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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

• Quantifier
• Subject
• Copula
• Predicate

 

            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.

 

1. Universal quantifier: The words ‘All and No’ are called universal quantifiers because they refer to every object in a certain set.

 

2. Particular quantifier: The word ‘Some’ is known as particular quantifier because it refers to at least one existing object in a certain set.

 

            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 non-standard form.

 

            Quality: The kind of affirmation made by the proposition

1. Affirmative: asserts a quality to a class
2. Negative: denies a quality to a class

When we put the quality and quantity of categorical proposition together we see that there are four and only four possible arrangements:

• Universal Affirmative
• Particular Affirmative
• Universal Negative
• Particular Negative

 

            Distribution: The quantity associated with either the subject or predicate term of a categorical proposition.

 

            Four-Fold 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

 

1. Universal Affirmative Proposition (denoted by A): It distributes only the subject i.e, the predicate is not interchangeable with the subject while maintaining the validity of the proposition.

            Example 1: All frogs are amphibians.

            This is proposition A since we cannot say ‘All amphibians are frogs’.

 

2. Universal Negative Proposition (denoted by E): It distributes both the subject and the predicate i.e. an entire class of predicate term is denied to the entire class of the subject term.

            Example 1: No book is pen.

 

3. Particular Affirmative Proposition (denoted by I): It distributes neither the subject nor the predicate.

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.

 

4. Particular Negative Proposition (denoted by O): It distributes only the predicate, e.g. Some cats are not wild. Here, the subject term ‘cats’ is used only for a part of its class and hence is undistributed while the predicate term ‘wild’ is denied in entirety to the subject term and hence is distributed.

 

            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.

 

1. Immediate Deductive Inference

In this method, conclusion is deduced from one of the given propositions, by any of the three ways -conversion, obvers ion and contraposition.

 

1. Conversion: The Conversion proceeds with interchanging the subject term and the predicate term i.e. the subject   term of the premise becomes the predicate term of the conclusion and the predicate term of the premise becomes the subject of the conclusion.

            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.

 

2. Obvers ion: In obvers ion, quality of the proposition is changed and the predicate term is replaced by its complement.

 

Table of Valid Obvers ions

 

           

	Obverted		Obverse
(E)	A: All dogs are mammals.	(E)	E: No dogs are non-mammals.
(A)	E: No writers are singers.	(A)	A: All writers are non-singers.
(O)	I: Some mangoes are fruits.	(O)	O: Some mangoes are not non-fruits.
(I)	O: Some politicians are not statesmen.	(I)	I: Some politicians are non-statesmen.

 

 

3. Contraposition: To obtain the contrapositive of a statement, we first replace the subject and predicate terms in the proposition and then exchange both these terms with their complements.

 

Table of Valid Contrapositions

 

           

	Proposition		Contrapositive
(A)	A: All snakes are reptiles	(A)	A: All non-reptiles are non-snakes.
(I)	I: Some snakes are reptiles.	(I)	I: Some non-reptiles are non-snakes.

 

          

Note: The valid converse, obverse or contrapositive of a given proposition always logically follows from the             proposition.

 

1. Mediate Deductive Inference (SYLLOGISM):

A syllogism is a deductive argument in which conclusion has to be drawn from two propositions referred to as the premises.

            Example:

1. All roses are flowers.
2. All flowers are beautiful.
3. All roses are beautiful.

 

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.

 

1. Major Term: Predicate of the conclusion is called major term of the proposition and is denoted by P (first letter of ‘Predicate’).

 

2. Minor Term: Subject of the conclusion is called minor term of the proposition and is denoted by S (first letter of ‘Subject’).

 

3. Middle Term: It is the term common to both the premises and is denoted by M (first letter of ‘Middle’).

 

            Example:

            Premises:

1. All cats are animals.
2. All lions are cats.

 

            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:

1. The conclusion does not contain the middle term.

            Example:

            Statements:     1. All women are boys.

