Symbolic Logic

DRAFT

Proposition : A proposition is a statement which can be classified as true or false.

Examples:- Washington DC is the capital of the United States of America.
Thames is the longest river in the world.
2+3=6.
The first statement is true, the second and third are false.
All of these are propositions.
The following are not propositions.
(i) Read this carefully.
(ii) What are you doing?
(iii) A+B=C

The truth value of a proposition is denoted by T if it is a true proposition and by F, if it is a false proposition.

Logical Operations:-

Conjunction:- Let p and q be two propositions. The proposition 'p and q' denoted by 'p^q' is true when both p and q are true and is false otherwise. The proposition p^q is called the conjunction of p and q.
Truth Table
p q p^q
T T T
T F F
F T F
F F F

Example:- Today is the 21st day of the month of May.
The statement is true only if it is both the 21st day of the month, and the month is May.
The proposition is false if it is not May, and if it is not the 21st day of the month. The proposition is false if it is the month of May but not the 21st day. Similarly, the proposition is false if it is the 21st day of the month but not the month of May.

Disjunction:- Let p and q be two propositions. The proposition 'p or q', denoted by 'pvq' is the proposition that is false when p and q are both false and true otherwise. The proposition 'pvq' is called disjunction.
Truth Table
p q pvq
T T T
T F T
F T T
F F F
Example:- Suppose p is 'It is a Sunday' and q is 'It is a rainy day'. The proposition 'pvq' is true on any day which is a Sunday or a rainy day (including rainy Sundays). It is false on days that are not Sundays and when it does not rain.

Negation:- Let p be a proposition. 'Not p' is another proposition called Negation of p. The negation of p is denoted by ~p. The proposition ~p is read as 'not p'.
Truth Table
p ~p
T F
F T
Example:- p is the proposition 'Today is a Sunday'. The negation ~p is that 'today is not a Sunday'. The negation can also be expressed as 'It is not true that today is a Sunday'.

Conditional Operator:- Let p and q be two propositions. The implication p-> is the proposition that is false when p is true and q is false and true otherwise. In this implication, p is called the hypothesis (or antecedent or premise) and q is called the conclusion (or consequence).
Truth Table
p q p->q
T T T
T F F
F T T
F F T
The proposition p->q arise in many mathematical reasoning and a wide variety of terminology is used to express p->q. It is also called
(1) If p, then q
(2) p implies q
(3) p only if q
(4) p is sufficient for q
(5) q if p
(6) q is necessary for p
(7) q whether p

Examples:-
1. If ABC is a right angled triangle with right angle at A, then a^2=b^2+c^2
2. If x=y and y=z, then x=z.
3. If an integer is a multiple of 4, then it is even.

Biconditional operator:- Let p and q be two propositions. The biconditional p<->q is the proposition that is true when p and q have the same truth values and is false otherwise.
Truth Table
p q p<->q
T T T
T F F
F T F
F F T
The bicondition p<->q is true when both p->q and q->p are true. Because of this, the terminology 'p if and only if q' is used for this bicondition. Other ways of expressing the biconditional p<->q are (1) p is necessary and sufficient for q, and (2)'if p then q' and conversely.

Examples:- 1. Two triangles are congruent if and only if the corresponding sides are equal.
2. Two lines are parallel if and only if they have the same slope.

Converse, Inverse and Contrapositive.

Let p and q be two propositions. p->q is a conditional proposition. The proposition q->p is called the converse of the proposition p->q. The proposition ~p->~q is called the inverse of the proposition. The proposition ~q->~p is called the contrapositive of the proposition p->q.
Truth Table
p q Conditional Converse Inverse Contrapositive
p->q q->p ~p->~q ~q->~p
T T T T T T
T F F T T F
F T T F F T
F F T T T T

Conditional proposition and its converse are not logically equivalent. But a conditional proposition p->q and its contrapositive ~q->~p are logically equivalent.

Example:- p: A is a square.
q: A is a rectangle.
p->q : If A is a square, then A is a rectangle.
q->p : If A is a rectangle, then A is a square.
p->q is true but q->p is false.

Example:- p : x^2 is odd.
q: x is odd.
~p: x^2 is even.
~q: x is even
It can be seen that the contrapositive is true. Since ~q->~p is true (that is, if x is even, x^2 is even), the original proposition p->q is also true.

Tautology and Contradiction:-

A compound proposition that is always true, no matter what the truth values of the propositions that occur is called Tautology.

A compound proposition that is always false is called a contradiction.
A proposition that is neither tautology nor a contradiction is called contingency.