table of Valid Obvers ions

Obverted 

Obverse 
(E) 
A: All dogs are mammals. 
(E) 
E: No dogs are nonmammals.

(A) 
E: No writers are singers. 
(A) 
A: All writers are nonsingers. 
(O) 
I: Some mangoes are fruits. 
(O) 
O: Some mangoes are not nonfruits.

(I) 
O: Some politicians are not statesmen. 
(I) 
I: Some politicians are nonstatesmen. 
Table of Valid Contrapositions

Proposition 

Contrapositive 
(A) 
A: All snakes are reptiles 
(A) 
A: All nonreptiles are nonsnakes. 
(I) 
I: Some snakes are reptiles. 
(I) 
I: Some nonreptiles are nonsnakes. 
Note: The valid converse, obverse or contrapositive of a given proposition always logically follows from the proposition.
A syllogism is a deductive argument in which conclusion has to be drawn from two propositions referred to as the premises.
Example:
Clearly, the propositions 1 and 2 are the premises and the proposition 3, which follows from the first two propositions, is called the conclusion.
Term: En Logic, a term is a word or a combination of words, which by itself can be used as a subject or predicate of a proposition.
Example:
Premises:
Conclusion:
All lions are animals.
Here ‘animals’ is the predicate of the conclusion and so, it is the major term P.
‘lions’ is the subject of the conclusion and so, it is the minor term, S.
‘cats’ is the term common to both the premises and so, it is the middle term, M.
Major And Minor Premises: Of the two premises, the major premise is that in which the middle term is the subject and the minor premise is that in which the middle term is the predicate.
Rules for deriving conclusion from two given premises:
Example:
Statements: 1. All women are boys.
Conclusions: 1. All boys are women.
Since both the conclusions 1 and 2 contain the middle term ‘boys’, so neither of them can follow.
Example:
Statements: 1. Some lions are cats.
Conclusions: 1. Some lions are animals.
Statement 1 is an Itype proposition which distributes neither the subject nor the predicate.
Statement 2 is an A type proposition which distributes the subject i.e. ‘cats’ only.
Conclusion 2 is an Atype proposition which distributes the subject ‘animals’ only
Since the term ‘animals’ is distributed in conclusion 2 without being distributed in the premises, so conclusion 2 cannot follow.
For the middle term to be distributed in a premise.
(i) M must be the subject if premise is an A proposition.
(ii) M must be subject or predicate if premise is an E proposition.
(iii) M must be predicate if premise is an O proposition.
Note: that in an I proposition, which distributes neither the subject nor the predicate, the middle term cannot be distributed.
Example:
Statements: 1. All bulbs are books.
Conclusions: 1. All books are bulbs.
In the premises, the middle term is ‘books’. Clearly, it is not distributed in the first premise which is an A proposition as it does not form its subject. Also, it is not distributed in the second premise which is an I proposition. Since the middle term is not distributed even once in the premises, so no conclusion follows.
(a) if both the premises are particular
Example:
Statements: 1. Some papers are pens.
Conclusions: 1. All papers are pencils.
Since both the premises are particular, so no definite conclusion follows.
(b) if both the premises are negative.
Example:
Statements: 1. No apple is orange.
Conclusions: 1. No apple is banana.
Since both the premises are negative, neither conclusion follows.
(c) If the major premise is particular and the minor premise is negative
Example:
Statements: 1. Some pens are pencils.
Conclusions: 1. No pens are books.
Here, the first premise containing the middle term ‘pens’ as the subject is the major premise and the second premise containing the middle term ‘pens’ as the predicate is the minor premise. Since the major premise is particular and the minor premise is negative, so no conclusion follows.
Example:
Statements: 1. All horses are cows.
Conclusions: 1. No buffaloes are cows.
Here, the first premise is an A proposition and so, the middle term ‘horses’ forming the subject is distributed. The second premise is an E proposition and so, the middle term’ horses’ forming the predicate is distributed. Since the middle term is distributed twice, so the conclusion cannot be universal.
Example:
Statements: 1. All Books are papers.
Conclusions: 1. No Books are pens.
Since one premise is negative, the conclusion must be negative. So, conclusion 2 cannot follow.
Example:
Statements: 1. Some dogs are cows.
Conclusions: 1. Some dogs are cats
Since one premise is particular, the conclusion must be particular. So, conclusion 2 cannot follow.
Example:
Statements: 1. All Boys are students.
Conclusions: 1. All Boys are brothers.
Since both the premises are affirmative, the conclusion must be affirmative. So, conclusion 2 cannot follow.
Example:
Statements: 1. All girls are students.
Conclusions: 1. All girls are singers.
Since both the premises are universal, the conclusion must be universal. So, conclusion 2 cannot follow.
Complementary Pair
A pair of statements such that if one is true, the other is false and when no definite conclusion can be drawn, either of them is bound to follow, is called a complementary pair. E and I type propositions together form a complementary pair and usually either of them follows, in a case where we cannot arrive at a definite conclusion, using the rules of syllogism.
Now we will study the various possible cases and draw all possible inferences in each case, along with verification through Venn diagrams.
Case 1: All mangoes are grapes. All grapes are fruits.
Immediate Deductive Inferences: The converse of first premise i.e. ‘Some grapes are mangoes’ and the converse of second premise i.e. ‘Some fruits are grapes’ both hold.
Mediate Deductive Inferences: Since both the premises are universal and affirmative, the conclusion must be universal affirmative. Also, the conclusion should not contain the middle term. So, it follows that ‘All mangoes are fruits’. The converse of this conclusion i.e. ‘Some fruits are mangoes’ also holds.
Venn Diagram 
Inferences 

