Entailment (⊢)
Shows that a conclusion logically follows from premises
P, Q ⊢ R means "from P and Q, we can derive R"Logical entailment (⊢) captures when one statement necessarily follows from another. P ⊢ Q means that whenever all statements in P are true, Q must also be true. This is the semantic foundation of valid reasoning - it's about logical necessity, not just syntactic manipulation.
Key Properties:
- • Semantic relationship (about truth)
- • Monotonic (adding premises preserves entailment)
- • Transitive (chains of entailment work)
- • Foundation of valid reasoning
Entailment vs Implication:
⊢ is a metalogical relation about logical necessity, while → is a logical connective within formulas. P ⊢ Q means Q necessarily follows from P; P → Q is a conditional statement.
P ⊢ Q means Q is a logical consequence of PBasic Entailment
Premise:
Entails:
Logical Consequence:
Modus Ponens Entailment
Premises:
Entailment:
Valid Inference:
Multiple Premises
Premise Set:
Entailment:
Complex Inference:
Key Point: Entailment is the semantic foundation of logic - it captures when conclusions necessarily follow from premises.
Example 1: Simple Deduction(Basic logical following)
Premise:
Entailment:
Explanation: Given the premises, the conclusion necessarily follows. There's no interpretation where the premises are true but the conclusion is false.
Example 2: Mathematical Reasoning(Formal proof)
Premises:
Entailment:
Explanation: Mathematical entailment: if x > 5 and y = x + 2, then y must be greater than 7. This is logically necessary.
Example 3: System Requirements(Software validation)
System Properties:
Security Entailment:
Explanation: In formal verification, we prove that system properties entail security requirements - if the properties hold, security necessarily follows.