Rules of Inference (original) (raw)
**Rules of Inference: Rules of inference are logical tools used to derive conclusions from premises. They form the foundation of logical reasoning, allowing us to build arguments, prove theorems, and solve problems in mathematics, computer science, and philosophy. Understanding these rules is crucial for constructing valid arguments and ensuring that conclusions follow logically from given information.
Table of Content
- What are Rules of Inference?
- Basic Rules of Inference
- Rule of Inference Table
- Rules of Inference
- Resolution Principle
- GATE Questions - Rule of Inference
What are Rules of Inference?
Rules of inference are essential tools in logic and mathematics, enabling the derivation of conclusions from premises. They form the backbone of logical reasoning, and proof techniques, and are extensively used in fields such as computer science, engineering, and mathematics.
Basic Rules of Inference
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
If p ∧ q, then q
**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.
Rule of Inference Table
| Rule of Inference | Description |
|---|---|
| Modus Ponens (MP) | If P implies Q, and P is true, then Q is true. |
| Modus Tollens (MT) | If P implies Q, and Q is false, then P is false. |
| Hypothetical Syllogism (HS) | If P implies Q and Q implies R, then P implies R. |
| Disjunctive Syllogism (DS) | If P or Q is true, and P is false, then Q is true. |
| Addition (Add) | If P is true, then P or Q is true. |
| Simplification (Simp) | If P and Q are true, then P is true |
| Conjunction (Conj) | If P and Q are true, then P and Q are true. |
**Rules of Inference
**Simple arguments can be used as building blocks to construct more complicated valid arguments. Certain simple arguments that have been established as valid are very important in terms of their usage. These arguments are called Rules of Inference. The most commonly used Rules of Inference are tabulated below -
| Rules of Inference | Tautology | Name |
|---|---|---|
| p, p\rightarrow q, \therefore q | (p ∧ (p → q)) → q | Modus Ponens |
| ¬q, p → q, ∴ ¬p | (¬q ∧ (p → q)) → ¬p | Modus Tollens |
| p → q, q → r, ∴ p → r | ((p → q) ∧ (q → r)) → (p → r) | Hypothetical Syllogism |
| ¬p, p ∨ q, ∴ q | (¬p ∧ (p ∨ q)) → q | Disjunctive Syllogism |
| p, ∴ (p ∨ q) | p → (p ∨ q) | Addition |
| (p ∧ q) → r, ∴ p → (q → r) | ((p ∧ q) → r) → (p → (q → r)) | Exportation |
| p ∨ q, ¬p ∨ r, ∴ q ∨ r | ((p ∨ q) ∧ (¬p ∨ r)) → (q ∨ r) | Resolution |
Similarly, we have Rules of Inference for quantified statements -
| Rule of Inference | Name |
|---|---|
| ∀xP(x) | Universal instantiation |
| P(c) for an arbitrary c | Universal generalization |
| ∃xP(x) | Existential instantiation |
| P(c) for some c | Existential generalization |
**Resolution Principle
**To understand the Resolution principle, first we need to know certain definitions.
- **Literal - A variable or negation of a variable. Eg- p, \neg q
- **Sum - Disjunction of literals. Eg- p\vee \neg q
- **Product - Conjunction of literals. Eg- p \wedge \neg q
- **Clause - A disjunction of literals i.e. it is a sum.
- **Resolvent - For any two clauses C_{1} and C_{2} , if there is a literal L_{1} in C_{1} that is complementary to a literal L_{2} in C_{2} , then removing both and joining the remaining clauses through a disjunction produces another clause C . C is called the resolvent of C_{1} and C_{2}
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
GATE Questions - Rule of Inference
**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.
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Conclusion - Rules of Inference
In logic, each rule of inference leads to a specific conclusion based on given premises. Modus Ponens establishes that if a statement P implies Q, and P is true, then Q must also be true. Conversely, Modus Tollens asserts that if P implies Q, and Q is false, then P must be false. Hypothetical Syllogism extends this reasoning by stating that if P implies Q and Q implies R, then P implies R. Disjunctive Syllogism states that if either P or Q is true, and P is false, then Q must be true. Addition indicates that if P is true, then P or Q is true. Simplification dictates that if both P and Q are true, then P must be true. Finally, Conjunction states that if both P and Q are true, then both P and Q are true. These rules collectively provide a framework for making logical deductions from given statements.