2. Some boys are students.

            Conclusions:    1. All boys are women.

2. Some boys are not students.

            Since both the conclusions 1 and 2 contain the middle term ‘boys’, so neither of them can follow.

 

2. No term can be distributed in the conclusion unless it is distributed in the premises.

            Example:

            Statements:     1. Some lions are cats.

2. All cats are animals.

            Conclusions:    1. Some lions are animals.

2. All animals are cats.

            Statement 1 is an I-type 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 A-type 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.

 

3. The middle term (M) should be distributed at least once in the premises. Otherwise, the conclusion 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.

2. Some books are black.

            Conclusions:   1. All books are bulbs.

2. Some bulbs are black.

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.

 

4. No conclusion follows

            (a) if both the premises are particular

            Example:

            Statements:     1. Some papers are pens.

2. Some pens are pencils.

            Conclusions:   1. All papers are pencils.

2. Some pencils are papers.

            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.

2. No orange is banana.

            Conclusions:    1. No apple is banana.

2. Some bananas are oranges.

                                    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.

2. No books are pens.

            Conclusions:   1. No pens are books.

2. Some pencils 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.

 

5. If the middle term is distributed twice, the conclusion cannot be universal.

            Example:

            Statements:     1. All horses are cows.

2. No buffaloes are horses.

            Conclusions:   1. No buffaloes are cows.

2. Some 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.

 

6. If one premise is negative, the conclusion must be negative.

            Example:                                                                       

            Statements:     1. All Books are papers.                                          

2. No paper is pen.

            Conclusions:   1. No Books are pens.                                            

2. Some pens are Books.

            Since one premise is negative, the conclusion must be negative. So, conclusion 2 cannot follow.

 

7. If one premise is particular, the conclusion must be particular.

            Example:

            Statements:     1. Some dogs are cows.

2. All cows are cats.

            Conclusions:   1. Some dogs are cats

2. All cats are dogs.

            Since one premise is particular, the conclusion must be particular. So, conclusion 2 cannot follow.

 

8. If both the premises are affirmative, the conclusion must be affirmative.

            Example:

            Statements:     1. All Boys are students.

2. All students are brothers.

            Conclusions:   1. All Boys are brothers.

2. Some Boys are not brothers.

            Since both the premises are affirmative, the conclusion must be affirmative. So, conclusion 2 cannot follow.

 

9. If both the premises are universal, the conclusion must be universal.

            Example:

            Statements:     1. All girls are students.

2. All students are singers.

            Conclusions:    1. All girls are singers.

2. Some singers are not students.

            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 A-type 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 I-type 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 A-type proposition, distributes the subject only while the second premise, being an I-type 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 A-type proposition, distributes subject only and the second premise, being an I-type 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 E-type 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 E-type proposition, distributes both the subject and the             predicate. Second premise, being an A-type 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   E-type 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. O-type, 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 E-type propositions form a complementary pair.

 

Important Points To Remember

 

While deriving logical conclusions, always remember that the following conclusions hold:

1. The converse of each of the given premises;
2. The conclusion that directly follows from the given premises in accordance with the rules of syllogism;
3. The converse of the derived conclusions.

 

 

Snap Test

 

 

1. Statements: All rivers are lakes. Some lakes are seas.

            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.

 

2. Statements: No boy student can play. Some boys students are athletes.

            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.

 

3. Statements: All mangoes are fruits. No fruit is a flower

            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.              

 

4. Statements: All bananas are mangoes. All oranges are mangoes.

            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,

 

5. Statements: Some dogs are cats. Some cats are lions.

            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.

 

6. Statements: No historian is geologist. No geologist is chemist.

            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.

 

7. Statements: All tigers are mammals. No mammal is crocodile.

            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.

 

8. Statements: All girls are students. All students are teachers,

            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.

 

9. Statements: Some apples are mangoes. All mangoes are green.

            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.

 

10. Statements: All fish are aquatic animals. No aquatic animal is a tiger.

            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.