Today, we are going to discuss on What is Propositional logic in Artificial Intelligence?
• In propositional logic, the most fundamental propositions are called primitive propositions.
• Primitive propositions cannot be decomposed. Propositions that can be decomposed are compound propositions.
• Primitive propositions can be denoted by some symbols, and these symbols are called atomic formulas. From atomic formulas we can construct various logic formulas corresponding to various compound propositions.
Recursive definition of logic formula
1) Suppose, P is a logic formula,
so, ￢P is also a logic formula.
2) Suppose, P and Q are logic formulas,
so, P∧Q, P∨Q, P ⇒ Q, and P⇔Q are also logic formulas.
Properties of Operators:
- P∧ Q= Q ∧ P, or
- P ∨ Q = Q ∨ P.
- (A ∨ B) ∨ C= A ∨ (B ∨ C)
- Identity element:
- P ∧ True = P,
- P ∨ True= True.
- L ∨ (M ∧ N) = (L ∨ M) ∧ (L ∨ N).
- DE Morgan’s Law:
- ¬ (A ∨ C) = (¬ A) ∧ (¬C).
- Double-negation elimination:
- ¬ (¬P) = P.
Rules of Inference in Artificial intelligence
Generating the conclusions from evidence and facts is termed as Inference in artificial intelligence. In AI, we need intelligent computers which can create new logic from old logic or by evidence and thus we need rules like inference rules.
Typically, Inference rules are known as the templates for generating valid arguments, which are applied to derive proofs in artificial intelligence, and the proof is a sequence of the conclusion that leads to the desired goal.
The implication among all the connectives plays an important role in inference rules.
Here are some terminologies related to inference rules:
- Implication: It is represented as P → Q. It is a Boolean expression.
- Converse: The converse of implication, which means the right-hand side proposition goes to the left-hand side and vice-versa.
- Contrapositive: The negation of converse is termed as contrapositive, and it can be represented as ¬ Q → ¬ P.
- Inverse:. Inverse is negation of implication can be represented as ¬ P → ¬ Q.
From the above term some of the compound statements are equivalent to each other, which we can prove using truth table:
Hence from the above truth table, we can prove that P → Q is equivalent to ¬ Q → ¬ P, and Q→ P is equivalent to ¬ P → ¬ Q.