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