Simplification
From a conjunction, conclude either conjunct
P ∧ Q ⊢ P (or Q)Simplification is one of the most straightforward inference rules in logic: if you know that both P and Q are true (P ∧ Q), then you can conclude that P is true, and you can also conclude that Q is true. This rule captures the intuitive principle that knowing multiple things allows you to know each thing individually.
Extracting Components:
From any conjunction, you can extract either component. This is like having a box containing two items - you can take out either item independently.
Logical Foundation:
This rule is based on the definition of conjunction: P ∧ Q is true if and only if both P and Q are true. So if the conjunction is true, each part must be true.
P ∧ Q ⊢ P and P ∧ Q ⊢ QBasic Simplification
Conjunction:
Extract P:
Alternative Extraction
Conjunction:
Extract Q:
Complex Simplification
Complex Conjunction:
Simplify to:
Key Point: Simplification extracts individual components from conjunctions - if we know P AND Q, we can conclude P or Q separately.
Example 1: Daily Observation(Extracting information)
Compound fact:
Simplified fact:
Explanation: Simplification allows us to extract individual facts from compound observations, which is essential for focused reasoning.
Example 2: Academic Records(Student achievement)
Combined achievement:
Individual achievement:
Explanation: In academic contexts, simplification helps extract specific achievements from overall performance records.
Example 3: System Status(Technical monitoring)
Overall status:
Component status:
Explanation: System monitoring relies on simplification to isolate individual component statuses from complex system states.