Topic 3: Entailment and Inference

The Heart of Logical Reasoning

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What is Entailment?

Entailment (⊨) is the fundamental relationship between knowledge and conclusions. It tells us when one sentence logically follows from another.

Definition:

α ⊨ β means "α entails β" or "β is a logical consequence of α". This means that in every model where α is true, β is also true.

α ⊨ β ⟺ M(α) ⊆ M(β)

Where M(α) is the set of models that make α true

Entailment Examples

Simple Examples:
Basic Entailment:
  • P ∧ Q ⊨ P
  • P ⊨ P ∨ Q
  • P → Q, P ⊨ Q
  • P ∨ Q, ¬P ⊨ Q
Complex Entailment:
  • ∀x P(x) ⊨ P(a)
  • P(a) ⊨ ∃x P(x)
  • P → Q, Q → R ⊨ P → R
  • P ∧ (Q ∨ R) ⊨ (P ∧ Q) ∨ (P ∧ R)
Non-Entailment Examples:
  • P ∨ Q ⊭ P (P might be false)
  • P → Q ⊭ Q (P might be false)
  • ∃x P(x) ⊭ P(a) (a might not be the existing x)

Inference: The Process

Inference is the process of deriving new sentences from existing knowledge using inference rules. It's how we actually compute entailment.

Inference Process:
  1. Start with known facts (KB)
  2. Apply inference rules
  3. Derive new sentences
  4. Add to knowledge base
  5. Repeat until goal reached
Goals:
  • Answer specific questions
  • Find contradictions
  • Discover new facts
  • Prove theorems
Key Point:

Inference is the computational process that implements entailment. While entailment is a semantic relationship, inference is the syntactic process we use to determine it.

Soundness and Completeness

These are the two most important properties of inference systems:

Soundness:

Only derive true conclusions

  • If KB ⊢ α, then KB ⊨ α
  • No false conclusions
  • Reliability guarantee
  • Essential for trust
Completeness:

Derive all true conclusions

  • If KB ⊨ α, then KB ⊢ α
  • No missing conclusions
  • Power guarantee
  • Essential for completeness
Why Both Matter:
  • Soundness alone: Safe but might miss truths
  • Completeness alone: Powerful but might derive falsehoods
  • Both together: Perfect inference system

Proof Systems

Proof systems are formal methods for demonstrating that one sentence follows from others using inference rules.

System Rules Soundness Completeness
Natural Deduction Introduction/Elimination ✓ ✓
Resolution Resolution rule only ✓ ✓
Tableaux Tree construction ✓ ✓
Hilbert System Axioms + Modus Ponens ✓ ✓

Common Inference Rules

Modus Ponens
P → Q, P ⊢ Q
If P implies Q, and P is true, then Q is true
Modus Tollens
P → Q, ¬Q ⊢ ¬P
If P implies Q, and Q is false, then P is false
Conjunction
P, Q ⊢ P ∧ Q
If P and Q are both true, then P∧Q is true
Resolution
P ∨ Q, ¬P ∨ R ⊢ Q ∨ R
Eliminate complementary literals
Universal Instantiation
∀x P(x) ⊢ P(a)
If P holds for all x, then P holds for specific a
Existential Generalization
P(a) ⊢ ∃x P(x)
If P holds for specific a, then P holds for some x

Example Proof

Let's work through a complete proof using the inference rules:

Given Knowledge Base:
  1. bird(tweety)
  2. ∀x (bird(x) → can_fly(x))
  3. ∀x (can_fly(x) → is_animal(x))
Goal: Prove is_animal(tweety)
4. can_fly(tweety) (Universal Instantiation from 2, Modus Ponens with 1)
5. is_animal(tweety) (Universal Instantiation from 3, Modus Ponens with 4)
Proof Complete!

We have successfully derived is_animal(tweety) from the given knowledge base using sound inference rules.

Model Checking vs Inference

There are two main approaches to determining entailment:

Model Checking:
  • Approach: Check all possible models
  • Method: Truth tables, enumeration
  • Advantage: Always works
  • Disadvantage: Exponential complexity
  • Use: Small problems, verification
Inference:
  • Approach: Apply inference rules
  • Method: Proof construction
  • Advantage: Can be efficient
  • Disadvantage: May not find all truths
  • Use: Large problems, automation
When to Use Which:
  • Model Checking: When you need to verify all possibilities
  • Inference: When you need to find specific conclusions efficiently
  • Both: Often used together for verification and discovery

Key Takeaways

Entailment:
  • α ⊨ β means β follows from α
  • Semantic relationship between sentences
  • Foundation of logical reasoning
Inference:
  • Process of deriving new sentences
  • Syntactic manipulation using rules
  • Computational implementation of entailment
System Properties:
  • Soundness: Only derive true conclusions
  • Completeness: Derive all true conclusions
  • Both together: Perfect inference system
Next Steps:

Now that we understand how to derive conclusions, we'll explore the specific language for representing knowledge in Topic 4: Propositional Logic.

Previous: Knowledge Base Structure Next: Propositional Logic