1. Some grapes are mangoes 2. Some fruits are grapes. 3. All mangoes are fruits. 
Case 2: All tigers are animals. All lions are animals.
Immediate Deductive Inferences: The converse of first premise i.e. ‘Some animals are tigers’ and the converse of second premise i.e. ‘Some animals are lions’ both hold.
Mediate Deductive Inferences: Both, being Atype propositions, distribute subject only. Thus, the middle term ‘animals’ is not distributed even once in the premises. So, no definite conclusion follows.
Venn diagram 
Inferences 

1. Some animals are tigers. 2. Some animals are lions. 3. Either ?No tiger tiger is lion? or ?Some tigers are lions? as E and Itype propositions form a complementary pair 
Case 3: All monkeys are apes. Some apes are mammals.
Immediate Deductive Inferences: The converse of the first premise i.e. ‘Some apes are monkeys’ and the converse of the second premise i.e. ‘Some mammals are apes’, both hold.
Mediate Deductive Inferences: First premise, being an Atype proposition, distributes the subject only while the second premise, being an Itype proposition, distributes neither subject nor predicate. Since the middle term ‘apes’ is not distributed even once in the premises, so no definite conclusion can be drawn.
Venn diagram 
Inferences 

1. Some apes are monkeys. 2. Some mammals are apes. 3. Either ?No monkey is mammal? or propositions form a complementary pair. 
Case 4: Some mathematicians are physicists. All physicists are biologists.
Immediate Deductive Inferences: The converse of the first premise i.e. ‘Some physicists are mathematicians’ and the converse of the second premise i.e. ‘Some biologists are physicists both hold.
Mediate Deductive Inferences: Since one premise is particular, the conclusion must be par ticular and should not contain the middle term. So, it follows that ‘Some mathematicians are biologists’. The converse of this conclusion i.e. ‘Some biologists are mathematicians’ also holds.
Venn diagram 
Inferences 

1. Some physicists are mathematics. 2. Some biologists are physicists. 3. Some mathematics are biologists. 4. Some biologists are mathematicians. 
Case 5: All computers are machines. Some computers are mobiles.
Immediate Deductive Inference: The converse of the first premise i.e. ‘Some machines are computers’ and the converse of the second premise i.e. ‘Some mobiles are computer’, both hold.
Mediate Deductive Inferences: Since one premise is particular, the conclusion must be particular and should not contain the middle term. So, it follows that ‘Some machines are mobiles’. The converse of this conclusion i.e. ‘Some mobiles are machines’ also holds.
Venn diagram 
Inferences 

