Rules of Inference
Rules of inference are standard logical patterns that allow us to derive a conclusion from one or more given premises in a logically valid way.
Rules of inference are important because these are the building blocks for formal proofs in mathematics, computer science, and logic. They ensure that every step in your reasoning is valid, so your conclusion is guaranteed to be true if your premises are true.
The two main categories of Rules of Inference are:
- Rules for Propositional Logic → Work with whole statements (propositions).
- Rules for Predicate Logic → Extend propositional rules to include quantifiers.
Rules of Inference for Propositional Logic
- These deal with statements (propositions) that are either true or false.
- They use logical connectives like ∧ (and), ∨(or), → (implies), and ¬ (not).
Let's discuss these in detail:
1. Modus Ponens (Law of Detachment)
If a conditional statement ("if-then" statement) is true, and its antecedent (the "if" part) is true, then its consequent (the "then" part) must also be true.
Form: If p → q and p, then q.
Example:
- Premise: If it rains, the ground will be wet.
- Premise: It is raining.
- Conclusion: The ground is wet.
2. Modus Tollens (Law of Contrapositive)
If a conditional statement is true, and its consequent is false, then its antecedent must also be false.
Form: If p → q and ¬q, then ¬p.
Example:
- Premise: If it rains, the ground will be wet.
- Premise: The ground is not wet.
- Conclusion: It is not raining.
3. Hypothetical Syllogism
If two conditional statements are true, where the consequent of the first is the antecedent of the second, then a third conditional statement combining the antecedent of the first and the consequent of the second is also true.
Form: If p → q and q → r, then p → r.
Example:
- Premise: If it rains, the ground will be wet.
- Premise: If the ground is wet, the plants will grow.
- Conclusion: If it rains, the plants will grow.
4. Disjunctive Syllogism
If a disjunction (an "or" statement) is true, and one of the disjuncts (the parts of the "or" statement) is false, then the other disjunct must be true.
Form: If p ∨ q and ¬p, then q.
Example:
- Premise: It is either raining or sunny.
- Premise: It is not raining.
- Conclusion: It is sunny.
5. Conjunction
If two statements are true, then their conjunction (an "and" statement) is also true.
Form: If p and q, then p ∧ q.
Example:
- Premise: It is raining.
- Premise: It is windy.
- Conclusion: It is raining and windy.
6. Simplification
If a conjunction (an "and" statement) is true, then each of its conjuncts is also true.
Form: If p ∧ q, then p
Example:
- Premise: It is raining and windy.
- Conclusion: It is raining.
7. Addition
If a statement is true, then the disjunction (an "or" statement) of that statement with any other statement is also true.
Form: If p, then p ∨ q
Example:
- Premise: It is raining.
- Conclusion: It is raining or sunny.
8. Absorption(Abs)
If a conditional statement (an "if-then" statement) is true, then the antecedent implies a conjunction of itself and the consequent.
Form: If P→Q, then P→(P∧Q)
Example:
- Premise: If it is raining, then the ground is wet.
- Conclusion: If it is raining, then it is raining and the ground is wet.
9. Resolution
If two disjunctions ("or" statements) are true, and one contains a proposition (P) while the other contains its negation (¬P), then the disjunction of the remaining parts is true.
Form: If P∨Q and ¬P∨R, then Q∨R.
Example:
- Premise 1: It is raining or it is snowing.
- Premise 2: It is not raining or it is cold.
- Conclusion: It is snowing or it is cold.
The above studied Rules of Inferences are tabulated below:
Rule of Inference | Form | Tautology | Description |
|---|---|---|---|
Modus Ponens (MP) | If p → q and p, then q. | p ∧ (p → q)) → ¬q | If P implies Q, and P is true, then Q is true. |
Modus Tollens (MT) | If p → q and ¬q, then ¬p. | (¬q ∧ (p → q)) → ¬p | If P implies Q, and Q is false, then P is false. |
Hypothetical Syllogism (HS) | If p → q and q → r, then p → r. | ((p → q) ∧ (q → r)) → (p → r) | If P implies Q and Q implies R, then P implies R. |
Disjunctive Syllogism (DS) | If p ∨ q, and ¬p, then q. | (¬p ∧ (p ∨ q)) → q | If P or Q is true, and P is false, then Q is true. |
Conjunction (Conj) | If p and q, then p ∧ q. | (p ∧ q) → (p ∧ q) or p → (q → (p ∧ q)) | If P and Q are true, then P and Q are true. |
Simplification (Simp) | If p ∧ q, then p | (p ∧ q) → p | If P and Q are true, then P is true |
Addition (Add) | If p, then p ∨ q | p → (p ∨ q) | If P is true, then P or Q is true. |
Absorption(Abs) | If p → q, then p → (p ∧ q) | (p → q) → (p → (p ∧ q)) | If P implies Q, then P implies P or Q is true. |
Resolution | If p ∨ q, and ¬p ∨ r, then q ∨ r. | p ∨ q, ¬p ∨ r ⇒ q ∨ r | If P or Q is true, and not P or R is true, then Q or R is true. |
Rules of Inference for Predicate Logic
Similarly, we have Rules of Inference for quantified statements or predicate logic, which is given in the table below.
