- Trending Categories
- Data Structure
- Networking
- RDBMS
- Operating System
- Java
- iOS
- HTML
- CSS
- Android
- Python
- C Programming
- C++
- C#
- MongoDB
- MySQL
- Javascript
- PHP

- Selected Reading
- UPSC IAS Exams Notes
- Developer's Best Practices
- Questions and Answers
- Effective Resume Writing
- HR Interview Questions
- Computer Glossary
- Who is Who

To deduce new statements from the statements whose truth that we already know, **Rules of Inference** are used.

Mathematical logic is often used for logical proofs. Proofs are valid arguments that determine the truth values of mathematical statements.

An argument is a sequence of statements. The last statement is the conclusion and all its preceding statements are called premises (or hypothesis). The symbol “$\therefore$”, (read therefore) is placed before the conclusion. A valid argument is one where the conclusion follows from the truth values of the premises.

Rules of Inference provide the templates or guidelines for constructing valid arguments from the statements that we already have.

**Argument**- Argument is a statement or premise which ends with a conclusion.**Validity**- A argument is a valid if and only if argument is true and conclusion can never be false.**Fallacy**- An incorrect reasoning resulting to invalid arguments.

An argument structure is defined as using Premises and Conclusion.

Premises - p_{1}, p_{2}, p_{3},...,p_{n}

Conclusion - q

$$\begin{matrix} P \\Q \\ \hline \therefore P \land Q \end{matrix}$$

If $ p_1 \land p_2 \land p_3 \land ,\dots \land p_n \rightarrow q $ is a tautology then the argument is considered as valid otherwise it is termed as invalid.

Rule of Inference | Name | Rule of Inference | Name |
---|---|---|---|

$$\begin{matrix}P \\\hline\therefore P \lor Q\end{matrix}$$ | Addition | $$\begin{matrix}P \lor Q \\\lnot P \\\hline\therefore Q\end{matrix}$$ | Disjunctive Syllogism |

$$\begin{matrix}P \\Q \\\hline\therefore P \land Q\end{matrix}$$ | Conjunction | $$\begin{matrix}P \rightarrow Q \\Q \rightarrow R \\\hline\therefore P \rightarrow R
\end{matrix}$$ | Hypothetical Syllogism |

$$\begin{matrix}P \land Q\\\hline\therefore P\end{matrix}$$ | Simplification | $$\begin{matrix}( P \rightarrow Q ) \land (R \rightarrow S) \\P \lor R \\
\hline\therefore Q \lor S\end{matrix}$$ | Constructive Dilemma |

$$\begin{matrix}P \rightarrow Q \\P \\\hline\therefore Q\end{matrix}$$ | Modus Ponens | $$\begin{matrix}(P \rightarrow Q) \land(R \rightarrow S) \\\lnot Q \lor \lnot S \\\hline\therefore \lnot P \lor \lnot R\end{matrix}$$ | Destructive Dilemma |

$$\begin{matrix}P \rightarrow Q \\\lnot Q \\\hline\therefore \lnot P\end{matrix}$$ | Modus Tollens |

Let's see how to rule the Rule of Inference in Statement calculus to deduce conclusion from the arguments or to check the validity of an argument. Consider the following statements:

If it rains, I shall not go to school.

If I don't go to school, I won't need to do homework.

Let's first identify the prepositions and use preposition variables for representation.

P - It rains.

Q - I shall go to school.

R - I need to do homework.

Here the hypotheses are following.

$ P \rightarrow \lnot Q $

$ \lnot Q \rightarrow \lnot R $

Now tautology is $ (P \rightarrow \lnot Q) \land ( \lnot Q \rightarrow \lnot R) \rightarrow P \rightarrow \lnot R $

This is Hypothetical Syllogism Rule of inference and we can deduce that if It rains, I won't need to do homework.

- Related Questions & Answers
- Inference Theory of the Predicate Calculus
- Rules Of Inference for Predicate Calculus
- Inference Theory of the Predicate Logic
- The Predicate Calculus
- Explain the inference rules for functional dependencies in DBMS
- What is the theory of computation?
- Expectation Theory, Liquidity Premium Theory, and Segmented Market Theory
- Theory of DC Servomotors
- What are the interpretations of the theory of computer architecture?
- Set Theory
- Type Inference in C++
- What is the theory of parallel universe and wormhole?
- PN Junction Theory of Semiconductor Diode
- Difference between Relational Algebra and Relational Calculus
- How can Keras be used in the training, evaluation and inference of the model?

Advertisements