1. Some animals are lions. 2. Some animals are monkeys. 3. Either ?No lion is monkey?s or ?Some Lions are monkey?s follow. 
Case 6: All lions are animals. Some monkeys are animals.
Immediate Deductive Inferences: The converse of the first premise i.e. ‘Some animals are lions’ and the converse of the second premise i.e. ‘Some animals are monkeys’, both hold.
Mediate Deductive Inferences: First premise, being an Atype proposition, distributes subject only and the second premise, being an Itype proposition, distributes neither subject nor predicate. So, the middle term ‘animals’ is not distributed even once in the premises. Hence, no definite conclusion can be drawn.
Venn diagram 
Inferences 

1. Some animals are lions. 2. Some animals are monkeys. 3. Either ?No lion is monkey? or ?Some Lions are monkeys? 
Case 7: Some students are physicists. Some physicists are mathematicians.
Immediate Deductive Inferences: The converse of the first premise i.e. ‘Some physicists are students’ and the converse of the second premise i.e. ‘Some mathematicians are physicists’, both hold.
Mediate Deductive Inferences: Since both premises are particular, no definite conclusion follows.
Venn diagram 
Inferences 

1. Some physicists are students. 2. Some mathematicians are physicists. 3. Either ?Some students are mathematicians? or ?No student is mathematician? follow; as I and Etype propositions form a complementary pair. 
Case 8: All apples are fruits. No fruit is cake.
Immediate Deductive Inferences: The converse of the first premise i.e. ‘Some fruits are apples’ and the converse of the second premise i.e. ‘No cake is fruit’, both hold.
Mediate Deductive Inference: Since both premises are universal, the conclusion must be universal. Since one premise is negative, the conclusion must be negative. So, it follows that ‘No apple is cake’. The converse of this conclusion of this conclusion i.e. ‘No cake is apple’ also holds.
Venn diagram 
Inferences 

1. Some fruits are apples. 2. No cake is fruit. 3. No apples is cakes. 4. No cake is apple. 
Case 9: No dog is ape. All apes are mammals.
Immediate Deductive Inferences: The converse of the first premise i.e. ‘No ape is dog’ and the converse of the second premise i.e. Some mammals are apes’, both hold.
Mediate Deductive Inference: First premise, being an Etype proposition, distributes both the subject and the predicate. Second premise, being an Atype proposition, distributes the subject.
Thus, the middle term ‘ape’ is distributed twice in the premises. So, the conclusion cannot be universal. Also, since one premise is negative, the conclusion must be negative. Thus, the conclusion must be particular negative i.e. O type. So, it follows that ‘some mammals are not dogs’.
Venn diagram 
Inferences 

1. No ape is dog. 2. Some mammals are apes. 3. Some mammals are not dogs. 4. Either ?Some dogs are mammals? or ?No dog is mammal? follow; as I and Etype propositions form a complementary pair. 
Case 10: Some boys are students. No student is teacher.
Immediate Deductive Inferences: The converse of the first premise i.e. ‘Some students are boys’ and the converse of the second premise i.e. ‘No teacher is student’, both hold.
Mediate Deductive Inferences: Since one premise is particular and the other negative, the conclusion must be particular negative i.e. Otype, So, it follows that ‘Some boys are not teachers’.
Venn diagram 
Inferences 

