Propositional logic

LOGIC

PREPOSITIONal

A statement is a declaratory sentence which is true or false but not both. In other words , a statement is a declarative sentence which has a definate truth table.

Preposition (statement)

Logical connectives or sentence connectives These are the words or symbols used to

combine two sentence to form a compound statement.

logic Name rank~ Negation 1

^ Conjunction 2

V Disjunction 3

=> Conditional 4

Biconditional 5

A B ^ V ~A => NOR

NAND XOR

EX-NOR

T T T T F T T F F F T

T F F T F F F F T T F

F T F T T T F F T T F

F F F F T T T T T F T

BASIC LOGICAL OPERATIONS

TAUTOLOGYi. A TAUTOLOGY IS A PREPOSITION

WHICH IS TRUE FOR ALL TRUTH VALUES OF ITS SUB-PREPOSITIONS OR COMPONENTS.

ii. A TAUTOLOGY IS ALSO CALLED LOGICALLY VALID OR LOGICALLY TRUE.

iii. ALL ENTRIES IN THE COLUMN OF TAUTOLOGY ARE TRUE.

For example:p^q=>q

P q p^q q p^q=>q

T T T T T

T F F F T

F T F T T

F F F F T

Contradiction CONTRADICTION IS A PREPOSITION WHICH IS

ALWAYS FALSE FOR ALL TRUTH VALUES OF ITS SUB-PREPOSITIONS OR COMPONENTS.

A CONTRADICTION IS ALSO CALLED LOGICALLY INVALID OR LOGICALLY FALSE

ALL ENTRIES IN THE COLUMN OF CONTRADICTION ARE FALSE.

FOR EXAMPLE(P v Q)^(~P)^(~Q)

P Q P V Q ~P ~Q (P v Q)^(~P)^(~Q)

T T T F F F

T F T F T F

F T T T F F

F F F T T F

Contingency

It is a preposition which is either true or false depending on the truth value of its

components or preposition..

FOR EXAMPLE~p ^ ~q

p q ~p ~q ~p ^ ~qT T F F FT F F T FF T T F FF F T T T

Logical equivalence Two statements are called logically

equivalent if the truth values of both the statements are always identical..

For example: If we take two statements p=>q and ~q

=>~p , then there truth table values must be equal to satisfy the condition of logical equivalence..

SINCE,THE TRUTH TABLE VALUES OF BOTH STATEMENTS IS SAME. THUS, THE TWO

STATEMENTS ARE LOGICALLY EQUIVALENT..

p q ~p ~q p=>q ~q=>~pT T F F T TT F F T F FF T T F T TF F T T T T

LOGICAL IMPLICATIONS

DIRECT IMPLICATION (p=>q) CONVERSE IMPLICATION (q=>p) INVERSE OR OPPOSITE IMPLICATION (~p=>~q) CONTRAPOSITIVE IMPLICATION (~q=>~p)

Algebra of preposition

1) Commutative law2) Associative law3) Distributive law4) De Morgan’s law5) Idempotent law6) Identity law

Idempotent law1. p V p p2. p ^ p p

p p p v p p v pp p ^ p p^ pp

T T T T T T

F F F F F F

Commutative law

• p v q = q v p• p ^ q = q ^ p

p q p v q q v p p ^ q q ^ pT T T T T T

T F T T F F

F T T T F F

F F F F F F

Associative law

• (p v q) v r p v (q v r) • (p ^ q) ^ r p ^ (q ^ r)

p q r p v q ( p v q) v r q V r p v (q v r)T T T T T T T

T T F T T T T

T F T T T T T

T F F T T F T

F T T T T T T

F T F T T T T

F F T F T T T

F F F F F F F

Distributive law• p ^ (q v r) (p ^ q) v (p ^ r) • p ^ (q v r) (p ^ q) v (p ^ r)

p q r q v r p^(q v r) p^q p^r (p^q)v(p^r)T T T T T T T T

T T F T T T F T

T F T T T F T T

T F F F F F F F

F T T T F F F F

F T F T F F F F

F F T T F F F F

F F F F F F F F

De Morgan’s law

• ~(p v q) ~p ^ ~q• ~(p ^ q) ~p v ~q

p q (p v q) ~(p v q) ~p ~q ~p ^ ~qT T T F F F F

T F T F F T F

F T T F T F F

F F F T T T T

Identity law

1) p ^ T p 2) T ^ p p 3) p v F p 4) F v p p

P T P ^ T

T T T

F T F

P F P v F

T F T

F F F

TRANSITIVE RULE

pq qr

-------------- pr

Rule of detachmentP

Pq----------

q

EXAMPLE TEST THE VALIDITY OF THE FOLLOWING

ARGUMENT…. IF A MAN IS A BACHELOR,HE IS WORRIED(A

PREMISE) IF A MAN IS WORRIED,HE DIES YOUNG(A

PREMISE)-------------------------------------------------------------------------

---------------------------- BACHELORS DIE YOUNG(CONCLUSION)

P: A man is a bachelorQ:he is worried

R: he dies young

The given argument in symbolic form can be written as:

pq (a premise) qr (a premise) -------------------- pr (conclusion)

The given argument is true by law of syllogism(law of transitive)…

p q r pq qr pr pq ^ qr (pq) ^ (qr) => pr

T T T T T T T T

T T F T F F F T

T F T F T T F T

T F F F T F F T

F T T T T T T T

F T F T F T F T

F F T T T T T T

F F F T T T T T

THANK YOU

PRESENTATION BY :

ASHWINI VIPAT