Implication and Iff in Propositional Logic

In logic and related fields like philosophy and mathematics, "if and only if" (abbreviated as "iff") is a biconditional logical connective between statements, where either both statements are said to be true or both are false. The outcome is that the truth of either one of the connected statements needs the truth of the other (i.e. either both statements are true, or both are false), although it is debatable whether the connective hence described is appropriately rendered by the "if and only if"—with its pre-existing meaning.

Biconditional Connective in Implication Iff

With biconditional connective, it means a statement of material equivalence, and can be likened to a basic material conditional ("only if", equal to "if ... then") joined with its reverse ("if"); and thus the name.

For example, M if and only if N implies that the only case in which M is true is if N is also true, while in the case of M if N, there could be other instances where M is true and N is false.

Implication If Then

Implication if then statements are usually a mathematical statement which is categorized in two parts i.e.: the assumptions or hypothesis, and the conclusion. Most mathematical statements you will notice have the form "If P, then Q" or "P implies Q" or "P ⇒ Q". The conditions that make up "P" is the hypothesis we make, and the conditions that make up "Q" are the conclusion.

If we need to prove that the statement "If P, then Q" is true, we would require beginning by making the assumptions "P" and then doing a little task to conclude that "Q" must also hold.

If we want to apply a statement of the form "If P, then Q", then we would require ensuring that the conditions "P" are met, before we jump to the conclusion "Q."

Examples of Implication If Then

if you look to apply the statement "x is even ⇒ x/2 is an integer", then you would require to verify that ‘x’ is even, before you come to the conclusion that x/2 is an integer.

In mathematical field, you will often come across statements in the form "P if and only if Q" or "P ⇔ Q". These statements are actually two "if/then" statements. The statement "P if and only if Q" will be equivalent to the statements "If P, then Q" and "If Q, then P." Another way to thinking about this type of statement is as equivalence between the statements P and Q: whenever P holds, Q holds, and whenever Q holds, P holds.

Solved Example on Implication If Then

Let’s consider an example and attempt to identify the contrapositive and converse of it.

Example: M: If two angle measurements of a triangle are equivalent then the triangle is isosceles.

Solution: The contrapositive statement is assigned as:

N: A triangle is not isosceles given that any two angle measurements of a triangle are not equal

Now, the converse statement would be:

N: If a triangle is isosceles then two angles of the triangle are equivalent

Fun Facts

• In daily use, a statement in the form "If P, then Q", sometimes implies "P if and only if Q."

• An example of everyday use of implication if then is when most people say "If you lend me Rs.1000, then I'll do your chores this month" they typically mean "I'll do your chores if and only if you lend me Rs. 1000." Specifically, if you don't lendRs.1000, they won't be doing your chores.

• In mathematics, the statement "P” means “Q" is very different from "P” if and only if “Q"

• If “P” is the statement "n is an integer" and “Q” be the statement "n/3 is a rational number." The statement "P” implies “Q" is the statement "If n is an integer, then n/3 is a rational number." This statement holds true.

• For the above point, on contrary, the statement "P” if and only if “Q" is the statement "n is an integer if and only if n/3 is a rational number," holds false.