Absorption
Absorb the antecedent into the consequent: if P then Q becomes if P then (P and Q)
P→Q ⊢ P→(P∧Q)Absorption is a fundamental equivalence rule that demonstrates a key insight about implications: if P implies Q, then P also implies the conjunction of P and Q. This might seem obvious, but it's a powerful tool for proof construction and logical manipulation, especially when working with complex conditional statements.
Why Absorption Works:
When P is true and P→Q holds, we know Q must be true (by modus ponens). But if both P and Q are true, then P∧Q is also true. Therefore, P→(P∧Q) is a valid conclusion. When P is false, both P→Q and P→(P∧Q) are vacuously true.
Practical Use:
Absorption is particularly useful in formal proofs where you need to strengthen a conclusion by explicitly stating that both the condition and its consequence hold together.
P→Q ⊢ P→(P∧Q)Basic Absorption
Original Implication:
Absorbed Form:
Complex Propositions
Original Form:
Absorbed Form:
Extended Example
Complex Implication:
After Absorption:
Key Point: Absorption strengthens an implication by making the antecedent part of the consequent.
Example 1: Academic Scenario
Given (Premise):
Therefore (Conclusion):
Explanation: The absorbed form explicitly states that when the condition is met, both the condition and its consequence are achieved together.
Example 2: Business Context
Given (Premise):
Therefore (Conclusion):
Explanation: In business planning, absorption helps clarify that implementing a strategy and achieving its results go hand in hand.