Inference rules are those rules which are used to describe certain conclusions. The inferred conclusions lead to the desired goal state.

In propositional logic, there are various inference rules which can be applied to prove the given statements and conclude them.

There are following laws/rules used in propositional logic:

• Modus Tollen: Let, P and Q be two propositional symbols:

Rule: Given, the negation of Q as (~Q).

If P→Q, then it will be (~P), i.e., the negation of P.

Example: If Aakash goes to the temple, then Aakash is a religious person. Aakash is not a religious person. Prove that Aakash doesn’t go to temple.

Solution: Let, P= Aakash goes to temple.

Q= Aakash is religious. Therefore, (~Q)= Aakash is not a religious person.

To prove: ~P→~Q

By using Modus Tollen rule, P→Q, i.e., ~P→~Q (because the value of Q is (~Q)).

Therefore, Aakash doesn’t go to the temple.

• Modus Ponen: Let, P and Q be two propositional symbols:

Rule: If P→Q is given, where P is positive, then Q value will also be positive.

Example: If Sheero is intelligent, then Sheero is smart. Sheero is intelligent. Prove that Sheero is smart.

Solution: Let, A= Sheero is intelligent.

B= Sheero is smart.

To prove: A→B.

By using Modus Ponen rule, A→B where A is positive. Hence, the value of B will be true. Therefore, Sheero is smart.

• Syllogism: It is a type of logical inference rule which concludes a result by using deducting reasoning approach.

Rule: If there are three variables say P, Q, and R where

P→Q and Q→R then P→R.

Example: Given a problem statement:

If Ram is the friend of Shyam and Shyam is the friend of Rahul, then Ram is the friend of Rahul.

Solution: Let, P= Ram is the friend of Shyam.

Q= Shyam is the friend of Rahul.

R= Ram is the friend of Rahul.

It can be represented as: If (P→Q) Ʌ (Q→R)= (P→R).

• Disjunctive Syllogism

Rule: If (~P) is given and (P V Q), then the output is Q.

Example: Sita is not beautiful or she is obedient.

Solution: Let, (~P)= Sita is beautiful.

Q= She is obedient.

P= Sita is not beautiful.

It can be represented as (P V Q) which results Sita is obedient.

Note: Logical equivalence rules can also be used as Inference rules in Proposition logic.

We can also apply the inference rules to the logical equivalence rules as well.

• Biconditional Elimination: If Aó B then (A→B) Ʌ (B→A) or

If (A→B) Ʌ (B→A) then A óB. (using one side implication rule).

• Contrapositive: If ¬(A óB) then ¬((A→B) Ʌ (B→A))

We can re-obtain it using De Morgan’s and Modus Ponen rule.

Using inference rules, we can also define a proof problem as follows:

• Initial State: It is the initial  knowledge base.
• Actions: It is the set of actions which contains all the inference rules applied over all the sentences that match the top half of the inference rule.
• Result: When we add the sentence at the bottom of the inference rule, it gives the result of the applied action.
• Goal: It is the state which contains that sentence we are trying to prove.

Note: It is more efficient to find a proof, as it removes irrelevant prepositions.