Rule of Inference | Form | Meaning |
|---|---|---|
Universal instantiation | ∀xP(x) ⇒ P(c) | If something is true for all x, it’s true for a particular case c. |
Universal generalization | P(c) ⇒ ∀x P(x) | If something is true for any arbitrary element, it’s true for all. |
Existential instantiation | ∃xP(x) ⇒ P(c) | If something exists, we can give it a name (c). |
Existential generalization | P(c)⇒ ∃x P(x) | If something is true for a particular c, it’s true for “some x”. |
Sample Problems on Rules of Inference
Example 1: Destructive Dilemma: Given: (P → Q) ∧ (R → S) ¬Q ∨ ¬S ∴ ¬P ∨ ¬R
Solution:
(P → Q) ∧ (R → S) (given)
¬Q ∨ ¬S (given)
P → Q (from 1, Simplification)
R → S (from 1, Simplification)
¬P ∨ ¬R (from 2, 3, and 4, Destructive Dilemma)
Answer: ¬P ∨ ¬R
Example 2: Constructive Dilemma: Given: (P → Q) ∧ (R → S) P ∨ R ∴ Q ∨ S
Solution:
(P → Q) ∧ (R → S) (given)
P ∨ R (given)
P → Q (from 1, Simplification)
R → S (from 1, Simplification)
Q ∨ S (from 2, 3, and 4, Constructive Dilemma)
Answer: Q ∨ S
Example 3: Resolution: Given: P ∨ Q ¬P ∨ R ∴ Q ∨ R
Solution:
P ∨ Q (given)
¬P ∨ R (given)
Q ∨ R (from 1 and 2, Resolution)
Answer: Q ∨ R
Example 4: Conjunction: Given: P Q ∴ P ∧ Q
Solution:
P (given)
Q (given)
P ∧ Q (from 1 and 2, Conjunction)
Answer: P ∧ Q
Example 5: Simplification:
Given: P ∧ Q ∴ P
Solution:
P ∧ Q (given)
P (from 1, Simplification)
Answer: P
Example 6: Addition: Given: P ∴ P ∨ Q
Solution:
P (given)
P ∨ Q (from 1, Addition)
Answer: P ∨ Q
Example 7: Disjunctive Syllogism:
Given: P ∨ Q ¬P ∴ Q
Solution:
P ∨ Q (given)
¬P (given)
Q (from 1 and 2, Disjunctive Syllogism)
Answer: Q
Example 8: Hypothetical Syllogism: Given: P → Q Q → R ∴ P → R
Solution:
P → Q (given)
Q → R (given)
P → R (from 1 and 2, Hypothetical Syllogism)
Answer: P → R
Example 9: Modus Tollens: Given: P → Q ¬Q ∴ ¬P
Solution:
P → Q (given)
¬Q (given)
¬P (from 1 and 2, Modus Tollens)
Answer: ¬P
Example 10: Modus Ponens: Given: P → Q P ∴ Q
Solution:
P → Q (given)
P (given)
Q (from 1 and 2, Modus Ponens)
Answer: Q
Rules of Inference - Unsolved Problems
1. Given: P → Q Q → R P Conclude: R
2. Given: P ∨ Q¬P R → ¬Q Conclude: R
3. Given: P → (Q ∧ R) P Conclude: R
4. Given: (P ∧ Q) → R ¬R P Conclude: ¬Q
5. Given: P → Q Q → R R → S ¬S Conclude: ¬P
6. Given: (P → Q) ∧ (R → S) P ∨ R ¬Q Conclude: S
7. Given: P → (Q ∨ R) ¬Q P Conclude: R
8. Given: (P ∧ Q) ∨ R ¬R ¬P Conclude: Q
9. Given: P → (Q → R) P Q Conclude: R
10. Given: (P → Q) ∧ (R → S) P ∨ R ¬Q ∧ ¬S Conclude: Contradiction
Also Check
GATE CS Corner Questions: Practicing the following questions will help you test your knowledge. All questions have been asked in GATE in previous years or in GATE Mock Tests.
