Understanding Logic > Syllogism: Theory, Rules, Tricks and Examples

Deductive Reasoning: Syllogism (Logic) Basics, Problems, 
Practice Questions, Answers and Explanations

LOGIC: 'Logic' is derived from the Greek word 'logos' meaning 'thought' or 'the word expressing thought'.

SYLLOGISM: A typical syllogism (Greek word, means 'inference' or 'deduction') is an argument that contains three parts: a major premise, a minor premise, and a conclusion. Our task is to determine the validity of the statement(s) on the basis of logic applied.

Format of question asked in competitive exams:

Directions: In these questions, two statements are being provided followed by two conclusions A and B. You have to study the two statements and then decide whether, from these two statements,

a. Only A follows
b. Only B follows
c. Both A and B follows
d. Either A or B follows
e. Neither A nor B follows.

1. Statement:

All balls are bats.
All bats are table.

Conclusion:

A. All balls are tables.
B. Some tables are balls.

2. Statement:

All balls are bats.
All bats are table.

Conclusion:

A. Some balls are tables.
B. Some tables are balls.

Note: Three and Four statements question is also asked. We'll discuss it next in this post.

:: THE THEORITICAL PART ::

Here we'll discuss the theoritical part of LOGIC and SYLLOGISM: INDEX

• Important Terms and Definitions
• Proposition
• Quantifier
• Subject
• Capula
• Predicate
• Four-Fold Classification
• Universal 
• Affirmative
• Negative
• Perticular / Individual
• Affirmative
• Negative
• Logical Deduction
• Definition
• Immediate Deductive Inference
• Conversion
• Obversion
• Contraposition
• Mediate Deductive Inference (SYLLOGISM)
• 'Terms' and premises
• Major Term (Predicate)
• Minor Term (Subject)
• Middle Term (COMMON)
• Rules for Deriving Conclusion
• Arguments / Statement Based Problems
• Two-Premise / Two-Statement Arguments
• Three-Premise / Three-Statement Arguments

Here we go in detail:

• Important Terms and Definitions
• A statement or argument (categorical) is called 'Proposition'.
• A proposition has this standard format: Quantifier + Subject + Copula + Predicate.
• Quantifier: Refers to object which specifies quantity.
• Universal Quanitifier: 'All' and 'No'
• Particular Quantifier: 'Some'
• Subject: About which something is said. It is denoted by 'S'
• Copula: Part which denotes relation between object and subject.
• Predicate: Part which is affirmed or denied about the subject.

• Four-Fold Classification: This classification is based on universal and particular quantities of porposition. The universal and individual affirmative quantifiers are said to be of types A and I respectively, from Latin AffIrmo, the universal and individual negative quantifiers of type E and O, from Latin NEgO. Aristotle’s theory was extended by logicians in the Middle Ages whose working language was Latin, whence this Latin mnemonics.

• With Universal Quanitifier: 
• 'All' - Universal Affirmative Proposition (Denoted by 'A')
• Ex. All mangoes are fruits.
• 'No' - Universal Negative Proposition (Denoted by 'E')
• Ex. No girl is dumb.
• With Particular / Individual Quantifier: 
• 'Some' - Particular Affirmative Proposition (Denoted by 'I')
• Ex. Some girls are extravagant.
• 'Some Not' - Particular Negative Proposition (Denoted by 'O')
• Some women are not thrifty.

We use above 'AEIO' rule in 'Analytical Method' of solving Syllogism questions.

☀ List of Quantifiers:

UNIVERSAL QUANTIFIERS (+ and -)	PARTICULAR / INDIVIDUAL QUANTIFIERS (+ and -)
All, Every, Any, None, Not a single, Only etc.	Some, Many, A few, Quite a few, Not many, Very little, Most of, Almost, Generally, Often, Frequently, etc.

☀ Venn Diagram for A, E, I, O Statements -

• LOGICAL DEDUCTION: Deductive reasoning, also deductive logic, logical deduction or, informally, "top-down" logic, is the process of reasoning from one or more statements (premises) to reach a logically certain conclusion. 

• 1. Immediate Deductive Inference: In this process conclusion can be deduced from any of the following three ways:

• Conversion:In conversion, the subject term and the predicate term are interchanged. In this given preposition is called convertend and, the drawn conclusion is called converse.
• Convertend → Subject Term ⇄ Predicate Term → Converse
• Here is table of Valid Conversion -

• Obversion: In obversion, the quality of proposition is changed and predicate term is replaced by its complement.
• Here is table of Valid Obversion

• Contraposition: In contraposition, subject and predicate terms of proposition replaced and then both subject and predicate exchanged with their complements.
• Here is table of Valid Contrapositions -

• Mediate Deductive Inference (SYLLOGISM): First introduced by Aristotle, a Syllogism is a deductive argument in which conclusion has to be drawn from two propositions referred to as the premises.

