1. Idea of the Connective IF–THEN
The connective IF–THEN is used to express a conditional relationship between two statements. It links a condition to a consequence and is written in the form "If P, then Q".
This connective does not claim that P happens; it only tells what must follow whenever P does happen. It is a key tool in mathematical proofs and logical arguments.
1.1. Understanding the conditional idea
When we use IF–THEN, we are describing a rule. The truth of the rule depends on whether the conclusion holds in every situation where the condition is true.
2. Formal Meaning of IF–THEN
If P and Q are statements, then the implication:
\(P \Rightarrow Q\)
means that whenever P is true, Q must also be true. If P is false, the implication makes no requirement about Q.
2.1. Hypothesis and Conclusion
- P is called the hypothesis (or condition).
- Q is called the conclusion.
The implication states that the conclusion must follow whenever the hypothesis holds.
3. Truth Behaviour of IF–THEN
The truth value of an implication is determined by checking only the situation where the hypothesis is true.
3.1. Intuitive truth patterns
- P true, Q true → true
- P true, Q false → false (the only false case)
- P false, Q true → true
- P false, Q false → true
The implication is false only when P happens but Q fails.
4. Examples
These examples show how IF–THEN works in everyday language and in mathematics.
4.1. Everyday examples
- "If it rains, then the ground becomes wet."
- "If the battery is low, then the device will shut down soon."
These express cause-and-effect style relationships.
4.2. Mathematical examples
- P: "x is a multiple of 4"
- Q: "x is even"
- Implication: "If x is a multiple of 4, then x is even"
This is always true because every number divisible by 4 is automatically divisible by 2.
5. Notes and Observations
These points help clarify common confusion around IF–THEN:
- The implication does not claim that P actually happens.
- It only states what must follow if P happens.
- The only situation where the implication fails is when P is true and Q is false.
- Implication is central to mathematical proofs, especially when showing that certain conditions lead to specific outcomes.