1. Some students are boys. 2. No teacher is student. 3. Some boys are not teachers. 4. Either ?Some boys are teachers? or ?No boy is teacher? follow; as I and Etype propositions form a complementary pair. 
Important Points To Remember
While deriving logical conclusions, always remember that the following conclusions hold:
Snap Test
Conclusions:
I. Some seas are rivers. 
II. All lakes are seas. 
(a) Only conclusion I follows
(b) Only conclusion II follows
(c) Either I or II follows
(d) Neither I nor II follows
(e) Both I and II follows
Ans. (d)
Explanation: The first premise is A type and distributes the subject. So, the middle term ‘lakes’ which forms its predicate, is not distributed. The second premise is I type and does not distribute either subject or predicate. So, the middle term ‘lakes forming its subject is not distributed. Since the middle term is not distributed even once in the premises, no definite conclusion follows.
Conclusions:
I. Boy athletes can play. 
II. Some athletes can play. 
(a) Only conclusion I follows
(b) Only conclusion II follows
(c) Either I or II follows
(d) Neither I nor II follows
(e) Both I and II follows
Ans. (d)
Explanation: Since one premise is negative, the conclusion must be negative. So, neither conclusion follows.
Conclusions:
I. Some mangoes are flowers. 
II. Some flowers are fruits. 
(a) Only conclusion I follows
(b) Only conclusion II follows
(c) Either I or II follows
(d) Neither I nor II follows
(e) Both I and II follows
Ans. (d)
Explanation: Since both the premises are universal and one premise is negative, the conclusion must be universal negative. So, neither I nor II follows.
Conclusions:
I. Some oranges are bananas. 
II. No orange is banana. 
(a) Only conclusion I follows
(b) Only conclusion II follows
(c) Either I or II follows
(d) Neither I nor II follows
(e) Both I and II follows
Ans. (c)
Explanation: Since the middle term ‘mangoes’ is not distributed even once in the premises, no definite conclusion follows. However, I and II involve only the extreme terms and form a complementary pair. So, either I or II follows,
Conclusions:
I. Some dogs are lions 
II. Some lions are dogs. 
(a) Only conclusion I follows
(b) Only conclusion II follows
(c) Either I or II follows
(d) Neither I nor II follows
(e) Both I and II
Ans. (d)
Explanation: Since both the premises are particular, no definite conclusion follows.
Conclusions:
I. No historian is chemist. 
II. All chemists are historians. 
(a) Only conclusion I follows
(b) Only conclusion II follows
(c) Either I or II follows
(d) Neither I nor II follows
(e) Both I and II follows
Ans. (d)
Explanation: Since both the premises are negative, no definite conclusion follows.
Conclusions:
I. No crocodile is tiger. 
II. Some mammals are tigers. 
(a) Only conclusion 5 follows
(b) Only conclusion II follows
(c) Either I or II follows
(d) Neither I nor II follows
(e) Both I and II follow
Ans. (e)
Explanation: Since both the premises are universal and one premise is negative, the conclusion must be universal negative. So, it follows that ‘No tiger is crocodile’. I is the converse of this conclusion and thus it follows. II is the converse of the first premise and so it also holds.
Conclusions:
I. All girls are teachers 
II. Some teachers are students. 
(a) Only conclusion I follows
(b) Only conclusion II follows
(c) Either I or II follows
(d) Neither I nor II follows
(e) Both I and II follow
Ans. (e)
Explanation: Since both the premises are universal and affirmative, the conclusion must be universal affirmative and should not contain the middle term. So, I follows II is the converse of the second premise and thus it also holds.
Conclusions:
I. All apples are green. 
II. All mangoes are apples. 
(a) Only conclusion I follows
(b) Only conclusion II follows
(c) Either I or II follows
(d) Neither I nor II follows
(e) Both I and II follows
Ans. (d)
Explanation: Since one premise is particular, the conclusion must be particular. So, neither I nor II follows.
Conclusions:
I. No tiger is a fish. 
II. No fish is a tiger. 
(a) Only conclusion I follows
(b) Only conclusion II follows
(c) Either I or II follows
(d) Neither I nor II follows
(e) Both I and II follow
Ans. (e)
Explanation: Since both the premises are universal and one premise is negative, the conclusion must be universal negative. Also, the conclusion should not contain the middle term. So, II follows; I is the converse of II and thus it also holds.
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