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Theory of Inference for the Statement Calculus
To deduce new statements from the statements whose truth that we already know, Rules of Inference are used.
What are Rules of Inference for?
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.
Important Definitions
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.
Argument Structure
An argument structure is defined as using Premises and Conclusion.
Premises - p1, p2, p3,...,pn
Conclusion - q
Example
$$\begin{matrix} P \Q \ \hline \therefore P \land Q \end{matrix}$$
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.
Table of Rules of Inference
| 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 |
Example
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.
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