• Term: term is a word or a combination of words, which can be used as a subject or predicate of a proposition.
• A valid categorical syllogism only has three terms: 
• Major Term, 
• Minor Term, and
• Middle Term

NOW IT COMES THE EXAM THING

:: RULES FOR DESRIVING CONCLUSION FROM GIVEN PREMISES ::

☀ Two-Statement Premises:

There are three methods to solve two-statement Syllogism questions:

• 1. Venn Diagram
• 2. AEIO (Analytical Method)
• 3. Distribution of Terms (Tick Method)

1. Venn Diagram: A Venn diagram (also known as a set diagram or logic diagram) is a diagram that shows all possible logical relations between a finite collection of different sets. Venn diagrams were conceived around 1880 by John Venn.

• Concept 1: The Diagram for All S is P -

• The Possible Conclusions are,
• All S is P.
• Some P is S.
• Some S is P

• Concept 2: The Diagram for All A is B and All B is C

• The Possible Conclusions are,
  • Between A and B,                                                             
  • All A is B.                                                             
  • Some A is B.                                                    .
  • Some B is A.                                                   
  • Between B and C,
  • All B is C.
  • Some B is C.
  • Some C is B. 
  • Between A and C,
  • All A is C.
  • Some A is C
  • Some C is A

• Concept 3: The Diagram for Some A is B

• The Possible Conclusions are,
• Some A is B.
• Some B is A.

• Concept 4: The Diagram for Some A is B and Some B is C

• The Possible Conclusions are,
• Between A and B,
• Some A is B.
• Some B is A.
• Between B and C,
• Some B is C.
• Some C is B.
• Between A and C,
• No Conclusion Possible.

• Concept 5: The Diagram for Some A is B; All B is C

• The Possible Conclusions are,
• Between A and B,
• Some A is B.
• Some B is A.
• Between B and C,
• All B is C.
• Some B is C.
• Some C is B.
• Between A and C,
• Some A is C.
• Some C is A.

• Concept 6 – The Diagram for All A is B and Some B is C

• The possible conclusions are,
• Between A and B,
• All A is B.
• Some A is B.
• Some B is A.
• Between B and C,
• Some B is C.
• Some C is B.
• Between A and C,
• No Conclusion Possible.

• Concept 7 – The Diagram for All B is A and All C is A

• The Possible Conclusions are,
• Between A and B,
• All B is A.                                              
• Some B is A.
• Some A is B.
• Between A and C,
• All C is A.
• Some C is A.
• Some A is C.
• Between A and C,
• No Conclusion Possible.

• Concept 8 – The Diagram for No A is B

• The Possible Conclusions are,
• No A is B.
• No B is A.
• Some A is not B.
• Some B is not A.

• Concept 9 – The Diagram for All A is B and No B is C

• The Possible Conclusions are,
• Between A and B
• All A is B.
• Some A is B.
• Some B is A.
• Between B and C
• No B is C.
• No C is B.
• Some B is not C.
• Some C is not B.
• Between A and C
• No A is C.
• Some A is Not C.

• Concept 10 – The Diagram for All A is B and No A is C

• The Possible Conclusions are,
• Between A and B,
• All A is B.
• Some A is B.
• Some B is A.
• Between B and C,
• Some B is not C.
• Between A and C,
• No A is C.
• No C is A.
• Some A is not C.
• Some C is not A.

• Concept 11 – The Diagram for Some A is B; No B is C

• The Possible Conclusions are,
• Between A and B,
• Some A is B.
• Some B is A.
• Between B and C,
• No B is C.
• No C is B.
• Some B is not C.
• Some C is not B.
• Between A and C,
• Some A is not C.

• Concept 12 – The Diagram for Some A is B; No A is C

• The Possible Conclusions are,
• Between A and B,
• Some A is B.
• Some B is A.
• Between B and C,
• Some B is not C.
• Between A and C,
• No A is C.
• No C is A.
• Some A is not C.
• Some C is not A.

2. Analytical Method: Content represented has been made specific for competitive exams; hence, may differ from original terminology discussed above in this post. i.e. 

I. Universal Affairmative = All (A)
II. Universal Negative = No (E)
III. Particular Affirmative = Some (I)
IV. Particular Negative = Some Not (O)

Rule	Statement 1	Statement 2	Conclusion
Rule of All+	All	All	All
	All	No	No
	All	Some	No conclusion
Rule of No+	No	All	Some not
	No	No	Some not
	No	Some	No conclusion
Rule of Some+	Some	All	Some
	Some	No	Some not
	Some	Some	No conclusion
Rule of Some Not+	Some not	All	No conclusion
	Some not	No	No conclusion
	Some not	Some	No conclusion
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Inorder to solve two premises apply above rules.