Overview
Disjunction introduction, also known as the addition rule, is a fundamental rule of inference in propositional logic. It asserts that if a statement P is true, then the statement ‘P or Q’ must also be true, regardless of the truth value of Q.
Key Concepts
The core idea is that if you know one part of a disjunction is true, the entire disjunction is true. This principle is often represented symbolically.
- From P, infer P ∨ Q.
- From Q, infer P ∨ Q.
Deep Dive
This rule is considered ‘trivial’ because it doesn’t require any complex reasoning. It simply expands a known truth into a more complex statement (a disjunction) that is guaranteed to be true if the original statement was true.
If P is true, then P ∨ Q is true.
If Q is true, then P ∨ Q is true.
Applications
Disjunction introduction is used in constructing proofs, particularly in systems of natural deduction. It allows logicians to introduce disjunctions as needed to reach their desired conclusions.
Challenges & Misconceptions
A common misconception is that this rule implies Q must also be true, which is incorrect. The rule only guarantees that at least one of the disjuncts is true.
FAQs
Q: What is the symbolic representation?
A: P ⊢ P ∨ Q
Q: Is this rule always valid?
A: Yes, in classical logic, it is a valid rule of inference.
Q: How is it different from disjunction elimination?
A: Disjunction elimination requires proving a conclusion from both P and Q, whereas introduction adds a disjunction.