In Artificial Intelligence (AI), especially in the context of knowledge representation and automated reasoning, rules of inference are fundamental tools used to derive new facts from known facts (or premises). They are used extensively in propositional logic and first-order logic to enable logical reasoning.
Rules of inference are logical rules that describe how to derive a conclusion from one or more premises. They are used in proof systems, expert systems, and automated theorem provers.
Each rule has the general form:
If premises A₁, A₂, ..., Aₙ are true, then conclusion B is true.
If P → Q and P are both true, then Q is true.
Example:
P: It is raining.
P → Q: If it is raining, then the ground is wet.
Therefore, Q: The ground is wet.
If P → Q and ¬Q are true, then ¬P is true.
Example:
P → Q: If it is raining, the ground is wet.
¬Q: The ground is not wet.
Therefore, ¬P: It is not raining.
If P → Q and Q → R, then P → R.
Example:
P → Q: If I study, I will pass.
Q → R: If I pass, I will be happy.
Therefore, P → R: If I study, I will be happy.
If P ∨ Q and ¬P, then Q.
Example:
P ∨ Q: It is raining or it is snowing.
¬P: It is not raining.
Therefore, Q: It is snowing.
From P, infer P ∨ Q.
Example:
P: I am tired.
Therefore, P ∨ Q: I am tired or I am hungry.
From P ∧ Q, infer P.
Example:
P ∧ Q: I am tired and hungry.
Therefore, P: I am tired.
From P and Q, infer P ∧ Q.
Example:
P: I am tired.
Q: I am hungry.
Therefore, P ∧ Q: I am tired and hungry.
From P ∨ Q and ¬P ∨ R, infer Q ∨ R.
It eliminates the variable P and combines the rest.
Automated reasoning: Used in inference engines to deduce new facts.
Expert systems: Encoded as rule-based logic to simulate expert behavior.
Theorem proving: Used to verify mathematical theorems or program correctness.
Natural language processing: Helps in semantic parsing and entailment tasks.
Knowledge base:
If it is raining, the ground is wet. (P → Q)
It is raining. (P)
Using Modus Ponens, infer:
The ground is wet. (Q)
This is how AI systems can infer conclusions from known facts.