Conjunction
If P is true and Q is true, then P and Q is true
P, Q ⊢ P∧QConjunction is one of the most straightforward inference rules. When you have established that P is true and separately established that Q is true, you can combine them into the single statement 'P and Q'. This rule formalizes how we combine separate pieces of evidence.
Building Complex Claims:
Conjunction allows us to build more complex statements from simpler ones. It's the logical foundation for accumulating evidence and combining facts.
Logical AND:
This rule corresponds to the logical AND operation. When both inputs are true, the combined output is also true. It's the constructive counterpart to Simplification.
P ∧ Q (both P and Q must be true)Simple Conjunction
Statement 1:
Statement 2:
Conjunction:
Multiple Conjunctions
Three conditions:
Meaning:
Complex Example
Nested:
Requirements:
Key Point: Conjunction is the strictest operator - ALL components must be true for the whole statement to be true.
Example 1: Daily Activities(Combined requirements)
Individual facts:
Combined with AND:
Explanation: Conjunction combines multiple true facts into a single compound statement, useful for describing situations that require multiple conditions.
Example 2: System Access(Security requirements)
Authentication steps:
Access granted when:
Explanation: In security systems, conjunction ensures that all required authentication factors are met before granting access.
Example 3: Mathematical Proofs(Logical reasoning)
Established facts:
Conclusion by conjunction:
Explanation: In formal proofs, conjunction introduction allows us to combine separately established facts into compound conclusions.