1. Idea of the Connective IF AND ONLY IF
The connective IF AND ONLY IF (often written as "iff") expresses a two-way logical relationship between two statements. It means that each statement implies the other.
This connective is used when two statements are tightly linked — one holds exactly when the other holds.
1.1. Why biconditionals matter
Biconditionals are commonly used in definitions and characterisations because they express an exact equivalence between two ideas.
2. Formal Meaning of IF AND ONLY IF
If P and Q are statements, then:
\(P \Leftrightarrow Q\)
means that both of the following are true:
- \(P \Rightarrow Q\)
- \(Q \Rightarrow P\)
2.1. Equivalence idea
The two statements rise and fall together. If one is true, the other must also be true; if one is false, the other must also be false.
3. Truth Behaviour of IF AND ONLY IF
The biconditional is the most symmetric connective because it checks whether two statements match in truth value.
3.1. Intuitive truth patterns
- P true, Q true → true
- P true, Q false → false
- P false, Q true → false
- P false, Q false → true
The biconditional is true precisely when P and Q agree.
4. Examples
These examples show how the biconditional works in logical statements and mathematics.
4.1. Everyday examples
- "You can unlock the door if and only if you enter the correct code."
- "The machine starts if and only if the key is turned."
Both directions express a tight linkage between conditions.
4.2. Mathematical examples
- P: "n is even"
- Q: "n is divisible by 2"
- Biconditional: "n is even if and only if it is divisible by 2"
This expresses the precise equivalence between the two ideas.
5. Notes and Observations
Important points about the IF AND ONLY IF connective:
- It combines two implications into one compact statement.
- It is true when both parts have the same truth value.
- Biconditionals are commonly used in definitions and theorems.
- They express exact equivalence, not just one-directional implication.