Conditional and Biconditional Statements

It is already mentioned in earlier classes that compound statements of different propositions can be obtained by conjunction, disjunction and negation of propositions.

We just recall these three basic logical connectivities and their truth table and subsequently we shall learn about conditional and biconditional statements.

Conjunction

If p and q are two propositions, then the compound proposition

"p and q" is known as conjunction of the proposition. It is denoted by

[The conjunction of two propositions p and q is true if both p and q are true and in all other cases it is false]

The truth table of p q is given below

The last column gives the truth table value of pq.

Disjunction

If p and q are two propositions, the compound proposition "p or q" is called the disjunction of p and q. It is denoted by p q.

[The disjunction p q of two proposition p and q are false if both p and q are false and in all other cases, it is true]

The truth table of p q is given below

The last column gives the truth value of p q.

Negation

Let p be any proposition. The proposition "not p" is called the negation of p. It is denoted by ~p.

[The negation of p is false if p is true and the negation of p is true if p is false]

Following is the truth table for ~p.

The Conditional Statements

If p and q are two propositions, then the compound proposition, "if p then q" is known as conditional statement or implication.

It is denoted by p q.

p is called the antecedent (or hypothesis).

q is called the consequent (or conclusion).

Note: p q can be read as

(i) If p then q

(ii) p is sufficient for q

(iii) q if p

(iv) q is necessary for p

(v) p only if q

The truth value of a conditional statement is given by the following rule.

[The conditional statement pq is false whenever p is true and q is false, otherwise pq is true.]

The following table shows the truth table of the conditional statement p q.

Truth table for the conditional statement p q is

We write each truth value of p q as follows.

Let us consider the logical statement "If x is prime then x² is not prime."

Let p: x is a prime number.

q: x² is not a prime number.

Here if p is true and q is true, then p q is true. That is if x is a prime number, then x² is not a prime number, for further reference, we state

C1

If both p and q has truth value T, then p q has truth value T.

(T T = T)

If p has the truth value T, q has the truth value T.

p: x is a prime number

q: x² is not a prime number

p q has the truth value T

C2

If p has the truth value T and q has the truth value F, then p q has the truth value F (T F = F).

If p has the truth value T, q has the truth value F.

p: x is a prime number

q: x² is a prime number

p q: If x is a prime number then x² is not a prime number. This statement is logically false p q has the truth value F.

C3

If p has the truth value F and q has the truth value T, then p q has the truth value T (F T = T).

If p has a truth value F and q has the truth value T, then

p: is not a prime number

q: x² is not a prime number

We know that whether x is prime or not, x² is not a prime number. This implies the truth value of p q is true. Therefore the truth value of p q is T.

C4

If both p and q have the truth value F, p q has the truth value

T(F F = T).

It should be noted that in the above example, the simple statement p and q depend on each other. In other words, if the truth value of one depends on the truth value of the other and

p q is true, then the conditional statement

p q is called implication.

Moreover, even though there is no connection between the simple proposition p and q, the conditional p q is true. For example,

let

p: e is a vowel

q: 3 + 4 = 7

Here the conditional p q is true because the consequent (q) is true.

It should be clear that the conditional statement, p q has truth value T, whenever the truth value of p is false.