Propositional Logic

 

PRPOSITIONAL LOGIC

            Propositional Logic is a procedure to provide reasoning through statements.

 

a.       Simple Proposition: it is one that is not the part of any other proposition. An atomic sentence is known as simple proposition. Propositional variable is a letter or symbol that assigns a simple proposition.

 

b.      Compound Proposition: If two or more than two simple sentences are joined together known as compound proposition.

 

Connectives: Connectives are known as the symbols through which multiple sentences can be combined together.

 

CONNECTIVE NAMES	Symbols	Definitions	Examples
Conjunction	Ù  or .	It is used to combine two simple propositions. It results in TRUE if both the statements are true else FALSE.	A=Ram is thin. B=He can climb the tree. Ram is thin and he can climb the tree. (A.B) (A  B)
Disjunction	+ or v	It is used to combine two simple propositions. It results in TRUE if either or both the statements are true else FALSE.	A=Ram is thin. B=He can climb the tree. Either Ram is thin or he can climb the tree. (A+B) (AvB)
Negation	~ or / or -	It is negative proposition of assertive sentence	A=Ram is thin. Ram is not thin. (~A) (A/)
Double Negation	~(~) or (/)/	It means complement of complement or double complement which results in the same proposition.	A=Ram is thin. Ram is thin. ~(~A) (A/)/
Implication or Conditional	àor =>	It is applicable to a compound statement by using if…then connective.  In this the condition may be termed as implies.	A=Ram is thin. B=He can climb the tree. If Ram is thin then he can climb the tree. (AàB) (A=>B)
Bi equivalence or bi conditional	ßà or ó	If both the statements are true or false then it is bi conditional else not.	A=Ram is thin. B=He can climb the tree. If and only if Ram is thin then he can climb the tree. (AßàB) (A<=>B)

 

Well framed formulae (wff): Proposition is another name of sentence or statement that may result in true or false. A proposition whether it is simple or compound is also referred as well framed formulae.

 

Truth Table:  It is the outcome of proposition whether it is true or false.

 

Truth tables of connectives:

 

CONJUNCTION:

 

A	B	A.B
0	0	0
0	1	0
1	0	0
1	1	1

 

DISJUNCTION:

 

A	B	A+B
0	0	0
0	1	1
1	0	1
1	1	1

 

NEGATION:

 

A	A/
0	1
1	0

 

DOUBLE NEGATION:

 

A	A/	(A/)/
0	1	0
1	0	1

 

IMPLICATION / CONDITIONAL:

 

A	B	AàB
0	0	1
0	1	1
1	0	0
1	1	1

BIEQUIVALENCE:

 

A	B	AóB
0	0	1
0	1	0
1	0	0
1	1	1

 

Antecedent and Consequent:

A=It is cold today.

B=You cannot go out.

The first part of sentence is called antecedent and second part of sentence is called consequent.

 

Converse: The conditional statement obtained after having interchange of antecedent and consequent is called converse statement.

Eg:

A=If 16 is an even number.

B=2 is its factor.

If 16 is an even number then 2 is its factor.

Converse is: If 2 is the factor then 16 is an even number.

Represented by: (BèA)

 

Inverse: The conditional statement obtained after negating the antecedent and consequent, then the proposition obtained is called inverse statement.

Eg:

A=You are tall.

B=You can climb the tree.

Inverse is: If you are not tall then you cannot climb the tree.

Represented by: (A^/èB^/)

 

Contrapositive: If consequent of a given proposition is negative antecedent and antecedent is the negative consequent, then the proposition obtained is called contrapositive statement.

Eg:

A=If 16 is an even number.

B=2 is its factor.

If 16 is an even number then 2 is its factor.

Converse is: If 16 is not an even number then 2 is not its factor.

Represented by: (A^/èB^/)

 

Tautology: If the output of any proposition results in  1 only then it is called tautology.

A	1	A+1
0	1	1
1	1	1

 

Contradiction: If the output of any proposition results in 0 only then it is called contradiction.

A	0	A.0
0	0	0
1	0	0

 

Contingency: If the output of any proposition results in combination of 0 and 1 then it is called contingency.

A	B	A+B	A(A+B)
0	0	0	0
0	1	1	0
1	0	1	1
1	1	1	1

 

Syllogism: In logical algebra, we need to draw conclusions from logical statements. The process of drawing conclusions are known as syllogism.